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Research ArticleThe (1198661015840119866 1119866)-Expansion Method and Its Applications forSolving Two Higher Order Nonlinear Evolution Equations

E M E Zayed and K A E Alurrfi

Department of Mathematics Faculty of Science Zagazig University PO Box 44519 Zagazig Egypt

Correspondence should be addressed to E M E Zayed emezayedhotmailcom

Received 19 March 2014 Accepted 23 April 2014 Published 17 June 2014

Academic Editor Oded Gottlieb

Copyright copy 2014 E M E Zayed and K A E Alurrfi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The two variable (1198661015840119866 1119866)-expansion method is employed to construct exact traveling wave solutions with parameters of two

higher order nonlinear evolution equations namely the nonlinear Klein-Gordon equations and the nonlinear Pochhammer-Chree equations When the parameters are replaced by special values the well-known solitary wave solutions of these equationsare rediscovered from the traveling waves This method can be thought of as the generalization of well-known original (119866

1015840119866)-

expansion method proposed by Wang et al It is shown that the two variable (1198661015840119866 1119866)-expansion method provides a more

powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics

1 Introduction

In the recent years investigations of exact solutions tononlinear PDEs play an important role in the study ofnonlinear physical phenomena Many powerful methodshave been presented such as the inverse scattering method[1] the Hirota bilinear transform method [2] the truncatedPainleve expansion method [3ndash6] the Backlund transformmethod [7 8] the exp-function method [9ndash13] the tanh-function method [14ndash17] the Jacobi elliptic function expan-sion method [18ndash20] the (119866

1015840119866)-expansion method [21ndash

30] the modified (1198661015840119866)-expansion method [31] and the

(1198661015840119866 1119866)-expansion method [32ndash34] The key idea of the

one variable (1198661015840119866)-expansion method is that the exact

solutions of nonlinear PDEs can be expressed by a polynomialin one variable (119866

1015840119866) in which 119866 = 119866(120585) satisfies the

second order linear ODE 11986610158401015840(120585) + 120582119866

1015840(120585) + 120583119866(120585) = 0

where 120582 and 120583 are constants and 1015840 = 119889119889120585 The key ideaof the two variable (119866

1015840119866 1119866)-expansion method is that

the exact traveling wave solutions of nonlinear PDEs can beexpressed by a polynomial in two variables (119866

1015840119866) and (1119866)

in which 119866 = 119866(120585) satisfies the second order linear ODE11986610158401015840(120585) + 120582119866(120585) = 120583 where 120582 and 120583 are constants The degree

of this polynomial can be determined by considering thehomogeneous balance between the highest-order derivatives

and the nonlinear terms appearing in the given nonlinearPDEs The coefficients of this polynomial can be obtainedby solving a set of algebraic equations resulted from theprocess of using this method Recently Li et al [32] haveapplied the (119866

1015840119866 1119866)-expansion method and determined

the exact solutions of Zakharov equations while Zayed et al[33 34] have used this method to find the exact solutionsof the combined KdV-mKdV equation and the Kadomtsev-Petviashvili equation

The objective of this paper is to apply the two variable(1198661015840119866 1119866)-expansion method to find the exact traveling

wave solutions of the higher order nonlinear Klein-Gordonequations [35]

119906119905119905

minus 1198962119906119909119909

+ 120572119906 minus 120573119906119899

+ 1205741199062119899minus1

= 0 119899 gt 2 (1)

and the higher order nonlinear Pochhammer-Chree equa-tions [36]

119906119905119905

minus 119906119909119909119905119905

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)119909119909

= 0 119899 ge 1 (2)

where 120572 120573 120574 and 119896 are constantsEquation (1) plays an important role in many scientific

applications such as the solid state physics the nonlin-ear optics and the quantum field theory (see [37ndash39])

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 746538 20 pageshttpdxdoiorg1011552014746538

2 Mathematical Problems in Engineering

Wazwaz [40 41] investigated the nonlinear Klein-Gordonequations and foundmany types of exact traveling wave solu-tions including compact solutions soliton solution solitarypatterns solutions and periodic solutions using the tanh-function method Zayed and Gepreel [35] have found theexact solutions of (1) using the (119866

1015840119866)-expansion method

Equation (2) represents nonlinear models of longitudinalwave propagation of elastic rods see [42ndash47] Zuo [36] hasdiscussed (2) using the extended (119866


and found the travelingwave solutions of these equationsTherest of this paper is organized as follows In Section 2 we givethe description of the two variable (119866

1015840119866 1119866)-expansion

method In Section 3 we apply this method to solve (1) and(2) In Section 4 some conclusions are given

2 Description of the Two Variable(1198661015840119866 1119866)-Expansion Method

Before we describe the main steps of this method we needthe following remarks (see [32ndash34])

Remark 1 If we consider the second order linear ODE

11986610158401015840

(120585) + 120582119866 (120585) = 120583 (3)

and set 120601 = 1198661015840119866 120595 = 1119866 then we get

1206011015840

= minus1206012

+ 120583120595 minus 120582 1205951015840

= minus120601120595 (4)

where 120582 and 120583 are constants

Remark 2 If 120582 lt 0 then the general solution of (3) has thefollowing form

119866 (120585) = 1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) +

120583

120582

(5)

where 1198601and 119860

2are arbitrary constants Consequently we

have

1205952

=minus120582

12058221205901

+ 1205832

(1206012

minus 2120583120595 + 120582) (6)

where 1205901

= 1198602

1minus 1198602

2

Remark 3 If 120582 gt 0 then the general solution of (3) has thefollowing form

119866 (120585) = 1198601sin (120585radic120582) + 119860

2cos (120585radic120582) +

120583

120582

(7)

and hence

1205952

=120582

12058221205902

minus 1205832

(1206012

minus 2120583120595 + 120582) (8)

where 1205902

= 1198602

1+ 1198602

2

Remark 4 If 120582 = 0 then the general solution of (3) has thefollowing form

119866 (120585) =

120583

2

1205852

+ 1198601120585 + 119860

2 (9)

and hence

1205952

=1

1198602

1minus 2120583119860

2

(1206012

minus 2120583120595) (10)

Suppose we have the following nonlinear evolution equation

119865 (119906 119906119905 119906119909 119906119909119909

) = 0 (11)

where 119865 is a polynomial in 119906(119909 119905) and its partial derivativesIn the following we give the main steps of the (119866

1015840119866 1119866)-

expansion method [32ndash34]

Step 1 The traveling wave transformation

119906 (119909 119905) = 119906 (120585) 120585 = 119909 minus 119862119905 (12)

where 119862 is a constant and reduces (11) to an ODE in thefollowing form

119875 (119906 1199061015840 11990610158401015840 ) = 0 (13)

where 119875 is a polynomial of 119906(120585) and its total derivatives withrespect to 120585

Step 2 Assuming that the solution of (13) can be expressedby a polynomial in the two variables 120601 and 120595 as follows

119906 (120585) =

119873

sum

119894=0

119886119894120601119894+

119873

sum

119894=1

119887119894120601119894minus1

120595 (14)

where 119886119894

(119894 = 0 1 2 119873) and 119887119894

(119894 = 1 2 119873) areconstants to be determined later

Step 3 Determine the positive integer 119873 in (14) by using thehomogeneous balance between the highest-order derivativesand the nonlinear terms in (13) In some nonlinear equationsthe balance number 119873 is not a positive integer In this casewe make the following transformations [48]

(a) when 119873 = 119902119901 where 119902119901 is a fraction in the lowestterms we let

119906 (120585) = V119902119901 (120585) (15)

then substitute (15) into (13) to get a new equation inthe new function V(120585) with a positive integer balancenumber

(b) when 119873 is a negative number we let

119906 (120585) = V119873 (120585) (16)

and substitute (16) into (13) to get a new equation inthe new function V(120585) with a positive integer balancenumber

Step 4 Substituting (14) into (13) along with (4) and (6) theleft-hand side of (13) can be converted into a polynomial in 120601

and 120595 in which the degree of 120595 is no longer than 1 Equatingeach coefficients of this polynomial to zero yields a system of

Mathematical Problems in Engineering 3

algebraic equations which can be solved by using the Mapleor Mathematica to get the values of 119886

119894 119887119894 119862 120583 119860

1 1198602 and 120582

where 120582 lt 0

Step 5 Similar to Step 4 substituting (14) into (13) along with(4) and (8) for 120582 gt 0 (or (4) and (10) for 120582 = 0) we obtain theexact solutions of (13) expressed by trigonometric functions(or by rational functions) respectively

3 Applications

In this section we will apply the method described inSection 2 to find the exact traveling wave solutions of (1) and(2) which are very important in themathematical physics andhave been paid attention to by many researchers

Example 5 In this example we start with the higher ordernonlinear Klein-Gordon equation (1) To this end we see thatthe traveling wave variable (12) permits us to convert (1) intothe following ODE

(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

+ 1205741199062119899minus1

= 0 119899 gt 2 (17)

Let us discuss the following two possibilities(I) If 120574 = 0By balancing between 119906

10158401015840 and 1199062119899minus1 in (17) we get 119873 =

1(119899 minus 1) According to Step 3 we use the transformation

119906 (120585) = V1(119899minus1) (120585) (18)

where V(120585) is a new function of 120585 Substituting (18) into (17)we get the new ODE

(1198622

minus 1198962) [

2 minus 119899

(119899 minus 1)2(V1015840)2

+1

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V3 + 120574V4

= 0

(19)

Determining the balance number 119873 of the new (19) we get119873 = 1 Consequently we get

V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (20)

where 1198860 1198861 and 119887

1are constants to be determined later

There are three cases to be discussed as follows

Case 1 (hyperbolic function solutions (120582 lt 0)) If 120582 lt 0substituting (20) into (19) and using (4) and (6) the left-handside of (19) becomes a polynomial in 120601 and 120595 Setting the

coefficients of this polynomial to be zero yields a system ofalgebraic equations in 119886

0 1198861 1198871 120583 120582 and 119862 as follows

1206014 1205741198864

1+

2 (1198622

minus 1198962) 1198862

1

(119899 minus 1)

+

2 (1198622

minus 1198962) 1198862

1

(119899 minus 1)2

+

12058221205741198874

1

(12058221205901

+ 1205832)2

minus

(1198622

minus 1198962) 1198991198862

1

(119899 minus 1)2

minus

61205821205741198862

11198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1205821198872

1

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1205821198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 1198991205821198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206013 minus 120573119886

3

1+ 4120574119886

01198863

1+

2 (1198622

minus 1198962) 11988601198861

(119899 minus 1)

+

312057312058211988611198872

1

(12058221205901

+ 1205832)

minus

12120582120574119886011988611198872

1

(12058221205901

+ 1205832)

minus

8120582212057412058311988611198873

1

(12058221205901

+ 1205832)2

+

4 (1198622

minus 1198962) 12058212058311988611198871

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 12058212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582120583119886

11198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206013120595 4120574119886

3

11198871

+

4 (1198622

minus 1198962) 11988611198871

(119899 minus 1)

+

4 (1198622

minus 1198962) 11988611198871

(119899 minus 1)2

minus

2 (1198622

minus 1198962) 11989911988611198871

(119899 minus 1)2

minus

412058212057411988611198873

1

(12058221205901

+ 1205832)

= 0

1206012 1205721198862

1minus 3120573119886

01198862

1+ 6120574119886

2

01198862

1+

212058231205741198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 1205821198862

1

(119899 minus 1)

+

4 (1198622

minus 1198962) 1205821198862

1

(119899 minus 1)2

minus

1205721205821198872

1

(12058221205901

+ 1205832)

+

212057312058221205831198873

1

(12058221205901

+ 1205832)2

minus

3 (1198622

minus 1198962) 12058221198872

1

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 12058221198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

61205821205741198862

01198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1198991205821198862

1

(119899 minus 1)2

minus

4120582312057412058321198874

1

(12058221205901

+ 1205832)3

minus

612058221205741198862

11198872

1

(12058221205901

+ 1205832)

+

312057312058211988601198872

1

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 11989912058221198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 12058212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

8120582212057412058311988601198873

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 120582212058321198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

+

(1198622

minus 1198962) 11989912058212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 12058212058311988601198871

(119899 minus 1) (12058221205901

+ 1205832)

= 0


1206012120595

2 (1198622

minus 1198962) 11988601198871

(119899 minus 1)

minus 31205731198862

11198871

minus

3 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)

minus

4 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)2

+ 1212057411988601198862

11198871

+

1205731205821198873

1

(12058221205901

+ 1205832)

minus

412058221205741205831198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 1198991205831198862

1

(119899 minus 1)2

minus

412058212057411988601198873

1

(12058221205901

+ 1205832)

+

5 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

11198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582120583119887

2

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601 minus 31205731198862

01198861

+ 41205741198863

01198861

+ 212057211988601198861

+

2 (1198622

minus 1198962) 12058211988601198861

(119899 minus 1)

+

3120573120582211988611198872

1

(12058221205901

+ 1205832)

minus

8120582312057412058311988611198873

1

(12058221205901

+ 1205832)2

minus

121205822120574119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601120595 212057211988611198871

minus 6120573119886011988611198871

+ 121205741198862

011988611198871

minus

3 (1198622

minus 1198962) 12058311988601198861

(119899 minus 1)

+

3 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)

+

4 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)2

minus

4120582212057411988611198873

1

(12058221205901

+ 1205832)

+

161205822120574120583211988611198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988611198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058211988611198871

(119899 minus 1)2

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

+

24120582120574120583119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 119899120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120595 41205741198863

01198871

minus 31205731198862

01198871

+ 212057211988601198871

+

12057312058221198873

1

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 12058211988601198871

(119899 minus 1)

minus

412058231205741205831198874

1

(12058221205901

+ 1205832)2

minus

4120582212057411988601198873

1

(12058221205901

+ 1205832)

minus

4 (1198622

minus 1198962) 1205821205831198862

1

(119899 minus 1)2

minus

4120573120582212058321198873

1

(12058221205901

+ 1205832)2

+

8120582312057412058331198874

1

(12058221205901

+ 1205832)3

+

21205721205821205831198872

1

(12058221205901

+ 1205832)

+

161205822120574120583211988601198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988601198872

1

(12058221205901

+ 1205832)

+

3 (1198622

minus 1198962) 12058221205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 12058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

01198872

1

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 119899120582120583119886

2

1

(119899 minus 1)2

minus

4 (1198622

minus 1198962) 120582212058331198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582120583211988601198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206010 1205721198862

0minus 1205731198863

0+ 1205741198864

0minus

12057212058221198872

1

(12058221205901

+ 1205832)

+

12058241205741198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 12058221198862

1

(119899 minus 1)2

+

212057312058231205831198873

1

(12058221205901

+ 1205832)2

minus

(1198622

minus 1198962) 12058231198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

3120573120582211988601198872

1

(12058221205901

+ 1205832)

minus

4120582412057412058321198874

1

(12058221205901

+ 1205832)3

minus

612058221205741198862

01198872

1

(12058221205901

+ 1205832)

minus

(1198622

minus 1198962) 11989912058221198862

1

(119899 minus 1)2

minus

8120582312057412058311988601198873

1

(12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 120582312058321198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

+

(1198622

minus 1198962) 120582212058311988601198871

(119899 minus 1) (12058221205901

+ 1205832)

+

(1198622

minus 1198962) 119899120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

(21)

On solving the above algebraic equations using the Maple orMathematica we get the following results

Result 1 Consider the following

1198860

= 0

1198861

= 0

1198871

=

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)

119862 = plusmn radic(119899 minus 1)

2120572

120582

+ 1198962

120572 =

1198991205732

(12058221205901

+ 1205832)

1205832120574(119899 + 1)

2

(22)

where 120583 = 0 120574 = 0 and 120573 = 0


From (5) (20) and (22) we deduce the traveling wavesolution of (17) as follows

119906 (120585)

= [

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)

times (1

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]

1(119899minus1)

(23)

where 120585 = 119909 plusmn 119905radic(119899 minus 1)2120572120582 + 119896

2

Result 2 (Figure 1) Consider the following

1198860

=

119899120573

2120574 (119899 + 1)

1198861

= plusmn

119899120573radicminus1120582

2120574 (119899 + 1)

1198871

= plusmn

119899120573radic12058221205901

+ 1205832

2120574120582 (119899 + 1)

120572 =

1198991205732

120574(119899 + 1)2


2120572

120582

+ 1198962

(24)

where 120574 = 0 and 120573 = 0In this result we deduce the traveling wave solution of

(17) as follows

119906 (120585)

=[[

[

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)

times (

1198601cosh (120585radicminus120582) + 119860

2sinh (120585radicminus120582)

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)

plusmn

119899120573

2120582120574 (119899 + 1)

times (

radic12058221205901

+ 1205832

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]]

]

1(119899minus1)

(25)


2

8000

6000

4000

2000

0000

8000

6000

4000

2000

0

minus4

minus2

0

2

4 4

2

0

minus2

minus4

xt

Figure 1The plot of solution (27) when 120582 = minus1 120574 = 1 120573 = 4 119899 = 3and 119896 = 4

In particular by setting 1198601

= 0 1198602

= 0 and 120583 = 0 in (25)we have the solitary solution

119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)

times (tanh (120585radicminus120582) + 119894 sech (120585radicminus120582)) ]

1(119899minus1)

119894 = radicminus1

(26)

while if 1198601

= 0 1198602

= 0 and 120583 = 0 then we have the solitarysolution

119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)

times (coth(120585radicminus120582) + csch(120585radicminus120582)) ]

1(119899minus1)

(27)

Case 2 (trigonometric function solution (120582 gt 0)) If 120582 gt 0substituting (20) into (19) and using (4) and (8) the left-handside of (19) becomes a polynomial in 120601 and 120595 Setting thecoefficients of this polynomial to be zero yields a system ofalgebraic equations in 119886

0 1198861 1198871 120583 120582 and119862which are omitted


here for simplicity On using the Maple or Mathematica wehave found the following results

1198860

= 0

1198861

= 0

1198871

=

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(1205832

minus 12058221205902)

1205832120574(119899 + 1)

2

(28)

where 120583 = 0 120574 = 0 and 120573 = 0From (7) (20) and (28) we deduce the traveling wave

solution of (17) as follows

119906 (120585) = [

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)

times (1

1198601sin(120585radic120582) + 119860

2cos(120585radic120582) + 120583120582

) ]

1(119899minus1)

(29)


2

Case 3 (rational function solutions (120582 = 0)) If 120582 = 0substituting (20) into (19) and using (4) and (10) the left-handside of (19) becomes a polynomial in 120601 and 120595 Setting thecoefficients of this polynomial to be zero yields a system ofalgebraic equations in 119886

0 1198861 1198871 120583 and 119862 which are omitted


1198860

= 0

1198861

= 0

1198871

=

119899120573 (21205831198602

minus 1198602

1)

120583120574 (119899 + 1)

119862 = plusmn radic1205732119899(119899 minus 1)

2(2120583119860

2minus 1198602

1)

1205832120574(119899 + 1)

2+ 1198962

120572 = 0

(30)


119906 (120585) = [

119899120573(21205831198602

minus 1198602

1)

120583120574(119899 + 1)

(1

(1205832)1205852

+ 1198601120585 + 119860

2

)]

1(119899minus1)

(31)

where 120585 = 119909 plusmn 119905radic1205732119899(119899 minus 1)

2(21205831198602

minus 1198602

1)1205832120574(119899 + 1)

2+ 1198962

(II) If 120574 = 0In this case (17) converts to

(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

= 0 119899 gt 2 (32)

By balancing between 11990610158401015840 and 119906

119899 in (32) we get 119873 = 2(119899 minus 1)According to Step 3 we use the transformation

119906 (120585) = V2(119899minus1) (120585) (33)


(1198622

minus 1198962) [

2 (3 minus 119899)

(119899 minus 1)2

(V1015840)2

+2

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V4 = 0

(34)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (35)

where 1198860 1198861 and 119887



Case 1 (hyperbolic function solutions (120582 lt 0)) (Figure 2)If 120582 lt 0 substituting (35) into (34) and using (4) and (6)the left-hand side of (34) becomes a polynomial in 120601 and 120595Setting the coefficients of this polynomial to be zero yieldsa system of algebraic equations in 119886

0 1198861 1198871 120583 120582 and 119862

which are omitted here for simplicity On using the Maple orMathematica we have found the following results

1198860

= 0

1198861

= 0

1198871

= plusmn radicminus1205721205901

(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(36)

where 120572 = 0 and 120573 = 0From (5) (35) and (36) we deduce the traveling wave

solutions of (32) as follows

119906 (120585) = [

[

plusmnradicminus1205721205901

(119899 + 1)

2120573

times (1198601sinh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)

+1198602cosh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(37)

where 120585 = 119909 minus 119862119905


14

12

10

08

06

04

02

minus4

minus2 minus2

minus4

0 0

2 2

4 4

xt

4 minus4

Figure 2 The plot of solution (38) when 120572 = 1 119862 = 1 120573 = 1 119899 = 3and 119896 = 2


= 0 and 1198602

= 0 in (37) wehave the solitary wave solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573

sech((119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(38)

while if 1198601

= 0 and 1198602

= 0 then we have the solitary wavesolutions

119906 (120585)

= [plusmnradicminus120572(119899 + 1)

2120573

csch((119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(39)




1198860

= 0

1198861

= 0

1198871

= plusmn radic1205721205902

(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(40)where 120572 = 0 and 120573 = 0

From (7) (35) and (40) we deduce the traveling wavesolutions of (32) as follows

119906 (120585) = [

[

plusmn radic1205721205902

(119899 + 1)

2120573

times (1198601sin(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)

+ 1198602cos(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(41)

where 120585 = 119909 minus 119862119905In particular by setting119860

1= 0 and119860

2= 0 in (41) we have

the periodic solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573

sec((119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(42)while if 119860

1= 0 and 119860

2= 0 then we have the periodic

solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573

csc((119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(43)

Case 3 (rational function solutions (120582 = 0)) If 120582 = 0substituting (35) into (34) and using (4) and (10) the left-hand side of (34) becomes a polynomial in 120601 and 120595 Settingthe coefficients of this polynomial to be zero yields a systemofalgebraic equations in 119886



1198860

= 0

1198861

= plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

1198871

= 1198871

119862 = 119862

120583 =

1198602

1(1198622

minus 1198962) (119899 + 1) minus 2120573119887

2

1(119899 minus 1)

2

21198601

(119899 + 1) (1198622

minus 1198962)

(44)

where 120573 = 0



119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905

Example 6 In this example we study the higher ordernonlinear Pochhammer-Chree equation (2) To this end wesee that the traveling wave variable (12) permits us to convert(2) into the following ODE

119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)

Integrating (46) twice with respect to 120585 and vanishing theconstants of integration we get

(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =

1119899 According to Step 3 we use the transformation

119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)

Balancing VV10158401015840 with V4 in (49) we get 119873 = 1 Consequentlywe get

V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn (1198601cosh(

119899120573

2

radicminus(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2

timesradicminus(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)

where 120585 = 119909 plusmn (radic120574[120572120574(119899 + 2)2

minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn tanh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn coth(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)

Note that the solutions (53) (54) and (55) are in agreementwith the solutions (16) (17) and (18) of [36] respectively


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)

In this result we deduce the traveling wave solution of (47)as follows119906 (120585)

=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)

times (1198601sinh (radicminus120582120585) + 119860

2cosh (radicminus120582120585) plusmn 120573

timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)

where 120585 = 119909 plusmn 119899radic(120572120582 + 1198992)119905


= 0 1198602

= 0 and 120573 = 0 in (57)we have the solitary wave solutions

119906 (120585) = [plusmnradicminus(119899 + 1)120582120572

(120582 + 1198992)120574

sech(radicminus120582120585)]

1119899

(58)

while if 1198601

= 0 1198602

= 0 and 120573 = 0 then we have the solitarywave solutions

119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574

csch(radicminus120582120585)]

1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

119862 = plusmn radic120572 minus

1205732

(119899 + 1)

120574(119899 + 2)2

(60)

In this result we deduce the traveling wave solution of (47)as follows

119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn (1198601cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(119899120573radicminus(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)

where 120585 = 119909 plusmn radic120572 minus (1205732(119899 + 1)120574(119899 + 2)

2)119905 120573 = 0

In particular by setting1198601

= 0 and1198602

= 0 in (61) we havethe solitary wave solutions

119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn tanh(119899120573radicminus(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sech(119899120573radicminus(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn coth(119899120573radicminus(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn csch(119899120573radicminus(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

119862 = plusmnradic120572 minus

1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602

= 0 and 120583 = 0 in(65) we have the same solitary wave solutions (62) while if1198601

= 0 1198602

= 0 and 120583 = 0 then we have the same solitarywave solutions (63)

Case 2 (trigonometric function solution (120582 gt 0)) If 120582 gt 0substituting (50) into (49) and using (4) and (8) the left-hand


side of (49) becomes a polynomial in 120601 and 120595 Setting thecoefficients of this polynomial to be zero yields a system ofalgebraic equations in 119886

0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

(66)On solving the above algebraic equations using the Maple orMathematica we get the following results



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn 119894

times ( (1198601cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602

= 0 in (68) wehave the periodic solutions

119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn 119894 tan(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602

= 0 then we have the periodicsolutions

119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn 119894 cot(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602

= 0 and 120573 = 0 in (72)we have the periodic solutions


(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt

Figure 3 The plot of solution (74) when 120572 = 10 120574 = minus1 120582 = 1 and119899 = 3

while if 1198601

= 0 1198602

= 0 and 120573 = 0 then we have the periodicsolutions


(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)

plusmn (radicminus1205902)

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn 119894 tan(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574

times (1 plusmn 119894 cot(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602

= 0 and 120583 = 0 in(80) we have the same periodic solutions (77) while if119860

1= 0

1198602

= 0 and 120583 = 0 then we have the same periodic solutions(78)

Case 3 (rational function solutions (120582 = 0)) If 120582 = 0substituting (50) into (49) and using (4) and (10) the left-hand side of (49) becomes a polynomial in 120601 and 120595 Settingthe coefficients of this polynomial to be zero yields a systemof algebraic equations in 119886

0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

119862 = plusmnradic120572

120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)

where 120585 = 119909 plusmn radic120572119905 and 120583 = (119899120573[21198991205731198602(119899 + 1) plusmn

2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)

In particular by setting 120573 = 0 in (83) we have thesolutions

119906 (120585) = [plusmnradicminus120572(119899 + 1)

1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)

which are equivalent to the solutions (50) obtained in [36](II) If 120574 = 0 120573 = 0In this case (47) converts to

(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)


119899+1 in (85) we get 119873 = 2119899According to Step 3 we use the transformation

119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)

where 120585 = 119909 minus 119862119905In particular by setting 119860

1= 0 and 119860

2= 0 in (90) we

have the solitary wave solutions

119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradicminus(119899 + 2)(119862

2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)

Note that the solutions (91) and (92) are in agreement withthe solutions (33) and (34) of [36] when 120585

0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we

have the periodic solutions

119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861

= plusmn radicminus2120572 (119899 + 2)

1198992120573

1198871

= 0

119862 = plusmn radic120572

120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)

where 120585 = 119909 plusmn radic120572119905Note that the solutions (98) are equivalent to the solutions

(40) obtained in [36]


1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)


119906 (120585) = [plusmnradicminus120572 (119899 + 2)

21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)

where 120585 = 119909 plusmn radic120572119905Finally if we set 120583 = 0 in (100) we get back to (98)

4 Conclusions

The two variable (1198661015840119866 1119866)-expansion method is used in

this paper to obtain some new solutions of two higher ordernonlinear evolution equations namely the nonlinear Klein-Gordon equations and the nonlinear Pochhammer-Chreeequations As the two parameters 119860

1and 119860

2take special

values we obtain the solitary wave solutions When 120583 = 0

and 119887119894

= 0 in (3) and (14) the two variable (1198661015840119866 1119866)-

expansion method reduces to the (1198661015840119866)-expansion method

So the two variable (1198661015840119866 1119866)-expansion method is an

extension of the (1198661015840119866)-expansion method The proposed

method in this paper is more effective and more general thanthe (119866

1015840119866)-expansionmethod because it gives exact solutions

in more general forms In summary the advantage of thetwo variable (119866

1015840119866 1119866)-expansionmethod over the (119866

1015840119866)-

expansion method is that the solutions obtained by using


the first method recover the solutions obtained by using thesecond one Finally all solutions obtained in this paper havebeen checked with the Maple by putting them back into theoriginal equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors wish to thank the referees for their comments onthis paper

References

[1] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressNew York NY USA 1991

[2] R Hirota ldquoExact solution of the korteweg-de vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[3] JWeissM Tabor andG Carnevale ldquoThe Painleve property forpartial differential equationsrdquo Journal of Mathematical Physicsvol 24 no 3 pp 522ndash526 1983

[4] N A Kudryashov ldquoExact soliton solutions of a generalizedevolution equation of wave dynamicsrdquo Journal of AppliedMathematics and Mechanics vol 52 no 3 pp 361ndash365 1988

[5] N A Kudryashov ldquoExact solutions of the generalized Kura-moto-Sivashinsky equationrdquo Physics Letters A vol 147 no 5-6pp 287ndash291 1990

[6] N A Kudryashov ldquoOn types of nonlinear nonintegrable equa-tions with exact solutionsrdquo Physics Letters A vol 155 no 4-5pp 269ndash275 1991

[7] M R Miura Backlund Transformation Springer Berlin Ger-many 1978

[8] C Rogers and W F Shadwick Backlund Transformations andTheir Applications Academic Press New York NY USA 1982

[9] J H He and X H Wu ldquoExp-function method for nonlinearwave equationsrdquo Chaos Solitons and Fractals vol 30 no 3 pp700ndash708 2006

[10] E Yusufoglu ldquoNew solitonary solutions for the MBBM equa-tions using Exp-function methodrdquo Physics Letters A vol 372no 4 pp 442ndash446 2008

[11] S Zhang ldquoApplication of Exp-function method to high-dimensional nonlinear evolution equationrdquo Chaos Solitons andFractals vol 38 no 1 pp 270ndash276 2008

[12] A Bekir ldquoThe exp-function method for Ostrovsky equationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 10 no 6 pp 735ndash739 2009

[13] A Bekir ldquoApplication of the exp-function method for nonlin-ear differential-difference equationsrdquo Applied Mathematics andComputation vol 215 no 11 pp 4049ndash4053 2010

[14] M A Abdou ldquoThe extended tanh method and its applicationsfor solving nonlinear physical modelsrdquo Applied Mathematicsand Computation vol 190 no 1 pp 988ndash996 2007

[15] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

[16] S Zhang and T C Xia ldquoA further improved tanh functionmethod exactly solving the (2 + 1)-dimensional dispersive longwave equationsrdquoAppliedMathematics E-Notes vol 8 pp 58ndash662008

[17] E Yusufoglu and A Bekir ldquoExact solutions of coupled non-linear Klein-Gordon equationsrdquo Mathematical and ComputerModelling vol 48 no 11-12 pp 1694ndash1700 2008

[18] Y Chen and Q Wang ldquoExtended Jacobi elliptic functionrational expansion method and abundant families of Jacobielliptic function solutions to (1+1)-dimensional dispersive longwave equationrdquo Chaos Solitons and Fractals vol 24 no 3 pp745ndash757 2005

[19] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[20] D Lu ldquoJacobi elliptic function solutions for two variant Boussi-nesq equationsrdquo Chaos Solitons and Fractals vol 24 no 5 pp1373ndash1385 2005

[21] M LWang X Li and J Zhang ldquoThe (1198661015840119866)-expansionmethod

and travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[22] S Zhang J L Tong and W Wang ldquoA generalized (1198661015840119866)-

expansion method for the mKdV equation with variable coeffi-cientsrdquo Physics Letters A vol 372 no 13 pp 2254ndash2257 2008

[23] E M E Zayed and K A Gepreel ldquoThe (119866

1015840

119866)-expansionmethod for finding traveling wave solutions of nonlinearpartial differential equations in mathematical physicsrdquo Journalof Mathematical Physics vol 50 no 1 pp 013502ndash013512 2009

[24] E M E Zayed ldquoThe (1198661015840119866)-expansionmethod and its applica-

tions to some nonlinear evolution equations in the mathemati-cal physicsrdquo Journal of AppliedMathematics andComputing vol30 no 1-2 pp 89ndash103 2009

[25] A Bekir ldquoApplication of the (1198661015840119866)-expansion method for

nonlinear evolution equationsrdquo Physics Letters A vol 372 no19 pp 3400ndash3406 2008

[26] B Ayhan and A Bekir ldquoThe (1198661015840119866)-expansion method for

the nonlinear lattice equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 9 pp 3490ndash34982012

[27] N A Kudryashov ldquoA note on the 119866

1015840

119866-expansion methodrdquoApplied Mathematics and Computation vol 217 no 4 pp 1755ndash1758 2010

[28] I Aslan ldquoA note on the (119866

1015840

119866)-expansion method againrdquoApplied Mathematics and Computation vol 217 no 2 pp 937ndash938 2010

[29] E M E Zayed ldquoTraveling wave solutions for higher dimen-sional nonlinear evolution equations using the (119866

1015840

119866)-expan-sion methodrdquo Journal of Applied Mathematics amp Informaticsvol 28 pp 383ndash395 2010

[30] N A Kudryashov ldquoMeromorphic solutions of nonlinear ordi-nary differential equationsrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 15 no 10 pp 2778ndash27902010

[31] S Zhang Y N Sun J M Ba and L Dong ldquoThe modified(119866

1015840

119866)-expansion method for nonlinear evolution equationsrdquoZeitschrift fur Naturforschung A vol 66 no 1-2 pp 33ndash39 2011

[32] L X Li E Q Li and M l Wang ldquoThe (119866

1015840

119866 1119866)-expansionmethod and its application to travelling wave solutions of the


Zakharov equationsrdquo Applied Mathematics vol 25 no 4 pp454ndash462 2010

[33] E M E Zayed and M A M Abdelaziz ldquoThe two-variable(119866

1015840

119866 1119866)-expansion method for solving the nonlinear KdV-mKdV equationrdquo Mathematical Problems in Engineering vol2012 Article ID 725061 14 pages 2012

[34] E M E Zayed S A Hoda Ibrahim and M A M Abde-laziz ldquoTraveling wave solutions of the nonlinear (3 + 1)-dimensional Kadomtsev-Petviashvili equation using the twovariables (119866

1015840

119866 1119866)-expansion methodrdquo Journal of AppliedMathematics vol 2012 Article ID 560531 8 pages 2012

[35] E M E Zayed and K A Gepreel ldquoThe modified (1198661015840119866)-

expansion method and its applications to construct exactsolutions for nonlinear PDEsrdquo WSEAS Transactions on Math-ematics vol 10 no 8 pp 270ndash278 2011

[36] J-M Zuo ldquoApplication of the extended (1198661015840119866)-expansion

method to solve the Pochhammer-Chree equationsrdquo AppliedMathematics and Computation vol 217 no 1 pp 376ndash383 2010

[37] E Infeld andGRowlandsNonlinearWaves Solitons andChaosCambridge University Press Cambridge UK 2000

[38] B Jang ldquoNew exact travelling wave solutions of nonlinearKlein-Gordon equationsrdquo Chaos Solitons and Fractals vol 41no 2 pp 646ndash654 2009

[39] Sirendaoreji ldquoAuxiliary equation method and new solutions ofKlein-Gordon equationsrdquo Chaos Solitons and Fractals vol 31no 4 pp 943ndash950 2007

[40] A M Wazwaz ldquoThe tanh and the sine-cosine methods forcompact and noncompact solutions of the nonlinear Klein-Gordon equationrdquo Applied Mathematics and Computation vol167 no 2 pp 1179ndash1195 2005

[41] A M Wazwaz ldquoCompactons solitons and periodic solutionsfor some forms of nonlinear Klein-Gordon equationsrdquo ChaosSolitons and Fractals vol 28 no 4 pp 1005ndash1013 2006

[42] A-M Wazwaz ldquoThe tanh-coth and the sine-cosine methodsfor kinks solitons and periodic solutions for the Pochhammer-Chree equationsrdquo Applied Mathematics and Computation vol195 no 1 pp 24ndash33 2008

[43] I L Bogolubsky ldquoSome examples of inelastic soliton interac-tionrdquo Computer Physics Communications vol 13 no 3 pp 149ndash155 1977

[44] P A Clarkson R J LeVeque and R Saxton ldquoSolitary-waveinteractions in elastic rodsrdquo Studies in AppliedMathematics vol75 no 2 pp 95ndash122 1986

[45] A Parker ldquoOn exact solutions of the regularized long-waveequation a direct approach to partially integrable equations ISolitarywave and solitonsrdquo Journal ofMathematical Physics vol36 no 7 pp 3498ndash3505 1995

[46] W Zhang and W Ma ldquoExplicit solitary-wave solutions to gen-eralized Pochhammer-Chree equationsrdquo Applied Mathematicsand Mechanics vol 20 no 6 pp 666ndash674 1999

[47] J Li and L Zhang ldquoBifurcations of traveling wave solutionsin generalized Pochhammer-Chree equationrdquo Chaos SolitonsAND Fractals vol 14 no 4 pp 581ndash593 2002

[48] B Li Y Chen and H Zhang ldquoExplicit exact solutions for newgeneral two-dimensional KdV-type and two-dimensional KdV-Burgers-type equations with nonlinear terms of any orderrdquoJournal of Physics A Mathematical and General vol 35 no 39pp 8253ndash8265 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Wazwaz [40 41] investigated the nonlinear Klein-Gordonequations and foundmany types of exact traveling wave solu-tions including compact solutions soliton solution solitarypatterns solutions and periodic solutions using the tanh-function method Zayed and Gepreel [35] have found theexact solutions of (1) using the (119866


Equation (2) represents nonlinear models of longitudinalwave propagation of elastic rods see [42ndash47] Zuo [36] hasdiscussed (2) using the extended (119866


and found the travelingwave solutions of these equationsTherest of this paper is organized as follows In Section 2 we givethe description of the two variable (119866

1015840119866 1119866)-expansion

method In Section 3 we apply this method to solve (1) and(2) In Section 4 some conclusions are given

2 Description of the Two Variable(1198661015840119866 1119866)-Expansion Method

Before we describe the main steps of this method we needthe following remarks (see [32ndash34])

Remark 1 If we consider the second order linear ODE

11986610158401015840

(120585) + 120582119866 (120585) = 120583 (3)

and set 120601 = 1198661015840119866 120595 = 1119866 then we get

1206011015840

= minus1206012

+ 120583120595 minus 120582 1205951015840

= minus120601120595 (4)

where 120582 and 120583 are constants

Remark 2 If 120582 lt 0 then the general solution of (3) has thefollowing form

119866 (120585) = 1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) +

120583

120582

(5)

where 1198601and 119860

2are arbitrary constants Consequently we

have

1205952

=minus120582

12058221205901

+ 1205832

(1206012

minus 2120583120595 + 120582) (6)

where 1205901

= 1198602

1minus 1198602

2

Remark 3 If 120582 gt 0 then the general solution of (3) has thefollowing form

119866 (120585) = 1198601sin (120585radic120582) + 119860

2cos (120585radic120582) +

120583

120582

(7)

and hence

1205952

=120582

12058221205902

minus 1205832

(1206012

minus 2120583120595 + 120582) (8)

where 1205902

= 1198602

1+ 1198602

2

Remark 4 If 120582 = 0 then the general solution of (3) has thefollowing form

119866 (120585) =

120583

2

1205852

+ 1198601120585 + 119860

2 (9)

and hence

1205952

=1

1198602

1minus 2120583119860

2

(1206012

minus 2120583120595) (10)

Suppose we have the following nonlinear evolution equation

119865 (119906 119906119905 119906119909 119906119909119909

) = 0 (11)

where 119865 is a polynomial in 119906(119909 119905) and its partial derivativesIn the following we give the main steps of the (119866

1015840119866 1119866)-

expansion method [32ndash34]

Step 1 The traveling wave transformation

119906 (119909 119905) = 119906 (120585) 120585 = 119909 minus 119862119905 (12)

where 119862 is a constant and reduces (11) to an ODE in thefollowing form

119875 (119906 1199061015840 11990610158401015840 ) = 0 (13)

where 119875 is a polynomial of 119906(120585) and its total derivatives withrespect to 120585

Step 2 Assuming that the solution of (13) can be expressedby a polynomial in the two variables 120601 and 120595 as follows

119906 (120585) =

119873

sum

119894=0

119886119894120601119894+

119873

sum

119894=1

119887119894120601119894minus1

120595 (14)

where 119886119894

(119894 = 0 1 2 119873) and 119887119894

(119894 = 1 2 119873) areconstants to be determined later

Step 3 Determine the positive integer 119873 in (14) by using thehomogeneous balance between the highest-order derivativesand the nonlinear terms in (13) In some nonlinear equationsthe balance number 119873 is not a positive integer In this casewe make the following transformations [48]

(a) when 119873 = 119902119901 where 119902119901 is a fraction in the lowestterms we let

119906 (120585) = V119902119901 (120585) (15)

then substitute (15) into (13) to get a new equation inthe new function V(120585) with a positive integer balancenumber

(b) when 119873 is a negative number we let

119906 (120585) = V119873 (120585) (16)

and substitute (16) into (13) to get a new equation inthe new function V(120585) with a positive integer balancenumber

Step 4 Substituting (14) into (13) along with (4) and (6) theleft-hand side of (13) can be converted into a polynomial in 120601

and 120595 in which the degree of 120595 is no longer than 1 Equatingeach coefficients of this polynomial to zero yields a system of



119894 119887119894 119862 120583 119860

1 1198602 and 120582

where 120582 lt 0


3 Applications



(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

+ 1205741199062119899minus1

= 0 119899 gt 2 (17)




119906 (120585) = V1(119899minus1) (120585) (18)


(1198622

minus 1198962) [

2 minus 119899

(119899 minus 1)2(V1015840)2

+1

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V3 + 120574V4

= 0

(19)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (20)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014 1205741198864

1+

2 (1198622

minus 1198962) 1198862

1

(119899 minus 1)

+

2 (1198622

minus 1198962) 1198862

1

(119899 minus 1)2

+

12058221205741198874

1

(12058221205901

+ 1205832)2

minus

(1198622

minus 1198962) 1198991198862

1

(119899 minus 1)2

minus

61205821205741198862

11198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1205821198872

1

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1205821198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 1198991205821198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206013 minus 120573119886

3

1+ 4120574119886

01198863

1+

2 (1198622

minus 1198962) 11988601198861

(119899 minus 1)

+

312057312058211988611198872

1

(12058221205901

+ 1205832)

minus

12120582120574119886011988611198872

1

(12058221205901

+ 1205832)

minus

8120582212057412058311988611198873

1

(12058221205901

+ 1205832)2

+

4 (1198622

minus 1198962) 12058212058311988611198871

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 12058212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582120583119886

11198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206013120595 4120574119886

3

11198871

+

4 (1198622

minus 1198962) 11988611198871

(119899 minus 1)

+

4 (1198622

minus 1198962) 11988611198871

(119899 minus 1)2

minus

2 (1198622

minus 1198962) 11989911988611198871

(119899 minus 1)2

minus

412058212057411988611198873

1

(12058221205901

+ 1205832)

= 0

1206012 1205721198862

1minus 3120573119886

01198862

1+ 6120574119886

2

01198862

1+

212058231205741198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 1205821198862

1

(119899 minus 1)

+

4 (1198622

minus 1198962) 1205821198862

1

(119899 minus 1)2

minus

1205721205821198872

1

(12058221205901

+ 1205832)

+

212057312058221205831198873

1

(12058221205901

+ 1205832)2

minus

3 (1198622

minus 1198962) 12058221198872

1

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 12058221198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

61205821205741198862

01198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1198991205821198862

1

(119899 minus 1)2

minus

4120582312057412058321198874

1

(12058221205901

+ 1205832)3

minus

612058221205741198862

11198872

1

(12058221205901

+ 1205832)

+

312057312058211988601198872

1

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 11989912058221198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 12058212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

8120582212057412058311988601198873

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 120582212058321198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

+

(1198622

minus 1198962) 11989912058212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 12058212058311988601198871

(119899 minus 1) (12058221205901

+ 1205832)

= 0


1206012120595

2 (1198622

minus 1198962) 11988601198871

(119899 minus 1)

minus 31205731198862

11198871

minus

3 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)

minus

4 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)2

+ 1212057411988601198862

11198871

+

1205731205821198873

1

(12058221205901

+ 1205832)

minus

412058221205741205831198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 1198991205831198862

1

(119899 minus 1)2

minus

412058212057411988601198873

1

(12058221205901

+ 1205832)

+

5 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

11198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582120583119887

2

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601 minus 31205731198862

01198861

+ 41205741198863

01198861

+ 212057211988601198861

+

2 (1198622

minus 1198962) 12058211988601198861

(119899 minus 1)

+

3120573120582211988611198872

1

(12058221205901

+ 1205832)

minus

8120582312057412058311988611198873

1

(12058221205901

+ 1205832)2

minus

121205822120574119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601120595 212057211988611198871

minus 6120573119886011988611198871

+ 121205741198862

011988611198871

minus

3 (1198622

minus 1198962) 12058311988601198861

(119899 minus 1)

+

3 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)

+

4 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)2

minus

4120582212057411988611198873

1

(12058221205901

+ 1205832)

+

161205822120574120583211988611198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988611198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058211988611198871

(119899 minus 1)2

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

+

24120582120574120583119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 119899120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120595 41205741198863

01198871

minus 31205731198862

01198871

+ 212057211988601198871

+

12057312058221198873

1

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 12058211988601198871

(119899 minus 1)

minus

412058231205741205831198874

1

(12058221205901

+ 1205832)2

minus

4120582212057411988601198873

1

(12058221205901

+ 1205832)

minus

4 (1198622

minus 1198962) 1205821205831198862

1

(119899 minus 1)2

minus

4120573120582212058321198873

1

(12058221205901

+ 1205832)2

+

8120582312057412058331198874

1

(12058221205901

+ 1205832)3

+

21205721205821205831198872

1

(12058221205901

+ 1205832)

+

161205822120574120583211988601198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988601198872

1

(12058221205901

+ 1205832)

+

3 (1198622

minus 1198962) 12058221205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 12058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

01198872

1

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 119899120582120583119886

2

1

(119899 minus 1)2

minus

4 (1198622

minus 1198962) 120582212058331198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582120583211988601198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206010 1205721198862

0minus 1205731198863

0+ 1205741198864

0minus

12057212058221198872

1

(12058221205901

+ 1205832)

+

12058241205741198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 12058221198862

1

(119899 minus 1)2

+

212057312058231205831198873

1

(12058221205901

+ 1205832)2

minus

(1198622

minus 1198962) 12058231198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

3120573120582211988601198872

1

(12058221205901

+ 1205832)

minus

4120582412057412058321198874

1

(12058221205901

+ 1205832)3

minus

612058221205741198862

01198872

1

(12058221205901

+ 1205832)

minus

(1198622

minus 1198962) 11989912058221198862

1

(119899 minus 1)2

minus

8120582312057412058311988601198873

1

(12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 120582312058321198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

+

(1198622

minus 1198962) 120582212058311988601198871

(119899 minus 1) (12058221205901

+ 1205832)

+

(1198622

minus 1198962) 119899120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

(21)



1198860

= 0

1198861

= 0

1198871

=

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(12058221205901

+ 1205832)

1205832120574(119899 + 1)

2

(22)

where 120583 = 0 120574 = 0 and 120573 = 0



119906 (120585)

= [

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)

times (1

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]

1(119899minus1)

(23)


2


1198860

=

119899120573

2120574 (119899 + 1)

1198861

= plusmn

119899120573radicminus1120582

2120574 (119899 + 1)

1198871

= plusmn

119899120573radic12058221205901

+ 1205832

2120574120582 (119899 + 1)

120572 =

1198991205732

120574(119899 + 1)2


2120572

120582

+ 1198962

(24)


(17) as follows

119906 (120585)

=[[

[

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)

times (

1198601cosh (120585radicminus120582) + 119860


1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)

plusmn

119899120573

2120582120574 (119899 + 1)

times (

radic12058221205901

+ 1205832

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]]

]

1(119899minus1)

(25)


2

8000

6000

4000

2000

0000

8000

6000

4000

2000

0

minus4

minus2

0

2

4 4

2

0

minus2

minus4

xt



= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)


(26)

while if 1198601

= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)

(27)





1198860

= 0

1198861

= 0

1198871

=

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(1205832

minus 12058221205902)

1205832120574(119899 + 1)

2

(28)



119906 (120585) = [

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)

times (1

1198601sin(120585radic120582) + 119860

2cos(120585radic120582) + 120583120582

) ]

1(119899minus1)

(29)


2




1198860

= 0

1198861

= 0

1198871

=

119899120573 (21205831198602

minus 1198602

1)

120583120574 (119899 + 1)


2(2120583119860

2minus 1198602

1)

1205832120574(119899 + 1)

2+ 1198962

120572 = 0

(30)


119906 (120585) = [

119899120573(21205831198602

minus 1198602

1)

120583120574(119899 + 1)

(1

(1205832)1205852

+ 1198601120585 + 119860

2

)]

1(119899minus1)

(31)


2(21205831198602

minus 1198602

1)1205832120574(119899 + 1)

2+ 1198962


(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

= 0 119899 gt 2 (32)



119906 (120585) = V2(119899minus1) (120585) (33)


(1198622

minus 1198962) [

2 (3 minus 119899)

(119899 minus 1)2

(V1015840)2

+2

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V4 = 0

(34)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (35)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(36)



119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sinh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)

+1198602cosh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(37)

where 120585 = 119909 minus 119862119905


14

12

10

08

06

04

02

minus4

minus2 minus2

minus4

0 0

2 2

4 4

xt

4 minus4



= 0 and 1198602


119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(38)

while if 1198601

= 0 and 1198602


119906 (120585)


2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(39)




1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(40)where 120572 = 0 and 120573 = 0


119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sin(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)

+ 1198602cos(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(41)


1= 0 and119860



119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(42)while if 119860

1= 0 and 119860


solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(43)




1198860

= 0

1198861

= plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

1198871

= 1198871

119862 = 119862

120583 =

1198602

1(1198622

minus 1198962) (119899 + 1) minus 2120573119887

2

1(119899 minus 1)

2

21198601

(119899 + 1) (1198622

minus 1198962)

(44)

where 120573 = 0



119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905


119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =


119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 119887119894 119862 120583 119860

1 1198602 and 120582

where 120582 lt 0


3 Applications



(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

+ 1205741199062119899minus1

= 0 119899 gt 2 (17)




119906 (120585) = V1(119899minus1) (120585) (18)


(1198622

minus 1198962) [

2 minus 119899

(119899 minus 1)2(V1015840)2

+1

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V3 + 120574V4

= 0

(19)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (20)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014 1205741198864

1+

2 (1198622

minus 1198962) 1198862

1

(119899 minus 1)

+

2 (1198622

minus 1198962) 1198862

1

(119899 minus 1)2

+

12058221205741198874

1

(12058221205901

+ 1205832)2

minus

(1198622

minus 1198962) 1198991198862

1

(119899 minus 1)2

minus

61205821205741198862

11198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1205821198872

1

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1205821198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 1198991205821198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206013 minus 120573119886

3

1+ 4120574119886

01198863

1+

2 (1198622

minus 1198962) 11988601198861

(119899 minus 1)

+

312057312058211988611198872

1

(12058221205901

+ 1205832)

minus

12120582120574119886011988611198872

1

(12058221205901

+ 1205832)

minus

8120582212057412058311988611198873

1

(12058221205901

+ 1205832)2

+

4 (1198622

minus 1198962) 12058212058311988611198871

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 12058212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582120583119886

11198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206013120595 4120574119886

3

11198871

+

4 (1198622

minus 1198962) 11988611198871

(119899 minus 1)

+

4 (1198622

minus 1198962) 11988611198871

(119899 minus 1)2

minus

2 (1198622

minus 1198962) 11989911988611198871

(119899 minus 1)2

minus

412058212057411988611198873

1

(12058221205901

+ 1205832)

= 0

1206012 1205721198862

1minus 3120573119886

01198862

1+ 6120574119886

2

01198862

1+

212058231205741198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 1205821198862

1

(119899 minus 1)

+

4 (1198622

minus 1198962) 1205821198862

1

(119899 minus 1)2

minus

1205721205821198872

1

(12058221205901

+ 1205832)

+

212057312058221205831198873

1

(12058221205901

+ 1205832)2

minus

3 (1198622

minus 1198962) 12058221198872

1

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 12058221198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

61205821205741198862

01198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 1198991205821198862

1

(119899 minus 1)2

minus

4120582312057412058321198874

1

(12058221205901

+ 1205832)3

minus

612058221205741198862

11198872

1

(12058221205901

+ 1205832)

+

312057312058211988601198872

1

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 11989912058221198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 12058212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

8120582212057412058311988601198873

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 120582212058321198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

+

(1198622

minus 1198962) 11989912058212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 12058212058311988601198871

(119899 minus 1) (12058221205901

+ 1205832)

= 0


1206012120595

2 (1198622

minus 1198962) 11988601198871

(119899 minus 1)

minus 31205731198862

11198871

minus

3 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)

minus

4 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)2

+ 1212057411988601198862

11198871

+

1205731205821198873

1

(12058221205901

+ 1205832)

minus

412058221205741205831198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 1198991205831198862

1

(119899 minus 1)2

minus

412058212057411988601198873

1

(12058221205901

+ 1205832)

+

5 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

11198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582120583119887

2

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601 minus 31205731198862

01198861

+ 41205741198863

01198861

+ 212057211988601198861

+

2 (1198622

minus 1198962) 12058211988601198861

(119899 minus 1)

+

3120573120582211988611198872

1

(12058221205901

+ 1205832)

minus

8120582312057412058311988611198873

1

(12058221205901

+ 1205832)2

minus

121205822120574119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601120595 212057211988611198871

minus 6120573119886011988611198871

+ 121205741198862

011988611198871

minus

3 (1198622

minus 1198962) 12058311988601198861

(119899 minus 1)

+

3 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)

+

4 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)2

minus

4120582212057411988611198873

1

(12058221205901

+ 1205832)

+

161205822120574120583211988611198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988611198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058211988611198871

(119899 minus 1)2

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

+

24120582120574120583119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 119899120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120595 41205741198863

01198871

minus 31205731198862

01198871

+ 212057211988601198871

+

12057312058221198873

1

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 12058211988601198871

(119899 minus 1)

minus

412058231205741205831198874

1

(12058221205901

+ 1205832)2

minus

4120582212057411988601198873

1

(12058221205901

+ 1205832)

minus

4 (1198622

minus 1198962) 1205821205831198862

1

(119899 minus 1)2

minus

4120573120582212058321198873

1

(12058221205901

+ 1205832)2

+

8120582312057412058331198874

1

(12058221205901

+ 1205832)3

+

21205721205821205831198872

1

(12058221205901

+ 1205832)

+

161205822120574120583211988601198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988601198872

1

(12058221205901

+ 1205832)

+

3 (1198622

minus 1198962) 12058221205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 12058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

01198872

1

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 119899120582120583119886

2

1

(119899 minus 1)2

minus

4 (1198622

minus 1198962) 120582212058331198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582120583211988601198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206010 1205721198862

0minus 1205731198863

0+ 1205741198864

0minus

12057212058221198872

1

(12058221205901

+ 1205832)

+

12058241205741198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 12058221198862

1

(119899 minus 1)2

+

212057312058231205831198873

1

(12058221205901

+ 1205832)2

minus

(1198622

minus 1198962) 12058231198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

3120573120582211988601198872

1

(12058221205901

+ 1205832)

minus

4120582412057412058321198874

1

(12058221205901

+ 1205832)3

minus

612058221205741198862

01198872

1

(12058221205901

+ 1205832)

minus

(1198622

minus 1198962) 11989912058221198862

1

(119899 minus 1)2

minus

8120582312057412058311988601198873

1

(12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 120582312058321198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

+

(1198622

minus 1198962) 120582212058311988601198871

(119899 minus 1) (12058221205901

+ 1205832)

+

(1198622

minus 1198962) 119899120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

(21)



1198860

= 0

1198861

= 0

1198871

=

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(12058221205901

+ 1205832)

1205832120574(119899 + 1)

2

(22)

where 120583 = 0 120574 = 0 and 120573 = 0



119906 (120585)

= [

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)

times (1

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]

1(119899minus1)

(23)


2


1198860

=

119899120573

2120574 (119899 + 1)

1198861

= plusmn

119899120573radicminus1120582

2120574 (119899 + 1)

1198871

= plusmn

119899120573radic12058221205901

+ 1205832

2120574120582 (119899 + 1)

120572 =

1198991205732

120574(119899 + 1)2


2120572

120582

+ 1198962

(24)


(17) as follows

119906 (120585)

=[[

[

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)

times (

1198601cosh (120585radicminus120582) + 119860


1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)

plusmn

119899120573

2120582120574 (119899 + 1)

times (

radic12058221205901

+ 1205832

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]]

]

1(119899minus1)

(25)


2

8000

6000

4000

2000

0000

8000

6000

4000

2000

0

minus4

minus2

0

2

4 4

2

0

minus2

minus4

xt



= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)


(26)

while if 1198601

= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)

(27)





1198860

= 0

1198861

= 0

1198871

=

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(1205832

minus 12058221205902)

1205832120574(119899 + 1)

2

(28)



119906 (120585) = [

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)

times (1

1198601sin(120585radic120582) + 119860

2cos(120585radic120582) + 120583120582

) ]

1(119899minus1)

(29)


2




1198860

= 0

1198861

= 0

1198871

=

119899120573 (21205831198602

minus 1198602

1)

120583120574 (119899 + 1)


2(2120583119860

2minus 1198602

1)

1205832120574(119899 + 1)

2+ 1198962

120572 = 0

(30)


119906 (120585) = [

119899120573(21205831198602

minus 1198602

1)

120583120574(119899 + 1)

(1

(1205832)1205852

+ 1198601120585 + 119860

2

)]

1(119899minus1)

(31)


2(21205831198602

minus 1198602

1)1205832120574(119899 + 1)

2+ 1198962


(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

= 0 119899 gt 2 (32)



119906 (120585) = V2(119899minus1) (120585) (33)


(1198622

minus 1198962) [

2 (3 minus 119899)

(119899 minus 1)2

(V1015840)2

+2

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V4 = 0

(34)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (35)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(36)



119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sinh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)

+1198602cosh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(37)

where 120585 = 119909 minus 119862119905


14

12

10

08

06

04

02

minus4

minus2 minus2

minus4

0 0

2 2

4 4

xt

4 minus4



= 0 and 1198602


119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(38)

while if 1198601

= 0 and 1198602


119906 (120585)


2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(39)




1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(40)where 120572 = 0 and 120573 = 0


119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sin(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)

+ 1198602cos(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(41)


1= 0 and119860



119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(42)while if 119860

1= 0 and 119860


solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(43)




1198860

= 0

1198861

= plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

1198871

= 1198871

119862 = 119862

120583 =

1198602

1(1198622

minus 1198962) (119899 + 1) minus 2120573119887

2

1(119899 minus 1)

2

21198601

(119899 + 1) (1198622

minus 1198962)

(44)

where 120573 = 0



119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905


119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =


119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1206012120595

2 (1198622

minus 1198962) 11988601198871

(119899 minus 1)

minus 31205731198862

11198871

minus

3 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)

minus

4 (1198622

minus 1198962) 1205831198862

1

(119899 minus 1)2

+ 1212057411988601198862

11198871

+

1205731205821198873

1

(12058221205901

+ 1205832)

minus

412058221205741205831198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 1198991205831198862

1

(119899 minus 1)2

minus

412058212057411988601198873

1

(12058221205901

+ 1205832)

+

5 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 1205821205831198872

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

11198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582120583119887

2

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601 minus 31205731198862

01198861

+ 41205741198863

01198861

+ 212057211988601198861

+

2 (1198622

minus 1198962) 12058211988601198861

(119899 minus 1)

+

3120573120582211988611198872

1

(12058221205901

+ 1205832)

minus

8120582312057412058311988611198873

1

(12058221205901

+ 1205832)2

minus

121205822120574119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 119899120582212058311988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120601120595 212057211988611198871

minus 6120573119886011988611198871

+ 121205741198862

011988611198871

minus

3 (1198622

minus 1198962) 12058311988601198861

(119899 minus 1)

+

3 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)

+

4 (1198622

minus 1198962) 12058211988611198871

(119899 minus 1)2

minus

4120582212057411988611198873

1

(12058221205901

+ 1205832)

+

161205822120574120583211988611198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988611198872

1

(12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058211988611198871

(119899 minus 1)2

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

8 (1198622

minus 1198962) 120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

+

24120582120574120583119886011988611198872

1

(12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 119899120582120583211988611198871

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

120595 41205741198863

01198871

minus 31205731198862

01198871

+ 212057211988601198871

+

12057312058221198873

1

(12058221205901

+ 1205832)

+

(1198622

minus 1198962) 12058211988601198871

(119899 minus 1)

minus

412058231205741205831198874

1

(12058221205901

+ 1205832)2

minus

4120582212057411988601198873

1

(12058221205901

+ 1205832)

minus

4 (1198622

minus 1198962) 1205821205831198862

1

(119899 minus 1)2

minus

4120573120582212058321198873

1

(12058221205901

+ 1205832)2

+

8120582312057412058331198874

1

(12058221205901

+ 1205832)3

+

21205721205821205831198872

1

(12058221205901

+ 1205832)

+

161205822120574120583211988601198873

1

(12058221205901

+ 1205832)2

minus

612057312058212058311988601198872

1

(12058221205901

+ 1205832)

+

3 (1198622

minus 1198962) 12058221205831198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

4 (1198622

minus 1198962) 12058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

121205821205741205831198862

01198872

1

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 119899120582120583119886

2

1

(119899 minus 1)2

minus

4 (1198622

minus 1198962) 120582212058331198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582120583211988601198871

(119899 minus 1) (12058221205901

+ 1205832)

minus

2 (1198622

minus 1198962) 11989912058212058331198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

1206010 1205721198862

0minus 1205731198863

0+ 1205741198864

0minus

12057212058221198872

1

(12058221205901

+ 1205832)

+

12058241205741198874

1

(12058221205901

+ 1205832)2

+

2 (1198622

minus 1198962) 12058221198862

1

(119899 minus 1)2

+

212057312058231205831198873

1

(12058221205901

+ 1205832)2

minus

(1198622

minus 1198962) 12058231198872

1

(119899 minus 1) (12058221205901

+ 1205832)

+

3120573120582211988601198872

1

(12058221205901

+ 1205832)

minus

4120582412057412058321198874

1

(12058221205901

+ 1205832)3

minus

612058221205741198862

01198872

1

(12058221205901

+ 1205832)

minus

(1198622

minus 1198962) 11989912058221198862

1

(119899 minus 1)2

minus

8120582312057412058311988601198873

1

(12058221205901

+ 1205832)2

minus

2 (1198622

minus 1198962) 120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

+

2 (1198622

minus 1198962) 120582312058321198872

1

(119899 minus 1) (12058221205901

+ 1205832)2

+

(1198622

minus 1198962) 120582212058311988601198871

(119899 minus 1) (12058221205901

+ 1205832)

+

(1198622

minus 1198962) 119899120582212058321198862

1

(119899 minus 1)2

(12058221205901

+ 1205832)

= 0

(21)



1198860

= 0

1198861

= 0

1198871

=

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(12058221205901

+ 1205832)

1205832120574(119899 + 1)

2

(22)

where 120583 = 0 120574 = 0 and 120573 = 0



119906 (120585)

= [

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)

times (1

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]

1(119899minus1)

(23)


2


1198860

=

119899120573

2120574 (119899 + 1)

1198861

= plusmn

119899120573radicminus1120582

2120574 (119899 + 1)

1198871

= plusmn

119899120573radic12058221205901

+ 1205832

2120574120582 (119899 + 1)

120572 =

1198991205732

120574(119899 + 1)2


2120572

120582

+ 1198962

(24)


(17) as follows

119906 (120585)

=[[

[

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)

times (

1198601cosh (120585radicminus120582) + 119860


1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)

plusmn

119899120573

2120582120574 (119899 + 1)

times (

radic12058221205901

+ 1205832

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]]

]

1(119899minus1)

(25)


2

8000

6000

4000

2000

0000

8000

6000

4000

2000

0

minus4

minus2

0

2

4 4

2

0

minus2

minus4

xt



= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)


(26)

while if 1198601

= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)

(27)





1198860

= 0

1198861

= 0

1198871

=

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(1205832

minus 12058221205902)

1205832120574(119899 + 1)

2

(28)



119906 (120585) = [

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)

times (1

1198601sin(120585radic120582) + 119860

2cos(120585radic120582) + 120583120582

) ]

1(119899minus1)

(29)


2




1198860

= 0

1198861

= 0

1198871

=

119899120573 (21205831198602

minus 1198602

1)

120583120574 (119899 + 1)


2(2120583119860

2minus 1198602

1)

1205832120574(119899 + 1)

2+ 1198962

120572 = 0

(30)


119906 (120585) = [

119899120573(21205831198602

minus 1198602

1)

120583120574(119899 + 1)

(1

(1205832)1205852

+ 1198601120585 + 119860

2

)]

1(119899minus1)

(31)


2(21205831198602

minus 1198602

1)1205832120574(119899 + 1)

2+ 1198962


(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

= 0 119899 gt 2 (32)



119906 (120585) = V2(119899minus1) (120585) (33)


(1198622

minus 1198962) [

2 (3 minus 119899)

(119899 minus 1)2

(V1015840)2

+2

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V4 = 0

(34)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (35)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(36)



119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sinh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)

+1198602cosh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(37)

where 120585 = 119909 minus 119862119905


14

12

10

08

06

04

02

minus4

minus2 minus2

minus4

0 0

2 2

4 4

xt

4 minus4



= 0 and 1198602


119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(38)

while if 1198601

= 0 and 1198602


119906 (120585)


2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(39)




1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(40)where 120572 = 0 and 120573 = 0


119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sin(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)

+ 1198602cos(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(41)


1= 0 and119860



119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(42)while if 119860

1= 0 and 119860


solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(43)




1198860

= 0

1198861

= plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

1198871

= 1198871

119862 = 119862

120583 =

1198602

1(1198622

minus 1198962) (119899 + 1) minus 2120573119887

2

1(119899 minus 1)

2

21198601

(119899 + 1) (1198622

minus 1198962)

(44)

where 120573 = 0



119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905


119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =


119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (120585)

= [

119899120573 (12058221205901

+ 1205832)

120583120582120574 (119899 + 1)

times (1

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]

1(119899minus1)

(23)


2


1198860

=

119899120573

2120574 (119899 + 1)

1198861

= plusmn

119899120573radicminus1120582

2120574 (119899 + 1)

1198871

= plusmn

119899120573radic12058221205901

+ 1205832

2120574120582 (119899 + 1)

120572 =

1198991205732

120574(119899 + 1)2


2120572

120582

+ 1198962

(24)


(17) as follows

119906 (120585)

=[[

[

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)

times (

1198601cosh (120585radicminus120582) + 119860


1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)

plusmn

119899120573

2120582120574 (119899 + 1)

times (

radic12058221205901

+ 1205832

1198601sinh (120585radicminus120582) + 119860

2cosh (120585radicminus120582) + 120583120582

)]]

]

1(119899minus1)

(25)


2

8000

6000

4000

2000

0000

8000

6000

4000

2000

0

minus4

minus2

0

2

4 4

2

0

minus2

minus4

xt



= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)


(26)

while if 1198601

= 0 1198602


119906 (120585) = [

119899120573

2120574 (119899 + 1)

plusmn

119899120573

2120574 (119899 + 1)


1(119899minus1)

(27)





1198860

= 0

1198861

= 0

1198871

=

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(1205832

minus 12058221205902)

1205832120574(119899 + 1)

2

(28)



119906 (120585) = [

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)

times (1

1198601sin(120585radic120582) + 119860

2cos(120585radic120582) + 120583120582

) ]

1(119899minus1)

(29)


2




1198860

= 0

1198861

= 0

1198871

=

119899120573 (21205831198602

minus 1198602

1)

120583120574 (119899 + 1)


2(2120583119860

2minus 1198602

1)

1205832120574(119899 + 1)

2+ 1198962

120572 = 0

(30)


119906 (120585) = [

119899120573(21205831198602

minus 1198602

1)

120583120574(119899 + 1)

(1

(1205832)1205852

+ 1198601120585 + 119860

2

)]

1(119899minus1)

(31)


2(21205831198602

minus 1198602

1)1205832120574(119899 + 1)

2+ 1198962


(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

= 0 119899 gt 2 (32)



119906 (120585) = V2(119899minus1) (120585) (33)


(1198622

minus 1198962) [

2 (3 minus 119899)

(119899 minus 1)2

(V1015840)2

+2

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V4 = 0

(34)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (35)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(36)



119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sinh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)

+1198602cosh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(37)

where 120585 = 119909 minus 119862119905


14

12

10

08

06

04

02

minus4

minus2 minus2

minus4

0 0

2 2

4 4

xt

4 minus4



= 0 and 1198602


119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(38)

while if 1198601

= 0 and 1198602


119906 (120585)


2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(39)




1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(40)where 120572 = 0 and 120573 = 0


119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sin(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)

+ 1198602cos(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(41)


1= 0 and119860



119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(42)while if 119860

1= 0 and 119860


solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(43)




1198860

= 0

1198861

= plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

1198871

= 1198871

119862 = 119862

120583 =

1198602

1(1198622

minus 1198962) (119899 + 1) minus 2120573119887

2

1(119899 minus 1)

2

21198601

(119899 + 1) (1198622

minus 1198962)

(44)

where 120573 = 0



119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905


119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =


119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198860

= 0

1198861

= 0

1198871

=

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)


2120572

120582

+ 1198962

120572 =

1198991205732

(1205832

minus 12058221205902)

1205832120574(119899 + 1)

2

(28)



119906 (120585) = [

119899120573 (1205832

minus 12058221205902)

120583120582120574 (119899 + 1)

times (1

1198601sin(120585radic120582) + 119860

2cos(120585radic120582) + 120583120582

) ]

1(119899minus1)

(29)


2




1198860

= 0

1198861

= 0

1198871

=

119899120573 (21205831198602

minus 1198602

1)

120583120574 (119899 + 1)


2(2120583119860

2minus 1198602

1)

1205832120574(119899 + 1)

2+ 1198962

120572 = 0

(30)


119906 (120585) = [

119899120573(21205831198602

minus 1198602

1)

120583120574(119899 + 1)

(1

(1205832)1205852

+ 1198601120585 + 119860

2

)]

1(119899minus1)

(31)


2(21205831198602

minus 1198602

1)1205832120574(119899 + 1)

2+ 1198962


(1198622

minus 1198962) 11990610158401015840

+ 120572119906 minus 120573119906119899

= 0 119899 gt 2 (32)



119906 (120585) = V2(119899minus1) (120585) (33)


(1198622

minus 1198962) [

2 (3 minus 119899)

(119899 minus 1)2

(V1015840)2

+2

(119899 minus 1)

VV10158401015840] + 120572V2 minus 120573V4 = 0

(34)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (35)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(36)



119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sinh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585)

+1198602cosh(

(119899 minus 1)

2radicminus

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(37)

where 120585 = 119909 minus 119862119905


14

12

10

08

06

04

02

minus4

minus2 minus2

minus4

0 0

2 2

4 4

xt

4 minus4



= 0 and 1198602


119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(38)

while if 1198601

= 0 and 1198602


119906 (120585)


2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(39)




1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(40)where 120572 = 0 and 120573 = 0


119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sin(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)

+ 1198602cos(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(41)


1= 0 and119860



119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(42)while if 119860

1= 0 and 119860


solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(43)




1198860

= 0

1198861

= plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

1198871

= 1198871

119862 = 119862

120583 =

1198602

1(1198622

minus 1198962) (119899 + 1) minus 2120573119887

2

1(119899 minus 1)

2

21198601

(119899 + 1) (1198622

minus 1198962)

(44)

where 120573 = 0



119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905


119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =


119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










14

12

10

08

06

04

02

minus4

minus2 minus2

minus4

0 0

2 2

4 4

xt

4 minus4



= 0 and 1198602


119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(38)

while if 1198601

= 0 and 1198602


119906 (120585)


2120573


2radicminus

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(39)




1198860

= 0

1198861

= 0

1198871


(119899 + 1)

2120573

119862 = 119862

120583 = 0

120582 =120572(119899 minus 1)

2

4 (1198622

minus 1198962)

(40)where 120572 = 0 and 120573 = 0


119906 (120585) = [

[


(119899 + 1)

2120573

times (1198601sin(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585)

+ 1198602cos(

(119899 minus 1)

2radic

120572

(1198622

minus 1198962)

120585))

minus1

]

]

2(119899minus1)

(41)


1= 0 and119860



119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(42)while if 119860

1= 0 and 119860


solutions

119906 (120585) = [plusmnradic120572(119899 + 1)

2120573


2radic

120572

(1198622

minus 1198962)

120585)]

2(119899minus1)

(43)




1198860

= 0

1198861

= plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

1198871

= 1198871

119862 = 119862

120583 =

1198602

1(1198622

minus 1198962) (119899 + 1) minus 2120573119887

2

1(119899 minus 1)

2

21198601

(119899 + 1) (1198622

minus 1198962)

(44)

where 120573 = 0



119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905


119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =


119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (120585) =[[

[

plusmn1

119899 minus 1

radic(1198622

minus 1198962) (119899 + 1)

2120573

times (

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

+ (1198871

(1205832)1205852

+ 1198601120585 + 119860

2

)]]

]

2(119899minus1)

(45)

where 120585 = 119909 minus 119862119905


119862211990610158401015840

minus 11986221199061015840101584010158401015840

minus (120572119906 + 120573119906119899+1

+ 1205741199062119899+1

)

10158401015840

= 0 (46)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

minus 1205741199062119899+1

= 0 (47)


10158401015840 and 1199062119899+1 in (47) we get 119873 =


119906 (120585) = V1119899 (120585) (48)


(1198622

minus 120572) 1198992V2 minus 119862

2119899VV10158401015840 minus 119862

2

(1 minus 119899) (V1015840)2

minus 1205731198992V3 minus 120574119899

2V4

= 0

(49)


V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (50)

where 1198860 1198861 and 119887





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

12059011205822

+ 1205832

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1+

11986221198991205821198872

1

12059011205822

+ 1205832

minus

119899212058221205741198874

1

(12059011205822

+ 1205832)2

+

611989921205821205741198862

11198872

1

12059011205822

+ 1205832

= 0

1206013

3119899212057312058211988611198872

1

12059011205822

+ 1205832

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

12059011205822

+ 1205832

+

121198992120582120574119886011988611198872

1

12059011205822

+ 1205832

minus

2119862211989912058212058311988611198871

12059011205822

+ 1205832

+

81198992120582212057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

12059011205822

+ 1205832

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

12059011205822

+ 1205832

minus

2119899212058231205741198874

1

(12059011205822

+ 1205832)2

+

11989921205721205821198872

1

12059011205822

+ 1205832

+

2119862211989912058221198872

1

12059011205822

+ 1205832

minus

119862211989921205821198872

1

12059011205822

+ 1205832

+

119862212058212058321198862

1

12059011205822

+ 1205832

+

611989921205821205741198862

01198872

1

12059011205822

+ 1205832

+

41198992120582312057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058221205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

11198872

1

12059011205822

+ 1205832

+

3119899212057312058211988601198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058321198872

1

(12059011205822

+ 1205832)2

minus

119862211989912058212058321198862

1

12059011205822

+ 1205832

minus

119862211989912058212058311988601198871

12059011205822

+ 1205832

+

81198992120582212057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

12059011205822

+ 1205832

+

11989921205731205821198873

1

12059011205822

+ 1205832

minus

311986221198991205821205831198872

1

12059011205822

+ 1205832

+

4119899212058221205741205831198874

1

(12059011205822

+ 1205832)2

+

4119899212058212057411988601198873

1

12059011205822

+ 1205832

minus

1211989921205821205741205831198862

11198872

1

12059011205822

+ 1205832

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

12059011205822

+ 1205832

minus

21198622120582212058311988611198871

12059011205822

+ 1205832

+

1211989921205822120574119886011988611198872

1

12059011205822

+ 1205832

minus

21198622119899120582212058311988611198871

12059011205822

+ 1205832

+

81198992120582312057412058311988611198873

1

(12059011205822

+ 1205832)2

= 0


120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

12059011205822

+ 1205832

+

41198622120582120583211988611198871

12059011205822

+ 1205832

minus

6119899212057312058212058311988611198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988611198873

1

(12059011205822

+ 1205832)2

+

41198622119899120582120583211988611198871

12059011205822

+ 1205832

minus

241198992120582120574120583119886011988611198872

1

12059011205822

+ 1205832

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

12059011205822

+ 1205832

+

119899212057312058221198873

1

12059011205822

+ 1205832

minus

41198992120573120582212058321198873

1

(12059011205822

+ 1205832)2

+

41198992120582212057411988601198873

1

12059011205822

+ 1205832

minus

81198992120582312057412058331198874

1

(12059011205822

+ 1205832)3

+

4119899212058231205741205831198874

1

(12059011205822

+ 1205832)2

minus

211989921205721205821205831198872

1

12059011205822

+ 1205832

+

41198622119899120582212058331198872

1

(12059011205822

+ 1205832)2

+

2119862211989912058212058331198862

1

12059011205822

+ 1205832

minus

3119862211989912058221205831198872

1

12059011205822

+ 1205832

+

2119862211989921205821205831198872

1

12059011205822

+ 1205832

minus

6119899212057312058212058311988601198872

1

12059011205822

+ 1205832

minus

1611989921205822120574120583211988601198873

1

(12059011205822

+ 1205832)2

minus

1211989921205821205741205831198862

01198872

1

12059011205822

+ 1205832

+

21198622119899120582120583211988601198871

12059011205822

+ 1205832

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

12059011205822

+ 1205832

minus

119899212058241205741198874

1

(12059011205822

+ 1205832)2

+

1198622120582212058321198862

1

12059011205822

+ 1205832

+

119862211989912058231198872

1

12059011205822

+ 1205832

+

119899212057212058221198872

1

12059011205822

+ 1205832

+

31198992120573120582211988601198872

1

12059011205822

+ 1205832

minus

1198622119899120582212058321198862

1

12059011205822

+ 1205832

+

41198992120582412057412058321198874

1

(12059011205822

+ 1205832)3

+

2119899212057312058231205831198873

1

(12059011205822

+ 1205832)2

+

6119899212058221205741198862

01198872

1

12059011205822

+ 1205832

minus

21198622119899120582312058321198872

1

(12059011205822

+ 1205832)2

minus

1198622119899120582212058311988601198871

12059011205822

+ 1205832

+

81198992120582312057412058311988601198873

1

(12059011205822

+ 1205832)2

= 0

(51)



1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(52)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+1198602sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sinh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cosh(

119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(53)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))]

1119899

(54)


while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










while if 1198601

= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(55)



1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

120583 = plusmn 120582120573radic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(56)


=[[

[

plusmn radic

1205901

(119899 + 1)

(120582 + 1198992) [120582120572120574(119899 + 2)

2minus 1205732

(119899 + 1) (120582 + 1198992)]

times (120582120572 (119899 + 2)



timesradic1205901

(119899 + 1) (120582 + 1198992)

120582120572120574(119899 + 2)2

minus 1205732

(119899 + 1) (120582 + 1198992)

)

minus1

)]]

]

1119899

(57)



= 0 1198602



(120582 + 1198992)120574


1119899

(58)

while if 1198601

= 0 1198602


119906 (120585) = [plusmnradic(119899 + 1)120582120572

(120582 + 1198992)120574


1119899

(59)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radic1205901

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(60)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602sinh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1


plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










plusmn radic1205901


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(61)


2)119905 120573 = 0


= 0 and1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(62)

while if 1198601

= 0 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(63)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732

(119899 + 1)])

12

times (21198992

(119899 + 2) 120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(64)


= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

) plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[11989941205901

+ 1205832] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12


(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cosh(119899120573radicminus

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(65)


2)119905 120573 = 0


= 0 1198602


= 0 1198602





0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 1198861 1198871 120583 120582 and 119862 as follows

1206014

11986221205821198872

1

1205832

minus 12058221205902

minus 11986221198991198862

1minus 11989921205741198864

1minus 11986221198862

1

+

11986221198991205821198872

1

1205832

minus 12058221205902

minus

119899212058221205741198874

1

(12058221205902

minus 1205832)2

+

611989921205821205741198862

11198872

1

1205832

minus 12058221205902

= 0

1206013

3119899212057312058211988611198872

1

1205832

minus 12058221205902

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

2119862212058212058311988611198871

1205832

minus 12058221205902

+

121198992120582120574119886011988611198872

1

1205832

minus 12058221205902

minus

2119862211989912058212058311988611198871

1205832

minus 12058221205902

+

81198992120582212057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

1206013120595

4119899212058212057411988611198873

1

1205832

minus 12058221205902

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus 2119862211988611198871

= 0

1206012 119862211989921198862

1minus 211986221205821198862

1minus 11989921205721198862

1minus 611989921205741198862

01198862

1minus 3119899212057311988601198862

1

+

119862212058221198872

1

1205832

minus 12058221205902

minus

2119899212058231205741198874

1

(12058221205902

minus 1205832)2

+

11989921205721205821198872

1

1205832

minus 12058221205902

+

2119862211989912058221198872

1

1205832

minus 12058221205902

minus

119862211989921205821198872

1

1205832

minus 12058221205902

+

119862212058212058321198862

1

1205832

minus 12058221205902

+

611989921205821205741198862

01198872

1

1205832

minus 12058221205902

+

41198992120582312057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058221205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

11198872

1

1205832

minus 12058221205902

+

3119899212057312058211988601198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058321198872

1

(12058221205902

minus 1205832)2

minus

119862211989912058212058321198862

1

1205832

minus 12058221205902

minus

119862211989912058212058311988601198871

1205832

minus 12058221205902

+

81198992120582212057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

minus

211986221205821205831198872

1

1205832

minus 12058221205902

+

11989921205731205821198873

1

1205832

minus 12058221205902

minus

311986221198991205821205831198872

1

1205832

minus 12058221205902

+

4119899212058221205741205831198874

1

(12058221205902

minus 1205832)2

+

4119899212058212057411988601198873

1

1205832

minus 12058221205902

minus

1211989921205821205741205831198862

11198872

1

1205832

minus 12058221205902

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

minus 2119862211989912058211988601198861

+

31198992120573120582211988611198872

1

1205832

minus 12058221205902

minus

21198622120582212058311988611198871

1205832

minus 12058221205902

+

1211989921205822120574119886011988611198872

1

1205832

minus 12058221205902

minus

21198622119899120582212058311988611198871

1205832

minus 12058221205902

+

81198992120582312057412058311988611198873

1

(12058221205902

minus 1205832)2

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 2119862212058211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus 119862211989912058211988611198871

minus 61198992120573119886011988611198871

+

41198992120582212057411988611198873

1

1205832

minus 12058221205902

+

41198622120582120583211988611198871

1205832

minus 12058221205902

minus

6119899212057312058212058311988611198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988611198873

1

(12058221205902

minus 1205832)2

+

41198622119899120582120583211988611198871

1205832

minus 12058221205902

minus

241198992120582120574120583119886011988611198872

1

1205832

minus 12058221205902

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+ 211986221205821205831198862

1minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

minus 119862211989912058211988601198871

minus 211986221198991205821205831198862

1

minus

2119862212058212058331198862

1

1205832

minus 12058221205902

+

119899212057312058221198873

1

1205832

minus 12058221205902

minus

41198992120573120582212058321198873

1

(12058221205902

minus 1205832)2

+

41198992120582212057411988601198873

1

1205832

minus 12058221205902

minus

81198992120582312057412058331198874

1

(1205832

minus 12058221205902)3

+

4119899212058231205741205831198874

1

(12058221205902

minus 1205832)2

minus

211989921205721205821205831198872

1

1205832

minus 12058221205902

+

41198622119899120582212058331198872

1

(12058221205902

minus 1205832)2

+

2119862211989912058212058331198862

1

1205832

minus 12058221205902

minus

3119862211989912058221205831198872

1

1205832

minus 12058221205902

+

2119862211989921205821205831198872

1

1205832

minus 12058221205902

minus

6119899212057312058212058311988601198872

1

1205832

minus 12058221205902

minus

1611989921205822120574120583211988601198873

1

(12058221205902

minus 1205832)2

minus

1211989921205821205741205831198862

01198872

1

1205832

minus 12058221205902

+

21198622119899120582120583211988601198871

1205832

minus 12058221205902

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0minus 119862212058221198862

1

+ 119862211989912058221198862

1minus

1198622119899212058221198872

1

1205832

minus 12058221205902

minus

119899212058241205741198874

1

(12058221205902

minus 1205832)2

+

1198622120582212058321198862

1

1205832

minus 12058221205902

+

119862211989912058231198872

1

1205832

minus 12058221205902

+

119899212057212058221198872

1

1205832

minus 12058221205902

+

31198992120573120582211988601198872

1

1205832

minus 12058221205902

minus

1198622119899120582212058321198862

1

1205832

minus 12058221205902

+

41198992120582412057412058321198874

1

(1205832

minus 12058221205902)3

+

2119899212057312058231205831198873

1

(12058221205902

minus 1205832)2

+

6119899212058221205741198862

01198872

1

1205832

minus 12058221205902

minus

21198622119899120582312058321198872

1

(12058221205902

minus 1205832)2

minus

1198622119899120582212058311988601198871

1205832

minus 12058221205902

+

81198992120582312057412058311988601198873

1

(12058221205902

minus 1205832)2

= 0




1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

119899 (119899 + 2) 120574

1198871

= 0

120583 = 0

120582 =

1198992

(119899 + 1) 1205732

4 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

119862 = plusmn

radic120574 [120572120574(119899 + 2)2

minus 1205732

(119899 + 1)]

(119899 + 2) 120574

(67)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574



119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(

119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

))]

]

1119899

(68)


minus 1205732(119899 + 1)](119899 + 2)120574)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [ minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(69)

while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


119899120573

2

radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(70)


1198860

= 0

1198861

= 0

1198871

= plusmn 120582120572 (119899 + 2)

times radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

120583 = plusmn 120582120573radic1205902

(119899 + 1) (120582 + 1198992)

1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2

120582 = 120582

119862 = plusmn 119899radic120572

120582 + 1198992

(71)


= [

[

plusmn radic

1205902

(119899 + 1)

(120582 + 1198992) [1205732

(119899 + 1) (120582 + 1198992) minus 120582120572120574(119899 + 2)

2]

times (120582120572 (119899 + 2)

times (1198601sin (radic120582120585) + 119860

2cos (radic120582120585)

plusmn120573radic1205902(119899 + 1)(120582 + 119899

2)

1205732(119899 + 1)(120582 + 119899

2) minus 120582120572120574(119899 + 2)

2)

minus1

)]

]

1119899

(72)



= 0 1198602



(120582 + 1198992)120574

sec(radic120582120585)]

1119899

(73)


100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100

200

300

400

500

minus4 minus4

minus2 minus2

0 0

2 2

4 4

xt


while if 1198601

= 0 1198602



(120582 + 1198992)120574

csc(radic120582120585)]

1119899

(74)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn

(119899 + 1) 120573radicminus1205902

2 (119899 + 2) 120574

120583 = 0

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(75)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus 1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)


times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585) + 1198602

times cos(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))

minus1

)]

]

1119899

(76)


2)119905 120573 = 0


= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn 119894 sec(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(77)


while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










while if 1198601

= 0 and 1198602


119906 (120585)

= [minus(119899 + 1) 120573

2 (119899 + 2) 120574


120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

plusmn119894 csc(119899120573radic(119899 + 1)

120572120574(119899 + 2)2

minus 1205732(119899 + 1)

120585))]

1119899

(78)


1198860

= minus(119899 + 1) 120573

2 (119899 + 2) 120574

1198861

= plusmn

radic(119899 + 1) [1205732

(119899 + 1) minus 120572120574(119899 + 2)2]

2119899 (119899 + 2) 120574

1198871

= plusmn ((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (21198992(119899 + 2)120574120573)

minus1

120583 = 120583

120582 =

11989921205732

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)


1205732

(119899 + 1)

120574(119899 + 2)2

(79)


119906 (120585)

= [

[

minus(119899 + 1) 120573

2 (119899 + 2) 120574


times ( (1198601cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

minus1198602sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585))

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

))

plusmn1

21198992

(119899 + 2) 120574120573

times (((119899 + 1)21205734

[1205832

minus 11989941205902] + 120572120574120583

2

(119899 + 2)2

times [120572120574(119899 + 2)2

minus 21205732(119899 + 1)])

12

times (1198601sin(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+ 1198602cos(119899120573radic

(119899 + 1)

120572120574(119899 + 2)2

minus 1205732

(119899 + 1)

120585)

+

120583

120582

)

minus1

)]

]

1119899

(80)


2)119905 120573 = 0


= 0 1198602


1= 0

1198602



0 1198861 1198871 and 120583 as follows

1206014 minus 119862

21198862

1minus 11986221198991198862

1minus 11989921205741198864

1minus

11986221198872

1

1198602

1minus 2120583119860

2

minus

11989921205741198874

1

(1198602

1minus 2120583119860

2)2

minus

11986221198991198872

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

11198872

1

1198602

1minus 2120583119860

2

= 0

1206013

2119862212058311988611198871

1198602

1minus 2120583119860

2

minus 11989921205731198863

1minus 2119862211989911988601198861

minus 4119899212057411988601198863

1

minus

3119899212057311988611198872

1

1198602

1minus 2120583119860

2

+

8119899212057412058311988611198873

1

(1198602

1minus 2120583119860

2)2

minus

121198992120574119886011988611198872

1

1198602

1minus 2120583119860

2

+

2119862211989912058311988611198871

1198602

1minus 2120583119860

2

= 0

1206013120595 minus 2119862

211988611198871

minus 2119862211989911988611198871

minus 411989921205741198863

11198871

minus

4119899212057411988611198873

1

1198602

1minus 2120583119860

2

= 0


1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1206012 minus 119899

21205721198862

1+ 119862211989921198862

1minus

11989921205721198872

1

1198602

1minus 2120583119860

2

minus 611989921205741198862

01198862

1

+

119862211989921198872

1

1198602

1minus 2120583119860

2

minus

119862212058321198862

1

1198602

1minus 2120583119860

2

minus 3119899212057311988601198862

1

minus

4119899212057412058321198874

1

(1198602

1minus 2120583119860

2)3

minus

2119862211989912058321198872

1

(1198602

1minus 2120583119860

2)2

minus

3119899212057311988601198872

1

1198602

1minus 2120583119860

2

+

119862211989912058321198862

1

1198602

1minus 2120583119860

2

minus

611989921205741198862

01198872

1

1198602

1minus 2120583119860

2

+

211989921205731205831198873

1

(1198602

1minus 2120583119860

2)2

+

8119899212057412058311988601198873

1

(1198602

1minus 2120583119860

2)2

+

119862211989912058311988601198871

1198602

1minus 2120583119860

2

= 0

1206012120595 2119862

21205831198862

1minus 2119862211989911988601198871

+

211986221205831198872

1

1198602

1minus 2120583119860

2

minus

11989921205731198873

1

1198602

1minus 2120583119860

2

+ 11986221198991205831198862

1minus 311989921205731198862

11198871

minus 12119899212057411988601198862

11198871

+

311986221198991205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057411988601198873

1

1198602

1minus 2120583119860

2

+

411989921205741205831198874

1

(1198602

1minus 2120583119860

2)2

+

1211989921205741205831198862

11198872

1

1198602

1minus 2120583119860

2

= 0

120601 21198622119899211988601198861

minus 2119899212057211988601198861

minus 311989921205731198862

01198861

minus 411989921205741198863

01198861

= 0

120601120595 21198622119899211988611198871

minus 2119899212057211988611198871

minus 1211989921205741198862

011988611198871

+ 3119862211989912058311988601198861

minus

41198622120583211988611198871

1198602

1minus 2120583119860

2

minus 61198992120573119886011988611198871

minus

41198622119899120583211988611198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988611198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988611198873

1

(1198602

1minus 2120583119860

2)2

+

241198992120574120583119886011988611198872

1

1198602

1minus 2120583119860

2

= 0

120595 21198622119899211988601198871

minus 2119899212057211988601198871

+

2119862212058331198862

1

1198602

1minus 2120583119860

2

minus 311989921205731198862

01198871

minus 411989921205741198863

01198871

+

8119899212057412058331198874

1

(1198602

1minus 2120583119860

2)3

+

4119862211989912058331198872

1

(1198602

1minus 2120583119860

2)2

+

211989921205721205831198872

1

1198602

1minus 2120583119860

2

minus

4119899212057312058321198873

1

(1198602

1minus 2120583119860

2)2

minus

2119862211989912058331198862

1

1198602

1minus 2120583119860

2

minus

2119862211989921205831198872

1

1198602

1minus 2120583119860

2

+

1211989921205741205831198862

01198872

1

1198602

1minus 2120583119860

2

minus

21198622119899120583211988601198871

1198602

1minus 2120583119860

2

+

6119899212057312058311988601198872

1

1198602

1minus 2120583119860

2

minus

161198992120574120583211988601198873

1

(1198602

1minus 2120583119860

2)2

= 0

1206010 119862211989921198862

0minus 11989921205721198862

0minus 11989921205731198863

0minus 11989921205741198864

0= 0

(81)


1198860

= 0 1198861

= 0

1198871

= (minus21198991205731198602

(119899 + 1)

plusmn2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1


120583 = (119899120573 [21198991205731198602

(119899 + 1) plusmn 2

times radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])

times (2120572120574 (119899 + 2))minus1

(82)


119906 (120585) = [ (( minus 21198991205731198602

(119899 + 1) plusmn 2

timesradic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1))

times (2119899120574(119899 + 2))minus1

)

times (1

(1205832)1205852

+ 1205851198601

+ 1198602

) ]

1119899

(83)


2radic119899212057321198602

2(119899 + 1)

2minus 120572120574119860

2

1(119899 + 2)

2(119899 + 1)])2120572120574(119899 + 2)



1198992120574

(1198601

1205851198601

+ 1198602

)]

1119899

(84)


(1198622

minus 120572) 119906 minus 119862211990610158401015840

minus 120573119906119899+1

= 0 (85)



119906 (120585) = V2119899 (120585) (86)


(1198622

minus 120572) 1198992V2 minus 2119862

2119899VV10158401015840 minus 2119862

2

(2 minus 119899) (V1015840)2

minus 1205731198992V4 = 0

(87)


1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1018

1018

1018

1017

0

0 0

minus4 minus4

minus2 minus2

2 2

4 4

t

x



V (120585) = 1198860

+ 1198861120601 (120585) + 119887

1120595 (120585) (88)

where 1198860 1198861 and 119887




0 1198861 1198871 120583 120582 and 119862


1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

120583 = 0

120582 = minus

1198992

(1198622

minus 120572)

41198622

119862 = 119862

(89)


119906 (120585)

=[[

[

plusmnradic

1205901

(119899 + 2) (120572 minus 1198622)

2120573

times ((1)

times (1198601sinh(

119899

2119862

radic1198622

minus 120572120585)

+1198602cosh (

119899

2119862

radic1198622

minus 120572120585))

minus1

)]]

]

2119899

(90)


1= 0 and 119860

2= 0 in (90) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sech(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(91)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[


2minus 120572)

2120573

csch(119899

2119862

radic1198622

minus 120572120585)]

]

2119899

(92)


0= 0 respectively




1198860

= 0

1198861

= 0

1198871

= plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

120583 = 0

120582 =

1198992

(120572 minus 1198622)

41198622

119862 = 119862

(93)



119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (120585)

=[[

[

plusmnradic

1205902

(119899 + 2) (1198622

minus 120572)

2120573

times ((1)

times (1198601sin(

119899

2119862

radic120572 minus 1198622120585)

+1198602cos(

119899

2119862

radic120572 minus 1198622120585))

minus1

)]]

]

2119899

(94)


1= 0 and 119860

2= 0 in (94) we


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

sec(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(95)

while if 1198601

= 0 and 1198602


119906 (120585) = [

[

plusmnradic(119899 + 2)(119862

2minus 120572)

2120573

csc(119899

2119862

radic120572 minus 1198622120585)]

]

2119899

(96)


0= 0 respectively





1198860

= 0

1198861


1198992120573

1198871

= 0


120583 = 0

(97)



1198992120573

(1198601

1198601120585 + 119860

2

)]

2119899

(98)




1198860

= 0

1198861


21198992120573

1198871

= plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573


120583 = 120583

(99)



21198992120573

(

120583120585 + 1198601

(1205832) 1205852

+ 1198601120585 + 119860

2

)

plusmnradic

minus

120572 (119899 + 2) [1198602

1minus 2120583119860

2]

21198992120573

times (1

(1205832) 1205852

+ 1198601120585 + 119860

2

) ]

2119899

(100)


4 Conclusions



1and 119860

2take special


and 119887119894









1015840119866)-






Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References


























1015840









1015840



1015840



1015840




1015840



1015840





1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1015840



1015840

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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