shape preserving interpolation using 2 rational cubic...

Research ArticleShape Preserving Interpolation Using 1198622 Rational Cubic Spline

Samsul Ariffin Abdul Karim1 and Kong Voon Pang2

1Fundamental and Applied Sciences Department Universiti Teknologi PETRONAS Bandar Seri Iskandar 32610 Seri IskandarPerak Darul Ridzuan Malaysia2School of Mathematical Sciences Universiti Sains Malaysia (USM) 11800 Minden Penang Malaysia

Correspondence should be addressed to Samsul Ariffin Abdul Karim samsul ariffinpetronascommy

Received 18 January 2016 Revised 10 April 2016 Accepted 21 April 2016

Academic Editor Francisco J Marcellan

Copyright copy 2016 S A Abdul Karim and K Voon Pang 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

This paper discusses the construction of new1198622 rational cubic spline interpolant with cubic numerator and quadratic denominator

The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic splineinterpolation The rational cubic spline has three parameters 120572

119894 120573119894 and 120574

119894 The sufficient conditions for the positivity are derived

on one parameter 120574119894while the other two parameters 120572

119894and 120573

119894are free parameters that can be used to change the final shape of

the resulting interpolating curves This will enable the user to produce many varieties of the positive interpolating curves Cubicspline interpolation with 119862

2 continuity is not able to preserve the shape of the positive data Notably our scheme is easy to useand does not require knots insertion and 119862

2 continuity can be achieved by solving tridiagonal systems of linear equations for theunknown first derivatives 119889

119894 119894 = 1 119899 minus 1 Comparisons with existing schemes also have been done in detail From all presented

numerical results the new 1198622 rational cubic spline gives very smooth interpolating curves compared to some established rational

cubic schemes An error analysis when the function to be interpolated is 119891(119905) isin 1198623[1199050 119905119899] is also investigated in detail

1 Introduction

Spline interpolation has been used extensively in manyresearch disciplines such as in car design and airplane fuse-lage Univariate and bivariate spline can be used to approx-imate or interpolate the given finite data sets Even thoughthe cubic spline has second-order parametric continuity 1198622it has some weakness such that the interpolating curves maygive few unwanted behavior of the original data due to theexisting wiggles along some interval This uncharacteristicbehavior may destroy the data If the given data is positivecubic spline may give some negative values along the wholeinterval where the interpolating curves will lie below 119909-axis For some application any negativity is unacceptableFor example the wind speed solar energy and rainfallreceived are always having positive values and any negativityvalues need to be avoided as it may destroy any importantinformation that may exist in the original data Similarly ifthe data is monotone then the resulting interpolating curvesalso must be monotone too Furthermore if the given data

is convex the rational cubic spline interpolation should beable to maintain the shape of the original data Thus shapepreserving interpolation is important in computer graphicsand computer aided geometric design (CAGD)

Due to the fact that cubic spline is not able to producecompletely the positive monotone and convex interpolatingcurves on entire given interval many researchers have pro-posed several methods and idea to preserve the positivitymonotonicity and convexity of the data Fritsch and Carlson[1] and Dougherty et al [2] have discussed the monotonicitypositivity and convexity preserving by using cubic splineinterpolation bymodifying the first derivative values inwhichthe shape violation is found Butt and Brodlie [3] and Brodlieand Butt [4] have used cubic spline interpolation to preservethe positivity and convexity of the finite data by insertingextra knots in the interval in which the positivity andorconvexity is not preserved by the cubic splineTheir methodsdid not give any extra freedom to the user in controlling thefinal shape of the interpolating curves In order to changethe final shape of the interpolating curves the user needs to

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 4875358 14 pageshttpdxdoiorg10115520164875358

2 Journal of Applied Mathematics

change the given data Another thing is that their methodsrequire the modification of the first derivative parametersSarfraz [5] Sarfraz et al [6 7] and Abbas [8] studied theuse of rational cubic interpolant for preserving the positivedata Meanwhile M Z Hussain and M Hussain [9] studiedpositivity preserving for curves and surfaces by utilizingrational cubic spline with quadratic denominator In worksby Hussain et al [10] and Sarfraz et al [11] the rationalcubic spline with quadratic denominator has been used forpositivity monotonicity and convexity preserving with 119862

2

continuity Hussain et al [10] have only one free parametermeanwhile Sarfraz et al [11] have no free parameter Abbas[8] and Abbas et al [12] have discussed the positivity byusing new 119862

2 rational cubic spline with two free parametersAnother paper concerning 119862

2 rational spline can be foundby Delbourgo [13] Gregory [14] and Delbourgo and Gregory[15] Karim and Kong [16ndash19] have proposed new 119862

1 rationalcubic spline (cubicquadratic) with three parameters wheretwo of them are free parameters The rational cubic splinehas been successfully applied to the local control of theinterpolating functions positivity monotonicity and con-vexity preserving as well as the derivative control includingan error analysis when the function to be interpolated is119891(119905) isin 119862

3[1199050 119905119899] Motivated by the works of Tian et al [20]

Abbas et al [12] Hussain et al [10] and Sarfraz et al [11] inthis paper the authors will proposed new 119862

2 rational cubicspline for positivity monotonicity and convexity preservingand data constrained modeling Under some circumstancesour 119862

2 rational cubic spline will give new 1198622 rational cubic

spline based on rational cubic spline defined by Tian etal [20] Numerical comparison between 119862

2 rational cubicspline and the works of Hussain et al [10] Abbas [8]Abbas et al [12] and Sarfraz et al [11] also has been madecomprehensively From all presented numerical results shapepreserving interpolation by using the new 119862

2 rational cubicspline gives comparable results with existing rational cubicspline schemesThemain scientific contribution of this paperis summarized as follows

(i) In this paper 1198622 rational cubic spline (cubic

quadratic) with three parameters has been used forpositivity monotonicity and convexity preservingand constrained data modeling while in works byKarim and Kong [17ndash19] M Z Hussain and MHussain [9] and Sarfraz [5] the degree of smoothnessattained is 1198621

(ii) Hussain et al [10] and Sarfraz et al [11] discussedthe positivity by using 119862

2 rational cubic spline(cubicquadratic) with two parameters with one or nofree parameter while our rational cubic spline has twofree parameters Even though Abbas et al [12] alsoproposed 119862

2 rational cubic spline (cubicquadratic)with two parameters their rational spline is differentform our rational cubic spline Furthermore it wasnoticed that schemes of Abbas et al [12] may notbe able to produce completely positive interpolatingcurves with 119862

2 continuity

(iii) When 120574119894

= 0 we may obtain the new 1198622 rational

cubic spline with two parameters an extension to theoriginal 1198621 rational cubic spline of Tian et al [20]Thus our1198622 rational cubic spline gives a larger class ofrational cubic spline which also includes the rationalcubic spline of Tian et al [20]

(iv) Our rational scheme is local while in Lamberti andManni [21] their scheme is global Furthermore ourrational scheme works well for both equally andunequally spaced data while the rational spline inter-polant by Duan et al [22] and Bao et al [23] onlyworks for equally spaced data

(v) Numerical comparisons between the 1198622 rational

cubic spline and the existing schemes such as Hussainet al [10] Sarfraz et al [11] and Abbas et al [12] forpositivity preserving and 119862

1 rational spline of Karimand Kong [17ndash19] also have been done comprehen-sively

(vi) Ourmethod also does not require any knots insertionMeanwhile the cubic spline interpolation by Butt andBrodlie [3] Brodlie andButt [4] and Fiorot andTabka[24] requires knots insertion in the interval where theinterpolating curves produce the negative values (liesbelow 119909-axis) for positive data nonmonotone inter-polating curves (for monotone data) and nonconvexinterpolating curves for convex data

(vii) This paper utilized the rational cubic spline mean-while in Dube and Tiwari [25] Pan and Wang [26]and Ibraheem et al [27] the rational trigonometricspline is used in place of standard rational cubicspline Thus no trigonometric functions are involvedTherefore the method is not computationally expen-sive

The remainder of the paper is organized as followsSection 2 introduces the new 119862

2 rational cubic spline withthree parameters with some discussion on the methods toestimate the first derivatives values as well as shape controls ofthe rational cubic spline interpolation Meanwhile Section 3discusses the positivity preserving by using 119862

2 rational cubicspline together with numerical demonstrations as well ascomparison with some existing schemes including erroranalysis Section 4 is devoted for research discussion Finallya summary and conclusions are given in Section 5

2 1198622 Rational Cubic Spline Interpolant

This section will introduce 1198622 rational cubic spline inter-

polant with three parameters Originally this rational cubicspline has been initiated by Karim and Kong [18] The maindifference is that in this paper the rational cubic spline has1198622 continuity while in work by Karim and Kong [18] it has

1198621 continuity We begin with the definition of 119862

2 rationalcubic spline interpolant given set of data points (119909

119894 119891119894) 119894 =

0 1 119899 such that 1199090lt 1199091lt sdot sdot sdot lt 119909

119899 Let ℎ

119894= 119909119894+1

minus 119909119894

Δ119894= (119891119894+1

minus 119891119894)ℎ119894and 120579 = (119909 minus 119909

119894)ℎ119894 where 0 le 120579 le 1 For

Journal of Applied Mathematics 3

119909 isin [119909119894 119909119894+1

] 119894 = 0 1 2 119899 minus 1 the rational cubic splineinterpolant with three parameters is defined as follows

119904 (119909) equiv 119904119894(119909)

=1198601198940(1 minus 120579)

3+ 1198601198941120579 (1 minus 120579)

2+ 11986011989421205792(1 minus 120579) + 119860

11989431205793

(1 minus 120579)2120572119894+ 120579 (1 minus 120579) (2120572

119894120573119894+ 120574119894) + 1205792120573119894

(1)

The following conditions will assure that the rational cubicspline interpolant in (1) has 1198622 continuity

119904 (119909119894) = 119891119894

119904 (119909119894+1

) = 119891119894+1

119904(1)

(119909119894) = 119889119894

119904(1)

(119909119894+1

) = 119889119894+1

119904(2)

(119909119894+) = 119904(2)

(119909119894minus)

(2)

where 119904(1)

(119909119894) and 119904

(2)(119909119894) denote the first- and second-order

derivative with respect to 119909 respectively Meanwhile thenotations 119904

(2)(119909119894+) and 119904

(2)(119909119894minus) correspond to the right and

left second derivatives values Furthermore 119889119894denotes the

derivative value which is given at the knot 119909119894 119894 = 0 1 2 119899

By using (2) the required 1198622 rational cubic spline inter-

polant with three parameters defined by (1) has the unknowns119860119894119895 119895 = 0 1 2 3 and is given as follows

1198601198940

= 120572119894119891119894

1198601198941

= (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894+ 120572119894ℎ119894119889119894

1198601198942

= (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

1198601198943

= 120573119894119891119894+1

(3)

The parameters 120572119894 120573119894gt 0 120574

119894ge 0 are used to control the final

shape of the interpolating curves From work by Karim andKong [19] the second-order derivative 119904

(2)(119909) is given as

119904(2)

(119909) =

sum3

119895=0119862119894119895(1 minus 120579)

3minus119895120579119895

ℎ119894[119876119894(120579)]3

(4)

where

1198621198940

= 21205722

119894(120574119894(Δ119894minus 119889119894) minus 120573119894(119889119894+1

minus Δ119894)

minus 2120572119894120573119894(119889119894minus Δ119894))

1198621198941

= 61205722

119894120573119894(Δ119894minus 119889119894)

1198621198942

= 61205721198941205732

119894(119889119894+1

minus Δ119894)

1198621198943

= 21205732

119894[120574119894(119889119894+1

minus Δ119894)

minus 120572119894(Δ119894minus 119889119894minus 2120573119894(119889119894+1

minus Δ119894))]

(5)

Now 1198622 continuity 119904(2)(119909

119894+) = 119904(2)

(119909119894minus) 119894 = 1 2 119899 minus 1

will give

2 [120574119894minus1

(119889119894minus Δ119894minus1

) minus 120572119894minus1

(Δ119894minus1

minus 119889119894minus1


(119889119894minus Δ119894minus1

))]

ℎ119894minus1

120573119894minus1

=2 (120574119894(Δ119894minus 119889119894) minus 120573119894(119889119894+1

minus Δ119894) minus 2120572

119894120573119894(119889119894minus Δ119894))

ℎ119894120572119894

(6)

Now (6) provides the following system of linear equationsthat can be used to compute the first derivative parameters119889119894 119894 = 1 2 119899 minus 1 such that

119886119894119889119894minus1

+ 119887119894119889119894+ 119888119894119889119894+1

= 119890119894 119894 = 1 2 119899 minus 1 (7)

with

119886119894= ℎ119894120572119894minus1

120572119894

119887119894= ℎ119894120572119894(120574119894minus1

+ 2120572119894minus1

120573119894minus1

) + ℎ119894minus1

120573119894minus1

(120574119894+ 2120572119894120573119894)

119888119894= ℎ119894minus1

120573119894minus1

120573119894

119890119894= ℎ119894120572119894(120574119894minus1

+ 120572119894minus1

+ 2120572119894minus1

120573119894minus1

) Δ119894minus1

+ ℎ119894minus1

120573119894minus1

(120574119894+ 120573119894+ 2120572119894120573119894) Δ119894

(8)

The system of linear equations given by (7) is strictly tridi-agonal and has a unique solution for the unknown derivativeparameters 119889

119894 119894 = 1 2 119899minus1 for all 120572

119894 120573119894gt 0 120574

119894ge 0The

system in (7) gives 119899 minus 1 linear equations for 119899 + 1 unknownderivative values Thus two more equations are required inorder to obtain the unique solution in (7) The following iscommon choices for the end points condition that is 119889

0and

119889119899

119904(1)

(1199090) = 1198890

119904(1)

(119909119899) = 119889119899

(9)

By using (3)1198622 rational cubic spline interpolant in (1) can bereformulated and is given as follows

119904 (119909) equiv 119904119894(119909) =

119875119894(120579)

119876119894(120579)

(10)

where

119875119894(120579)

= 120572119894119891119894(1 minus 120579)

3

+ ((2120572119894120573119894+ 120572119894+ 120574119894) 119891119894+ 120572119894ℎ119894119889119894) 120579 (1 minus 120579)

2

+ ((2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

) 1205792(1 minus 120579)

+ 120573119894119891119894+1

1205793

119876119894(120579) = (1 minus 120579)

2120572119894+ 120579 (1 minus 120579) (2120572

119894120573119894+ 120574119894) + 1205792120573119894

(11)

The data-dependent sufficient conditions on parameters 120574119894

will be developed in order to preserve the positivity data con-strained monotonicity and convexity on the entire interval


[119909119894 119909119894+1

] 119894 = 0 1 2 119899 minus 1 The remaining parameters 120572119894

and120573119894can be used to refine the resulting interpolating curves

Thus the rational cubic spline provides greater flexibility tothe user in controlling the final shape of the interpolatingcurves

Theorem 1 (1198622 rational cubic spline interpolant) The ratio-nal cubic spline with three parameters defined by (1) is 119862

2isin

[1199090 119909119899] if there exist positive parameters 120572

119894 120573119894gt 0 120574

119894ge 0

and 119889119894 119894 = 1 2 119899 minus 1 that satisfy (7)

The choice of the end point derivatives 1198890and 119889

119899

depended on the original data which are chosen as follows

Choice 1 (Geometric Mean Method (GMM)) Consider

1198890=

0 Δ0= 0 or Δ

20= 0

Δ(1+ℎ0ℎ1)

0Δ(minusℎ0ℎ1)

21otherwise

119889119899

=

0 Δ119899minus1

= 0 or Δ119899119899minus2

= 0

Δ(1+ℎ119899minus1ℎ119899minus2)

119899minus1Δ(minusℎ119899minus1ℎ119899minus2)

119899119899minus2otherwise

(12)

Choice 2 (Arithmetic Mean Method (AMM)) Consider

1198890= Δ0+ (Δ0minus Δ1) (

ℎ0

ℎ0+ ℎ1

)

119889119899= Δ119899minus1

+ (Δ119899minus1

minus Δ119899minus2

) (ℎ119899minus1

ℎ119899minus1

+ ℎ119899minus2

)

(13)

Delbourgo and Gregory [29] give more details about themethod that can be used to estimate the first derivative valueIn this paper the AMMwill be used to estimate the end pointderivatives 119889

0and 119889

119899 respectively

Some observation and shape control analysis of the new1198622 rational cubic spline interpolant defined by (10) are given

as follows

(1) When 120572119894

gt 0 120573119894

gt 0 and 120574119894

= 0 the rationalinterpolant in (10) reduces to the rational spline of theform cubicquadratic by Tian et al [20] and we mayobtain 119862

2 rational cubic spline with two parametersan extension to 119862

1 rational cubic spline originallyproposed by Tian et al [20] Thus by rewriting 119862

2

condition in (7) we can obtain 1198622 rational cubic of

Tian et al [20](2) When 120572

119894= 120573119894

= 1 120574119894

= 0 the rational cubicinterpolant in (1) is just a standard cubic Hermitespline with 119862

1 continuity that may not be able tocompletely preserve the positivity of the data [18]

119904 (119909) = (1 minus 120579)2(1 minus 2120579) 119891

119894+ 1205792(3 minus 2120579) 119891

119894+1

+ 120579 (1 minus 120579)2119889119894minus 1205792(1 minus 120579) 119889

119894+1

(14)

for 119894 = 0 1 119899 minus 1

Table 1 A data from work by Sarfraz et al [6]

119894 0 1 1 3 4119909119894

0 2 3 9 11119891119894

05 15 7 9 13ℎ119894

2 1 6 2Δ119894

05 55 13 2119889119894(1198621) minus2833 3833 47619 15833 24167

119889119894(1198622) minus2833 43223 49828 12146 24167

(3) Furthermore the rational interpolant in (1) can bewritten as [18]

119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

+ℎ119894120579 (1 minus 120579) [120572

119894(119889119894minus Δ119894) (1 minus 120579) + 120573

119894(Δ119894minus 119889119894+1

) 120579]

119876119894(120579)

(15)

(4) Obviously when 120572119894

rarr 0 120573119894

rarr 0 or 120574119894

rarr infinthe rational interpolant in (10) converges to followingstraight line

lim120572119894 120573119894rarr0120574119894rarrinfin

119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(16)

Either the decrease of the parameters 120572119894and 120573

119894or the

increase of 120574119894will reduce the rational cubic spline to a

linear interpolant Figure 1 shows this example We test shapecontrol analysis by using the data from work by Sarfraz et al[6] given in Table 1

To show the difference between 1198622 rational cubic spline

with three parameters and 1198621 rational cubic spline of Karim

and Kong [17ndash19] we choose 120572119894

= 120573119894

= 1 120574119894

= 2

for both cases The main difference is that to generate 1198621

rational interpolating curves the first derivative parameter119889119894 119894 = 0 1 2 119899 is calculated by using Arithmetic Mean

Method (AMM) meanwhile to generate 1198622 rational cubic

spline with three parameters the first derivative parameter119889119894 119894 = 1 2 119899minus1 is calculated by solving (7) with suitable

choices of the end point derivatives 1198890and 119889

119899

Remark 2 If Δ119894minus119889119894= 0 or 119889

119894+1minusΔ119894= 0 then 119889

119894= 119889119894+1

= Δ119894

In this case the rational interpolant in (10) will be linear in thecorresponding interval or region that is

119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(17)

Figure 2 shows the shape control using the rational cubicspline for the given data in Table 1

Figure 2(f) shows the combination of Figures 2(a) 2(d)and 2(e) Meanwhile Figure 2(g) shows the combination ofFigures 2(a) 2(b) and 2(c) respectively

Clearly the curves approach to the straight line if 120572119894rarr

0 120573119894rarr 0 or 120574

119894rarr infin Furthermore decreases in the value

of 120573119894will pull the curves upward and vice versa This shape

control will be useful for shape preserving interpolation aswell as local control of the interpolating curves

Meanwhile from Figure 1(d) it can be seen clearly that1198622 rational cubic spline interpolation (shown as red color) is


x-axis

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12

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4

2

0

1086420

minus2

(a)

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12

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2

0

1086420

minus2

(b)

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1086420

(c)

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1086420

minus2

(d)

Figure 1 Interpolating curve for data in Table 1 (a) Default 1198621 cubic spline with 120572119894= 120573119894= 1 and 120574

119894= 0 (b) 1198621 rational cubic spline with

120572119894= 120573119894= 1 and 120574

119894= 2 (c) 1198622 rational cubic spline by using derivative calculated from tridiagonal equation (7) with 120572

119894= 120573119894= 1 and 120574

119894= 2

(d) The graphs of (b) 1198621 rational cubic spline (blue) and (c) 1198622 rational cubic spline (red)

smoother than 1198621 rational cubic spline interpolation (shown

as black color)Thus the new1198622 rational cubic spline provides

good alternative to the existing 1198621 and 119862

2 rational cubicspline

Remark 3 Since the constructed 1198622 rational cubic spline

has three parameters then how do we choose the parametervalues To answer this question the choices of the parameterstotally depend on the data that are under considerations bythe main user The main difference between our 119862

2 rationalcubic spline and119862

2 cubic spline interpolation is that in orderto change the final shape of the interpolating curves ourschemes are only required to change the parameters valueswithout the need to change the data points itself But thereare no free parameters in the descriptions of 1198622 cubic splineinterpolation

3 Positivity Preserving Using 1198622 Rational

Cubic Spline Interpolation

In this section the positivity preserving by using the pro-posed 119862

2 rational cubic spline interpolation defined by (10)will be discussed in detail We follow the same idea ofKarim and Kong [18] and Abbas et al [12] such that simple

data-dependent conditions for positivity are derived on oneparameter 120574

119894while the remaining parameters 120572

119894and 120573

119894are

free to be utilized The main objective is that in orderto preserve the positivity of the positive data the rationalcubic spline interpolant must be positive on the entire giveninterval The simple way to achieve it is by finding theautomated choice of the shape parameter 120574

119894 We begin by

giving the definition of strictly positive dataGiven the strictly positive set of data (119909

119894 119891119894) 119894 =

0 1 119899 1199090lt 1199091lt sdot sdot sdot lt 119909

119899 such that

119891119894gt 0 119894 = 0 1 119899 (18)

Now from (10) the rational cubic spline will preserve thepositivity of the data if and only if 119875

119894(120579) gt 0 and 119876

119894(120579) gt 0

Since for all 120572119894 120573119894gt 0 and 120574

119894ge 0 the denominator 119876

119894(120579) gt

0 119894 = 0 1 119899minus1Thus 119904(119909) gt 0 if and only if119875119894(120579) gt 0 119894 =

0 1 119899 minus 1The cubic polynomial 119875119894(120579) 119894 = 0 1 119899 minus 1

can be written as follows [18]

119875119894(120579) = 119861

1198941205793+ 1198621198941205792+ 119863119894120579 + 119864119894 (19)


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(a)

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(b)

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(c)

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(d)

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(e)

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(f)

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(g)

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(h)

Figure 2 Continued


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1086420

(i)

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(j)

Figure 2 Shape control of rational cubic interpolating curves with various values of shape parameters list in Table 2

Table 2 Shape parameters value for Figure 2

Figure 2 119894 0 1 2 3

Figure 2(a)120572119894

01 01 01 01120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(b)120572119894

1 1 1 1120573119894

01 01 01 01120574119894

1 1 1 1

Figure 2(c)120572119894

1 1 1 1120573119894

1 1 1 1120574119894

100 100 100 100

Figure 2(d)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

100 100 100 100

Figure 2(e)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

1000 1000 1000 1000

Figure 2(h)120572119894

001 001 001 001120573119894

001 001 001 001120574119894

1 1 1 1

Figure 2(i)120572119894

001 001 001 001120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(j)120572119894

1 1 1 1120573119894

001 001 001 001120574119894

1 1 1 1

where

119861119894= 120572119894ℎ119894119889119894+ 120573119894ℎ119894119889119894+1

+ 2120572119894120573119894119891119894+ 120574119894119891119894minus 2120572119894120573119894119891119894+1

minus 120574119894119891119894+1

119862119894= minus2120572

119894ℎ119894119889119894minus 120573119894ℎ119894119889119894+1

+ 120572119894119891119894minus 4120572119894120573119894119891119894minus 2120574119894119891119894

+ 120573119894119891119894+1

+ 2120572119894120573119894119891119894+1

+ 120574119894119891119894+1

119863119894= 120572119894ℎ119894119889119894minus 2120572119894119891119894+ 2120572119894120573119894119891119894+ 120574119894119891

119864119894= 120572119894119891119894

(20)

From Schmidt and Hess [30] with variable substitution 120579 =

119904(1 + 119904) 119904 ge 0 119875119894(120579) 119894 = 0 1 119899 minus 1 can be rewritten as

follows

119875119894(119904) = 119860119904

3+ 1198611199042+ 119862119904 + 119863 (21)

where119860 = 120573

119894119891119894+1

119861 = (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

119862 = (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894minus 120572119894ℎ119894119889119894

119863 = 120572119894119891119894

(22)

Theorem 4 For strictly positive data defined in (18) 1198622

rational cubic interpolant defined over the interval [1199090 119909119899] is

positive if in each subinterval [119909119894 119909119894+1

] 119894 = 0 1 119899 minus 1the involving parameters 120572

119894 120573119894 and 120574

119894satisfy the following

sufficient conditions

120572119894 120573119894gt 0

120574119894gt Max0 minus120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

]

(23)

Remark 5 The sufficient condition for positivity preservingby using 119862

2 rational cubic interpolant is different from thesufficient condition for positivity preserving by using 119862

1

rational cubic interpolant of Karim and Kong [18] To achieve1198622 continuity the derivative parameters 119889

119894 119894 = 1 2 119899minus1

must be calculated from1198622 condition given in (7)meanwhile

to achieve 1198621 continuity the derivative parameters 119889

119894 119894 =

0 1 2 119899 are estimated by using standard approximationmethods such as Arithmetic Mean Method (AMM)


Table 3 A positive data from work by Hussain et al [10]

119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25

Table 4 Numerical results

119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570

Remark 6 The sufficient condition in (23) can be rewrittenas

120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)

The following is an algorithm that can be used to generate1198622 positivity-preserving curves

Algorithm 7 (1198622 positivity preserving) Consider the follow-ing

(1) Input the data points 119909119894 119891119894119899

119894=0 1198890 and 119889

119899

(2) For 119894 = 0 1 119899minus1 calculate the parameter 120574119894using

(24) with suitable choices of 120572119894gt 0 120573

119894gt 0 and 120582

119894gt

0

(3) For 119894 = 1 2 119899minus1 calculate the value of derivative119889119894 by solving (7) by using LU decomposition and so

forth(4) For 119894 = 0 1 119899minus1 construct the piecewise positive

1198622 rational interpolating curves defined by (10)

31 Numerical Demonstrations In this section severalnumerical results for positivity preserving by using 119862

2

rational interpolating curves will be shown includingcomparison with existing rational cubic spline schemes Twosets of positive data taken from works by Hussain et al [10]and Sarfraz et al [28] were used

Figures 3 and 4 show the positivity preserving by using1198622rational cubic spline for data in Tables 3 and 5 respectivelyFigures 3(a) and 4(a) show the default cubic Hermite splinepolynomial for data in Tables 3 and 5 respectively Numericalresults of Figure 3(d) were obtained by using the parametersfrom the data in Table 4 Meanwhile numerical results of

Table 5 A Positive data from work by Sarfraz et al [28]

119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10


119894 0 1 2 3 4 5 6119889119894(1198621) minus965 minus635 325 0 minus395 565 835

Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894

01 1685 01 01 485 01119889119894(1198622) minus965 minus486 334 minus048 minus4057 525 835

Figure 4(e) were generated by using the data in Table 6 Itcan be seen clearly that the positivity preserving by using our1198622 rational cubic spline gives more smooth results compared

with the works of Hussain et al [10] and Abbas et al [12]The graphical results in Figure 3(e) were very smooth andvisually pleasing Meanwhile for the positive data given inTable 5 our 1198622 rational cubic spline gives comparable resultswith the works of Abbas et al [12] Hussain et al [10] andSarfraz et al [11] The final resulting positive curves by usingthe proposed 119862

2 rational cubic spline interpolant are slightlydifferent between the works of Abbas et al [12] Hussainet al [10] and Sarfraz et al [11] Finally Figure 5 showsthe examples of positive interpolating by using Fritsch andCarlson [1] cubic spline schemes that are well documented inMatlab as PCHIP It can be seen clearly that shape preservingby using PCHIP does not give smooth results and is notvisually pleasing enough Some of the interpolating curvestend to overshot on some interval that the interpolatingcurves are tight when compared with our work in this paperFor instance in Figure 4(b) the interpolating curves are verytight and not visually pleasing

Error Analysis In this section the error analysis for thefunction to be interpolated is 119891(119905) isin 119862

3[1199050 119905119899] using

our 1198622 rational cubic spline which will be discussed in

detail Note that the constructed rational cubic spline withthree parameters is a local interpolant and without loss ofgenerality we may just consider the error on the subinterval119868119894= [119905119894 119905119894+1

] By using Peano Kernel Theorem [31] the errorof interpolation in each subinterval 119868

119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)

Figure 3 Comparison of (a) cubic Hermite spline curve (b) Hussain et al [10] (c) Abbas et al [12] and positivity-preserving rational cubicspline with values of shape parameters (d) 120572

119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)

Figure 4 Comparison of (a) cubic Hermite spline curve (b) Sarfraz et al [11] (c) Abbas et al [12] and positivity-preserving rational cubicspline with values of shape parameters (d) 120572

119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1

minus 120591)The absolute error in each subinterval 119868

119894= [119909119894 119909119894+1

] isgiven as follows

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)

In order to enable us to derive the error analysis of1198622 rationalcubic spline interpolation we need to study the propertiesof the kernel functions 119903(120591 119905) and 119904(120591 119905) and evaluate thefollowing definite integrals

int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)

To simplify the integrals in (29) we begin by finding the rootsof 119903(119905 119905) = 0 119903(120591 119905) = 0 and 119904(120591 119905) = 0 respectively It is easyto see that the roots of 119903(119905 119905) = 0 in [0 1] are 120579 = 0 1 and120579lowast= 1 minus 120573

119894120588119894 where 120588

119894= 2120572119894120573119894+ 120574119894 Meanwhile the roots of

119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where

119867 = radic(120588119894minus 120573119894) (120572119894+ 120588119894120579) minus 120572

119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588

119894) Thus the following three cases can be

obtained

Case 1 For 0 le 120573119894120588119894le 1 0 lt 120579 lt 120579

lowast then (28) takes thefollowing form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579

lowastlt 120579 lt 1 then (28) takes the

following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894

gt 1 0 lt 120579 lt 1 then (28) takes thefollowing form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)

Theorem 8 The error for the interpolating rational cubicspline interpolant defined by (1) in each subinterval 119868

119894=

[119905119894 119905119894+1

] when 119891(119905) isin 1198623[1199050 119905119899] is given by

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579

lowastle 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574

119894= 0 the interpolation

function defined by (1) is reduced to the standard cubicHermite interpolation with

1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894

= 196 This is standard error for cubic Hermitepolynomial spline

4 Discussions

From all numerical results presented in Section 31 it canbe seen clearly that the proposed 119862

2 rational cubic spline


works very well and it is comparable with existing schemessuch as Hussain et al [10] Sarfraz et al [11] and Abbaset al [12] for positivity preserving Furthermore similar tothe construction of rational cubic of Abbas et al [8] our 119862

2

rational cubic spline interpolation also has three parameterswhere two are free parameters But based on our numericalexperiments it was noticed that schemes of Abbas et al[12] may not be able to produce completely 119862

2 positiveinterpolating curves Meanwhile the sufficient condition forpositivity monotonicity and convexity preserving and dataconstrained are derived on the remaining parameter One ofthe advantages by using our 119862

2 rational cubic spline is thatwhen 120574

119894= 0 we may obtain the new variant of 1198622 rational

cubic spline of Hussain and Ali [32] M Z Hussain and MHussain [9] and Tian et al [20] for positivity- and convexity-preserving interpolation Thus our 119862

2 rational cubic splinehas a larger spline class compared to the works by Abbas etal [12] Furthermore the error analysis when the function tobe interpolated is 119891(119905) isin 119862

3[1199050 119905119899] also has been derived in

detail

5 Conclusions

In this paper the new 1198622 rational cubic spline with three

parameters has been introduced It is an extension to theworkof Karim and Pang [16] To achieve 119862

2 continuity at the joinknots 119909

119894 119894 = 1 2 119899 minus 1 the first derivative value 119889

119894 is

calculated by solving systems of linear equation (tridiagonal)that is strictly positive and the solution is unique Shapecontrol of the new 119862

2 rational cubic interpolation withnumerical examples also was presented Finally 119862

2 rationalcubic spline has been used for positivity preserving includinga comparison with existing schemes From the numericalresults clearly our 1198622 rational cubic spline gives comparableresults with existing schemes Finally work in parametricshape preserving is underway by the authors

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Samsul Ariffin Abdul Karim would like to acknowledgeUniversiti Teknologi PETRONAS (UTP) for the financialsupport received in the form of a research grant ShortTerm Internal Research Funding (STIRF) no 0153AA-D91including the Mathematica Software

References

[1] F N Fritsch and R E Carlson ldquoMonotone piecewise cubicinterpolationrdquo SIAM Journal on Numerical Analysis vol 17 no2 pp 238ndash246 1980

[2] R L Dougherty A S Edelman and J M Hyman ldquoNonneg-ativity- monotonicity- or convexity-preserving cubic and

quintic Hermite interpolationrdquo Mathematics of Computationvol 52 no 186 pp 471ndash494 1989

[3] S Butt and KW Brodlie ldquoPreserving positivity using piecewisecubic interpolationrdquo Computers and Graphics vol 17 no 1 pp55ndash64 1993

[4] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquo Computers and Graphics vol 15 no 1 pp15ndash23 1991

[5] M Sarfraz ldquoVisualization of positive and convex data by arational cubic spline interpolationrdquo Information Sciences vol146 no 1ndash4 pp 239ndash254 2002

[6] M Sarfraz S Butt andM Z Hussain ldquoVisualization of shapeddata by a rational cubic spline interpolationrdquo Computers ampGraphics vol 25 no 5 pp 833ndash845 2001

[7] M SarfrazM ZHussain andANisar ldquoPositive datamodelingusing spline functionrdquo Applied Mathematics and Computationvol 216 no 7 pp 2036ndash2049 2010

[8] M Abbas Shape preserving data visualization for curves andsurfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012

[9] M Z Hussain and M Hussain ldquoVisualization of data subjectto positive constraintrdquo Journal of Information and ComputingSciences vol 1 no 3 pp 149ndash160 2006

[10] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011

[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving C2 rational splinerdquo inProceeding of 5th International Conference on Information Visu-alisation pp 528ndash533 London UK July 2011

[12] M Abbas A A Majid M N H Awang and J M AlildquoPositivity-preserving C2 rational cubic spline interpolationrdquoScienceAsia vol 39 no 2 pp 208ndash213 2013

[13] R Delbourgo ldquoAccurate C2 rational interpolants in tensionrdquoSIAM Journal on Numerical Analysis vol 30 no 2 pp 595ndash6071993

[14] J A Gregory ldquoShape preserving spline interpolationrdquoComputer-Aided Design vol 18 no 1 pp 53ndash57 1986

[15] R Delbourgo and J A Gregory ldquo1198622 rational quadratic splineinterpolation to monotonic datardquo IMA Journal of NumericalAnalysis vol 3 no 2 pp 141ndash152 1983

[16] S A A Karim andK V Pang ldquoLocal control of the curves usingrational cubic splinerdquo Journal of AppliedMathematics vol 2014Article ID 872637 12 pages 2014

[17] S A A Karim and K V Pang ldquoMonotonicity-preserving usingrational cubic spline interpolationrdquo Research Journal of AppliedSciences vol 9 no 4 pp 214ndash223 2014

[18] S A A Karim and V P Kong ldquoShape preserving interpolationusing rational cubic splinerdquoResearch Journal of Applied SciencesEngineering and Technology vol 8 no 2 pp 167ndash168 2014

[19] S A A Karim and V P Kong ldquoConvexity-preserving usingrational cubic spline interpolationrdquo Research Journal of AppliedSciences Engineering and Technology vol 8 no 3 pp 312ndash3202014

[20] M Tian Y Zhang J Zhu and Q Duan ldquoConvexity-preservingpiecewise rational cubic interpolationrdquo Journal of Informationand Computational Science vol 2 no 4 pp 799ndash803 2005

[21] P Lamberti and C Manni ldquoShape-preserving 1198622 functional

interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001


[22] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-

polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005

[23] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009

[24] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[25] M Dube and P Tiwari ldquoConvexity preserving 1198622 rational

quadratic trigonometric splinerdquo in Proceedings of the Interna-tional Conference of Numerical Analysis and AppliedMathemat-ics (ICNAAM rsquo12) vol 1479 pp 995ndash998 September 2012

[26] Y-J Pan and G-J Wang ldquoConvexity-preserving interpolationof trigonometric polynomial curves with a shape parameterrdquoJournal of ZhejiangUniversity SCIENCEA vol 8 no 8 pp 1199ndash1209 2007

[27] F IbraheemM Hussain M Z Hussain and A A Bhatti ldquoPos-itive data visualization using trigonometric functionrdquo Journalof Applied Mathematics vol 2012 Article ID 247120 19 pages2012

[28] M Sarfraz M Z Hussain and F S Chaudary ldquoShape preserv-ing cubic spline for data visualizationrdquo Computer Graphics andCADCAM vol 1 pp 185ndash193 2005

[29] R Delbourgo and J A Gregory ldquoThe determination of deriva-tive parameters for amonotonic rational quadratic interpolantrdquoIMA Journal of Numerical Analysis vol 5 no 4 pp 397ndash4061985

[30] J W Schmidt and W Hess ldquoPositivity of cubic polynomialson intervals and positive spline interpolationrdquo BIT NumericalMathematics vol 28 no 2 pp 340ndash352 1988

[31] M H Schultz Spline Analysis Prentice-Hall Englewood-CliffsNJ USA 1973

[32] M Z Hussain and J M Ali ldquoPositivity preserving piecewiserational cubic interpolationrdquoMatematika vol 22 no 2 pp 147ndash153 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


change the given data Another thing is that their methodsrequire the modification of the first derivative parametersSarfraz [5] Sarfraz et al [6 7] and Abbas [8] studied theuse of rational cubic interpolant for preserving the positivedata Meanwhile M Z Hussain and M Hussain [9] studiedpositivity preserving for curves and surfaces by utilizingrational cubic spline with quadratic denominator In worksby Hussain et al [10] and Sarfraz et al [11] the rationalcubic spline with quadratic denominator has been used forpositivity monotonicity and convexity preserving with 119862

2

continuity Hussain et al [10] have only one free parametermeanwhile Sarfraz et al [11] have no free parameter Abbas[8] and Abbas et al [12] have discussed the positivity byusing new 119862

2 rational cubic spline with two free parametersAnother paper concerning 119862

2 rational spline can be foundby Delbourgo [13] Gregory [14] and Delbourgo and Gregory[15] Karim and Kong [16ndash19] have proposed new 119862

1 rationalcubic spline (cubicquadratic) with three parameters wheretwo of them are free parameters The rational cubic splinehas been successfully applied to the local control of theinterpolating functions positivity monotonicity and con-vexity preserving as well as the derivative control includingan error analysis when the function to be interpolated is119891(119905) isin 119862

3[1199050 119905119899] Motivated by the works of Tian et al [20]

Abbas et al [12] Hussain et al [10] and Sarfraz et al [11] inthis paper the authors will proposed new 119862

2 rational cubicspline for positivity monotonicity and convexity preservingand data constrained modeling Under some circumstancesour 119862

2 rational cubic spline will give new 1198622 rational cubic

spline based on rational cubic spline defined by Tian etal [20] Numerical comparison between 119862

2 rational cubicspline and the works of Hussain et al [10] Abbas [8]Abbas et al [12] and Sarfraz et al [11] also has been madecomprehensively From all presented numerical results shapepreserving interpolation by using the new 119862

2 rational cubicspline gives comparable results with existing rational cubicspline schemesThemain scientific contribution of this paperis summarized as follows

(i) In this paper 1198622 rational cubic spline (cubic

quadratic) with three parameters has been used forpositivity monotonicity and convexity preservingand constrained data modeling while in works byKarim and Kong [17ndash19] M Z Hussain and MHussain [9] and Sarfraz [5] the degree of smoothnessattained is 1198621

(ii) Hussain et al [10] and Sarfraz et al [11] discussedthe positivity by using 119862

2 rational cubic spline(cubicquadratic) with two parameters with one or nofree parameter while our rational cubic spline has twofree parameters Even though Abbas et al [12] alsoproposed 119862

2 rational cubic spline (cubicquadratic)with two parameters their rational spline is differentform our rational cubic spline Furthermore it wasnoticed that schemes of Abbas et al [12] may notbe able to produce completely positive interpolatingcurves with 119862

2 continuity

(iii) When 120574119894

= 0 we may obtain the new 1198622 rational

cubic spline with two parameters an extension to theoriginal 1198621 rational cubic spline of Tian et al [20]Thus our1198622 rational cubic spline gives a larger class ofrational cubic spline which also includes the rationalcubic spline of Tian et al [20]

(iv) Our rational scheme is local while in Lamberti andManni [21] their scheme is global Furthermore ourrational scheme works well for both equally andunequally spaced data while the rational spline inter-polant by Duan et al [22] and Bao et al [23] onlyworks for equally spaced data

(v) Numerical comparisons between the 1198622 rational

cubic spline and the existing schemes such as Hussainet al [10] Sarfraz et al [11] and Abbas et al [12] forpositivity preserving and 119862

1 rational spline of Karimand Kong [17ndash19] also have been done comprehen-sively

(vi) Ourmethod also does not require any knots insertionMeanwhile the cubic spline interpolation by Butt andBrodlie [3] Brodlie andButt [4] and Fiorot andTabka[24] requires knots insertion in the interval where theinterpolating curves produce the negative values (liesbelow 119909-axis) for positive data nonmonotone inter-polating curves (for monotone data) and nonconvexinterpolating curves for convex data

(vii) This paper utilized the rational cubic spline mean-while in Dube and Tiwari [25] Pan and Wang [26]and Ibraheem et al [27] the rational trigonometricspline is used in place of standard rational cubicspline Thus no trigonometric functions are involvedTherefore the method is not computationally expen-sive

The remainder of the paper is organized as followsSection 2 introduces the new 119862

2 rational cubic spline withthree parameters with some discussion on the methods toestimate the first derivatives values as well as shape controls ofthe rational cubic spline interpolation Meanwhile Section 3discusses the positivity preserving by using 119862

2 rational cubicspline together with numerical demonstrations as well ascomparison with some existing schemes including erroranalysis Section 4 is devoted for research discussion Finallya summary and conclusions are given in Section 5

2 1198622 Rational Cubic Spline Interpolant

This section will introduce 1198622 rational cubic spline inter-

polant with three parameters Originally this rational cubicspline has been initiated by Karim and Kong [18] The maindifference is that in this paper the rational cubic spline has1198622 continuity while in work by Karim and Kong [18] it has

1198621 continuity We begin with the definition of 119862

2 rationalcubic spline interpolant given set of data points (119909

119894 119891119894) 119894 =

0 1 119899 such that 1199090lt 1199091lt sdot sdot sdot lt 119909

119899 Let ℎ

119894= 119909119894+1

minus 119909119894

Δ119894= (119891119894+1

minus 119891119894)ℎ119894and 120579 = (119909 minus 119909

119894)ℎ119894 where 0 le 120579 le 1 For


119909 isin [119909119894 119909119894+1


119904 (119909) equiv 119904119894(119909)

=1198601198940(1 minus 120579)

3+ 1198601198941120579 (1 minus 120579)

2+ 11986011989421205792(1 minus 120579) + 119860

11989431205793

(1 minus 120579)2120572119894+ 120579 (1 minus 120579) (2120572

119894120573119894+ 120574119894) + 1205792120573119894

(1)


119904 (119909119894) = 119891119894

119904 (119909119894+1

) = 119891119894+1

119904(1)

(119909119894) = 119889119894

119904(1)

(119909119894+1

) = 119889119894+1

119904(2)

(119909119894+) = 119904(2)

(119909119894minus)

(2)

where 119904(1)

(119909119894) and 119904



(2)(119909119894+) and 119904






1198601198940

= 120572119894119891119894

1198601198941

= (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894+ 120572119894ℎ119894119889119894

1198601198942

= (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

1198601198943

= 120573119894119891119894+1

(3)




(2)(119909) is given as

119904(2)

(119909) =

sum3

119895=0119862119894119895(1 minus 120579)

3minus119895120579119895

ℎ119894[119876119894(120579)]3

(4)

where

1198621198940

= 21205722

119894(120574119894(Δ119894minus 119889119894) minus 120573119894(119889119894+1

minus Δ119894)

minus 2120572119894120573119894(119889119894minus Δ119894))

1198621198941

= 61205722

119894120573119894(Δ119894minus 119889119894)

1198621198942

= 61205721198941205732

119894(119889119894+1

minus Δ119894)

1198621198943

= 21205732

119894[120574119894(119889119894+1

minus Δ119894)


minus Δ119894))]

(5)

Now 1198622 continuity 119904(2)(119909

119894+) = 119904(2)

(119909119894minus) 119894 = 1 2 119899 minus 1

will give

2 [120574119894minus1

(119889119894minus Δ119894minus1

) minus 120572119894minus1

(Δ119894minus1



(119889119894minus Δ119894minus1

))]

ℎ119894minus1

120573119894minus1

=2 (120574119894(Δ119894minus 119889119894) minus 120573119894(119889119894+1

minus Δ119894) minus 2120572

119894120573119894(119889119894minus Δ119894))

ℎ119894120572119894

(6)


119886119894119889119894minus1

+ 119887119894119889119894+ 119888119894119889119894+1

= 119890119894 119894 = 1 2 119899 minus 1 (7)

with

119886119894= ℎ119894120572119894minus1

120572119894

119887119894= ℎ119894120572119894(120574119894minus1

+ 2120572119894minus1

120573119894minus1

) + ℎ119894minus1

120573119894minus1

(120574119894+ 2120572119894120573119894)

119888119894= ℎ119894minus1

120573119894minus1

120573119894

119890119894= ℎ119894120572119894(120574119894minus1

+ 120572119894minus1

+ 2120572119894minus1

120573119894minus1

) Δ119894minus1

+ ℎ119894minus1

120573119894minus1

(120574119894+ 120573119894+ 2120572119894120573119894) Δ119894

(8)


119894 119894 = 1 2 119899minus1 for all 120572

119894 120573119894gt 0 120574

119894ge 0The


0and

119889119899

119904(1)

(1199090) = 1198890

119904(1)

(119909119899) = 119889119899

(9)


119904 (119909) equiv 119904119894(119909) =

119875119894(120579)

119876119894(120579)

(10)

where

119875119894(120579)

= 120572119894119891119894(1 minus 120579)

3

+ ((2120572119894120573119894+ 120572119894+ 120574119894) 119891119894+ 120572119894ℎ119894119889119894) 120579 (1 minus 120579)

2

+ ((2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

) 1205792(1 minus 120579)

+ 120573119894119891119894+1

1205793

119876119894(120579) = (1 minus 120579)

2120572119894+ 120579 (1 minus 120579) (2120572

119894120573119894+ 120574119894) + 1205792120573119894

(11)




[119909119894 119909119894+1





2isin


119894 120573119894gt 0 120574

119894ge 0



119899



1198890=

0 Δ0= 0 or Δ

20= 0

Δ(1+ℎ0ℎ1)

0Δ(minusℎ0ℎ1)

21otherwise

119889119899

=

0 Δ119899minus1

= 0 or Δ119899119899minus2

= 0




(12)


1198890= Δ0+ (Δ0minus Δ1) (

ℎ0

ℎ0+ ℎ1

)

119889119899= Δ119899minus1

+ (Δ119899minus1


) (ℎ119899minus1

ℎ119899minus1

+ ℎ119899minus2

)

(13)


0and 119889

119899 respectively


as follows

(1) When 120572119894

gt 0 120573119894

gt 0 and 120574119894




2



119894= 120573119894

= 1 120574119894



119904 (119909) = (1 minus 120579)2(1 minus 2120579) 119891

119894+ 1205792(3 minus 2120579) 119891

119894+1


119894+1

(14)

for 119894 = 0 1 119899 minus 1


119894 0 1 1 3 4119909119894

0 2 3 9 11119891119894

05 15 7 9 13ℎ119894

2 1 6 2Δ119894

05 55 13 2119889119894(1198621) minus2833 3833 47619 15833 24167

119889119894(1198622) minus2833 43223 49828 12146 24167


119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

+ℎ119894120579 (1 minus 120579) [120572

119894(119889119894minus Δ119894) (1 minus 120579) + 120573

119894(Δ119894minus 119889119894+1

) 120579]

119876119894(120579)

(15)


rarr 0 120573119894

rarr 0 or 120574119894



119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(16)


119894or the






= 120573119894

= 1 120574119894

= 2






119899


119894+1minusΔ119894= 0 then 119889

119894= 119889119894+1

= Δ119894


119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(17)




0 120573119894rarr 0 or 120574






x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(d)



120572119894= 120573119894= 1 and 120574


119894= 120573119894= 1 and 120574

119894= 2
















119894and 120573

119894are


119894 We begin by


119894 119891119894) 119894 =


119899 such that

119891119894gt 0 119894 = 0 1 119899 (18)


119894(120579) gt 0 and 119876

119894(120579) gt 0



119894(120579) gt




119875119894(120579) = 119861

1198941205793+ 1198621198941205792+ 119863119894120579 + 119864119894 (19)


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

14

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(d)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(e)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(f)

x-axis

y-axis

12

14

10

8

6

4

2

0

1086420

(g)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(h)

Figure 2 Continued


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(i)

x-axis

y-axis

15

10

5

0

1086420

(j)



Figure 2 119894 0 1 2 3

Figure 2(a)120572119894

01 01 01 01120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(b)120572119894

1 1 1 1120573119894

01 01 01 01120574119894

1 1 1 1

Figure 2(c)120572119894

1 1 1 1120573119894

1 1 1 1120574119894

100 100 100 100

Figure 2(d)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

100 100 100 100

Figure 2(e)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

1000 1000 1000 1000

Figure 2(h)120572119894

001 001 001 001120573119894

001 001 001 001120574119894

1 1 1 1

Figure 2(i)120572119894

001 001 001 001120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(j)120572119894

1 1 1 1120573119894

001 001 001 001120574119894

1 1 1 1

where

119861119894= 120572119894ℎ119894119889119894+ 120573119894ℎ119894119889119894+1

+ 2120572119894120573119894119891119894+ 120574119894119891119894minus 2120572119894120573119894119891119894+1

minus 120574119894119891119894+1

119862119894= minus2120572

119894ℎ119894119889119894minus 120573119894ℎ119894119889119894+1

+ 120572119894119891119894minus 4120572119894120573119894119891119894minus 2120574119894119891119894

+ 120573119894119891119894+1

+ 2120572119894120573119894119891119894+1

+ 120574119894119891119894+1

119863119894= 120572119894ℎ119894119889119894minus 2120572119894119891119894+ 2120572119894120573119894119891119894+ 120574119894119891

119864119894= 120572119894119891119894

(20)



follows

119875119894(119904) = 119860119904

3+ 1198611199042+ 119862119904 + 119863 (21)

where119860 = 120573

119894119891119894+1

119861 = (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

119862 = (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894minus 120572119894ℎ119894119889119894

119863 = 120572119894119891119894

(22)





119894 120573119894 and 120574



120572119894 120573119894gt 0

120574119894gt Max0 minus120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

]

(23)



1


119894 119894 = 1 2 119899minus1



119894 119894 =




119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25


119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570


120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)




119894=0 1198890 and 119889

119899



119894gt 0 and 120582

119894gt

0





2




119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10



Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894






3[1199050 119905119899] using




119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119909 isin [119909119894 119909119894+1


119904 (119909) equiv 119904119894(119909)

=1198601198940(1 minus 120579)

3+ 1198601198941120579 (1 minus 120579)

2+ 11986011989421205792(1 minus 120579) + 119860

11989431205793

(1 minus 120579)2120572119894+ 120579 (1 minus 120579) (2120572

119894120573119894+ 120574119894) + 1205792120573119894

(1)


119904 (119909119894) = 119891119894

119904 (119909119894+1

) = 119891119894+1

119904(1)

(119909119894) = 119889119894

119904(1)

(119909119894+1

) = 119889119894+1

119904(2)

(119909119894+) = 119904(2)

(119909119894minus)

(2)

where 119904(1)

(119909119894) and 119904



(2)(119909119894+) and 119904






1198601198940

= 120572119894119891119894

1198601198941

= (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894+ 120572119894ℎ119894119889119894

1198601198942

= (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

1198601198943

= 120573119894119891119894+1

(3)




(2)(119909) is given as

119904(2)

(119909) =

sum3

119895=0119862119894119895(1 minus 120579)

3minus119895120579119895

ℎ119894[119876119894(120579)]3

(4)

where

1198621198940

= 21205722

119894(120574119894(Δ119894minus 119889119894) minus 120573119894(119889119894+1

minus Δ119894)

minus 2120572119894120573119894(119889119894minus Δ119894))

1198621198941

= 61205722

119894120573119894(Δ119894minus 119889119894)

1198621198942

= 61205721198941205732

119894(119889119894+1

minus Δ119894)

1198621198943

= 21205732

119894[120574119894(119889119894+1

minus Δ119894)


minus Δ119894))]

(5)

Now 1198622 continuity 119904(2)(119909

119894+) = 119904(2)

(119909119894minus) 119894 = 1 2 119899 minus 1

will give

2 [120574119894minus1

(119889119894minus Δ119894minus1

) minus 120572119894minus1

(Δ119894minus1



(119889119894minus Δ119894minus1

))]

ℎ119894minus1

120573119894minus1

=2 (120574119894(Δ119894minus 119889119894) minus 120573119894(119889119894+1

minus Δ119894) minus 2120572

119894120573119894(119889119894minus Δ119894))

ℎ119894120572119894

(6)


119886119894119889119894minus1

+ 119887119894119889119894+ 119888119894119889119894+1

= 119890119894 119894 = 1 2 119899 minus 1 (7)

with

119886119894= ℎ119894120572119894minus1

120572119894

119887119894= ℎ119894120572119894(120574119894minus1

+ 2120572119894minus1

120573119894minus1

) + ℎ119894minus1

120573119894minus1

(120574119894+ 2120572119894120573119894)

119888119894= ℎ119894minus1

120573119894minus1

120573119894

119890119894= ℎ119894120572119894(120574119894minus1

+ 120572119894minus1

+ 2120572119894minus1

120573119894minus1

) Δ119894minus1

+ ℎ119894minus1

120573119894minus1

(120574119894+ 120573119894+ 2120572119894120573119894) Δ119894

(8)


119894 119894 = 1 2 119899minus1 for all 120572

119894 120573119894gt 0 120574

119894ge 0The


0and

119889119899

119904(1)

(1199090) = 1198890

119904(1)

(119909119899) = 119889119899

(9)


119904 (119909) equiv 119904119894(119909) =

119875119894(120579)

119876119894(120579)

(10)

where

119875119894(120579)

= 120572119894119891119894(1 minus 120579)

3

+ ((2120572119894120573119894+ 120572119894+ 120574119894) 119891119894+ 120572119894ℎ119894119889119894) 120579 (1 minus 120579)

2

+ ((2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

) 1205792(1 minus 120579)

+ 120573119894119891119894+1

1205793

119876119894(120579) = (1 minus 120579)

2120572119894+ 120579 (1 minus 120579) (2120572

119894120573119894+ 120574119894) + 1205792120573119894

(11)




[119909119894 119909119894+1





2isin


119894 120573119894gt 0 120574

119894ge 0



119899



1198890=

0 Δ0= 0 or Δ

20= 0

Δ(1+ℎ0ℎ1)

0Δ(minusℎ0ℎ1)

21otherwise

119889119899

=

0 Δ119899minus1

= 0 or Δ119899119899minus2

= 0




(12)


1198890= Δ0+ (Δ0minus Δ1) (

ℎ0

ℎ0+ ℎ1

)

119889119899= Δ119899minus1

+ (Δ119899minus1


) (ℎ119899minus1

ℎ119899minus1

+ ℎ119899minus2

)

(13)


0and 119889

119899 respectively


as follows

(1) When 120572119894

gt 0 120573119894

gt 0 and 120574119894




2



119894= 120573119894

= 1 120574119894



119904 (119909) = (1 minus 120579)2(1 minus 2120579) 119891

119894+ 1205792(3 minus 2120579) 119891

119894+1


119894+1

(14)

for 119894 = 0 1 119899 minus 1


119894 0 1 1 3 4119909119894

0 2 3 9 11119891119894

05 15 7 9 13ℎ119894

2 1 6 2Δ119894

05 55 13 2119889119894(1198621) minus2833 3833 47619 15833 24167

119889119894(1198622) minus2833 43223 49828 12146 24167


119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

+ℎ119894120579 (1 minus 120579) [120572

119894(119889119894minus Δ119894) (1 minus 120579) + 120573

119894(Δ119894minus 119889119894+1

) 120579]

119876119894(120579)

(15)


rarr 0 120573119894

rarr 0 or 120574119894



119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(16)


119894or the






= 120573119894

= 1 120574119894

= 2






119899


119894+1minusΔ119894= 0 then 119889

119894= 119889119894+1

= Δ119894


119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(17)




0 120573119894rarr 0 or 120574






x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(d)



120572119894= 120573119894= 1 and 120574


119894= 120573119894= 1 and 120574

119894= 2
















119894and 120573

119894are


119894 We begin by


119894 119891119894) 119894 =


119899 such that

119891119894gt 0 119894 = 0 1 119899 (18)


119894(120579) gt 0 and 119876

119894(120579) gt 0



119894(120579) gt




119875119894(120579) = 119861

1198941205793+ 1198621198941205792+ 119863119894120579 + 119864119894 (19)


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

14

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(d)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(e)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(f)

x-axis

y-axis

12

14

10

8

6

4

2

0

1086420

(g)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(h)

Figure 2 Continued


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(i)

x-axis

y-axis

15

10

5

0

1086420

(j)



Figure 2 119894 0 1 2 3

Figure 2(a)120572119894

01 01 01 01120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(b)120572119894

1 1 1 1120573119894

01 01 01 01120574119894

1 1 1 1

Figure 2(c)120572119894

1 1 1 1120573119894

1 1 1 1120574119894

100 100 100 100

Figure 2(d)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

100 100 100 100

Figure 2(e)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

1000 1000 1000 1000

Figure 2(h)120572119894

001 001 001 001120573119894

001 001 001 001120574119894

1 1 1 1

Figure 2(i)120572119894

001 001 001 001120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(j)120572119894

1 1 1 1120573119894

001 001 001 001120574119894

1 1 1 1

where

119861119894= 120572119894ℎ119894119889119894+ 120573119894ℎ119894119889119894+1

+ 2120572119894120573119894119891119894+ 120574119894119891119894minus 2120572119894120573119894119891119894+1

minus 120574119894119891119894+1

119862119894= minus2120572

119894ℎ119894119889119894minus 120573119894ℎ119894119889119894+1

+ 120572119894119891119894minus 4120572119894120573119894119891119894minus 2120574119894119891119894

+ 120573119894119891119894+1

+ 2120572119894120573119894119891119894+1

+ 120574119894119891119894+1

119863119894= 120572119894ℎ119894119889119894minus 2120572119894119891119894+ 2120572119894120573119894119891119894+ 120574119894119891

119864119894= 120572119894119891119894

(20)



follows

119875119894(119904) = 119860119904

3+ 1198611199042+ 119862119904 + 119863 (21)

where119860 = 120573

119894119891119894+1

119861 = (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

119862 = (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894minus 120572119894ℎ119894119889119894

119863 = 120572119894119891119894

(22)





119894 120573119894 and 120574



120572119894 120573119894gt 0

120574119894gt Max0 minus120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

]

(23)



1


119894 119894 = 1 2 119899minus1



119894 119894 =




119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25


119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570


120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)




119894=0 1198890 and 119889

119899



119894gt 0 and 120582

119894gt

0





2




119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10



Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894






3[1199050 119905119899] using




119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[119909119894 119909119894+1





2isin


119894 120573119894gt 0 120574

119894ge 0



119899



1198890=

0 Δ0= 0 or Δ

20= 0

Δ(1+ℎ0ℎ1)

0Δ(minusℎ0ℎ1)

21otherwise

119889119899

=

0 Δ119899minus1

= 0 or Δ119899119899minus2

= 0




(12)


1198890= Δ0+ (Δ0minus Δ1) (

ℎ0

ℎ0+ ℎ1

)

119889119899= Δ119899minus1

+ (Δ119899minus1


) (ℎ119899minus1

ℎ119899minus1

+ ℎ119899minus2

)

(13)


0and 119889

119899 respectively


as follows

(1) When 120572119894

gt 0 120573119894

gt 0 and 120574119894




2



119894= 120573119894

= 1 120574119894



119904 (119909) = (1 minus 120579)2(1 minus 2120579) 119891

119894+ 1205792(3 minus 2120579) 119891

119894+1


119894+1

(14)

for 119894 = 0 1 119899 minus 1


119894 0 1 1 3 4119909119894

0 2 3 9 11119891119894

05 15 7 9 13ℎ119894

2 1 6 2Δ119894

05 55 13 2119889119894(1198621) minus2833 3833 47619 15833 24167

119889119894(1198622) minus2833 43223 49828 12146 24167


119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

+ℎ119894120579 (1 minus 120579) [120572

119894(119889119894minus Δ119894) (1 minus 120579) + 120573

119894(Δ119894minus 119889119894+1

) 120579]

119876119894(120579)

(15)


rarr 0 120573119894

rarr 0 or 120574119894



119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(16)


119894or the






= 120573119894

= 1 120574119894

= 2






119899


119894+1minusΔ119894= 0 then 119889

119894= 119889119894+1

= Δ119894


119904 (119909) = (1 minus 120579) 119891119894+ 120579119891119894+1

(17)




0 120573119894rarr 0 or 120574






x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(d)



120572119894= 120573119894= 1 and 120574


119894= 120573119894= 1 and 120574

119894= 2
















119894and 120573

119894are


119894 We begin by


119894 119891119894) 119894 =


119899 such that

119891119894gt 0 119894 = 0 1 119899 (18)


119894(120579) gt 0 and 119876

119894(120579) gt 0



119894(120579) gt




119875119894(120579) = 119861

1198941205793+ 1198621198941205792+ 119863119894120579 + 119864119894 (19)


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

14

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(d)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(e)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(f)

x-axis

y-axis

12

14

10

8

6

4

2

0

1086420

(g)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(h)

Figure 2 Continued


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(i)

x-axis

y-axis

15

10

5

0

1086420

(j)



Figure 2 119894 0 1 2 3

Figure 2(a)120572119894

01 01 01 01120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(b)120572119894

1 1 1 1120573119894

01 01 01 01120574119894

1 1 1 1

Figure 2(c)120572119894

1 1 1 1120573119894

1 1 1 1120574119894

100 100 100 100

Figure 2(d)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

100 100 100 100

Figure 2(e)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

1000 1000 1000 1000

Figure 2(h)120572119894

001 001 001 001120573119894

001 001 001 001120574119894

1 1 1 1

Figure 2(i)120572119894

001 001 001 001120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(j)120572119894

1 1 1 1120573119894

001 001 001 001120574119894

1 1 1 1

where

119861119894= 120572119894ℎ119894119889119894+ 120573119894ℎ119894119889119894+1

+ 2120572119894120573119894119891119894+ 120574119894119891119894minus 2120572119894120573119894119891119894+1

minus 120574119894119891119894+1

119862119894= minus2120572

119894ℎ119894119889119894minus 120573119894ℎ119894119889119894+1

+ 120572119894119891119894minus 4120572119894120573119894119891119894minus 2120574119894119891119894

+ 120573119894119891119894+1

+ 2120572119894120573119894119891119894+1

+ 120574119894119891119894+1

119863119894= 120572119894ℎ119894119889119894minus 2120572119894119891119894+ 2120572119894120573119894119891119894+ 120574119894119891

119864119894= 120572119894119891119894

(20)



follows

119875119894(119904) = 119860119904

3+ 1198611199042+ 119862119904 + 119863 (21)

where119860 = 120573

119894119891119894+1

119861 = (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

119862 = (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894minus 120572119894ℎ119894119889119894

119863 = 120572119894119891119894

(22)





119894 120573119894 and 120574



120572119894 120573119894gt 0

120574119894gt Max0 minus120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

]

(23)



1


119894 119894 = 1 2 119899minus1



119894 119894 =




119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25


119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570


120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)




119894=0 1198890 and 119889

119899



119894gt 0 and 120582

119894gt

0





2




119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10



Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894






3[1199050 119905119899] using




119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

minus2

(d)



120572119894= 120573119894= 1 and 120574


119894= 120573119894= 1 and 120574

119894= 2
















119894and 120573

119894are


119894 We begin by


119894 119891119894) 119894 =


119899 such that

119891119894gt 0 119894 = 0 1 119899 (18)


119894(120579) gt 0 and 119876

119894(120579) gt 0



119894(120579) gt




119875119894(120579) = 119861

1198941205793+ 1198621198941205792+ 119863119894120579 + 119864119894 (19)


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(a)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

14

(b)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(c)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(d)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(e)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(f)

x-axis

y-axis

12

14

10

8

6

4

2

0

1086420

(g)

x-axis

y-axis

12

10

8

6

4

2

0

1086420

(h)

Figure 2 Continued


x-axis

y-axis

12

10

8

6

4

2

0

1086420

(i)

x-axis

y-axis

15

10

5

0

1086420

(j)



Figure 2 119894 0 1 2 3

Figure 2(a)120572119894

01 01 01 01120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(b)120572119894

1 1 1 1120573119894

01 01 01 01120574119894

1 1 1 1

Figure 2(c)120572119894

1 1 1 1120573119894

1 1 1 1120574119894

100 100 100 100

Figure 2(d)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

100 100 100 100

Figure 2(e)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

1000 1000 1000 1000

Figure 2(h)120572119894

001 001 001 001120573119894

001 001 001 001120574119894

1 1 1 1

Figure 2(i)120572119894

001 001 001 001120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(j)120572119894

1 1 1 1120573119894

001 001 001 001120574119894

1 1 1 1

where

119861119894= 120572119894ℎ119894119889119894+ 120573119894ℎ119894119889119894+1

+ 2120572119894120573119894119891119894+ 120574119894119891119894minus 2120572119894120573119894119891119894+1

minus 120574119894119891119894+1

119862119894= minus2120572

119894ℎ119894119889119894minus 120573119894ℎ119894119889119894+1

+ 120572119894119891119894minus 4120572119894120573119894119891119894minus 2120574119894119891119894

+ 120573119894119891119894+1

+ 2120572119894120573119894119891119894+1

+ 120574119894119891119894+1

119863119894= 120572119894ℎ119894119889119894minus 2120572119894119891119894+ 2120572119894120573119894119891119894+ 120574119894119891

119864119894= 120572119894119891119894

(20)



follows

119875119894(119904) = 119860119904

3+ 1198611199042+ 119862119904 + 119863 (21)

where119860 = 120573

119894119891119894+1

119861 = (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

119862 = (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894minus 120572119894ℎ119894119889119894

119863 = 120572119894119891119894

(22)





119894 120573119894 and 120574



120572119894 120573119894gt 0

120574119894gt Max0 minus120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

]

(23)



1


119894 119894 = 1 2 119899minus1



119894 119894 =




119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25


119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570


120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)




119894=0 1198890 and 119889

119899



119894gt 0 and 120582

119894gt

0





2




119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10



Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894






3[1199050 119905119899] using




119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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1086420

(a)

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1086420

14

(b)

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1086420

(c)

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12

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0

1086420

(d)

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2

0

1086420

(e)

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0

1086420

(f)

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14

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2

0

1086420

(g)

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12

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1086420

(h)

Figure 2 Continued


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10

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4

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0

1086420

(i)

x-axis

y-axis

15

10

5

0

1086420

(j)



Figure 2 119894 0 1 2 3

Figure 2(a)120572119894

01 01 01 01120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(b)120572119894

1 1 1 1120573119894

01 01 01 01120574119894

1 1 1 1

Figure 2(c)120572119894

1 1 1 1120573119894

1 1 1 1120574119894

100 100 100 100

Figure 2(d)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

100 100 100 100

Figure 2(e)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

1000 1000 1000 1000

Figure 2(h)120572119894

001 001 001 001120573119894

001 001 001 001120574119894

1 1 1 1

Figure 2(i)120572119894

001 001 001 001120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(j)120572119894

1 1 1 1120573119894

001 001 001 001120574119894

1 1 1 1

where

119861119894= 120572119894ℎ119894119889119894+ 120573119894ℎ119894119889119894+1

+ 2120572119894120573119894119891119894+ 120574119894119891119894minus 2120572119894120573119894119891119894+1

minus 120574119894119891119894+1

119862119894= minus2120572

119894ℎ119894119889119894minus 120573119894ℎ119894119889119894+1

+ 120572119894119891119894minus 4120572119894120573119894119891119894minus 2120574119894119891119894

+ 120573119894119891119894+1

+ 2120572119894120573119894119891119894+1

+ 120574119894119891119894+1

119863119894= 120572119894ℎ119894119889119894minus 2120572119894119891119894+ 2120572119894120573119894119891119894+ 120574119894119891

119864119894= 120572119894119891119894

(20)



follows

119875119894(119904) = 119860119904

3+ 1198611199042+ 119862119904 + 119863 (21)

where119860 = 120573

119894119891119894+1

119861 = (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

119862 = (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894minus 120572119894ℎ119894119889119894

119863 = 120572119894119891119894

(22)





119894 120573119894 and 120574



120572119894 120573119894gt 0

120574119894gt Max0 minus120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

]

(23)



1


119894 119894 = 1 2 119899minus1



119894 119894 =




119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25


119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570


120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)




119894=0 1198890 and 119889

119899



119894gt 0 and 120582

119894gt

0





2




119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10



Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894






3[1199050 119905119899] using




119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x-axis

y-axis

12

10

8

6

4

2

0

1086420

(i)

x-axis

y-axis

15

10

5

0

1086420

(j)



Figure 2 119894 0 1 2 3

Figure 2(a)120572119894

01 01 01 01120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(b)120572119894

1 1 1 1120573119894

01 01 01 01120574119894

1 1 1 1

Figure 2(c)120572119894

1 1 1 1120573119894

1 1 1 1120574119894

100 100 100 100

Figure 2(d)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

100 100 100 100

Figure 2(e)120572119894

5 5 5 5120573119894

5 5 5 5120574119894

1000 1000 1000 1000

Figure 2(h)120572119894

001 001 001 001120573119894

001 001 001 001120574119894

1 1 1 1

Figure 2(i)120572119894

001 001 001 001120573119894

1 1 1 1120574119894

1 1 1 1

Figure 2(j)120572119894

1 1 1 1120573119894

001 001 001 001120574119894

1 1 1 1

where

119861119894= 120572119894ℎ119894119889119894+ 120573119894ℎ119894119889119894+1

+ 2120572119894120573119894119891119894+ 120574119894119891119894minus 2120572119894120573119894119891119894+1

minus 120574119894119891119894+1

119862119894= minus2120572

119894ℎ119894119889119894minus 120573119894ℎ119894119889119894+1

+ 120572119894119891119894minus 4120572119894120573119894119891119894minus 2120574119894119891119894

+ 120573119894119891119894+1

+ 2120572119894120573119894119891119894+1

+ 120574119894119891119894+1

119863119894= 120572119894ℎ119894119889119894minus 2120572119894119891119894+ 2120572119894120573119894119891119894+ 120574119894119891

119864119894= 120572119894119891119894

(20)



follows

119875119894(119904) = 119860119904

3+ 1198611199042+ 119862119904 + 119863 (21)

where119860 = 120573

119894119891119894+1

119861 = (2120572119894120573119894+ 120573119894+ 120574119894) 119891119894+1

minus 120573119894ℎ119894119889119894+1

119862 = (2120572119894120573119894+ 120572119894+ 120574119894) 119891119894minus 120572119894ℎ119894119889119894

119863 = 120572119894119891119894

(22)





119894 120573119894 and 120574



120572119894 120573119894gt 0

120574119894gt Max0 minus120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

]

(23)



1


119894 119894 = 1 2 119899minus1



119894 119894 =




119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25


119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570


120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)




119894=0 1198890 and 119889

119899



119894gt 0 and 120582

119894gt

0





2




119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10



Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894






3[1199050 119905119899] using




119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 0 1 2 3119909119894

00 10 170 180119891119894

025 10 1110 25


119894 0 1 2 3119889119894(1198621) minus7296 8796 123429 154570

Δ119894

075 14429 139120572119894

05 05 05120573119894

05 05 05120574119894

1384 314 025119889119894(1198622) minus7296 2108 825421 154570


120572119894 120573119894gt 0

120574119894= 120582119894+Max0 minus 120572

119894[ℎ119894119889119894+ (2120573119894+ 1) 119891

119894

119891119894

]

120573119894[ℎ119894119889119894+1

minus (2120572119894+ 1) 119891

119894+1

119891119894+1

] 120582119894gt 0

(24)




119894=0 1198890 and 119889

119899



119894gt 0 and 120582

119894gt

0





2




119894 0 1 2 3 4 5 6119909119894

2 3 7 8 9 13 14119891119894

10 2 3 7 2 3 10



Δ119894

minus8 225 40 minus05 025 7120572119894

25 25 25 25 25 25120573119894

25 25 25 25 25 25120574119894






3[1199050 119905119899] using




119894= [119905119894 119905119894+1

] is defined as

119877 [119891] = 119891 (119905) minus 119875119894(119905) =

1

2int

119905119894+1

119905119894

119891(3)

119877119905[(119905 minus 120591)

2

+] 119889120591 (25)

where

119877119905[(119905 minus 120591)

2

+] =

119903 (120591 119905) 119905119894lt 120591 lt 119905

119904 (120591 119905) 119905 lt 120591 lt 119905119894+1

(26)


x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x-axis

y-axis

25

20

15

10

5

0

minus5

00 05 10 15

(a)

x-axis

y-axis

25

20

15

10

5

0

30

0 02 04 06 08 1 12 14 16 18 2

(b)

x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(c)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(d)

x-axis

y-axis

25

20

15

10

5

0

00 05 10 15

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 2


x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x-axis

y-axis

10

8

6

4

2

0

minus2

1086420 1412

(a)

x-axis

y-axis

11

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14 16

(b)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

0

1

2 4 6 8 10 12 14

(c)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(d)

x-axis

y-axis

10

8

6

4

2

0

1086420 1412

(e)


119894= 120573119894= 05 (e) 120572

119894= 120573119894= 25


x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x-axis

y-axis

25

20

15

10

5

0

0 02 04 06 08 1 12 14 16 18

(a)

x-axis

y-axis

10

9

8

7

6

5

4

3

2

1

0

2 4 6 8 10 12 14

(b)

Figure 5

with

119903 (120591 119905) = (119905 minus 120591)2+ 119904 (120591 119905)

119904 (120591 119905)

= minus

[(2120572119894120573119894+ 120573119894+ 120574119894) 1205772minus 2ℎ119894120573119894120577 1205792(1 minus 120579) + 120573

11989412057721205793]

119876119894(120579)

(27)

with 120577 = (119905119894+1


119894= [119909119894 119909119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 (28)


int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591 = int

119905

119905119894

|119903 (120591 119909)| 119889120591

+ int

119905119894+1

119905

|119904 (120591 119909)| 119889120591

(29)


119894120588119894 where 120588


119903(120591 119905) = 0 are

120591119896= 119909 minus

120579ℎ119894(120579120588119894+ (minus1)

119896+1119867)

120572119894+ 120579120588119894

119896 = 1 2 (30)

where


119894120588119894120579 (31)

The roots of 119904(120591 119905) = 0 are 1205913= 119905119894+1

1205914= 119905119894+1

minus 2120573119894ℎ119894(1 minus

120579)(120573119894+ (1 minus 120579)120588


obtained



1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119866119894(120591)

(32)

with

119866119894(120591) = int

119905

119905119894

minus119903 (120591 119905) 119889120591 + int

1205914

119905

minus119904 (120591 119905) 119889120591

+ int

119905119894+1

1205914

119904 (120591 119905) 119889120591

(33)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205771(120572119894 120573119894 120574119894 120579) (34)


where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

1205771(120572119894 120573119894 120574119894 120579) = minus

ℎ3

1198941205793

3+

120573119894ℎ3

1198941205793

3119876119894(120579)

minus

2 (120588119894+ 120573119894) (8ℎ3

1198941205792(1 minus 120579)

4)

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+

81205732

119894(ℎ3

1198941205792(1 minus 120579)

2)

3 (120588119894(1 minus 120579) + 120573

119894)2

119876119894(120579)

minus16120573119894ℎ3

1198941205793(1 minus 120579)

3

3 (120588119894(1 minus 120579) + 120573

119894)3

119876119894(120579)

+ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

[120588119894minus 2120573119894]

(35)

Case 2 For 0 le 120573119894120588119894le 1 120579


following form

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171198662(120591)

(36)

with

1198662(120591) = int

1205912

119905119894

minus119903 (120591 119905) 119889120591 + int

119905

1205912

119903 (120591 119905) 119889120591

+ int

119905119894+1

119905

119904 (120591 119905) 119889120591

(37)

Hence

1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205772(120572119894 120573119894 120574119894 120579) (38)

where

1205772(120572119894 120573119894 120574119894 120579) =

2ℎ3

1198941205793(120579120588119894minus 119867)3

3

minus2120573119894ℎ3

1198941205793[1198663(120579)]3

3119876119894(120579)

minus2 (120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

3119876119894(120579)

+2120573119894ℎ3

119894(1 minus 120579) 120579

2[1198663(120579)]3

119876119894(120579)

(39)

with 1198663(120579) = (1 minus 120579) + 120579(120579120588

119894minus 119867)(120572

119894+ 120588119894120579)

Case 3 For 120573119894120588119894


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)100381710038171003817100381710038171205773(120572119894 120573119894 120574119894 120579)

(40)

where

1205773(120572119894 120573119894 120574119894 120579) =

ℎ3

1198941205793

3minus

(120588119894+ 120573119894) ℎ3

119894(1 minus 120579) 120579

2

3119876119894(120579)

+120573119894ℎ3

119894(1 minus 120579) 120579

2

119876119894(120579)

minus120573119894ℎ3

1198941205793

3119876119894(120579)

(41)


119894=

[119905119894 119905119894+1


1003816100381610038161003816119891 (119905) minus 119875119894(119905)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905119894+1

119905119894

119877119905[(119905 minus 120591)

2

+] 119889120591

=1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119905

119905119894

119903 (120591 119905) 119889120591 + int

119905119894+1

119905

119904 (120591 119905) 119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

(42)

with

119888119894= max 120577 (120572

119894 120573119894 120574119894 120579)

120577 (120572119894 120573119894 120574119894 120579) =

1205771(120572119894 120573119894 120574119894 120579) 0 le 120579 le 120579

lowast

1205772(120572119894 120573119894 120574119894 120579) 120579


1205773(120572119894 120573119894 120574119894 120579) 0 le 120579 le 1

(43)

Remark 9 When 120572119894= 1 120573

119894= 1 and 120574



1205771(120572119894 120573119894 120574119894 120579) =

41205792(1 minus 120579)

3

(3 (3 minus 2120579)2)

0 le 120579 le1

2

1205772(120572119894 120573119894 120574119894 120579) =

41205793(1 minus 120579)

2

(3 (1 + 2120579)2)

1

2le 120579 le 1

1205773(120572119894 120573119894 120574119894 120579) = 0 0 le 120579 le 1

(44)

and 119888119894


4 Discussions





2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2








detail

5 Conclusions





119894 is




Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









shape preserving interpolation using 2 rational cubic...

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