research article analytical solution of general bagley

Research ArticleAnalytical Solution of General Bagley-Torvik Equation

William Labecca Osvaldo Guimaratildees and Joseacute Roberto C Piqueira

Escola Politecnica da Universidade de Sao Paulo Avenida Prof Luciano Gualberto Travessa 3No 158 05508-900 Sao Paulo SP Brazil

Correspondence should be addressed to Jose Roberto C Piqueira piqueiralacuspbr

Received 21 October 2015 Accepted 18 November 2015

Academic Editor Ivan D Rukhlenko

Copyright copy 2015 William Labecca et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerningempirical dataThere are several works treating this equation by using numerical methods and analytic formulations However theanalytical solutions presented in the literature consider particular cases of boundary and initial conditions with inhomogeneoustermoften expressed in polynomial formHere by using Laplace transformmethodology the general inhomogeneous case is solvedwithout restrictions in boundary and initial conditions The generalized Mittag-Leffler functions with three parameters are usedand the solutions presented are expressed in terms of Wimanrsquos functions and their derivatives

1 Introduction

Bagley-Torvik equations (BTE) firstly appeared in their sem-inal work [1] where they proposed to model viscoelasticbehavior of geological strata metals and glasses by usingfractional differential equations showing that this approachis effective in describing structures containing elastic andviscoelastic components

Initially in [2] inhomogeneous BTE was studied withan analytical solution being proposed Since then therewere several works to solve BTE starting with numericalprocedures for a reformulated BTE as a system of functionaldifferential equations of order 12 [3ndash5]

Following a numerical way for solving BTE a general-ization of Taylorrsquos and Besselrsquos collocation method [6 7] andthe use of evolutionary computation [8] provided acceptablesolutions from engineering point of view

Approximation techniques were successfully applied toBTE mainly by using enhanced homotopic perturbationmethods [9 10] fractional iteration techniques [11] and cubicpolynomial spline functions [12]

Analytical exact solutions for BTE were obtained in [13]for the particular initial condition 119906(0) = 1199061015840(0) = 0 con-sidering the boundary condition given by 119906(0) = 119906(1) = 1Besides by using a modified generalized Laguerre spectral

method for fractional differential equations BTE was solvedin [14] for some specific conditions

The aim of this work is to provide an analytical solutionfor the most generic case of inhomogeneous BTE in termsof the derivatives of Wimanrsquos functions [2] considering thatthree-parameter generalized Mittag-Leffler function intro-duced in [15] can be better explored

The approach presented here consists of using the Laplacetransform of Prabhakarrsquos function in order to solve the mostgeneral BTE carrying out a subsequent transformation of theachieved solution in terms of the Mittag-Leffler functions

In the next section the BTE solution by using the referredto method is presented followed by a section with the solu-tion expressed by using Wimanrsquos functions and their deriva-tives motivated by the fact that Wimanrsquos functions can beimplemented in computational software like119872119886119905ℎ119890119898119886119905119894119888119886copyeasing the solution of BTE in practical applications

2 Laplace Transformation Solution of BTEMittag-Leffler Functions

Themost general inhomogeneous BTE [1] is given by

119860D2119906 (119909) + 119861D

32119906 (119909) + 119862119906 (119909) = 120601 (119909) (1)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 591715 4 pageshttpdxdoiorg1011552015591715

2 Mathematical Problems in Engineering

which by using operational notation can be expressed as

[119860D2+ 119861D

32+ 119862] 119906 (119909) = 120601 (119909) (2)

Here the domain of 120601(119909) is considered to be thetime consequently it is assumed that 119905 isin [0infin) TheRiemann-Liouville fractional integral is adopted and there-fore Caputorsquos fractional derivative 119888119863]

0can be simply denoted

byD [16 17]The method to be used to solve BTE is the traditional

Laplace transform that due to the fractional nature of theexponents results from Prabhakar functions that is three-parameter Mittag-Leffler generalized functions [2 15]

Considering Caputorsquos fractional derivative [18] theLaplace transform of the fractional exponent term is

L D32119906 (119909) = 119904

32119880 (119904) minus 119904

12119906 (0) minus 119904

minus121199061015840

(0) (3)

implying that the Laplace transform of the BTE is written as

119880 (119904) (1198601199042+ 11986111990432+ 119862)

= (119860119904 + 11986111990412) 119906 (0) + (119861119904

minus12+ 119860) 119906

1015840

(0) + Φ (119904)

(4)

Defining 119861119860 = 120582 and 119862119860 = 120583 and considering thebinomial development

(1 +

12058211990432

1199042+ 120583

)

minus1

= (1 +

11986111990432

1198601199042+ 119862

)

minus1

=

infin

sum

119896=0

(minus1)119896 12058211989611990431198962

(1199042+ 120583)119896

(5)

the Laplace transform of BTErsquos solution is

119880 (119904) =

infin

sum

119896=0

(minus1)119896120582119896 11990431198962

(1199042+ 120583)119896+1[(119904 + 120582119904

12) 119906 (0)

+ (120582119904minus12+ 1) 119906

1015840

(0) +

1

119860

Φ (119904)]

(6)

In order to invert the Laplace transform to obtain thegeneral BTErsquos solution119880(119904) is rewritten expanding the sums

119880 (119904) = 119906 (0)

infin

sum

119896=0

(minus1)119896120582119896 11990431198962+1

(1199042+ 120583)119896+1

+ 119906 (0)

infin

sum

119896=0

(minus1)119896120582119896+1 11990431198962+12

(1199042+ 120583)119896+1

+ 1199061015840

(0)

infin

sum

119896=0

(minus1)119896120582119896+1 11990431198962minus12

(1199042+ 120583)119896+1

+ 1199061015840

(0)

infin

sum

119896=0

(minus1)119896120582119896 11990431198962

(1199042+ 120583)119896+1

+

1

119860

Φ (119904)

infin

sum

119896=0

(minus1)119896120582119896 11990431198962

(1199042+ 120583)119896+1

(7)

As shown in [2 15] it is possible to define three-parameterMittag-Leffler general functions as

119864120574

120572120573(119905) fl

infin

sum

119896=0

(120574)119896(119905119896)

Γ (120572119896 + 120573) 119896

(120572 120573 120574 isin C R (120572) R (120573) R (120574) gt 0)

(8)

with (120583)119896= Γ(120583 + 119896)Γ(120583) being the Pochhammer symbol

When 120574 = 1 these functions are called Wimanrsquos functionsHere the main fact to be used is related to the inverse of

the Laplace transform of the main terms that appear in (7)given by

Lminus1

119904120572120574minus120573

(119904120572+ 120582)120574 = 119909

120573minus1119864120574

120572120573(minus120582119909120572) (9)

Consequently the first four terms of (7) can be invertedand written as

Lminus1

11990431198962+1

(1199042+ 120583)119896+1 = 119909

(12)119896119864119896+1

2(12)(119896+2)(minus1205831199092)

Lminus1

11990431198962+12

(1199042+ 120583)119896+1 = 119909

(12)(119896+1)119864119896+1

2(12)(119896+3)(minus1205831199092)

Lminus1

11990431198962minus12

(1199042+ 120583)119896+1 = 119909

(12)(119896+3)119864119896+1

2(12)(119896+5)(minus1205831199092)

Lminus1

11990431198962

(1199042+ 120583)119896+1 = 119909

(12)(119896+2)119864119896+1

2(12)(119896+4)(minus1205831199092)

(10)

The inhomogeneous term is calculated by using convolu-tion

Lminus1

Φ (119904) 11990431198962

(1199042+ 120583)119896+1

= 120601 (119909) lowast [119909(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus1205831199092)]

(11)

and consequently the general solution of BTE is given by

119906 (119909) = 119906 (0)

infin

sum

119896=0

(minus120582)1198961199091198962119864119896+1

2(12)(119896+2)(minus1205831199092) + 119906 (0)

sdot

infin

sum

119896=0

(minus120582)119896119909(12)(119896+1)

119864119896+1

2(12)(119896+3)(minus1205831199092) + 1199061015840

(0)

sdot

infin

sum

119896=0

(minus120582)119896119909(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus1205831199092) + 1199061015840

(0)

sdot

infin

sum

119896=0

(minus120582)119896119909(12)(119896+3)

119864119896+1

2(12)(119896+5)(minus1205831199092) +

1

119860

sdot

infin

sum

119896=0

(minus120582)119896

sdot int

119909

0

120601 (119909 minus 120585) 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus1205831205852) 119889120585

(12)

Mathematical Problems in Engineering 3

In order to simplify the BTE general solution expressedby (12) a factor 120588(119904) can be defined

120588 (119904) fl1

2

1 minus (minus1)[4119904] (13)

with [119896] equiv int(119896) Therefore the BTE solution is expressed as

119906 (119909) =

infin

sum

119896=0

(minus

119861

119860

)

119896

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 119909(12)(119896+119904minus1)

119864119896+1

2(12)(119896+119904+1)(minus

1198621199092

119860

) +

1

119860

sdot int

119909

0

120601 (119909 minus 120585)

sdot 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus

119862

119860

1205852)119889120585

(14)

Operator 119863120588(119904) represents the integer order classicalderivative that is119863120588 equiv (119889119889119909)120588 with the property1198630119906(119909) equiv119906(119909)

Expression (14) is a general analytical solution of BTE interms of three-parameter Mittag-Leffler functions 119864120574

120572120573(119909)

However the solution established in [2] for the special casewith 119906(0) = 1199061015840(0) = 0 is given by using Wimanrsquos functionsderivatives that is two-parameter Mittag-Leffler functions

As Wimanrsquos functions and derivatives are part of severalsoftware packages as119872119886119905ℎ119890119898119886119905119894119888119886copy for instance and three-parameter Mittag-Leffler are not the results will be modifiedto be compatible with this fact easing the practical approachof the problem

3 BTE Solution via WimanrsquosFunctions and Derivatives

Here the solution of BTE expressed by (14) is modified inorder to be written in terms of Wimanrsquos functions and theirderivativesThe explicit forms of these functions are given by

119864(119896)

120582120583(119910) equiv

119889119896

119889119910119896119864120582120583(119910) =

infin

sum

119895=0

(119895 + 119896)119910119895

119895Γ (120582119895 + 120582119896 + 120583)

(119896 = 0 1 2 )

(15)

For the sake of clearness the work will be divided intotwo parts homogeneous solution terms and inhomogeneoussolution term

31 Homogeneous Solution Terms Taking the homogeneouspart of (14) denoted by ℎ(119909)

ℎ (119909) =

infin

sum

119896=0

(minus

119861

119860

)

119896

sdot

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0119909(12)(119896+119904minus1)

119864119896+1

2(12)(119896+119904+1)

sdot (minus

1198621199092

119860

) =

infin

sum

119896=0

(minus

119861

119860

)

119896

sdot

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0119909(12)(119896+119904minus1)

sdot

infin

sum

119903=0

(119896 + 119903) (minus1)119903(119862119860)

1199031199092119903

119896Γ (2119903 + 1198962 + 1199042 + 12)

(16)

If the 119903 and 119896 sums are uniformly convergent

ℎ (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903 4

sum

119904=1

119863120588(119904)119906 (119909)|119909=01199092119903+(12)(119904minus1)

infin

sum

119896=0

(119896 + 119903)

119896Γ [1199032 + 1198962 + (12) (3119903 + 119904 + 1)]

(minus

119861radic119909

119860

)

119896

(17)

Observing that the last sum can be written as the 119903thderivative of Wimanrsquos function the homogeneous part of thesolution is

ℎ (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903 4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 1199092119903+(12)(119904minus1)

119864(119903)

12(12)(3119903+119904+1)(minus

119861radic119909

119860

)

(18)

32 Inhomogeneous Solution Term Calling the inhomoge-neous term of the BTE solution 120595(119909) its expression is

120595 (119909) =

infin

sum

119896=0

(minus

119861

119860

)

1198961

119860

sdot int

119909

0

120601 (119909 minus 120585) 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus

119862

119860

1205852)119889120585

=

infin

sum

119896=0

(minus

119861

119860

)

1198961

119860

int

119909

0

120601 (119909 minus 120585) 1205851198962+1

sdot

infin

sum

119903=0

(minus1)119903(119896 + 1)

119903

119903Γ (2119903 + 1198962 + 2)

(

119862

119860

1205852)

119903

119889120585

(19)

Considering the convergence of the integral and uniformconvergence of the sums their order can be changed and itcan be written as

120595 (119909) = int

119909

0

120601 (119909 minus 120585)

infin

sum

119903=0

(minus1)119903

119860119903

(

119862

119860

)

119903

1205852119903+1

sdot

infin

sum

119896=0

(minus1)119896Γ (119896 + 119903 + 1) 120585

1198962

Γ (119896 + 1) Γ (2119903 + 1198962 + 2)

(

119861

119860

)

119896

119889120585

= int

119909

0

120601 (119909 minus 120585)

1

119860

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1

sdot

infin

sum

119896=0

(119896 + 119903)

119896

(minus119861radic120585119860)

119896

Γ [(12) 119903 + (12) 119896 + ((32) 119903 + 2)]

119889120585

(20)


It can be noticed that the last sum of (20) can be written asthe 119903-order derivative ofWimanrsquos function and consequently

120595 (119909) = int

119909

0

120601 (119909 minus 120585)

1

119860

sdot

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(21)

Then defining

119866 (120585) =

1

119860

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585) (22)

it follows that

120595 (119909) = int

119909

0

120601 (119909 minus 120585) 119866 (120585) 119889120585 (23)

according to the BTE analytical solution obtained in [2] for119906(0) = 119906

1015840(0) = 0

33 Complete BTE Solution Considering that 119906(119909) = ℎ(119909) +120595(119909) the general analytical solution of BTE in terms ofWimanrsquos function is

119906 (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 1199092119903+(12)(119904minus1)

119864(119903)

12(12)(3119903+119904+1)(minus

119861radic119909

119860

) +

1

119860

sdot int

119909

0

120601 (119909 minus 120585) 1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(24)

with 120588(119904) fl (12)1 minus (minus1)[4119904] and 119864(119903)

120572120573(119911) =

(119889119903119889119911119903)119864120572120573(119911)

4 Conclusion

A general analytical solution of BTE defined by Caputorsquosfractional derivatives was obtained in terms of Wimanrsquosfunctions and their derivativesThe reasoning was conductedby using Laplace transform and Mittag-Leffler functions

The final expression permits the calculations in practicalcases as it is built consideringWimanrsquos functions that are partof the most usual numerical packages

Conflict of Interests

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

References

[1] P J Torvik andR L Bagley ldquoOn the appearance of the fractionalderivative in the behavior of real materialsrdquo Journal of AppliedMechanics vol 51 no 2 pp 294ndash298 1984

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] A H Bhrawy T M Taha and J A T Machado ldquoA reviewof operational matrices and spectral techniques for fractionalcalculusrdquoNonlinearDynamics vol 81 no 3 pp 1023ndash1052 2015

[4] K Diethelm and N J Ford ldquoNumerical solution of the Bagley-Torvik equationrdquoBIT NumericalMathematics vol 42 no 3 pp490ndash507 2002

[5] Z H Wang and X Wang ldquoGeneral solution of the Bagley-Torvik equation with fractional-order derivativerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 15 no5 pp 1279ndash1285 2010

[6] Y Cenesiz Y Keskin and A Kurnaz ldquoThe solution of theBagley-Torvik equationwith the generalizedTAYlor collocationmethodrdquo IEEE Engineering and Applied Mathematics vol 347no 2 pp 452ndash466 2010

[7] S Yuzbası ldquoNumerical solution of the Bagley-Torvik equationby the Bessel collocationmethodrdquoMathematical Methods in theApplied Sciences vol 36 no 3 pp 300ndash312 2013

[8] M A Raja J A Khan and IMQureshi ldquoSolution of fractionalorder system of Bagley-Torvik equation using evolutionarycomputational intelligencerdquo Mathematical Problems in Engi-neering vol 2011 Article ID 675075 18 pages 2011

[9] F Abidi and K Omrani ldquoThe homotopy analysis method forsolving the Fornberg-Whitham equation and comparison withAdomianrsquos decomposition methodrdquo Computers amp Mathematicswith Applications vol 59 no 8 pp 2743ndash2750 2010

[10] M Zolfaghari R Ghaderi A SheikholEslami et al ldquoAppli-cation of the enhanced homotopy perturbation method tosolve the fractional-order BagleyndashTorvik differential equationrdquoPhysica Scripta vol 2009 article T136 Article ID 014032 2009

[11] T Mekkaoui and Z Hammouch ldquoApproximate analytical solu-tions to the Bagley-Torvik equation by the fractional iterationmethodrdquo Annals of the University of Craiova-Mathematics andComputer Science Series vol 39 no 2 pp 251ndash256 2012

[12] W K Zahra and S M Elkholy ldquoCubic spline solution offractional Bagley-Torvik equationrdquo Electronic Journal of Mathe-matical Analysis andApplications vol 1 no 2 pp 230ndash241 2013

[13] D Baleanu A H Bhrawy and T M Taha ldquoA modifiedgeneralized Laguerre spectral method for fractional differentialequations on the half linerdquo Abstract and Applied Analysis vol2013 Article ID 413529 12 pages 2013

[14] S Stanek ldquoTwo-point boundary value problems for the gen-eralized Bagley-Torvik fractional differential equationrdquo CentralEuropean Journal of Mathematics vol 11 no 3 pp 574ndash5932013

[15] T R Prabhakar ldquoA singular integral equationwith a generalizedMittag-Leffler function in the kernelrdquo Yokohama MathematicalJournal vol 19 pp 7ndash15 1971

[16] K M Kolwankar and A D Gangal ldquoFractional differentiabilityof nowhere differentiable functions and dimensionsrdquo Chaosvol 6 no 4 pp 505ndash513 1996

[17] T Kisela Fractional differential equations and their applications[MS thesis] University of lrsquoAquila amp Brno University ofTechnology Brno Czech Republic 2008

[18] A M O Anwar F Jarad D Baleanu and F Ayaz ldquoFractionalCaputo heat equation within the double Laplace transformrdquoRomanian Journal of Physics vol 58 no 1-2 pp 15ndash22 2013

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Stochastic AnalysisInternational Journal of


which by using operational notation can be expressed as

[119860D2+ 119861D

32+ 119862] 119906 (119909) = 120601 (119909) (2)

Here the domain of 120601(119909) is considered to be thetime consequently it is assumed that 119905 isin [0infin) TheRiemann-Liouville fractional integral is adopted and there-fore Caputorsquos fractional derivative 119888119863]

0can be simply denoted

byD [16 17]The method to be used to solve BTE is the traditional

Laplace transform that due to the fractional nature of theexponents results from Prabhakar functions that is three-parameter Mittag-Leffler generalized functions [2 15]

Considering Caputorsquos fractional derivative [18] theLaplace transform of the fractional exponent term is

L D32119906 (119909) = 119904

32119880 (119904) minus 119904

12119906 (0) minus 119904

minus121199061015840

(0) (3)

implying that the Laplace transform of the BTE is written as

119880 (119904) (1198601199042+ 11986111990432+ 119862)

= (119860119904 + 11986111990412) 119906 (0) + (119861119904

minus12+ 119860) 119906

1015840

(0) + Φ (119904)

(4)

Defining 119861119860 = 120582 and 119862119860 = 120583 and considering thebinomial development

(1 +

12058211990432

1199042+ 120583

)

minus1

= (1 +

11986111990432

1198601199042+ 119862

)

minus1

=

infin

sum

119896=0

(minus1)119896 12058211989611990431198962

(1199042+ 120583)119896

(5)

the Laplace transform of BTErsquos solution is

119880 (119904) =

infin

sum

119896=0

(minus1)119896120582119896 11990431198962

(1199042+ 120583)119896+1[(119904 + 120582119904

12) 119906 (0)

+ (120582119904minus12+ 1) 119906

1015840

(0) +

1

119860

Φ (119904)]

(6)

In order to invert the Laplace transform to obtain thegeneral BTErsquos solution119880(119904) is rewritten expanding the sums

119880 (119904) = 119906 (0)

infin

sum

119896=0

(minus1)119896120582119896 11990431198962+1

(1199042+ 120583)119896+1

+ 119906 (0)

infin

sum

119896=0

(minus1)119896120582119896+1 11990431198962+12

(1199042+ 120583)119896+1

+ 1199061015840

(0)

infin

sum

119896=0

(minus1)119896120582119896+1 11990431198962minus12

(1199042+ 120583)119896+1

+ 1199061015840

(0)

infin

sum

119896=0

(minus1)119896120582119896 11990431198962

(1199042+ 120583)119896+1

+

1

119860

Φ (119904)

infin

sum

119896=0

(minus1)119896120582119896 11990431198962

(1199042+ 120583)119896+1

(7)

As shown in [2 15] it is possible to define three-parameterMittag-Leffler general functions as

119864120574

120572120573(119905) fl

infin

sum

119896=0

(120574)119896(119905119896)

Γ (120572119896 + 120573) 119896

(120572 120573 120574 isin C R (120572) R (120573) R (120574) gt 0)

(8)

with (120583)119896= Γ(120583 + 119896)Γ(120583) being the Pochhammer symbol

When 120574 = 1 these functions are called Wimanrsquos functionsHere the main fact to be used is related to the inverse of

the Laplace transform of the main terms that appear in (7)given by

Lminus1

119904120572120574minus120573

(119904120572+ 120582)120574 = 119909

120573minus1119864120574

120572120573(minus120582119909120572) (9)

Consequently the first four terms of (7) can be invertedand written as

Lminus1

11990431198962+1

(1199042+ 120583)119896+1 = 119909

(12)119896119864119896+1

2(12)(119896+2)(minus1205831199092)

Lminus1

11990431198962+12

(1199042+ 120583)119896+1 = 119909

(12)(119896+1)119864119896+1

2(12)(119896+3)(minus1205831199092)

Lminus1

11990431198962minus12

(1199042+ 120583)119896+1 = 119909

(12)(119896+3)119864119896+1

2(12)(119896+5)(minus1205831199092)

Lminus1

11990431198962

(1199042+ 120583)119896+1 = 119909

(12)(119896+2)119864119896+1

2(12)(119896+4)(minus1205831199092)

(10)

The inhomogeneous term is calculated by using convolu-tion

Lminus1

Φ (119904) 11990431198962

(1199042+ 120583)119896+1

= 120601 (119909) lowast [119909(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus1205831199092)]

(11)

and consequently the general solution of BTE is given by

119906 (119909) = 119906 (0)

infin

sum

119896=0

(minus120582)1198961199091198962119864119896+1

2(12)(119896+2)(minus1205831199092) + 119906 (0)

sdot

infin

sum

119896=0

(minus120582)119896119909(12)(119896+1)

119864119896+1

2(12)(119896+3)(minus1205831199092) + 1199061015840

(0)

sdot

infin

sum

119896=0

(minus120582)119896119909(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus1205831199092) + 1199061015840

(0)

sdot

infin

sum

119896=0

(minus120582)119896119909(12)(119896+3)

119864119896+1

2(12)(119896+5)(minus1205831199092) +

1

119860

sdot

infin

sum

119896=0

(minus120582)119896

sdot int

119909

0

120601 (119909 minus 120585) 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus1205831205852) 119889120585

(12)



120588 (119904) fl1

2

1 minus (minus1)[4119904] (13)


119906 (119909) =

infin

sum

119896=0

(minus

119861

119860

)

119896

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 119909(12)(119896+119904minus1)

119864119896+1

2(12)(119896+119904+1)(minus

1198621199092

119860

) +

1

119860

sdot int

119909

0

120601 (119909 minus 120585)

sdot 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus

119862

119860

1205852)119889120585

(14)



120572120573(119909)





119864(119896)

120582120583(119910) equiv

119889119896

119889119910119896119864120582120583(119910) =

infin

sum

119895=0

(119895 + 119896)119910119895

119895Γ (120582119895 + 120582119896 + 120583)

(119896 = 0 1 2 )

(15)



ℎ (119909) =

infin

sum

119896=0

(minus

119861

119860

)

119896

sdot

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0119909(12)(119896+119904minus1)

119864119896+1

2(12)(119896+119904+1)

sdot (minus

1198621199092

119860

) =

infin

sum

119896=0

(minus

119861

119860

)

119896

sdot

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0119909(12)(119896+119904minus1)

sdot

infin

sum

119903=0

(119896 + 119903) (minus1)119903(119862119860)

1199031199092119903

119896Γ (2119903 + 1198962 + 1199042 + 12)

(16)


ℎ (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903 4

sum

119904=1

119863120588(119904)119906 (119909)|119909=01199092119903+(12)(119904minus1)

infin

sum

119896=0

(119896 + 119903)

119896Γ [1199032 + 1198962 + (12) (3119903 + 119904 + 1)]

(minus

119861radic119909

119860

)

119896

(17)


ℎ (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903 4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 1199092119903+(12)(119904minus1)

119864(119903)

12(12)(3119903+119904+1)(minus

119861radic119909

119860

)

(18)


120595 (119909) =

infin

sum

119896=0

(minus

119861

119860

)

1198961

119860

sdot int

119909

0

120601 (119909 minus 120585) 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus

119862

119860

1205852)119889120585

=

infin

sum

119896=0

(minus

119861

119860

)

1198961

119860

int

119909

0

120601 (119909 minus 120585) 1205851198962+1

sdot

infin

sum

119903=0

(minus1)119903(119896 + 1)

119903

119903Γ (2119903 + 1198962 + 2)

(

119862

119860

1205852)

119903

119889120585

(19)


120595 (119909) = int

119909

0

120601 (119909 minus 120585)

infin

sum

119903=0

(minus1)119903

119860119903

(

119862

119860

)

119903

1205852119903+1

sdot

infin

sum

119896=0

(minus1)119896Γ (119896 + 119903 + 1) 120585

1198962

Γ (119896 + 1) Γ (2119903 + 1198962 + 2)

(

119861

119860

)

119896

119889120585

= int

119909

0

120601 (119909 minus 120585)

1

119860

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1

sdot

infin

sum

119896=0

(119896 + 119903)

119896

(minus119861radic120585119860)

119896

Γ [(12) 119903 + (12) 119896 + ((32) 119903 + 2)]

119889120585

(20)



120595 (119909) = int

119909

0

120601 (119909 minus 120585)

1

119860

sdot

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(21)

Then defining

119866 (120585) =

1

119860

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585) (22)

it follows that

120595 (119909) = int

119909

0

120601 (119909 minus 120585) 119866 (120585) 119889120585 (23)


1015840(0) = 0


119906 (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 1199092119903+(12)(119904minus1)

119864(119903)

12(12)(3119903+119904+1)(minus

119861radic119909

119860

) +

1

119860

sdot int

119909

0

120601 (119909 minus 120585) 1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(24)


120572120573(119911) =

(119889119903119889119911119903)119864120572120573(119911)

4 Conclusion





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588 (119904) fl1

2

1 minus (minus1)[4119904] (13)


119906 (119909) =

infin

sum

119896=0

(minus

119861

119860

)

119896

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 119909(12)(119896+119904minus1)

119864119896+1

2(12)(119896+119904+1)(minus

1198621199092

119860

) +

1

119860

sdot int

119909

0

120601 (119909 minus 120585)

sdot 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus

119862

119860

1205852)119889120585

(14)



120572120573(119909)





119864(119896)

120582120583(119910) equiv

119889119896

119889119910119896119864120582120583(119910) =

infin

sum

119895=0

(119895 + 119896)119910119895

119895Γ (120582119895 + 120582119896 + 120583)

(119896 = 0 1 2 )

(15)



ℎ (119909) =

infin

sum

119896=0

(minus

119861

119860

)

119896

sdot

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0119909(12)(119896+119904minus1)

119864119896+1

2(12)(119896+119904+1)

sdot (minus

1198621199092

119860

) =

infin

sum

119896=0

(minus

119861

119860

)

119896

sdot

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0119909(12)(119896+119904minus1)

sdot

infin

sum

119903=0

(119896 + 119903) (minus1)119903(119862119860)

1199031199092119903

119896Γ (2119903 + 1198962 + 1199042 + 12)

(16)


ℎ (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903 4

sum

119904=1

119863120588(119904)119906 (119909)|119909=01199092119903+(12)(119904minus1)

infin

sum

119896=0

(119896 + 119903)

119896Γ [1199032 + 1198962 + (12) (3119903 + 119904 + 1)]

(minus

119861radic119909

119860

)

119896

(17)


ℎ (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903 4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 1199092119903+(12)(119904minus1)

119864(119903)

12(12)(3119903+119904+1)(minus

119861radic119909

119860

)

(18)


120595 (119909) =

infin

sum

119896=0

(minus

119861

119860

)

1198961

119860

sdot int

119909

0

120601 (119909 minus 120585) 120585(12)(119896+2)

119864119896+1

2(12)(119896+4)(minus

119862

119860

1205852)119889120585

=

infin

sum

119896=0

(minus

119861

119860

)

1198961

119860

int

119909

0

120601 (119909 minus 120585) 1205851198962+1

sdot

infin

sum

119903=0

(minus1)119903(119896 + 1)

119903

119903Γ (2119903 + 1198962 + 2)

(

119862

119860

1205852)

119903

119889120585

(19)


120595 (119909) = int

119909

0

120601 (119909 minus 120585)

infin

sum

119903=0

(minus1)119903

119860119903

(

119862

119860

)

119903

1205852119903+1

sdot

infin

sum

119896=0

(minus1)119896Γ (119896 + 119903 + 1) 120585

1198962

Γ (119896 + 1) Γ (2119903 + 1198962 + 2)

(

119861

119860

)

119896

119889120585

= int

119909

0

120601 (119909 minus 120585)

1

119860

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1

sdot

infin

sum

119896=0

(119896 + 119903)

119896

(minus119861radic120585119860)

119896

Γ [(12) 119903 + (12) 119896 + ((32) 119903 + 2)]

119889120585

(20)



120595 (119909) = int

119909

0

120601 (119909 minus 120585)

1

119860

sdot

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(21)

Then defining

119866 (120585) =

1

119860

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585) (22)

it follows that

120595 (119909) = int

119909

0

120601 (119909 minus 120585) 119866 (120585) 119889120585 (23)


1015840(0) = 0


119906 (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 1199092119903+(12)(119904minus1)

119864(119903)

12(12)(3119903+119904+1)(minus

119861radic119909

119860

) +

1

119860

sdot int

119909

0

120601 (119909 minus 120585) 1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(24)


120572120573(119911) =

(119889119903119889119911119903)119864120572120573(119911)

4 Conclusion





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120595 (119909) = int

119909

0

120601 (119909 minus 120585)

1

119860

sdot

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(21)

Then defining

119866 (120585) =

1

119860

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585) (22)

it follows that

120595 (119909) = int

119909

0

120601 (119909 minus 120585) 119866 (120585) 119889120585 (23)


1015840(0) = 0


119906 (119909) =

infin

sum

119903=0

(minus1)119903

119903

(

119862

119860

)

119903

4

sum

119904=1

119863120588(119904)119906 (119909)|119909=0

sdot 1199092119903+(12)(119904minus1)

119864(119903)

12(12)(3119903+119904+1)(minus

119861radic119909

119860

) +

1

119860

sdot int

119909

0

120601 (119909 minus 120585) 1205852119903+1119864(119903)

122+(32)119903(minus

119861

119860

radic120585)119889120585

(24)


120572120573(119911) =

(119889119903119889119911119903)119864120572120573(119911)

4 Conclusion





References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article analytical solution of general bagley

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