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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 531984 7 pageshttpdxdoiorg1011552013531984

Research ArticleA Novel Integral Operator Transform and Its Application toSome FODE and FPDE with Some Kind of Singularities

Abdon Atangana1 and Adem Kilicman2

1 Institute for Groundwater Studies Faculty of Natural and Agricultural Sciences University of the Free StateBloemfontein 9300 South Africa

2Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia 43400 Serdang Selangor Malaysia

Correspondence should be addressed to Adem Kilicman kilicmanyahoocom

Received 14 April 2013 Accepted 11 July 2013

Academic Editor Hossein Jafari

Copyright copy 2013 A Atangana and A KilicmanThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We introduced a novel integral transform operator We proved the existence and the uniqueness of the relatively new operatorWe presented some useful properties of the new operator We presented the application of this operator for solving some kind offractional ordinary and partial differential equation containing some kind of singularity

1 Introduction

Mathematical notation aside the motivation behind integraltransforms is easy to understand There are many classes ofproblems that are difficult to solve or at least quite unwieldyalgebraically in their original representations An integraltransform ldquomapsrdquo an equation from its original ldquodomainrdquo intoanother domain [1ndash3]Manipulating and solving the equationin the target domain can be much easier than manipulationand solution in the original domain The solution is thenmapped back to the original domain with the inverse ofthe integral transform There exist few integral transformoperators in the literature [1ndash3] which are commonly usedto solve partial fractional and fractional ordinary differentialequations

The Fourier transform named after Joseph Fourier is amathematical transform with many applications in physicsand engineering [4ndash11] Very commonly it transforms amathematical function of time 119891(119905) into a new functionsometimes denoted by 119865 whose argument is frequency withunits of cycles per second or (hertz) or radians per secondThe new function is then known as the Fourier transformandor the frequency spectrumof the function119891The Fouriertransform is also a reversible operation Thus given thefunction 119891 one can determine the original function 119891 see in[8]

The Laplace transform is an integral transform perhapssecond only to the Fourier transform in its utility in solvingphysical problems [12ndash17] The Laplace transform is particu-larly useful in solving linear ordinary differential equations orpartial fractional differential equations such as those arisingin the analysis of groundwater pollution model [13] andelectronic circuits [14]

In mathematics the Mellin transform [15] is an integraltransform that may be regarded as the multiplicative versionof the two-sided Laplace transform This integral transformis closely connected to the theory of Dirichlet series and isoften used in number theory and the theory of asymptoticexpansions it is closely related to the Laplace transform andthe Fourier transform and the theory of the gamma functionand allied special functions

The Mellin transform is widely used in computer sciencebecause of its scale invariance property [18] The magnitudeof the Mellin transform of a scaled function is identical tothe magnitude of the original function [18] This scale invari-ance property is analogous to the Fourier transformrsquos shiftinvariance propertyThemagnitude of a Fourier transform ofa time-shifted function is identical to the original functionThis property is useful in image recognition An image of anobject is easily scaled when the object is moved towards oraway from the camera [19]

2 Mathematical Problems in Engineering

In mathematics the Sumudu transform is an integraltransform similar to the Laplace transform introduced in theearly 1990s by Watugala to solve differential equations andcontrol engineering problems [20ndash27] It is equivalent to theLaplace-Carson transform with the substitution 119901 = 1119906

However there exists some kind of fractional ordinaryand partial differential equations with some kind of singu-larities that cannot be solved directly via the above integraltransform operators In particular the following kind offractional ordinary and partial differential equations

0119863120572

119909119910 (119909) +

1

119909119899119910 (119909) = 119891 (119909) (1)

or

0119863120572

119909119906 (119909 119905) +

1

119909119899119906 (119909 119905) = ℎ (119909 119905) (2)

where 120572 gt 00119863120572

119909 is the fractional derivative (Riemann-

Liouville or Caputo) and 119899 ge 1To solve the above equations some scholars make use

of the Frobenius method to obtain the solutions in seriesform The Laplace transform of the product of two functionsis different from the product of the Laplace transform ofthe two functions The Fourier transform of the productof two functions is equivalent to the convolution of theFourier transform of the two functions This renders itvery difficult to apply directly either the Laplace transformor the Fourier transform operators to solve this type ofequation Therefore some scholars multiply 119909119899 on both sidesof the above equations and then apply the Fourier or theLaplace transform It is therefore worth to define an integraltransform similar to Laplace or Laplace-Carson transform totransform such equation to an ordinary or partial differentialequation without any additional transformation

The aim of this work is to further introduce an integraltransform operator that can be used to solve some kind ofordinary partial and fractional ordinary partial differentialequation with some kind of singularities We will start withthe definition and present some theorems

2 Definitions and Theorems

Definition 1 Let 119891(119909) be a continuous function over an openinterval (0infin) such that its Laplace transform is 119899 timedifferentiable then the new integral transform of order 119899 of119891 is defined as follows

119872119899 (119904) = 119872119899 [119891 (119909)] (119904) = int

infin

0

119909119899119890minus119909119904119891 (119909) 119889119909 (3)

and the inverse of the new integral transform of order 119899 isdefined as

119891 (119909) = 119872minus1

119899[119872119899[119891 (119909)]]

=(minus1)119899

2120587119894int120572+119894infin

120572minus119894infin

119890119904119909 [(minus1)119899 [

1

Γ (119899 minus 1)

times int119904

0

(119904 minus 119905)119899minus1119872119899 (119905) 119889119905

+119899minus1

sum119896=0

119904119896

119896119910119896]]119889119904

119910119896=120597119896119865 (0)

120597119904119896

(4)

where 119865(119904) is the Laplace transform of 119891(119909) Before wecontinue we will prove that the above definition is indeedthe inverse operator transform of order 119899 In fact from thedefinition of new transform of order 119899 of a function 119891(119909) wehave that

119872119899(119904) = 119872

119899[119891 (119909)] (119904)

= intinfin

0

119909119899119890minus119909119904119891 (119909) 119889119909 = (minus1)119899 119889119899119865 (119904)

119889119904119899

(5)

thus

1

Γ (119899 minus 1)int119904

0

(119904 minus 119905)119899minus1119872119899(119905) 119889119905

= (minus1)119899 [119865 (119904) minus

119899minus1

sum119896=0

119904119896

119896119910119896]

(6)

It follows that

(minus1)119899

2120587119894int120572+119894infin


119890119904119909 [(minus1)119899 [

1

Γ (119899 minus 1)

times int119904

0

(119904 minus 119905)119899minus1119872119899 (119905) 119889119905

+119899minus1

sum119896=0

119904119896

119896119910119896]]119889119904

=(minus1)119899

2120587119894int120572+119894infin


119890119904119909 [(minus1)119899[119865 (119904)]] 119889119904

119872minus1119899[119872119899[119891 (119909)]]

=(minus1)2119899

2120587119894int120572+119894infin


119890119904119909 [[119865 (119904)]] 119889119904 = 119891 (119909)

(7)

Therefore the inverse of the new integral transform is welldefined Our next concern is to prove the uniqueness and theexistence of the new integral transform

Mathematical Problems in Engineering 3

Theorem 2 Let 119891(119909) and 119892(119909) be continuous functionsdefined for 119909 ge 0 and having new transforms of order 119899 119865(119901)and 119866(119901) respectively If 119865(119901) = 119866(119901) then 119891(119909) = 119892(119909)

Proof From the definition of the inverse of the new transformof order 119899 if 120572 is sufficiently large then the integral expres-sion by

119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119865 (119901)] 119889119901 (8)

for the inverse of the new integral transform of order 119899 canbe used to obtain

119891 (119909) =(minus1)2119899

2120587119894int120572+119894infin


119890119901119909 [119865 (119901)] 119889119901 (9)

By hypothesis we have that 119865(119901) = 119866(119901) then replacing thisin the above expression we have the following

119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119866 (119901)] 119889119901 (10)

which boils down to

119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119866 (119901)] 119889119901 = 119892 (119909) (11)

and this proves the uniqueness of the new integral transformof order 119899

Theorem 3 If 119891(119905) is a piecewise continuous on every finiteinterval in [0 119905

0) and satisfies

1003816100381610038161003816119905119899119891 (119905)

1003816100381610038161003816 le 119872119890120572119905 (12)

for all 119905 isin [1199050infin) then119872

119899[119891(119909)](119904) exists for all 119904 gt 120572

Proof To prove the theoremwemust show that the improperintegral converges for 119904 gt 119886 Splitting the improper integralinto two parts we have

intinfin

0

119905119899119890minus119904119905119891 (119905) 119889119905

= int1199050

0

119905119899119890minus119904119905119891 (119905) 119889119905 + intinfin

1199050

119905119899119890minus119904119905119891 (119905) 119889119905

(13)

The first integral on the right side exists by hypothesis 1hence the existence of the new integral transform of order 119899119872119899(119904) depends on the convergence of the second integral By

hypothesis 2 we have10038161003816100381610038161003816119905119899119890minus119904119905119891 (119905)

10038161003816100381610038161003816 le 119872119890120572119905119890minus119904119905 = 119872119890(120572minus119904)119905 (14)

Now

intinfin

1199050

119872119890(120572minus119904)119905119889119905 = 119872119890(120572minus119904)1199050

120572 minus 119904 (15)

this converges for 120572 lt 119904 Then by the comparison test forimproper integrals theorem119872

119899(119904) exists for 120572 lt 119904

Remark 4 There is a relationship between the Laplace trans-form and the new integral transform of order 119899 as follows

119871 (119891 (119909)) (119904) = 119872119899(1

119909119899119891 (119909)) (119904)

119871 (119891 (119909)) (119904) = 1198720(119891 (119909)) (119904)

119872119899(119891 (119909)) (119904) = (minus1)

119899 119889119899 [119865 (119904)]

119889119904119899

(16)

where 119865(119904) is the Laplace transform of 119891(119909)

Remark 5 There is a relationship between the Laplace-Carson transform and then new integral transform of order119899 as follows

119871119888(119891 (119909)) (119904) = 119872

1(119891 (119909)) (119904) (17)

Theorem 6 A function 119891(119909) which is continuous on [0infin)and satisfies the growth condition 119891(119909) can be recovered fromthe Laplace transform 119865(119901) as follows

119891 (119909) = lim119899rarrinfin

(minus1)119899

119899(119899

119909)119899+1

119872119899(119899

119904) (18)

Evidently themain difficulty in usingTheorem 6 for computingthe inverse Laplace transform is the repeated symbolic differen-tiation of 119865(119901)

3 Some Properties of the NewIntegral Transform

In this section we consider some of the properties of thenew integral transform that will enable us to find furthertransform pairs 119891(119909)119872

119899(119904) without having to compute

consider the following

(I) 119872119899 [119904 + 119888] = 119872119899 [119890minus119888119909119891 (119909)]

(II) 119872119899[119891 (119886119909)] (119904) =

1

119886119872119899[119904

119886]

(III) int120572+119894infin


119890119904119909119872119899(119904) 119889119904 = 119909

119899119891 (119909)

(IV) 119872119899 [119886119891 (119909) + 119887119892 (119909)] (119904)

= [119886119872119899(119891 (119909)) + 119887119872

119899(119892 (119909))] (119904)

(V) 119872119899 [119891 (119909)

119909119899] (119904) = 119871 [119891 (119909)] (119904)

(VI) 119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119866 (119904))

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(VII) 119872119899[119889119899119891 (119909)

119889119909119899] (119904)

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(19)


Let us verify the above properties We will start with I bydefinition we have the following

119872119899[119890minus119888119909119891 (119909)]

= intinfin

0

[119909119899119890minus119888119909119890minus119904119909119891 (119909)] 119889119909

= intinfin

0

[119909119899119890minus(119888+119904)119909119891 (119909)] 119889119909 = 119872119899 [119904 + 119888]

(20)

and then the first property is verifiedFor II we have the following by definition

119872119899[119891 (119886119909)] (119904)

= intinfin

0

[119909119899119890minus119909119904119891 (119886119909)] 119889119909 = (minus1)

119899 119889119899

119889119904119899[119871 [119891 (119886119909)] (119904)]

(21)

Now using the property of the Laplace transform119871[119891(119886119909)](119904) = (1119886)119865(119904119886) from this we can furtherobtain

119872119899[119891 (119886119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[1

119886119865 (

119904

119886)]

=1

119886(minus1)119899 119889119899

119889119904119899[119865 (

119904

119886)] =

1

119886119872119899[119904

119886]

(22)

and then the property number II is verifiedFor number III we have the following Let 119892(119909) = 119909119899119891(119909)

then

int120572+119894infin


119890119904119909119872119899(119904) 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119909119899119891 (119909) 119889119909] 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119892 (119909) 119889119909] 119889119904

(23)

By the theorem of inverse Laplace transform we obtain

int120572+119894infin


119890119904119909119872119899(119904) 119889119904 = 119892 (119909) = 119909

119899119891 (119909) (24)

numbers IV and V are obvious to be verified For number VIwe have the following by definition

119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= intinfin

0

[119909119899119890minus119904119909119891 (119909) lowast ℎ (119909)]

= (minus1)119899 119889119899

119889119904119899[119871 (119891 (119909) lowast ℎ (119909)) (119904)]

(25)

now using the property of Laplace transform of the convolu-tion we obtain the following

119871 (119891 (119909) lowast ℎ (119909)) (119904) = 119865 (119904) sdot 119866 (119904) (26)

and then using the property of the derivative of order 119899 forthe product of two functions we obtain

119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[119865 (119904) sdot 119866 (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119866 (119904))

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(27)

and then the property number VI is verifiedFor number VII by definition we have the following

119872119899[119889119899119891 (119909)

119889119909119899] (119904)

= intinfin

0

[119909119899119890minus119904119909119889119899119891 (119909)

119889119909119899] 119889119909

= (minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

(28)

now using the property of the Laplace transform

119871(119889119899119891 (119909)

119889119909119899) (119904) = 119904

119899119865 (119904) minus119899minus1

sum119896=0

119904119899minus119896minus1119889119896119891 (0)

119889119909119896(29)

now deriving the above expression 119899 times we obtain thefollowing expression

(minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(30)

that is

119872119899[119889119899119891 (119909)

119889119909119899] (119904) = (minus1)

119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896 (31)

This completes the proof of number VI

4 Application to FODE and FPDE

Recently the differential equations of fractional order deriva-tive with singularities have been the focus of many studiesdue to their frequent appearance in various applications influidmechanics viscoelasticity biology physics engineeringand groundwater models in particular the monitoring ofthe flow through the geological formation and the pollutionmigration Consequently considerable attention has beengiven to the solutions of fractional differential equations andintegral equations with singularity of physical interest Thereexists in the literature some integral transform method thatcan be used to derive exact and approximate solutions forsuch equations see for instance Laplace transform method[4ndash11] the Fourier transform method [12ndash17] the Mellin


transform method [18 19] the Sumudu transform method[20ndash27] the Adomian decomposition method [28 29] andthe homotopy decompositionmethod [30ndash33] In this sectionwe present the application of the proposed integral operatorto the Cauchy-type of fractional ordinary differential andpartial differential equationsWe will start with the fractionalordinary differential equation Here we consider the Cauchy-type equation of the following form

119863120572119903119903Φ (119903) +

1

119903119899Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (32)

To solve the above equation we apply on both sides the newintegral transform of order 119899 to obtain the following

(minus1)119899 119889119899

119889119904119899119863120572119903119903Φ (119904) + Φ (119904) = 0 (33)

The new integral transform has gotten rid of the singularitythe new equation is just an ordinary fractional differentialequation which can be solved with for instance the homo-topy decomposition method Let us find the exact solution ofthe above equation for 119899 = 1 given below as

119863120572119903119903Φ (119903) +

1

119903Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (34)

We will make use of the new integral transform to deriveanalytical solution of (34) Applying the new transform oforder 1 on both sides of the above equation we obtain thefollowing expression

119889 [119871 (Φ) (119904)]

119889119904+ (

120572

119904+1

119904120572) (119871 (Φ) (119904))

=119897

sum119898=2

119889119898(119898 minus 1) 119904

119898minus2minus120572

(35)

where 119889119898= 119863120572minus1198980+ Φ(0+) (119898 = 2 119897) Now one can derive

the solution of the ordinary order differential equation withrespect to the Laplace transform of Θ(s) = 119871(Φ(119903))

Θ (119904) = 119904minus120572 exp[minus 119904

1minus120572

1 minus 120572]

times [1198861+119897

sum119898=2

119889119898(119898 minus 1) int 119904

119898minus2 exp[minus 1199041minus120572

1 minus 120572]119889119904]

(36)

with 1198861an arbitrary real constant that will be obtained via the

initial conditionWe next expand the exponential function inthe integrand in a series and using term-by-term integrationwe arrive at the following expression

Θ (119904) = 119888Θ1(119904) +

119897

sum119898=2

119889119898(119898 minus 1)Θ

lowast

119898(119904) (37)

with of course

Θ1(119904) = 119904

minus120572 exp[minus 1199041minus120572

1 minus 120572]

Θlowast119898(119904) = 119904

minus120572 exp[ 1199041minus120572

120572 minus 1]

timesinfin

sum119895=0

(1

1 minus 120572)119895 119904(1minus120572)119895+119898minus1

[(1 minus 120572) 119895 + 119898 minus 1] 119895

(38)

Now applying the inverse Laplace transform on Θ1(119904) and

using the fact that

119904minus[120572+(120572minus1)119895] = 119871[119903120572+(120572minus1)119895minus1

Γ (120572 + (120572 minus 1) 119895)] (39)

we obtain

Φ1(119903) = 119903

120572minus1oΨ1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1] (40)

with oΨ1[] the generalized Wright function for 119901 = 1 and

119902 = 2 [34ndash37] We next expand the exponential functionexp[minus1199041minus120572(1 minus 120572)] in power series multiplying the resultingtwo series in addition to this if we consider the number119887119896(120572119898) defined for 120572 gt 0 119898 = 2 119897 (120572 = (119901+119898minus1)119901 119901 notin

N) and 119896 isin N0

119887119896(120572119898) =

119897

sum119901119895=0119896119901+119895=119896

(minus1)119902

119901119895 (1 minus 120572) 119902 + 119898 minus 1 (41)

The above family of number possesses satisfies the followingrecursive formula

119887119896(120572119898)

119887119896+1(120572119898)

=120572 minus 119898

120572 minus 1+ 119896 (42)

which produces the explicit expression for 119887119896(120572119898) in the

form of

119887119896(120572119898) =

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1) Γ [((120572 minus 119898) (120572 minus 1)) + 119896] 119896 isin N

0

(43)

Now having the above expression on hand we can derive that

Θlowast119898(119904) = 119904

119898minus120572minus1(infin

sum119895=0

(1

1 minus 120572)119895 119904(1minus120572)119901

119901)

times (infin

sum119901=0

(1

1 minus 120572)119901 (minus1)119901

[(1 minus 120572) 119901 + 119898 minus 1]

119904(1minus120572)119895

119901)

=infin

sum119896=0

119887119896(120572119898) (

1

1 minus 120572)119896

times 119904(1minus120572)119896+119898minus120572minus1 (119898 = 2 119897)

(44)


However remembering (40) with 120573 = (120572 minus 1)119896 + 120572 + 1 minus 119898we can further derive the following expression forΦlowast

119898(119903) as

Φlowast119898(119903) =

infin

sum119896=0

119887119896 (120572119898) (

1

1 minus 120572)119896

timesΓ (119896 + 1)

Γ [120572 + 1 minus 119898 + (120572 minus 1) 119896]

119909(120572minus1)119896+120572minus119898

119896

(45)

or in the simplified version we have

Φlowast119898(119903) =

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)Φ119898(119903) (46)

where

Φ119898(119903)

= 119903120572minus1198981Ψ2

times [(1 1)

(120572 + 1 minus 119898 120572 minus 1) (120572 minus 119898

120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(47)

It follows that the solution of the Cauchy-type equation is inthe form of

Φ (119903)

= 1198861119903120572minus1 oΨ

1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1]

+ 1198862

119897

sum119898=2

119887119898(119898 minus 1)

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(48)

We will examine the solution of the following fractionalpartial differential equation of the following form

119862

0119863120572

119905119906 (119909 119905) =

1

119909

1205972119906 (119909 119905)

1205971199092 0 lt 120572 le 1 (49)

with initial and boundary conditions of the form

119906 (119909 0) = 0 119906 (1199090 119905) = ℎ (119905)

120597119909119906 (0 119905) = 119906 (0 119905) = 0 (119905 ge 0)

(50)

To solve the above problem the first step consists of applyingthe new integral transform on both sides of (49) to obtain

120597119904

119862

0119863120572

119905119880 (119904 119905) = minus119904

2119880 (119904 119905) (51)

where 119904 is the Laplace variableThenext step in this derivationis to apply the Fourier transform in time to obtain

(119894119901)1205721205971199041198801(119904 119901) = minus1199042119880

1(119904 119901) (52)

where119901 is the Fourier variable It follows that the solution ofthe above equation is simply given as

1198801(119904 119901) = 119888 (119901) exp[minus119904

3

3(119894119901)minus120572] (53)

The next step is to put exponential function in series form asfollows

exp[minus1199043

3(119894119901)minus120572]

=infin

sum119896=0

((minus11990433) (119894119901)minus120572)119896

119896=infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896

(54)

Then we first apply the inverse Laplace in both sides of theabove equation to obtain

1198801(119909 119901) = 119871minus1(119888 (119901)

infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896) (55)

Making use of the linearity to the inverse Laplace transformwe obtain

1198801(119909 119901) =

infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119888(119901) (119894119901)minus119896120572

119896 (56)

And finally making use of the inverse Fourier transform andits linearity we obtain

119906 (119909 119905) =infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119865minus1 [119888(119901) (119894119901)minus119896120572]

119896 (57)

This produces the solution of (49)

5 Conclusion

We introduced a new integral operator transform We pre-sented its existence and uniqueness We presented someproperties and its application for solving some kind ofordinary and partial fractional differential equations thatarise in many fields of sciences

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

A Atangana wrote the first draft and A Kilicman correctedthe final versionAll authors read and approved the final draft

Acknowledgments

The authors would like to thank the referee for some valuablecomments and helpful suggestions Special thanks go to theeditor for his valuable time spent to evaluate this paper


References

[1] A D Polyanin and A V Manzhirov Handbook of IntegralEquations CRC Press Boca Raton Fla USA 1998

[2] R K M Thambynayagam The Diffusion Handbook AppliedSolutions for EngineersMcGraw-Hill NewYork NYUSA 2011

[3] M Hazewinkel ldquoIntegral transformrdquo in Encyclopedia of Mathe-matics Springer 2001

[4] B Boashash Time-Frequency Signal Analysis and Processing AComprehensive Reference Elsevier Science Oxford UK 2003

[5] S Bochner and K Chandrasekharan Fourier TransformsPrinceton University Press Princeton NJ USA 1949

[6] R N Bracewell the Fourier Transform and Its ApplicationsMcGraw-Hill Boston Mass USA 3rd edition 2000

[7] G A Campbell and R M Foster Fourier Integrals for PracticalApplications D Van Nostrand Company New York NY USA1948

[8] E U Condon ldquoImmersion of the Fourier transform in acontinuous group of functional transformationsrdquo Proceedings oftheNational Academy of Sciences of theUSA vol 23 pp 158ndash1641937

[9] J Duoandikoetxea Fourier Analysis vol 29 The AmericanMathematical Society Providence RI USA 2001

[10] L Grafakos Classical and Modern Fourier Analysis Prentice-Hall 2004

[11] E Hewitt and K A Ross Abstract Harmonic Analysis Vol IIStructure and Analysis for Compact Groups Analysis on LocallyCompact Abelian Groups Springer New York NY USA 1970

[12] L Schwartz ldquoTransformation de Laplace des distributionsrdquoSeminaire Mathematique de lrsquoUniversite de Lund vol 1952 pp196ndash206 1952 (French)

[13] AAtangana andAKilicman ldquoAnalytical solutions of the space-time fractional derivative of advection dispersion equationrdquoMathematical Problems in Engineering vol 2013 Article ID8531279 2013

[14] W M Siebert Circuits Signals and Systems MIT Press Cam-bridge Mass USA 1986

[15] A Atangana ldquoA note on the triple laplace transform and itsapplications to some kind of third-order differential equationrdquoAbstract and Applied Analysis vol 2013 Article ID 769102 10pages 2013

[16] D V Widder ldquoWhat is the Laplace transformrdquo The AmericanMathematical Monthly vol 52 pp 419ndash425 1945

[17] J Williams Laplace Transforms (Problem Solvers) vol 10George Allen and Unwin 1973

[18] P Flajolet X Gourdon and P Dumas ldquoMellin transforms andasymptotics harmonic sumsrdquo Theoretical Computer Sciencevol 144 no 1-2 pp 3ndash58 1995

[19] J Galambos and I Simonelli Products of Random VariablesApplications to Problems of Physics and to Arithmetical Func-tions vol 4 Marcel Dekker New York NY USA 2004

[20] G K Watugala ldquoSumudu transform a new integral trans-form to solve differential equations and control engineeringproblemsrdquo International Journal of Mathematical Education inScience and Technology vol 24 no 1 pp 35ndash43 1993

[21] S Weerakoon ldquoApplication of Sumudu transform to partialdifferential equationsrdquo International Journal of MathematicalEducation in Science and Technology vol 25 no 2 pp 277ndash2831994

[22] M G M Hussain and F B M Belgacem ldquoTransient solutionsofMaxwellrsquos equations based on sumudu transformrdquo Progress inElectromagnetics Research vol 74 pp 273ndash289 2007

[23] F Oberhettinger and L Badii Tables of Laplace TransformsSpringer Berlin Germany 1973

[24] V A Ditkin and A P Prudnikov Integral Transforms andOperational Calculus Pergamon Press Oxford UK 1965

[25] W Balser From Divergent Power Series to Analytic Functionsvol 1582 Springer Berlin Germany 1994

[26] A Atangana and A Kilicma ldquoThe use of sumudu transformfor solving certain nonlinear fractional heat-like equationsrdquoAbstract and Applied Analysis vol 2013 Article ID 737481 p12 2013

[27] SWeerakoon ldquoThe ldquoSumudu transformrdquo and the Laplace trans-form replyrdquo International Journal of Mathematical Education inScience and Technology vol 28 no 1 p 160 1997

[28] M Y Ongun ldquoThe Laplace Adomian Decomposition Methodfor solving a model for HIV infection of 1198621198634+119879 cellsrdquo Mathe-matical and Computer Modelling vol 53 no 5-6 pp 597ndash6032011

[29] A Atangana ldquoNew class of boundary value problemsrdquo Informa-tion Sciences Letters vol 1 no 2 pp 67ndash76 2012

[30] A Atangana and J F Botha ldquoAnalytical solution of groundwaterflow equation via homotopy decompositionmethodrdquo Journal ofEarth Science and Climatic Change vol 3 article 115 2012

[31] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013

[32] A Atangana and E Alabaraoye ldquoSolving a system of fractionalpartial differential equations arising in the model of HIVinfection of CD4+ cells and attractor one-dimensional Keller-Segel equationsrdquo in Advances in Difference Equations vol 2013article 94 2013

[33] A Atangana A Ahmed andN Bilick ldquoA generalized version ofa low velocity impact between a rigid sphere and a transverselyisotropic strain-hardening plate supported by a rigid substrateusing the concept of non-integer derivativesrdquo Abstract andApplied Analysis vol 2013 Article ID 671321 9 pages 2013

[34] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[35] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press New York NY USA 1999

[36] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[37] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierAmsterdam The Netherlands 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In mathematics the Sumudu transform is an integraltransform similar to the Laplace transform introduced in theearly 1990s by Watugala to solve differential equations andcontrol engineering problems [20ndash27] It is equivalent to theLaplace-Carson transform with the substitution 119901 = 1119906

However there exists some kind of fractional ordinaryand partial differential equations with some kind of singu-larities that cannot be solved directly via the above integraltransform operators In particular the following kind offractional ordinary and partial differential equations

0119863120572

119909119910 (119909) +

1

119909119899119910 (119909) = 119891 (119909) (1)

or

0119863120572

119909119906 (119909 119905) +

1

119909119899119906 (119909 119905) = ℎ (119909 119905) (2)

where 120572 gt 00119863120572

119909 is the fractional derivative (Riemann-

Liouville or Caputo) and 119899 ge 1To solve the above equations some scholars make use

of the Frobenius method to obtain the solutions in seriesform The Laplace transform of the product of two functionsis different from the product of the Laplace transform ofthe two functions The Fourier transform of the productof two functions is equivalent to the convolution of theFourier transform of the two functions This renders itvery difficult to apply directly either the Laplace transformor the Fourier transform operators to solve this type ofequation Therefore some scholars multiply 119909119899 on both sidesof the above equations and then apply the Fourier or theLaplace transform It is therefore worth to define an integraltransform similar to Laplace or Laplace-Carson transform totransform such equation to an ordinary or partial differentialequation without any additional transformation

The aim of this work is to further introduce an integraltransform operator that can be used to solve some kind ofordinary partial and fractional ordinary partial differentialequation with some kind of singularities We will start withthe definition and present some theorems

2 Definitions and Theorems

Definition 1 Let 119891(119909) be a continuous function over an openinterval (0infin) such that its Laplace transform is 119899 timedifferentiable then the new integral transform of order 119899 of119891 is defined as follows

119872119899 (119904) = 119872119899 [119891 (119909)] (119904) = int

infin

0

119909119899119890minus119909119904119891 (119909) 119889119909 (3)

and the inverse of the new integral transform of order 119899 isdefined as

119891 (119909) = 119872minus1

119899[119872119899[119891 (119909)]]

=(minus1)119899

2120587119894int120572+119894infin


119890119904119909 [(minus1)119899 [

1

Γ (119899 minus 1)

times int119904

0

(119904 minus 119905)119899minus1119872119899 (119905) 119889119905

+119899minus1

sum119896=0

119904119896

119896119910119896]]119889119904

119910119896=120597119896119865 (0)

120597119904119896

(4)

where 119865(119904) is the Laplace transform of 119891(119909) Before wecontinue we will prove that the above definition is indeedthe inverse operator transform of order 119899 In fact from thedefinition of new transform of order 119899 of a function 119891(119909) wehave that

119872119899(119904) = 119872

119899[119891 (119909)] (119904)

= intinfin

0

119909119899119890minus119909119904119891 (119909) 119889119909 = (minus1)119899 119889119899119865 (119904)

119889119904119899

(5)

thus

1

Γ (119899 minus 1)int119904

0

(119904 minus 119905)119899minus1119872119899(119905) 119889119905

= (minus1)119899 [119865 (119904) minus

119899minus1

sum119896=0

119904119896

119896119910119896]

(6)

It follows that

(minus1)119899

2120587119894int120572+119894infin


119890119904119909 [(minus1)119899 [

1

Γ (119899 minus 1)

times int119904

0

(119904 minus 119905)119899minus1119872119899 (119905) 119889119905

+119899minus1

sum119896=0

119904119896

119896119910119896]]119889119904

=(minus1)119899

2120587119894int120572+119894infin


119890119904119909 [(minus1)119899[119865 (119904)]] 119889119904

119872minus1119899[119872119899[119891 (119909)]]

=(minus1)2119899

2120587119894int120572+119894infin


119890119904119909 [[119865 (119904)]] 119889119904 = 119891 (119909)

(7)

Therefore the inverse of the new integral transform is welldefined Our next concern is to prove the uniqueness and theexistence of the new integral transform




119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119865 (119901)] 119889119901 (8)


119891 (119909) =(minus1)2119899

2120587119894int120572+119894infin


119890119901119909 [119865 (119901)] 119889119901 (9)


119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119866 (119901)] 119889119901 (10)

which boils down to

119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119866 (119901)] 119889119901 = 119892 (119909) (11)



0) and satisfies

1003816100381610038161003816119905119899119891 (119905)

1003816100381610038161003816 le 119872119890120572119905 (12)


119899[119891(119909)](119904) exists for all 119904 gt 120572


intinfin

0

119905119899119890minus119904119905119891 (119905) 119889119905

= int1199050

0

119905119899119890minus119904119905119891 (119905) 119889119905 + intinfin

1199050

119905119899119890minus119904119905119891 (119905) 119889119905

(13)



10038161003816100381610038161003816 le 119872119890120572119905119890minus119904119905 = 119872119890(120572minus119904)119905 (14)

Now

intinfin

1199050

119872119890(120572minus119904)119905119889119905 = 119872119890(120572minus119904)1199050

120572 minus 119904 (15)


119899(119904) exists for 120572 lt 119904


119871 (119891 (119909)) (119904) = 119872119899(1

119909119899119891 (119909)) (119904)

119871 (119891 (119909)) (119904) = 1198720(119891 (119909)) (119904)

119872119899(119891 (119909)) (119904) = (minus1)

119899 119889119899 [119865 (119904)]

119889119904119899

(16)



119871119888(119891 (119909)) (119904) = 119872

1(119891 (119909)) (119904) (17)


119891 (119909) = lim119899rarrinfin

(minus1)119899

119899(119899

119909)119899+1

119872119899(119899

119904) (18)






(I) 119872119899 [119904 + 119888] = 119872119899 [119890minus119888119909119891 (119909)]

(II) 119872119899[119891 (119886119909)] (119904) =

1

119886119872119899[119904

119886]



119890119904119909119872119899(119904) 119889119904 = 119909

119899119891 (119909)

(IV) 119872119899 [119886119891 (119909) + 119887119892 (119909)] (119904)

= [119886119872119899(119891 (119909)) + 119887119872

119899(119892 (119909))] (119904)

(V) 119872119899 [119891 (119909)

119909119899] (119904) = 119871 [119891 (119909)] (119904)

(VI) 119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119866 (119904))

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(VII) 119872119899[119889119899119891 (119909)

119889119909119899] (119904)

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(19)



119872119899[119890minus119888119909119891 (119909)]

= intinfin

0

[119909119899119890minus119888119909119890minus119904119909119891 (119909)] 119889119909

= intinfin

0

[119909119899119890minus(119888+119904)119909119891 (119909)] 119889119909 = 119872119899 [119904 + 119888]

(20)


119872119899[119891 (119886119909)] (119904)

= intinfin

0

[119909119899119890minus119909119904119891 (119886119909)] 119889119909 = (minus1)

119899 119889119899

119889119904119899[119871 [119891 (119886119909)] (119904)]

(21)


119872119899[119891 (119886119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[1

119886119865 (

119904

119886)]

=1

119886(minus1)119899 119889119899

119889119904119899[119865 (

119904

119886)] =

1

119886119872119899[119904

119886]

(22)


then

int120572+119894infin


119890119904119909119872119899(119904) 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119909119899119891 (119909) 119889119909] 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119892 (119909) 119889119909] 119889119904

(23)


int120572+119894infin


119890119904119909119872119899(119904) 119889119904 = 119892 (119909) = 119909

119899119891 (119909) (24)


119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= intinfin

0

[119909119899119890minus119904119909119891 (119909) lowast ℎ (119909)]

= (minus1)119899 119889119899

119889119904119899[119871 (119891 (119909) lowast ℎ (119909)) (119904)]

(25)


119871 (119891 (119909) lowast ℎ (119909)) (119904) = 119865 (119904) sdot 119866 (119904) (26)


119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[119865 (119904) sdot 119866 (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119866 (119904))

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(27)


119872119899[119889119899119891 (119909)

119889119909119899] (119904)

= intinfin

0

[119909119899119890minus119904119909119889119899119891 (119909)

119889119909119899] 119889119909

= (minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

(28)


119871(119889119899119891 (119909)

119889119909119899) (119904) = 119904

119899119865 (119904) minus119899minus1

sum119896=0

119904119899minus119896minus1119889119896119891 (0)

119889119909119896(29)


(minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(30)

that is

119872119899[119889119899119891 (119909)

119889119909119899] (119904) = (minus1)

119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896 (31)






119863120572119903119903Φ (119903) +

1

119903119899Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (32)


(minus1)119899 119889119899

119889119904119899119863120572119903119903Φ (119904) + Φ (119904) = 0 (33)


119863120572119903119903Φ (119903) +

1

119903Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (34)


119889 [119871 (Φ) (119904)]

119889119904+ (

120572

119904+1

119904120572) (119871 (Φ) (119904))

=119897

sum119898=2

119889119898(119898 minus 1) 119904


(35)



Θ (119904) = 119904minus120572 exp[minus 119904

1minus120572

1 minus 120572]

times [1198861+119897

sum119898=2

119889119898(119898 minus 1) int 119904


1 minus 120572]119889119904]

(36)



Θ (119904) = 119888Θ1(119904) +

119897

sum119898=2

119889119898(119898 minus 1)Θ

lowast

119898(119904) (37)

with of course

Θ1(119904) = 119904


1 minus 120572]

Θlowast119898(119904) = 119904

minus120572 exp[ 1199041minus120572

120572 minus 1]

timesinfin

sum119895=0

(1


[(1 minus 120572) 119895 + 119898 minus 1] 119895

(38)


using the fact that


Γ (120572 + (120572 minus 1) 119895)] (39)

we obtain

Φ1(119903) = 119903

120572minus1oΨ1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1] (40)




119887119896(120572119898) =

119897

sum119901119895=0119896119901+119895=119896

(minus1)119902

119901119895 (1 minus 120572) 119902 + 119898 minus 1 (41)


119887119896(120572119898)

119887119896+1(120572119898)

=120572 minus 119898

120572 minus 1+ 119896 (42)


form of

119887119896(120572119898) =

Γ [(120572 minus 119898) (120572 minus 1)]


0

(43)


Θlowast119898(119904) = 119904


sum119895=0

(1

1 minus 120572)119895 119904(1minus120572)119901

119901)

times (infin

sum119901=0

(1

1 minus 120572)119901 (minus1)119901

[(1 minus 120572) 119901 + 119898 minus 1]

119904(1minus120572)119895

119901)

=infin

sum119896=0

119887119896(120572119898) (

1

1 minus 120572)119896


(44)



119898(119903) as

Φlowast119898(119903) =

infin

sum119896=0

119887119896 (120572119898) (

1

1 minus 120572)119896

timesΓ (119896 + 1)

Γ [120572 + 1 minus 119898 + (120572 minus 1) 119896]

119909(120572minus1)119896+120572minus119898

119896

(45)


Φlowast119898(119903) =

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)Φ119898(119903) (46)

where

Φ119898(119903)

= 119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(47)


Φ (119903)

= 1198861119903120572minus1 oΨ

1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1]

+ 1198862

119897

sum119898=2

119887119898(119898 minus 1)

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(48)


119862

0119863120572

119905119906 (119909 119905) =

1

119909

1205972119906 (119909 119905)

1205971199092 0 lt 120572 le 1 (49)


119906 (119909 0) = 0 119906 (1199090 119905) = ℎ (119905)

120597119909119906 (0 119905) = 119906 (0 119905) = 0 (119905 ge 0)

(50)


120597119904

119862

0119863120572

119905119880 (119904 119905) = minus119904

2119880 (119904 119905) (51)


(119894119901)1205721205971199041198801(119904 119901) = minus1199042119880

1(119904 119901) (52)


1198801(119904 119901) = 119888 (119901) exp[minus119904

3

3(119894119901)minus120572] (53)


exp[minus1199043

3(119894119901)minus120572]

=infin

sum119896=0

((minus11990433) (119894119901)minus120572)119896

119896=infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896

(54)


1198801(119909 119901) = 119871minus1(119888 (119901)

infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896) (55)


1198801(119909 119901) =

infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119888(119901) (119894119901)minus119896120572

119896 (56)


119906 (119909 119905) =infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119865minus1 [119888(119901) (119894119901)minus119896120572]

119896 (57)


5 Conclusion






Acknowledgments



References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119865 (119901)] 119889119901 (8)


119891 (119909) =(minus1)2119899

2120587119894int120572+119894infin


119890119901119909 [119865 (119901)] 119889119901 (9)


119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119866 (119901)] 119889119901 (10)

which boils down to

119891 (119909) =(minus1)119899

2120587119894int120572+119894infin


119890119901119909 [(minus1)119899119866 (119901)] 119889119901 = 119892 (119909) (11)



0) and satisfies

1003816100381610038161003816119905119899119891 (119905)

1003816100381610038161003816 le 119872119890120572119905 (12)


119899[119891(119909)](119904) exists for all 119904 gt 120572


intinfin

0

119905119899119890minus119904119905119891 (119905) 119889119905

= int1199050

0

119905119899119890minus119904119905119891 (119905) 119889119905 + intinfin

1199050

119905119899119890minus119904119905119891 (119905) 119889119905

(13)



10038161003816100381610038161003816 le 119872119890120572119905119890minus119904119905 = 119872119890(120572minus119904)119905 (14)

Now

intinfin

1199050

119872119890(120572minus119904)119905119889119905 = 119872119890(120572minus119904)1199050

120572 minus 119904 (15)


119899(119904) exists for 120572 lt 119904


119871 (119891 (119909)) (119904) = 119872119899(1

119909119899119891 (119909)) (119904)

119871 (119891 (119909)) (119904) = 1198720(119891 (119909)) (119904)

119872119899(119891 (119909)) (119904) = (minus1)

119899 119889119899 [119865 (119904)]

119889119904119899

(16)



119871119888(119891 (119909)) (119904) = 119872

1(119891 (119909)) (119904) (17)


119891 (119909) = lim119899rarrinfin

(minus1)119899

119899(119899

119909)119899+1

119872119899(119899

119904) (18)






(I) 119872119899 [119904 + 119888] = 119872119899 [119890minus119888119909119891 (119909)]

(II) 119872119899[119891 (119886119909)] (119904) =

1

119886119872119899[119904

119886]



119890119904119909119872119899(119904) 119889119904 = 119909

119899119891 (119909)

(IV) 119872119899 [119886119891 (119909) + 119887119892 (119909)] (119904)

= [119886119872119899(119891 (119909)) + 119887119872

119899(119892 (119909))] (119904)

(V) 119872119899 [119891 (119909)

119909119899] (119904) = 119871 [119891 (119909)] (119904)

(VI) 119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119866 (119904))

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(VII) 119872119899[119889119899119891 (119909)

119889119909119899] (119904)

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(19)



119872119899[119890minus119888119909119891 (119909)]

= intinfin

0

[119909119899119890minus119888119909119890minus119904119909119891 (119909)] 119889119909

= intinfin

0

[119909119899119890minus(119888+119904)119909119891 (119909)] 119889119909 = 119872119899 [119904 + 119888]

(20)


119872119899[119891 (119886119909)] (119904)

= intinfin

0

[119909119899119890minus119909119904119891 (119886119909)] 119889119909 = (minus1)

119899 119889119899

119889119904119899[119871 [119891 (119886119909)] (119904)]

(21)


119872119899[119891 (119886119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[1

119886119865 (

119904

119886)]

=1

119886(minus1)119899 119889119899

119889119904119899[119865 (

119904

119886)] =

1

119886119872119899[119904

119886]

(22)


then

int120572+119894infin


119890119904119909119872119899(119904) 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119909119899119891 (119909) 119889119909] 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119892 (119909) 119889119909] 119889119904

(23)


int120572+119894infin


119890119904119909119872119899(119904) 119889119904 = 119892 (119909) = 119909

119899119891 (119909) (24)


119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= intinfin

0

[119909119899119890minus119904119909119891 (119909) lowast ℎ (119909)]

= (minus1)119899 119889119899

119889119904119899[119871 (119891 (119909) lowast ℎ (119909)) (119904)]

(25)


119871 (119891 (119909) lowast ℎ (119909)) (119904) = 119865 (119904) sdot 119866 (119904) (26)


119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[119865 (119904) sdot 119866 (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119866 (119904))

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(27)


119872119899[119889119899119891 (119909)

119889119909119899] (119904)

= intinfin

0

[119909119899119890minus119904119909119889119899119891 (119909)

119889119909119899] 119889119909

= (minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

(28)


119871(119889119899119891 (119909)

119889119909119899) (119904) = 119904

119899119865 (119904) minus119899minus1

sum119896=0

119904119899minus119896minus1119889119896119891 (0)

119889119909119896(29)


(minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(30)

that is

119872119899[119889119899119891 (119909)

119889119909119899] (119904) = (minus1)

119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896 (31)






119863120572119903119903Φ (119903) +

1

119903119899Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (32)


(minus1)119899 119889119899

119889119904119899119863120572119903119903Φ (119904) + Φ (119904) = 0 (33)


119863120572119903119903Φ (119903) +

1

119903Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (34)


119889 [119871 (Φ) (119904)]

119889119904+ (

120572

119904+1

119904120572) (119871 (Φ) (119904))

=119897

sum119898=2

119889119898(119898 minus 1) 119904


(35)



Θ (119904) = 119904minus120572 exp[minus 119904

1minus120572

1 minus 120572]

times [1198861+119897

sum119898=2

119889119898(119898 minus 1) int 119904


1 minus 120572]119889119904]

(36)



Θ (119904) = 119888Θ1(119904) +

119897

sum119898=2

119889119898(119898 minus 1)Θ

lowast

119898(119904) (37)

with of course

Θ1(119904) = 119904


1 minus 120572]

Θlowast119898(119904) = 119904

minus120572 exp[ 1199041minus120572

120572 minus 1]

timesinfin

sum119895=0

(1


[(1 minus 120572) 119895 + 119898 minus 1] 119895

(38)


using the fact that


Γ (120572 + (120572 minus 1) 119895)] (39)

we obtain

Φ1(119903) = 119903

120572minus1oΨ1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1] (40)




119887119896(120572119898) =

119897

sum119901119895=0119896119901+119895=119896

(minus1)119902

119901119895 (1 minus 120572) 119902 + 119898 minus 1 (41)


119887119896(120572119898)

119887119896+1(120572119898)

=120572 minus 119898

120572 minus 1+ 119896 (42)


form of

119887119896(120572119898) =

Γ [(120572 minus 119898) (120572 minus 1)]


0

(43)


Θlowast119898(119904) = 119904


sum119895=0

(1

1 minus 120572)119895 119904(1minus120572)119901

119901)

times (infin

sum119901=0

(1

1 minus 120572)119901 (minus1)119901

[(1 minus 120572) 119901 + 119898 minus 1]

119904(1minus120572)119895

119901)

=infin

sum119896=0

119887119896(120572119898) (

1

1 minus 120572)119896


(44)



119898(119903) as

Φlowast119898(119903) =

infin

sum119896=0

119887119896 (120572119898) (

1

1 minus 120572)119896

timesΓ (119896 + 1)

Γ [120572 + 1 minus 119898 + (120572 minus 1) 119896]

119909(120572minus1)119896+120572minus119898

119896

(45)


Φlowast119898(119903) =

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)Φ119898(119903) (46)

where

Φ119898(119903)

= 119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(47)


Φ (119903)

= 1198861119903120572minus1 oΨ

1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1]

+ 1198862

119897

sum119898=2

119887119898(119898 minus 1)

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(48)


119862

0119863120572

119905119906 (119909 119905) =

1

119909

1205972119906 (119909 119905)

1205971199092 0 lt 120572 le 1 (49)


119906 (119909 0) = 0 119906 (1199090 119905) = ℎ (119905)

120597119909119906 (0 119905) = 119906 (0 119905) = 0 (119905 ge 0)

(50)


120597119904

119862

0119863120572

119905119880 (119904 119905) = minus119904

2119880 (119904 119905) (51)


(119894119901)1205721205971199041198801(119904 119901) = minus1199042119880

1(119904 119901) (52)


1198801(119904 119901) = 119888 (119901) exp[minus119904

3

3(119894119901)minus120572] (53)


exp[minus1199043

3(119894119901)minus120572]

=infin

sum119896=0

((minus11990433) (119894119901)minus120572)119896

119896=infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896

(54)


1198801(119909 119901) = 119871minus1(119888 (119901)

infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896) (55)


1198801(119909 119901) =

infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119888(119901) (119894119901)minus119896120572

119896 (56)


119906 (119909 119905) =infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119865minus1 [119888(119901) (119894119901)minus119896120572]

119896 (57)


5 Conclusion






Acknowledgments



References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872119899[119890minus119888119909119891 (119909)]

= intinfin

0

[119909119899119890minus119888119909119890minus119904119909119891 (119909)] 119889119909

= intinfin

0

[119909119899119890minus(119888+119904)119909119891 (119909)] 119889119909 = 119872119899 [119904 + 119888]

(20)


119872119899[119891 (119886119909)] (119904)

= intinfin

0

[119909119899119890minus119909119904119891 (119886119909)] 119889119909 = (minus1)

119899 119889119899

119889119904119899[119871 [119891 (119886119909)] (119904)]

(21)


119872119899[119891 (119886119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[1

119886119865 (

119904

119886)]

=1

119886(minus1)119899 119889119899

119889119904119899[119865 (

119904

119886)] =

1

119886119872119899[119904

119886]

(22)


then

int120572+119894infin


119890119904119909119872119899(119904) 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119909119899119891 (119909) 119889119909] 119889119904

= int120572+119894infin


119890119904119909 [intinfin

0

119890minus119909119904119892 (119909) 119889119909] 119889119904

(23)


int120572+119894infin


119890119904119909119872119899(119904) 119889119904 = 119892 (119909) = 119909

119899119891 (119909) (24)


119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= intinfin

0

[119909119899119890minus119904119909119891 (119909) lowast ℎ (119909)]

= (minus1)119899 119889119899

119889119904119899[119871 (119891 (119909) lowast ℎ (119909)) (119904)]

(25)


119871 (119891 (119909) lowast ℎ (119909)) (119904) = 119865 (119904) sdot 119866 (119904) (26)


119872119899[119891 (119909) lowast ℎ (119909)] (119904)

= (minus1)119899 119889119899

119889119904119899[119865 (119904) sdot 119866 (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119866 (119904))

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(27)


119872119899[119889119899119891 (119909)

119889119909119899] (119904)

= intinfin

0

[119909119899119890minus119904119909119889119899119891 (119909)

119889119909119899] 119889119909

= (minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

(28)


119871(119889119899119891 (119909)

119889119909119899) (119904) = 119904

119899119865 (119904) minus119899minus1

sum119896=0

119904119899minus119896minus1119889119896119891 (0)

119889119909119896(29)


(minus1)119899 119889119899

119889119904119899[119871(

119889119899119891 (119909)

119889119909119899) (119904)]

= (minus1)119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896

(30)

that is

119872119899[119889119899119891 (119909)

119889119909119899] (119904) = (minus1)

119899

119899

sum119896=0

119862119896119899

119889119896 (119904119899)

119889119904119896times119889119899minus119896 (119865 (119904))

119889119904119899minus119896 (31)






119863120572119903119903Φ (119903) +

1

119903119899Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (32)


(minus1)119899 119889119899

119889119904119899119863120572119903119903Φ (119904) + Φ (119904) = 0 (33)


119863120572119903119903Φ (119903) +

1

119903Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (34)


119889 [119871 (Φ) (119904)]

119889119904+ (

120572

119904+1

119904120572) (119871 (Φ) (119904))

=119897

sum119898=2

119889119898(119898 minus 1) 119904


(35)



Θ (119904) = 119904minus120572 exp[minus 119904

1minus120572

1 minus 120572]

times [1198861+119897

sum119898=2

119889119898(119898 minus 1) int 119904


1 minus 120572]119889119904]

(36)



Θ (119904) = 119888Θ1(119904) +

119897

sum119898=2

119889119898(119898 minus 1)Θ

lowast

119898(119904) (37)

with of course

Θ1(119904) = 119904


1 minus 120572]

Θlowast119898(119904) = 119904

minus120572 exp[ 1199041minus120572

120572 minus 1]

timesinfin

sum119895=0

(1


[(1 minus 120572) 119895 + 119898 minus 1] 119895

(38)


using the fact that


Γ (120572 + (120572 minus 1) 119895)] (39)

we obtain

Φ1(119903) = 119903

120572minus1oΨ1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1] (40)




119887119896(120572119898) =

119897

sum119901119895=0119896119901+119895=119896

(minus1)119902

119901119895 (1 minus 120572) 119902 + 119898 minus 1 (41)


119887119896(120572119898)

119887119896+1(120572119898)

=120572 minus 119898

120572 minus 1+ 119896 (42)


form of

119887119896(120572119898) =

Γ [(120572 minus 119898) (120572 minus 1)]


0

(43)


Θlowast119898(119904) = 119904


sum119895=0

(1

1 minus 120572)119895 119904(1minus120572)119901

119901)

times (infin

sum119901=0

(1

1 minus 120572)119901 (minus1)119901

[(1 minus 120572) 119901 + 119898 minus 1]

119904(1minus120572)119895

119901)

=infin

sum119896=0

119887119896(120572119898) (

1

1 minus 120572)119896


(44)



119898(119903) as

Φlowast119898(119903) =

infin

sum119896=0

119887119896 (120572119898) (

1

1 minus 120572)119896

timesΓ (119896 + 1)

Γ [120572 + 1 minus 119898 + (120572 minus 1) 119896]

119909(120572minus1)119896+120572minus119898

119896

(45)


Φlowast119898(119903) =

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)Φ119898(119903) (46)

where

Φ119898(119903)

= 119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(47)


Φ (119903)

= 1198861119903120572minus1 oΨ

1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1]

+ 1198862

119897

sum119898=2

119887119898(119898 minus 1)

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(48)


119862

0119863120572

119905119906 (119909 119905) =

1

119909

1205972119906 (119909 119905)

1205971199092 0 lt 120572 le 1 (49)


119906 (119909 0) = 0 119906 (1199090 119905) = ℎ (119905)

120597119909119906 (0 119905) = 119906 (0 119905) = 0 (119905 ge 0)

(50)


120597119904

119862

0119863120572

119905119880 (119904 119905) = minus119904

2119880 (119904 119905) (51)


(119894119901)1205721205971199041198801(119904 119901) = minus1199042119880

1(119904 119901) (52)


1198801(119904 119901) = 119888 (119901) exp[minus119904

3

3(119894119901)minus120572] (53)


exp[minus1199043

3(119894119901)minus120572]

=infin

sum119896=0

((minus11990433) (119894119901)minus120572)119896

119896=infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896

(54)


1198801(119909 119901) = 119871minus1(119888 (119901)

infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896) (55)


1198801(119909 119901) =

infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119888(119901) (119894119901)minus119896120572

119896 (56)


119906 (119909 119905) =infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119865minus1 [119888(119901) (119894119901)minus119896120572]

119896 (57)


5 Conclusion






Acknowledgments



References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119863120572119903119903Φ (119903) +

1

119903119899Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (32)


(minus1)119899 119889119899

119889119904119899119863120572119903119903Φ (119904) + Φ (119904) = 0 (33)


119863120572119903119903Φ (119903) +

1

119903Φ (119903) = 0 119897 minus 1 lt 120572 le 119897 (34)


119889 [119871 (Φ) (119904)]

119889119904+ (

120572

119904+1

119904120572) (119871 (Φ) (119904))

=119897

sum119898=2

119889119898(119898 minus 1) 119904


(35)



Θ (119904) = 119904minus120572 exp[minus 119904

1minus120572

1 minus 120572]

times [1198861+119897

sum119898=2

119889119898(119898 minus 1) int 119904


1 minus 120572]119889119904]

(36)



Θ (119904) = 119888Θ1(119904) +

119897

sum119898=2

119889119898(119898 minus 1)Θ

lowast

119898(119904) (37)

with of course

Θ1(119904) = 119904


1 minus 120572]

Θlowast119898(119904) = 119904

minus120572 exp[ 1199041minus120572

120572 minus 1]

timesinfin

sum119895=0

(1


[(1 minus 120572) 119895 + 119898 minus 1] 119895

(38)


using the fact that


Γ (120572 + (120572 minus 1) 119895)] (39)

we obtain

Φ1(119903) = 119903

120572minus1oΨ1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1] (40)




119887119896(120572119898) =

119897

sum119901119895=0119896119901+119895=119896

(minus1)119902

119901119895 (1 minus 120572) 119902 + 119898 minus 1 (41)


119887119896(120572119898)

119887119896+1(120572119898)

=120572 minus 119898

120572 minus 1+ 119896 (42)


form of

119887119896(120572119898) =

Γ [(120572 minus 119898) (120572 minus 1)]


0

(43)


Θlowast119898(119904) = 119904


sum119895=0

(1

1 minus 120572)119895 119904(1minus120572)119901

119901)

times (infin

sum119901=0

(1

1 minus 120572)119901 (minus1)119901

[(1 minus 120572) 119901 + 119898 minus 1]

119904(1minus120572)119895

119901)

=infin

sum119896=0

119887119896(120572119898) (

1

1 minus 120572)119896


(44)



119898(119903) as

Φlowast119898(119903) =

infin

sum119896=0

119887119896 (120572119898) (

1

1 minus 120572)119896

timesΓ (119896 + 1)

Γ [120572 + 1 minus 119898 + (120572 minus 1) 119896]

119909(120572minus1)119896+120572minus119898

119896

(45)


Φlowast119898(119903) =

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)Φ119898(119903) (46)

where

Φ119898(119903)

= 119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(47)


Φ (119903)

= 1198861119903120572minus1 oΨ

1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1]

+ 1198862

119897

sum119898=2

119887119898(119898 minus 1)

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(48)


119862

0119863120572

119905119906 (119909 119905) =

1

119909

1205972119906 (119909 119905)

1205971199092 0 lt 120572 le 1 (49)


119906 (119909 0) = 0 119906 (1199090 119905) = ℎ (119905)

120597119909119906 (0 119905) = 119906 (0 119905) = 0 (119905 ge 0)

(50)


120597119904

119862

0119863120572

119905119880 (119904 119905) = minus119904

2119880 (119904 119905) (51)


(119894119901)1205721205971199041198801(119904 119901) = minus1199042119880

1(119904 119901) (52)


1198801(119904 119901) = 119888 (119901) exp[minus119904

3

3(119894119901)minus120572] (53)


exp[minus1199043

3(119894119901)minus120572]

=infin

sum119896=0

((minus11990433) (119894119901)minus120572)119896

119896=infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896

(54)


1198801(119909 119901) = 119871minus1(119888 (119901)

infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896) (55)


1198801(119909 119901) =

infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119888(119901) (119894119901)minus119896120572

119896 (56)


119906 (119909 119905) =infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119865minus1 [119888(119901) (119894119901)minus119896120572]

119896 (57)


5 Conclusion






Acknowledgments



References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898(119903) as

Φlowast119898(119903) =

infin

sum119896=0

119887119896 (120572119898) (

1

1 minus 120572)119896

timesΓ (119896 + 1)

Γ [120572 + 1 minus 119898 + (120572 minus 1) 119896]

119909(120572minus1)119896+120572minus119898

119896

(45)


Φlowast119898(119903) =

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)Φ119898(119903) (46)

where

Φ119898(119903)

= 119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(47)


Φ (119903)

= 1198861119903120572minus1 oΨ

1[(120572 120572 minus 1) |

119909120572minus1

120572 minus 1]

+ 1198862

119897

sum119898=2

119887119898(119898 minus 1)

Γ [(120572 minus 119898) (120572 minus 1)]

(119898 minus 1)119903120572minus1198981Ψ2

times [(1 1)


120572 minus 1 1)

|119903120572minus1

120572 minus 1]

(48)


119862

0119863120572

119905119906 (119909 119905) =

1

119909

1205972119906 (119909 119905)

1205971199092 0 lt 120572 le 1 (49)


119906 (119909 0) = 0 119906 (1199090 119905) = ℎ (119905)

120597119909119906 (0 119905) = 119906 (0 119905) = 0 (119905 ge 0)

(50)


120597119904

119862

0119863120572

119905119880 (119904 119905) = minus119904

2119880 (119904 119905) (51)


(119894119901)1205721205971199041198801(119904 119901) = minus1199042119880

1(119904 119901) (52)


1198801(119904 119901) = 119888 (119901) exp[minus119904

3

3(119894119901)minus120572] (53)


exp[minus1199043

3(119894119901)minus120572]

=infin

sum119896=0

((minus11990433) (119894119901)minus120572)119896

119896=infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896

(54)


1198801(119909 119901) = 119871minus1(119888 (119901)

infin

sum119896=0

(minus11990433)119896

(119894119901)minus119896120572

119896) (55)


1198801(119909 119901) =

infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119888(119901) (119894119901)minus119896120572

119896 (56)


119906 (119909 119905) =infin

sum119896=0

119871minus1 [(minus11990433)119896

] 119865minus1 [119888(119901) (119894119901)minus119896120572]

119896 (57)


5 Conclusion






Acknowledgments



References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a novel integral operator …downloads.hindawi.com/journals/mpe/2013/531984.pdfa...

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