Research ArticleOn Generalized Fractional Kinetic Equations InvolvingGeneralized Bessel Function of the First Kind

Dinesh Kumar1 S D Purohit2 A Secer3 and A Atangana4

1 Department of Mathematics amp Statistics J N Vyas University Jodhpur 342001 India2Department of Basic Sciences (Mathematics) College of Technology and Engineering M P University of Agriculture and TechnologyUdaipur Rajasthan 313001 India

3 Department of Mathematical Engineering Yildiz Technical University Davutpasa 34210 Istanbul Turkey4 Institute for Groundwater Studies University of the Free State Bloemfontein 9300 South Africa

Correspondence should be addressed to A Atangana abdonatanganayahoofr

Received 18 July 2014 Revised 1 September 2014 Accepted 2 September 2014

Academic Editor Hossein Jafari

Copyright copy 2015 Dinesh Kumar et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the firstkindThemanifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractionalkinetic equation in the paper The results obtained here are quite general in nature and capable of yielding a very large number ofknown and (presumably) new results

1 Introduction and Preliminaries

Bessel functions are playing the important role in studyingsolutions of differential equations and they are associatedwith a wide range of problems in important areas of math-ematical physics like problems of acoustics radiophysicshydrodynamics and atomic and nuclear physics These con-siderations have led various workers in the field of specialfunctions to explore the possible extensions and applicationsof the Bessel functions Among many properties of Besselfunctions they also have investigated some possible exten-sions of the Bessel functions

The generalized Bessel function of the first kind 120596119901(119911) isdefined for 119911 isin C 0 and 119887 119888 119901 isin C(R(119901) gt minus1) by thefollowing series [1 page 10 (115)] (for recent work see also[2ndash6])

120596119901119887119888 (119911) = 120596119901 (119911) =

infin

sum

119896=0

(minus1)119896119888119896

119896Γ (119901 + (119887 + 1) 2 + 119896)

(

119911

2

)

2119896+119901

(1)

where C denotes the set of complex numbers and Γ(119886) is thefamiliar Gamma function

The special cases of series (1) can be obtained as follows

(i) If we put 119887 = 119888 = 1 in (1) then we obtain the familiarBessel function of the first kind [7] of order 119901 for119911 119901 isin C withR(119901) gt minus1 defined and represented bythe following expressions (see also [1 8])

119869119901 (119911) =

infin

sum

119896=0

(minus1)119896

119896Γ (119901 + 119896 + 1)

(

119911

2

)

2119896+119901

119911 isin C (2)

(ii) Putting 119887 = 1 and 119888 = minus1 in series (1) we get themodified Bessel function of the first kind of order 119901defined by (see [1 7])

119868119901 (119911) =

infin

sum

119896=0

1

119896Γ (119901 + 119896 + 1)

(

119911

2

)

2119896+119901

119911 isin C (3)

the series given by (3) is also a special case of Galuersquosgeneralized modified Bessel function [9] dependingon parameters 119886 = 0 1 2 and 119901 gt minus1 given asfollows

119886119868119901(119911) =

infin

sum

119896=0

(1199112)2119896+119901

119896Γ (119901 + 119886119896 + 1)

119911 isin C (4)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 289387 7 pageshttpdxdoiorg1011552015289387

2 Mathematical Problems in Engineering

(iii) Letting 119887 = 2 and 119888 = 1 in series (1) we have thespherical Bessel function of the first kind of order 119901defined by (see [1])

119895119901 (119911) =radic

120587

2

infin

sum

119896=0

(minus1)119896

119896Γ (119901 + 119896 + 32)

(

119911

2

)

2119896+119901

119911 isin C

(5)

Furthermore Deniz et al [10] considered the function120593119901119887119888(119911) defined in terms of the generalized Bessel function120596119901(119911) by the transformation

120593119901119887119888 (119911) = 2119901Γ(119901 +

119887 + 1

2

) 1199111minus1199012

120596119901 (radic119911)

= 119911 +

infin

sum

119896=1

(minus119888)119896

4119896(])119896

119911119896+1

119896

(6)

where ] = 119901+ (119887+ 1)2 notin 119885minus

0= 0 minus1 minus2 and (119886)119896 is the

Pochhammer symbol defined (for 119886 isin C) by

(119886)119896 =

1 (119896 = 0)

119886 (119886 + 1) sdot sdot sdot (119886 + 119896 minus 1) (119896 isin 119873 = 1 2 3 )

=

Γ (119886 + 119896)

Γ (119886)

(119886 isin C 119885minus

0)

(7)

Fractional differential equations appear more and morefrequently for modeling of relevant systems in several fieldsof applied sciencesThese equations play important roles notonly in mathematics but also in physics dynamical systemscontrol systems and engineering to create the mathematicalmodel ofmany physical phenomena In particular the kineticequations describe the continuity of motion of substance andare the basic equations of mathematical physics and naturalscience Therefore in literature we found several papers thatanalyze extensions and generalizations of these equationsinvolving various fractional calculus operators One may forinstance refer to such type of works by [11ndash23]

Haubold and Mathai [13] have established a functionaldifferential equation between rate of change of reaction thedestruction rate and the production rate as follows

119889119873

119889119905

= minus119889 (119873119905) + 119901 (119873119905) (8)

where 119873 = 119873(119905) is the rate of reaction 119889 = 119889(119873) is the rateof destruction 119901 = 119901(119873) is the rate of production and 119873119905

denotes the function defined by119873119905(119905lowast) = 119873(119905 minus 119905

lowast) 119905lowast gt 0

Haubold and Mathai studied a special case of (8) whenspatial fluctuations or inhomogeneities in the quantity 119873(119905)

are neglected is given by the equation

119889119873119894

119889119905

= minus119888119894119873119894 (119905) (9)

together with the initial condition that 119873119894(119905 = 0) = 1198730 isthe number of density of species 119894 at time 119905 = 0 119888119894 gt 0 If we

decline the index 119894 and integrate the standard kinetic equation(9) we have

119873(119905) minus 1198730 = minus1198880119863minus1

119905119873(119905) (10)

where0119863minus1

119905is the special case of the Riemann-Liouville

integral operator0119863minus]119905

defined as

0119863minus]119905119891 (119905) =

1

Γ (])int

119905

0

(119905 minus 119904)]minus1

119891 (119904) 119889119904 119905 gt 0 R (]) gt 0

(11)

Haubold and Mathai [13] have given the fractional general-ization of the standard kinetic equation (10) as

119873(119905) minus 1198730 = minus119888]0119863minus]119905119873(119905) (12)

and have provided the solution of (12) as follows

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896

Γ (]119896 + 1)(119888119905)

]119896 (13)

Further Saxena and Kalla [17] considered the followingfractional kinetic equation

119873(119905) minus 1198730119891 (119905) = minus119888](0119863minus]119905119873) (119905) (R (]) gt 0)

(14)

where119873(119905) denotes the number density of a given species attime 119905 1198730 = 119873(0) is the number density of that species attime 119905 = 0 119888 is a constant and 119891 isin 119871(0infin)

By applying the Laplace transform to (14) we have

119871 [119873 (119905)] (119901)

= 1198730

119865 (119901)

1 + 119888]119901minus]

= 1198730 (

infin

sum

119899=0

(minus119888])119899119901minus119899]

)119865 (119901) (119899 isin 1198730

10038161003816100381610038161003816100381610038161003816

119888

119901

10038161003816100381610038161003816100381610038161003816

lt 1)

(15)

where the Laplace transform [24] is defined by

119865 (119901) = 119871 [119891 (119905)] = int

infin

0

119890minus119901119905

119891 (119905) 119889119905 R (119901) gt 0 (16)

The aim of this paper is to develop a new and furthergeneralized form of the fractional kinetic equation involvinggeneralized Bessel function of the first kind The manifoldgenerality of the generalized Bessel function of the first kindis discussed in terms of the solution of the above fractionalkinetic equation Moreover the results obtained here arequite capable of yielding a very large number of known and(presumably) new results

2 Solution of Generalized FractionalKinetic Equations

In this section we will investigate the solution of the general-ized fractional kinetic equations The results are as follows

Mathematical Problems in Engineering 3

Theorem 1 If 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C andR(119897) gt minus1 thenfor the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119905) = minus119889]0119863minus]119905119873(119905) (17)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(18)

where 119864]2119896+119897+1(sdot) is the generalized Mittag-Leffler function[25]

Proof The Laplace transform of the Riemann-Liouville frac-tional integral operator is given by [26 27]

119871 0119863minus]119905119891 (119905) 119901 = 119901

minus]119865 (119901) (19)

where 119865(119901) is defined in (16) Now applying the Laplacetransform to both sides of (17) we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119905) 119901] minus 119889]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901)

= 1198730 int

infin

0

119890minus119901119905infin

sum

119896=0

(minus119888)119896

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

minus 119889]119901minus]119873(119901)

119873 (119901) [1 + 119889]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

119896Γ (119897 + 119896 + (119887 + 1) 2)

int

infin

0

119890minus119901119905

1199052119896+119897

119889119905

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896 + 119897 + 1)

1199012119896+119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times

119901minus(2119896+119897+1)

infin

sum

119903=0

(1)119903 [minus (119901119889)minus]]

119903

(119903)

(20)

Taking Laplace inverse of (20) and using 119871minus1119901minus] = 119905]minus1

Γ(])R(]) gt 0 we have

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119889]119903119901minus(2119896+119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

(2119896+119897+]119903)

Γ (]119903 + 2119896 + 119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

1199052119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896 + 119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896 + 119897 + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(21)

This completes the proof of Theorem 1

If we set 119887 = 119888 = 1 in (17) then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) given by (2) and we arrive at the following result

Corollary 2 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119905) = minus119889]0119863minus]119905119873(119905) (22)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(23)

4 Mathematical Problems in Engineering

Further taking 119887 = 1 and 119888 = minus1 in (17) then weobtain result of generalized fractional kinetic equation havingmodified Bessel function of the first kind

Corollary 3 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119905) = minus119889]0119863minus]119905119873(119905) (24)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(25)

Letting 119887 = 2 and 119888 = 1 in (17) then generalized Besselfunction 120596119897119887119888(119911) reduces to the spherical Bessel function ofthe first kind 119895119897(119911) given by (5) and we obtain the followinginteresting result

Corollary 4 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119905) = minus119889]0119863minus]119905119873(119905) (26)

there holds the solution of (22)

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(27)

Theorem 5 If 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C andR(119897) gt minus1 thenfor the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (28)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(29)

where 119864]2119896]+]119897+1(sdot) is the generalized Mittag-Leffler function

Proof The Laplace transform of the Riemann-Liouville frac-tional integral operator is given by [26]

119871 0119863minus]119905119891 (119905) 119901 = 119901

minus]119865 (119901) (30)

where 119865(119901) is defined in (16) Now applying the Laplacetransform to both sides of (28) we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119889

]119871 [0119863minus]119905119873(119905) 119901]

(31)

119873(119901)

= 1198730 int

infin

0

119890minus119901119905infin

sum

119896=0

(minus119888)119896

119896Γ (119897 + 119896 + (119887 + 1) 2)

times (

119889]119905]

2

)

2119896+119897

minus 119889]119901minus]119873(119901)

119873 (119901) [1 + 119889]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

int

infin

0

119890minus119901119905

1199052119896]+]119897+1minus1

119889119905

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 [minus (119901119889)minus]]

119903

(119903)

(32)

Taking Laplace inverse of (32) andusing119871minus1119901minus] = 119905]minus1

Γ(])R(]) gt 0 we have

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119889]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

](2119896+119897+119903)

Γ (]119903 + 2119896] + ]119897 + 1)

Mathematical Problems in Engineering 5

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

119905](2119896+119897)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896] + ]119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(33)

This completes the proof of Theorem 5

If we set 119887 = 119888 = 1 in Theorem 5 then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) and we arrive at the special case of (28)

Corollary 6 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (34)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(35)

On taking 119887 = 1 and 119888 = minus1 in (28) then generalizedBessel function 120596119897119887119888(119911) reduces to Bessel function of the firstkind 119869119897(119911) and we get the following result

Corollary 7 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (36)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(37)

Further if we put 119887 = 2 and 119888 = 1 in (28) then we arriveat the following interesting result

Corollary 8 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (38)

the following result holds

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(39)

where 119895119897(119911) is the spherical Bessel function of the first kind

Theorem 9 If 119886 gt 0 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C 119886 = 119889 andR(119897) gt minus1 then for the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119886

]0119863minus]119905119873(119905) (40)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(41)

Proof Applying the Laplace transform to both sides of (40)we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119886

]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901) [1 + 119886]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 (minus1)119903119901minus]119903119886]119903

(119903)

(42)

6 Mathematical Problems in Engineering

Taking Laplace inverse of (42) we arrive at

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119886]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119886]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(43)

This completes the proof of Theorem 9

Remark 10 The special cases forTheorem 9 can be developedon similar lines to that of Corollaries 6ndash8 but we do not statehere due to lack of space

3 Conclusion

In this paper we have studied a new fractional generalizationof the standard kinetic equation and derived solutions for itIt is not difficult to obtain several further analogous fractionalkinetic equations and their solutions as those exhibited hereby main results Moreover by the use of close relationshipsof the generalized Bessel function of the first kind 120596119901(119911)

with many special functions we can easily construct variousknown and new fractional kinetic equations

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for the very carefulreading and the valuable suggestions

References

[1] A Baricz Generalized Bessel Functions of the First Kind vol1994 of Lecture Notes in Mathematics Springer Berlin Ger-many 2010

[2] A Baricz ldquoGeometric properties of generalized Bessel func-tionsrdquo Publicationes Mathematicae Debrecen vol 73 no 1-2 pp155ndash178 2008

[3] J Choi P Agarwal S Mathur and S D Purohit ldquoCertain newintegral formulas involving the generalized Bessel functionsrdquoBulletin of the KoreanMathematical Society vol 51 no 4 ArticleID 9951003 2014

[4] P Malik S R Mondal and A Swaminathan ldquoFractional inte-gration of generalized bessel function of the first kindrdquo in Pro-ceedings of the ASME International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference (IDETCCIE rsquo11) pp 409ndash418 Washington DCUSA August 2011

[5] S RMondal andK SNisar ldquoMarichev-Saigo-Maeda fractionalintegration operators involving generalized Bessel functionsrdquoMathematical Problems in Engineering vol 2014 Article ID274093 11 pages 2014

[6] D Baleanu P Agarwal and S D Purohit ldquoCertain fractionalintegral formulas involving the product of generalized Besselfunctionsrdquo The Scientific World Journal vol 2013 Article ID567132 9 pages 2013

[7] G N Watson A Treatise on the Theory of Bessel FunctionsCambridge University Press The Macmillan Cambridge UK1944

[8] J K Prajapat ldquoCertain geometric properties of normalizedBessel functionsrdquo Applied Mathematics Letters vol 24 no 12pp 2133ndash2139 2011

[9] L Galue ldquoA generalized Bessel functionrdquo Integral Transformsand Special Functions vol 14 no 5 pp 395ndash401 2003

[10] E Deniz H Orhan andHM Srivastava ldquoSome sufficient con-ditions for univalence of certain families of integral operatorsinvolving generalized Bessel functionsrdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 883ndash917 2011

[11] G M Zaslavsky ldquoFractional kinetic equation for Hamiltonianchaosrdquo Physica D vol 76 no 1ndash3 pp 110ndash122 1994

[12] A I Saichev andGM Zaslavsky ldquoFractional kinetic equationssolutions and applicationsrdquo Chaos vol 7 no 4 pp 753ndash7641997

[13] H J Haubold and A M Mathai ldquoThe fractional kinetic equa-tion and thermonuclear functionsrdquo Astrophysics and Space Sci-ence vol 273 no 1ndash4 pp 53ndash63 2000

[14] R K Saxena A M Mathai and H J Haubold ldquoOn fractionalkinetic equationsrdquo Astrophysics and Space Science vol 282 no1 pp 281ndash287 2002

[15] R K Saxena A M Mathai and H J Haubold ldquoOn generalizedfractional kinetic equationsrdquo Physica A vol 344 no 3-4 pp657ndash664 2004

[16] R K Saxena AMMathai andH J Haubold ldquoSolution of gen-eralized fractional reaction-diffusion equationsrdquo Astrophysicsand Space Science vol 305 no 3 pp 305ndash313 2006

[17] R K Saxena and S L Kalla ldquoOn the solutions of certain frac-tional kinetic equationsrdquo Applied Mathematics and Computa-tion vol 199 no 2 pp 504ndash511 2008

[18] V B L Chaurasia and S C Pandey ldquoOn the new computablesolution of the generalized fractional kinetic equations involv-ing the generalized function for the fractional calculus and

Mathematical Problems in Engineering 7

related functionsrdquo Astrophysics and Space Science vol 317 no3-4 pp 213ndash219 2008

[19] V G Gupta B Sharma and F B M Belgacem ldquoOn the solu-tions of generalized fractional kinetic equationsrdquoAppliedMath-ematical Sciences vol 5 no 19 pp 899ndash910 2011

[20] A Chouhan and S Sarswat ldquoOn solution of generalized Kineticequation of fractional orderrdquo International journal ofMathemat-ical Sciences and Applications vol 2 no 2 pp 813ndash818 2012

[21] A Chouhan S D Purohit and S Saraswat ldquoAn alternativemethod for solving generalized differential equations of frac-tional orderrdquo Kragujevac Journal of Mathematics vol 37 no 2pp 299ndash306 2013

[22] A Gupta and C L Parihar ldquoOn solutions of generalized kineticequations of fractional orderrdquo Boletim da Sociedade Paranaensede Matematica vol 32 no 1 pp 181ndash189 2014

[23] R K Saxena and D Kumar ldquoSolution of fractional kineticequation associated with aleph functionrdquo Submitted

[24] M R Spiegel Theory and Problems of Laplace TransformsSchaums Outline Series McGraw-Hill New York NY USA1965

[25] G Mittag-Leffler ldquoSur la representation analytique drsquounebranche uniforme drsquoune fonction monogenerdquo Acta Mathemat-ica vol 29 no 1 pp 101ndash181 1905

[26] A Erdelyi W Magnus F Oberhettinger and F G TricomiTables of Integral Transforms vol 1 McGraw-Hill New YorkNY USA 1954

[27] HM Srivastava andR K Saxena ldquoOperators of fractional inte-gration and their applicationsrdquo Applied Mathematics and Com-putation vol 118 no 1 pp 1ndash52 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

(iii) Letting 119887 = 2 and 119888 = 1 in series (1) we have thespherical Bessel function of the first kind of order 119901defined by (see [1])

119895119901 (119911) =radic

120587

2

infin

sum

119896=0

(minus1)119896

119896Γ (119901 + 119896 + 32)

(

119911

2

)

2119896+119901

119911 isin C

(5)

Furthermore Deniz et al [10] considered the function120593119901119887119888(119911) defined in terms of the generalized Bessel function120596119901(119911) by the transformation

120593119901119887119888 (119911) = 2119901Γ(119901 +

119887 + 1

2

) 1199111minus1199012

120596119901 (radic119911)

= 119911 +

infin

sum

119896=1

(minus119888)119896

4119896(])119896

119911119896+1

119896

(6)

where ] = 119901+ (119887+ 1)2 notin 119885minus

0= 0 minus1 minus2 and (119886)119896 is the

Pochhammer symbol defined (for 119886 isin C) by

(119886)119896 =

1 (119896 = 0)

119886 (119886 + 1) sdot sdot sdot (119886 + 119896 minus 1) (119896 isin 119873 = 1 2 3 )

=

Γ (119886 + 119896)

Γ (119886)

(119886 isin C 119885minus

0)

(7)

Fractional differential equations appear more and morefrequently for modeling of relevant systems in several fieldsof applied sciencesThese equations play important roles notonly in mathematics but also in physics dynamical systemscontrol systems and engineering to create the mathematicalmodel ofmany physical phenomena In particular the kineticequations describe the continuity of motion of substance andare the basic equations of mathematical physics and naturalscience Therefore in literature we found several papers thatanalyze extensions and generalizations of these equationsinvolving various fractional calculus operators One may forinstance refer to such type of works by [11ndash23]

Haubold and Mathai [13] have established a functionaldifferential equation between rate of change of reaction thedestruction rate and the production rate as follows

119889119873

119889119905

= minus119889 (119873119905) + 119901 (119873119905) (8)

where 119873 = 119873(119905) is the rate of reaction 119889 = 119889(119873) is the rateof destruction 119901 = 119901(119873) is the rate of production and 119873119905

denotes the function defined by119873119905(119905lowast) = 119873(119905 minus 119905

lowast) 119905lowast gt 0

Haubold and Mathai studied a special case of (8) whenspatial fluctuations or inhomogeneities in the quantity 119873(119905)

are neglected is given by the equation

119889119873119894

119889119905

= minus119888119894119873119894 (119905) (9)

together with the initial condition that 119873119894(119905 = 0) = 1198730 isthe number of density of species 119894 at time 119905 = 0 119888119894 gt 0 If we

decline the index 119894 and integrate the standard kinetic equation(9) we have

119873(119905) minus 1198730 = minus1198880119863minus1

119905119873(119905) (10)

where0119863minus1

119905is the special case of the Riemann-Liouville

integral operator0119863minus]119905

defined as

0119863minus]119905119891 (119905) =

1

Γ (])int

119905

0

(119905 minus 119904)]minus1

119891 (119904) 119889119904 119905 gt 0 R (]) gt 0

(11)

Haubold and Mathai [13] have given the fractional general-ization of the standard kinetic equation (10) as

119873(119905) minus 1198730 = minus119888]0119863minus]119905119873(119905) (12)

and have provided the solution of (12) as follows

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896

Γ (]119896 + 1)(119888119905)

]119896 (13)

Further Saxena and Kalla [17] considered the followingfractional kinetic equation

119873(119905) minus 1198730119891 (119905) = minus119888](0119863minus]119905119873) (119905) (R (]) gt 0)

(14)

where119873(119905) denotes the number density of a given species attime 119905 1198730 = 119873(0) is the number density of that species attime 119905 = 0 119888 is a constant and 119891 isin 119871(0infin)

By applying the Laplace transform to (14) we have

119871 [119873 (119905)] (119901)

= 1198730

119865 (119901)

1 + 119888]119901minus]

= 1198730 (

infin

sum

119899=0

(minus119888])119899119901minus119899]

)119865 (119901) (119899 isin 1198730

10038161003816100381610038161003816100381610038161003816

119888

119901

10038161003816100381610038161003816100381610038161003816

lt 1)

(15)

where the Laplace transform [24] is defined by

119865 (119901) = 119871 [119891 (119905)] = int

infin

0

119890minus119901119905

119891 (119905) 119889119905 R (119901) gt 0 (16)

The aim of this paper is to develop a new and furthergeneralized form of the fractional kinetic equation involvinggeneralized Bessel function of the first kind The manifoldgenerality of the generalized Bessel function of the first kindis discussed in terms of the solution of the above fractionalkinetic equation Moreover the results obtained here arequite capable of yielding a very large number of known and(presumably) new results

2 Solution of Generalized FractionalKinetic Equations

In this section we will investigate the solution of the general-ized fractional kinetic equations The results are as follows

Mathematical Problems in Engineering 3

Theorem 1 If 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C andR(119897) gt minus1 thenfor the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119905) = minus119889]0119863minus]119905119873(119905) (17)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(18)

where 119864]2119896+119897+1(sdot) is the generalized Mittag-Leffler function[25]

Proof The Laplace transform of the Riemann-Liouville frac-tional integral operator is given by [26 27]

119871 0119863minus]119905119891 (119905) 119901 = 119901

minus]119865 (119901) (19)

where 119865(119901) is defined in (16) Now applying the Laplacetransform to both sides of (17) we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119905) 119901] minus 119889]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901)

= 1198730 int

infin

0

119890minus119901119905infin

sum

119896=0

(minus119888)119896

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

minus 119889]119901minus]119873(119901)

119873 (119901) [1 + 119889]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

119896Γ (119897 + 119896 + (119887 + 1) 2)

int

infin

0

119890minus119901119905

1199052119896+119897

119889119905

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896 + 119897 + 1)

1199012119896+119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times

119901minus(2119896+119897+1)

infin

sum

119903=0

(1)119903 [minus (119901119889)minus]]

119903

(119903)

(20)

Taking Laplace inverse of (20) and using 119871minus1119901minus] = 119905]minus1

Γ(])R(]) gt 0 we have

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119889]119903119901minus(2119896+119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

(2119896+119897+]119903)

Γ (]119903 + 2119896 + 119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

1199052119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896 + 119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896 + 119897 + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(21)

This completes the proof of Theorem 1

If we set 119887 = 119888 = 1 in (17) then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) given by (2) and we arrive at the following result

Corollary 2 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119905) = minus119889]0119863minus]119905119873(119905) (22)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(23)

4 Mathematical Problems in Engineering

Further taking 119887 = 1 and 119888 = minus1 in (17) then weobtain result of generalized fractional kinetic equation havingmodified Bessel function of the first kind

Corollary 3 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119905) = minus119889]0119863minus]119905119873(119905) (24)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(25)

Letting 119887 = 2 and 119888 = 1 in (17) then generalized Besselfunction 120596119897119887119888(119911) reduces to the spherical Bessel function ofthe first kind 119895119897(119911) given by (5) and we obtain the followinginteresting result

Corollary 4 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119905) = minus119889]0119863minus]119905119873(119905) (26)

there holds the solution of (22)

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(27)

Theorem 5 If 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C andR(119897) gt minus1 thenfor the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (28)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(29)

where 119864]2119896]+]119897+1(sdot) is the generalized Mittag-Leffler function

Proof The Laplace transform of the Riemann-Liouville frac-tional integral operator is given by [26]

119871 0119863minus]119905119891 (119905) 119901 = 119901

minus]119865 (119901) (30)

where 119865(119901) is defined in (16) Now applying the Laplacetransform to both sides of (28) we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119889

]119871 [0119863minus]119905119873(119905) 119901]

(31)

119873(119901)

= 1198730 int

infin

0

119890minus119901119905infin

sum

119896=0

(minus119888)119896

119896Γ (119897 + 119896 + (119887 + 1) 2)

times (

119889]119905]

2

)

2119896+119897

minus 119889]119901minus]119873(119901)

119873 (119901) [1 + 119889]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

int

infin

0

119890minus119901119905

1199052119896]+]119897+1minus1

119889119905

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 [minus (119901119889)minus]]

119903

(119903)

(32)

Taking Laplace inverse of (32) andusing119871minus1119901minus] = 119905]minus1

Γ(])R(]) gt 0 we have

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119889]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

](2119896+119897+119903)

Γ (]119903 + 2119896] + ]119897 + 1)

Mathematical Problems in Engineering 5

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

119905](2119896+119897)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896] + ]119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(33)

This completes the proof of Theorem 5

If we set 119887 = 119888 = 1 in Theorem 5 then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) and we arrive at the special case of (28)

Corollary 6 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (34)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(35)

On taking 119887 = 1 and 119888 = minus1 in (28) then generalizedBessel function 120596119897119887119888(119911) reduces to Bessel function of the firstkind 119869119897(119911) and we get the following result

Corollary 7 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (36)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(37)

Further if we put 119887 = 2 and 119888 = 1 in (28) then we arriveat the following interesting result

Corollary 8 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (38)

the following result holds

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(39)

where 119895119897(119911) is the spherical Bessel function of the first kind

Theorem 9 If 119886 gt 0 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C 119886 = 119889 andR(119897) gt minus1 then for the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119886

]0119863minus]119905119873(119905) (40)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(41)

Proof Applying the Laplace transform to both sides of (40)we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119886

]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901) [1 + 119886]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 (minus1)119903119901minus]119903119886]119903

(119903)

(42)

6 Mathematical Problems in Engineering

Taking Laplace inverse of (42) we arrive at

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119886]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119886]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(43)

This completes the proof of Theorem 9

Remark 10 The special cases forTheorem 9 can be developedon similar lines to that of Corollaries 6ndash8 but we do not statehere due to lack of space

3 Conclusion

In this paper we have studied a new fractional generalizationof the standard kinetic equation and derived solutions for itIt is not difficult to obtain several further analogous fractionalkinetic equations and their solutions as those exhibited hereby main results Moreover by the use of close relationshipsof the generalized Bessel function of the first kind 120596119901(119911)

with many special functions we can easily construct variousknown and new fractional kinetic equations

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for the very carefulreading and the valuable suggestions

References

[1] A Baricz Generalized Bessel Functions of the First Kind vol1994 of Lecture Notes in Mathematics Springer Berlin Ger-many 2010

[2] A Baricz ldquoGeometric properties of generalized Bessel func-tionsrdquo Publicationes Mathematicae Debrecen vol 73 no 1-2 pp155ndash178 2008

[3] J Choi P Agarwal S Mathur and S D Purohit ldquoCertain newintegral formulas involving the generalized Bessel functionsrdquoBulletin of the KoreanMathematical Society vol 51 no 4 ArticleID 9951003 2014

[4] P Malik S R Mondal and A Swaminathan ldquoFractional inte-gration of generalized bessel function of the first kindrdquo in Pro-ceedings of the ASME International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference (IDETCCIE rsquo11) pp 409ndash418 Washington DCUSA August 2011

[5] S RMondal andK SNisar ldquoMarichev-Saigo-Maeda fractionalintegration operators involving generalized Bessel functionsrdquoMathematical Problems in Engineering vol 2014 Article ID274093 11 pages 2014

[6] D Baleanu P Agarwal and S D Purohit ldquoCertain fractionalintegral formulas involving the product of generalized Besselfunctionsrdquo The Scientific World Journal vol 2013 Article ID567132 9 pages 2013

[7] G N Watson A Treatise on the Theory of Bessel FunctionsCambridge University Press The Macmillan Cambridge UK1944

[8] J K Prajapat ldquoCertain geometric properties of normalizedBessel functionsrdquo Applied Mathematics Letters vol 24 no 12pp 2133ndash2139 2011

[9] L Galue ldquoA generalized Bessel functionrdquo Integral Transformsand Special Functions vol 14 no 5 pp 395ndash401 2003

[10] E Deniz H Orhan andHM Srivastava ldquoSome sufficient con-ditions for univalence of certain families of integral operatorsinvolving generalized Bessel functionsrdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 883ndash917 2011

[11] G M Zaslavsky ldquoFractional kinetic equation for Hamiltonianchaosrdquo Physica D vol 76 no 1ndash3 pp 110ndash122 1994

[12] A I Saichev andGM Zaslavsky ldquoFractional kinetic equationssolutions and applicationsrdquo Chaos vol 7 no 4 pp 753ndash7641997

[13] H J Haubold and A M Mathai ldquoThe fractional kinetic equa-tion and thermonuclear functionsrdquo Astrophysics and Space Sci-ence vol 273 no 1ndash4 pp 53ndash63 2000

[14] R K Saxena A M Mathai and H J Haubold ldquoOn fractionalkinetic equationsrdquo Astrophysics and Space Science vol 282 no1 pp 281ndash287 2002

[15] R K Saxena A M Mathai and H J Haubold ldquoOn generalizedfractional kinetic equationsrdquo Physica A vol 344 no 3-4 pp657ndash664 2004

[16] R K Saxena AMMathai andH J Haubold ldquoSolution of gen-eralized fractional reaction-diffusion equationsrdquo Astrophysicsand Space Science vol 305 no 3 pp 305ndash313 2006

[17] R K Saxena and S L Kalla ldquoOn the solutions of certain frac-tional kinetic equationsrdquo Applied Mathematics and Computa-tion vol 199 no 2 pp 504ndash511 2008

[18] V B L Chaurasia and S C Pandey ldquoOn the new computablesolution of the generalized fractional kinetic equations involv-ing the generalized function for the fractional calculus and

Mathematical Problems in Engineering 7

related functionsrdquo Astrophysics and Space Science vol 317 no3-4 pp 213ndash219 2008

[19] V G Gupta B Sharma and F B M Belgacem ldquoOn the solu-tions of generalized fractional kinetic equationsrdquoAppliedMath-ematical Sciences vol 5 no 19 pp 899ndash910 2011

[20] A Chouhan and S Sarswat ldquoOn solution of generalized Kineticequation of fractional orderrdquo International journal ofMathemat-ical Sciences and Applications vol 2 no 2 pp 813ndash818 2012

[21] A Chouhan S D Purohit and S Saraswat ldquoAn alternativemethod for solving generalized differential equations of frac-tional orderrdquo Kragujevac Journal of Mathematics vol 37 no 2pp 299ndash306 2013

[22] A Gupta and C L Parihar ldquoOn solutions of generalized kineticequations of fractional orderrdquo Boletim da Sociedade Paranaensede Matematica vol 32 no 1 pp 181ndash189 2014

[23] R K Saxena and D Kumar ldquoSolution of fractional kineticequation associated with aleph functionrdquo Submitted

[24] M R Spiegel Theory and Problems of Laplace TransformsSchaums Outline Series McGraw-Hill New York NY USA1965

[25] G Mittag-Leffler ldquoSur la representation analytique drsquounebranche uniforme drsquoune fonction monogenerdquo Acta Mathemat-ica vol 29 no 1 pp 101ndash181 1905

[26] A Erdelyi W Magnus F Oberhettinger and F G TricomiTables of Integral Transforms vol 1 McGraw-Hill New YorkNY USA 1954

[27] HM Srivastava andR K Saxena ldquoOperators of fractional inte-gration and their applicationsrdquo Applied Mathematics and Com-putation vol 118 no 1 pp 1ndash52 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Theorem 1 If 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C andR(119897) gt minus1 thenfor the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119905) = minus119889]0119863minus]119905119873(119905) (17)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(18)

where 119864]2119896+119897+1(sdot) is the generalized Mittag-Leffler function[25]

Proof The Laplace transform of the Riemann-Liouville frac-tional integral operator is given by [26 27]

119871 0119863minus]119905119891 (119905) 119901 = 119901

minus]119865 (119901) (19)

where 119865(119901) is defined in (16) Now applying the Laplacetransform to both sides of (17) we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119905) 119901] minus 119889]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901)

= 1198730 int

infin

0

119890minus119901119905infin

sum

119896=0

(minus119888)119896

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

minus 119889]119901minus]119873(119901)

119873 (119901) [1 + 119889]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

119896Γ (119897 + 119896 + (119887 + 1) 2)

int

infin

0

119890minus119901119905

1199052119896+119897

119889119905

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896 + 119897 + 1)

1199012119896+119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times

119901minus(2119896+119897+1)

infin

sum

119903=0

(1)119903 [minus (119901119889)minus]]

119903

(119903)

(20)

Taking Laplace inverse of (20) and using 119871minus1119901minus] = 119905]minus1

Γ(])R(]) gt 0 we have

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119889]119903119901minus(2119896+119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

(2119896+119897+]119903)

Γ (]119903 + 2119896 + 119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)1198962minus(2119896+119897)

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

1199052119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896 + 119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896 + 119897 + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(21)

This completes the proof of Theorem 1

If we set 119887 = 119888 = 1 in (17) then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) given by (2) and we arrive at the following result

Corollary 2 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119905) = minus119889]0119863minus]119905119873(119905) (22)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(23)

4 Mathematical Problems in Engineering

Further taking 119887 = 1 and 119888 = minus1 in (17) then weobtain result of generalized fractional kinetic equation havingmodified Bessel function of the first kind

Corollary 3 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119905) = minus119889]0119863minus]119905119873(119905) (24)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(25)

Letting 119887 = 2 and 119888 = 1 in (17) then generalized Besselfunction 120596119897119887119888(119911) reduces to the spherical Bessel function ofthe first kind 119895119897(119911) given by (5) and we obtain the followinginteresting result

Corollary 4 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119905) = minus119889]0119863minus]119905119873(119905) (26)

there holds the solution of (22)

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(27)

Theorem 5 If 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C andR(119897) gt minus1 thenfor the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (28)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(29)

where 119864]2119896]+]119897+1(sdot) is the generalized Mittag-Leffler function

Proof The Laplace transform of the Riemann-Liouville frac-tional integral operator is given by [26]

119871 0119863minus]119905119891 (119905) 119901 = 119901

minus]119865 (119901) (30)

where 119865(119901) is defined in (16) Now applying the Laplacetransform to both sides of (28) we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119889

]119871 [0119863minus]119905119873(119905) 119901]

(31)

119873(119901)

= 1198730 int

infin

0

119890minus119901119905infin

sum

119896=0

(minus119888)119896

119896Γ (119897 + 119896 + (119887 + 1) 2)

times (

119889]119905]

2

)

2119896+119897

minus 119889]119901minus]119873(119901)

119873 (119901) [1 + 119889]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

int

infin

0

119890minus119901119905

1199052119896]+]119897+1minus1

119889119905

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 [minus (119901119889)minus]]

119903

(119903)

(32)

Taking Laplace inverse of (32) andusing119871minus1119901minus] = 119905]minus1

Γ(])R(]) gt 0 we have

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119889]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

](2119896+119897+119903)

Γ (]119903 + 2119896] + ]119897 + 1)

Mathematical Problems in Engineering 5

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

119905](2119896+119897)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896] + ]119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(33)

This completes the proof of Theorem 5

If we set 119887 = 119888 = 1 in Theorem 5 then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) and we arrive at the special case of (28)

Corollary 6 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (34)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(35)

On taking 119887 = 1 and 119888 = minus1 in (28) then generalizedBessel function 120596119897119887119888(119911) reduces to Bessel function of the firstkind 119869119897(119911) and we get the following result

Corollary 7 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (36)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(37)

Further if we put 119887 = 2 and 119888 = 1 in (28) then we arriveat the following interesting result

Corollary 8 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (38)

the following result holds

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(39)

where 119895119897(119911) is the spherical Bessel function of the first kind

Theorem 9 If 119886 gt 0 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C 119886 = 119889 andR(119897) gt minus1 then for the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119886

]0119863minus]119905119873(119905) (40)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(41)

Proof Applying the Laplace transform to both sides of (40)we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119886

]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901) [1 + 119886]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 (minus1)119903119901minus]119903119886]119903

(119903)

(42)

6 Mathematical Problems in Engineering

Taking Laplace inverse of (42) we arrive at

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119886]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119886]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(43)

This completes the proof of Theorem 9

Remark 10 The special cases forTheorem 9 can be developedon similar lines to that of Corollaries 6ndash8 but we do not statehere due to lack of space

3 Conclusion

In this paper we have studied a new fractional generalizationof the standard kinetic equation and derived solutions for itIt is not difficult to obtain several further analogous fractionalkinetic equations and their solutions as those exhibited hereby main results Moreover by the use of close relationshipsof the generalized Bessel function of the first kind 120596119901(119911)

with many special functions we can easily construct variousknown and new fractional kinetic equations

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for the very carefulreading and the valuable suggestions

References

[1] A Baricz Generalized Bessel Functions of the First Kind vol1994 of Lecture Notes in Mathematics Springer Berlin Ger-many 2010

[2] A Baricz ldquoGeometric properties of generalized Bessel func-tionsrdquo Publicationes Mathematicae Debrecen vol 73 no 1-2 pp155ndash178 2008

[3] J Choi P Agarwal S Mathur and S D Purohit ldquoCertain newintegral formulas involving the generalized Bessel functionsrdquoBulletin of the KoreanMathematical Society vol 51 no 4 ArticleID 9951003 2014

[4] P Malik S R Mondal and A Swaminathan ldquoFractional inte-gration of generalized bessel function of the first kindrdquo in Pro-ceedings of the ASME International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference (IDETCCIE rsquo11) pp 409ndash418 Washington DCUSA August 2011

[5] S RMondal andK SNisar ldquoMarichev-Saigo-Maeda fractionalintegration operators involving generalized Bessel functionsrdquoMathematical Problems in Engineering vol 2014 Article ID274093 11 pages 2014

[6] D Baleanu P Agarwal and S D Purohit ldquoCertain fractionalintegral formulas involving the product of generalized Besselfunctionsrdquo The Scientific World Journal vol 2013 Article ID567132 9 pages 2013

[7] G N Watson A Treatise on the Theory of Bessel FunctionsCambridge University Press The Macmillan Cambridge UK1944

[8] J K Prajapat ldquoCertain geometric properties of normalizedBessel functionsrdquo Applied Mathematics Letters vol 24 no 12pp 2133ndash2139 2011

[9] L Galue ldquoA generalized Bessel functionrdquo Integral Transformsand Special Functions vol 14 no 5 pp 395ndash401 2003

[10] E Deniz H Orhan andHM Srivastava ldquoSome sufficient con-ditions for univalence of certain families of integral operatorsinvolving generalized Bessel functionsrdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 883ndash917 2011

[11] G M Zaslavsky ldquoFractional kinetic equation for Hamiltonianchaosrdquo Physica D vol 76 no 1ndash3 pp 110ndash122 1994

[12] A I Saichev andGM Zaslavsky ldquoFractional kinetic equationssolutions and applicationsrdquo Chaos vol 7 no 4 pp 753ndash7641997

[13] H J Haubold and A M Mathai ldquoThe fractional kinetic equa-tion and thermonuclear functionsrdquo Astrophysics and Space Sci-ence vol 273 no 1ndash4 pp 53ndash63 2000

[14] R K Saxena A M Mathai and H J Haubold ldquoOn fractionalkinetic equationsrdquo Astrophysics and Space Science vol 282 no1 pp 281ndash287 2002

[15] R K Saxena A M Mathai and H J Haubold ldquoOn generalizedfractional kinetic equationsrdquo Physica A vol 344 no 3-4 pp657ndash664 2004

[16] R K Saxena AMMathai andH J Haubold ldquoSolution of gen-eralized fractional reaction-diffusion equationsrdquo Astrophysicsand Space Science vol 305 no 3 pp 305ndash313 2006

[17] R K Saxena and S L Kalla ldquoOn the solutions of certain frac-tional kinetic equationsrdquo Applied Mathematics and Computa-tion vol 199 no 2 pp 504ndash511 2008

[18] V B L Chaurasia and S C Pandey ldquoOn the new computablesolution of the generalized fractional kinetic equations involv-ing the generalized function for the fractional calculus and

Mathematical Problems in Engineering 7

related functionsrdquo Astrophysics and Space Science vol 317 no3-4 pp 213ndash219 2008

[19] V G Gupta B Sharma and F B M Belgacem ldquoOn the solu-tions of generalized fractional kinetic equationsrdquoAppliedMath-ematical Sciences vol 5 no 19 pp 899ndash910 2011

[20] A Chouhan and S Sarswat ldquoOn solution of generalized Kineticequation of fractional orderrdquo International journal ofMathemat-ical Sciences and Applications vol 2 no 2 pp 813ndash818 2012

[21] A Chouhan S D Purohit and S Saraswat ldquoAn alternativemethod for solving generalized differential equations of frac-tional orderrdquo Kragujevac Journal of Mathematics vol 37 no 2pp 299ndash306 2013

[22] A Gupta and C L Parihar ldquoOn solutions of generalized kineticequations of fractional orderrdquo Boletim da Sociedade Paranaensede Matematica vol 32 no 1 pp 181ndash189 2014

[23] R K Saxena and D Kumar ldquoSolution of fractional kineticequation associated with aleph functionrdquo Submitted

[24] M R Spiegel Theory and Problems of Laplace TransformsSchaums Outline Series McGraw-Hill New York NY USA1965

[25] G Mittag-Leffler ldquoSur la representation analytique drsquounebranche uniforme drsquoune fonction monogenerdquo Acta Mathemat-ica vol 29 no 1 pp 101ndash181 1905

[26] A Erdelyi W Magnus F Oberhettinger and F G TricomiTables of Integral Transforms vol 1 McGraw-Hill New YorkNY USA 1954

[27] HM Srivastava andR K Saxena ldquoOperators of fractional inte-gration and their applicationsrdquo Applied Mathematics and Com-putation vol 118 no 1 pp 1ndash52 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Further taking 119887 = 1 and 119888 = minus1 in (17) then weobtain result of generalized fractional kinetic equation havingmodified Bessel function of the first kind

Corollary 3 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119905) = minus119889]0119863minus]119905119873(119905) (24)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(25)

Letting 119887 = 2 and 119888 = 1 in (17) then generalized Besselfunction 120596119897119887119888(119911) reduces to the spherical Bessel function ofthe first kind 119895119897(119911) given by (5) and we obtain the followinginteresting result

Corollary 4 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119905) = minus119889]0119863minus]119905119873(119905) (26)

there holds the solution of (22)

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896 + 119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119905

2

)

2119896+119897

times 119864]2119896+119897+1 (minus119889]119905])

(27)

Theorem 5 If 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C andR(119897) gt minus1 thenfor the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (28)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(29)

where 119864]2119896]+]119897+1(sdot) is the generalized Mittag-Leffler function

Proof The Laplace transform of the Riemann-Liouville frac-tional integral operator is given by [26]

119871 0119863minus]119905119891 (119905) 119901 = 119901

minus]119865 (119901) (30)

where 119865(119901) is defined in (16) Now applying the Laplacetransform to both sides of (28) we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119889

]119871 [0119863minus]119905119873(119905) 119901]

(31)

119873(119901)

= 1198730 int

infin

0

119890minus119901119905infin

sum

119896=0

(minus119888)119896

119896Γ (119897 + 119896 + (119887 + 1) 2)

times (

119889]119905]

2

)

2119896+119897

minus 119889]119901minus]119873(119901)

119873 (119901) [1 + 119889]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

int

infin

0

119890minus119901119905

1199052119896]+]119897+1minus1

119889119905

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 [minus (119901119889)minus]]

119903

(119903)

(32)

Taking Laplace inverse of (32) andusing119871minus1119901minus] = 119905]minus1

Γ(])R(]) gt 0 we have

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119889]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

](2119896+119897+119903)

Γ (]119903 + 2119896] + ]119897 + 1)

Mathematical Problems in Engineering 5

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

119905](2119896+119897)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896] + ]119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(33)

This completes the proof of Theorem 5

If we set 119887 = 119888 = 1 in Theorem 5 then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) and we arrive at the special case of (28)

Corollary 6 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (34)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(35)

On taking 119887 = 1 and 119888 = minus1 in (28) then generalizedBessel function 120596119897119887119888(119911) reduces to Bessel function of the firstkind 119869119897(119911) and we get the following result

Corollary 7 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (36)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(37)

Further if we put 119887 = 2 and 119888 = 1 in (28) then we arriveat the following interesting result

Corollary 8 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (38)

the following result holds

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(39)

where 119895119897(119911) is the spherical Bessel function of the first kind

Theorem 9 If 119886 gt 0 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C 119886 = 119889 andR(119897) gt minus1 then for the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119886

]0119863minus]119905119873(119905) (40)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(41)

Proof Applying the Laplace transform to both sides of (40)we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119886

]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901) [1 + 119886]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 (minus1)119903119901minus]119903119886]119903

(119903)

(42)

6 Mathematical Problems in Engineering

Taking Laplace inverse of (42) we arrive at

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119886]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119886]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(43)

This completes the proof of Theorem 9

Remark 10 The special cases forTheorem 9 can be developedon similar lines to that of Corollaries 6ndash8 but we do not statehere due to lack of space

3 Conclusion

In this paper we have studied a new fractional generalizationof the standard kinetic equation and derived solutions for itIt is not difficult to obtain several further analogous fractionalkinetic equations and their solutions as those exhibited hereby main results Moreover by the use of close relationshipsof the generalized Bessel function of the first kind 120596119901(119911)

with many special functions we can easily construct variousknown and new fractional kinetic equations

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for the very carefulreading and the valuable suggestions

References

[1] A Baricz Generalized Bessel Functions of the First Kind vol1994 of Lecture Notes in Mathematics Springer Berlin Ger-many 2010

[2] A Baricz ldquoGeometric properties of generalized Bessel func-tionsrdquo Publicationes Mathematicae Debrecen vol 73 no 1-2 pp155ndash178 2008

[3] J Choi P Agarwal S Mathur and S D Purohit ldquoCertain newintegral formulas involving the generalized Bessel functionsrdquoBulletin of the KoreanMathematical Society vol 51 no 4 ArticleID 9951003 2014

[4] P Malik S R Mondal and A Swaminathan ldquoFractional inte-gration of generalized bessel function of the first kindrdquo in Pro-ceedings of the ASME International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference (IDETCCIE rsquo11) pp 409ndash418 Washington DCUSA August 2011

[5] S RMondal andK SNisar ldquoMarichev-Saigo-Maeda fractionalintegration operators involving generalized Bessel functionsrdquoMathematical Problems in Engineering vol 2014 Article ID274093 11 pages 2014

[6] D Baleanu P Agarwal and S D Purohit ldquoCertain fractionalintegral formulas involving the product of generalized Besselfunctionsrdquo The Scientific World Journal vol 2013 Article ID567132 9 pages 2013

[7] G N Watson A Treatise on the Theory of Bessel FunctionsCambridge University Press The Macmillan Cambridge UK1944

[8] J K Prajapat ldquoCertain geometric properties of normalizedBessel functionsrdquo Applied Mathematics Letters vol 24 no 12pp 2133ndash2139 2011

[9] L Galue ldquoA generalized Bessel functionrdquo Integral Transformsand Special Functions vol 14 no 5 pp 395ndash401 2003

[10] E Deniz H Orhan andHM Srivastava ldquoSome sufficient con-ditions for univalence of certain families of integral operatorsinvolving generalized Bessel functionsrdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 883ndash917 2011

[11] G M Zaslavsky ldquoFractional kinetic equation for Hamiltonianchaosrdquo Physica D vol 76 no 1ndash3 pp 110ndash122 1994

[12] A I Saichev andGM Zaslavsky ldquoFractional kinetic equationssolutions and applicationsrdquo Chaos vol 7 no 4 pp 753ndash7641997

[13] H J Haubold and A M Mathai ldquoThe fractional kinetic equa-tion and thermonuclear functionsrdquo Astrophysics and Space Sci-ence vol 273 no 1ndash4 pp 53ndash63 2000

[14] R K Saxena A M Mathai and H J Haubold ldquoOn fractionalkinetic equationsrdquo Astrophysics and Space Science vol 282 no1 pp 281ndash287 2002

[15] R K Saxena A M Mathai and H J Haubold ldquoOn generalizedfractional kinetic equationsrdquo Physica A vol 344 no 3-4 pp657ndash664 2004

[16] R K Saxena AMMathai andH J Haubold ldquoSolution of gen-eralized fractional reaction-diffusion equationsrdquo Astrophysicsand Space Science vol 305 no 3 pp 305ndash313 2006

[17] R K Saxena and S L Kalla ldquoOn the solutions of certain frac-tional kinetic equationsrdquo Applied Mathematics and Computa-tion vol 199 no 2 pp 504ndash511 2008

[18] V B L Chaurasia and S C Pandey ldquoOn the new computablesolution of the generalized fractional kinetic equations involv-ing the generalized function for the fractional calculus and

Mathematical Problems in Engineering 7

related functionsrdquo Astrophysics and Space Science vol 317 no3-4 pp 213ndash219 2008

[19] V G Gupta B Sharma and F B M Belgacem ldquoOn the solu-tions of generalized fractional kinetic equationsrdquoAppliedMath-ematical Sciences vol 5 no 19 pp 899ndash910 2011

[20] A Chouhan and S Sarswat ldquoOn solution of generalized Kineticequation of fractional orderrdquo International journal ofMathemat-ical Sciences and Applications vol 2 no 2 pp 813ndash818 2012

[21] A Chouhan S D Purohit and S Saraswat ldquoAn alternativemethod for solving generalized differential equations of frac-tional orderrdquo Kragujevac Journal of Mathematics vol 37 no 2pp 299ndash306 2013

[22] A Gupta and C L Parihar ldquoOn solutions of generalized kineticequations of fractional orderrdquo Boletim da Sociedade Paranaensede Matematica vol 32 no 1 pp 181ndash189 2014

[23] R K Saxena and D Kumar ldquoSolution of fractional kineticequation associated with aleph functionrdquo Submitted

[24] M R Spiegel Theory and Problems of Laplace TransformsSchaums Outline Series McGraw-Hill New York NY USA1965

[25] G Mittag-Leffler ldquoSur la representation analytique drsquounebranche uniforme drsquoune fonction monogenerdquo Acta Mathemat-ica vol 29 no 1 pp 101ndash181 1905

[26] A Erdelyi W Magnus F Oberhettinger and F G TricomiTables of Integral Transforms vol 1 McGraw-Hill New YorkNY USA 1954

[27] HM Srivastava andR K Saxena ldquoOperators of fractional inte-gration and their applicationsrdquo Applied Mathematics and Com-putation vol 118 no 1 pp 1ndash52 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

119905](2119896+119897)

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (]119903 + 2119896] + ]119897 + 1)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119889]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(33)

This completes the proof of Theorem 5

If we set 119887 = 119888 = 1 in Theorem 5 then generalized Besselfunction 120596119897119887119888(119911) reduces to Bessel function of the first kind119869119897(119911) and we arrive at the special case of (28)

Corollary 6 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119869119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (34)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(35)

On taking 119887 = 1 and 119888 = minus1 in (28) then generalizedBessel function 120596119897119887119888(119911) reduces to Bessel function of the firstkind 119869119897(119911) and we get the following result

Corollary 7 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119868119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (36)

the following result holds

119873(119905) = 1198730

infin

sum

119896=0

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + 1)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(37)

Further if we put 119887 = 2 and 119888 = 1 in (28) then we arriveat the following interesting result

Corollary 8 If 119889 gt 0 ] gt 0 119897 119905 isin C andR(119897) gt minus1 then forthe solution of the equation

119873(119905) minus 1198730119895119897 (119889]119905]) = minus119889

]0119863minus]119905119873(119905) (38)

the following result holds

119873(119905) = 1198730radic

120587

2

infin

sum

119896=0

(minus1)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + 32)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119889]119905])

(39)

where 119895119897(119911) is the spherical Bessel function of the first kind

Theorem 9 If 119886 gt 0 119889 gt 0 ] gt 0 119888 119887 119897 119905 isin C 119886 = 119889 andR(119897) gt minus1 then for the solution of the equation

119873(119905) minus 1198730120596119897119887119888 (119889]119905]) = minus119886

]0119863minus]119905119873(119905) (40)

there holds the formula

119873(119905) = 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(41)

Proof Applying the Laplace transform to both sides of (40)we get

119871 [119873 (119905) 119901]

= 1198730119871 [120596119897119887119888 (119889]119905]) 119901] minus 119886

]119871 [0119863minus]119905119873(119905) 119901]

119873 (119901) [1 + 119886]119901minus]]

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

119896Γ (119897 + 119896 + (119887 + 1) 2)

Γ (2119896] + ]119897 + 1)1199012119896]+]119897+1

119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119901minus(2119896]+]119897+1)

infin

sum

119903=0

(1)119903 (minus1)119903119901minus]119903119886]119903

(119903)

(42)

6 Mathematical Problems in Engineering

Taking Laplace inverse of (42) we arrive at

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119886]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119886]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(43)

This completes the proof of Theorem 9

Remark 10 The special cases forTheorem 9 can be developedon similar lines to that of Corollaries 6ndash8 but we do not statehere due to lack of space

3 Conclusion

In this paper we have studied a new fractional generalizationof the standard kinetic equation and derived solutions for itIt is not difficult to obtain several further analogous fractionalkinetic equations and their solutions as those exhibited hereby main results Moreover by the use of close relationshipsof the generalized Bessel function of the first kind 120596119901(119911)

with many special functions we can easily construct variousknown and new fractional kinetic equations

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for the very carefulreading and the valuable suggestions

References

[1] A Baricz Generalized Bessel Functions of the First Kind vol1994 of Lecture Notes in Mathematics Springer Berlin Ger-many 2010

[2] A Baricz ldquoGeometric properties of generalized Bessel func-tionsrdquo Publicationes Mathematicae Debrecen vol 73 no 1-2 pp155ndash178 2008

[3] J Choi P Agarwal S Mathur and S D Purohit ldquoCertain newintegral formulas involving the generalized Bessel functionsrdquoBulletin of the KoreanMathematical Society vol 51 no 4 ArticleID 9951003 2014

[4] P Malik S R Mondal and A Swaminathan ldquoFractional inte-gration of generalized bessel function of the first kindrdquo in Pro-ceedings of the ASME International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference (IDETCCIE rsquo11) pp 409ndash418 Washington DCUSA August 2011

[5] S RMondal andK SNisar ldquoMarichev-Saigo-Maeda fractionalintegration operators involving generalized Bessel functionsrdquoMathematical Problems in Engineering vol 2014 Article ID274093 11 pages 2014

[6] D Baleanu P Agarwal and S D Purohit ldquoCertain fractionalintegral formulas involving the product of generalized Besselfunctionsrdquo The Scientific World Journal vol 2013 Article ID567132 9 pages 2013

[7] G N Watson A Treatise on the Theory of Bessel FunctionsCambridge University Press The Macmillan Cambridge UK1944

[8] J K Prajapat ldquoCertain geometric properties of normalizedBessel functionsrdquo Applied Mathematics Letters vol 24 no 12pp 2133ndash2139 2011

[9] L Galue ldquoA generalized Bessel functionrdquo Integral Transformsand Special Functions vol 14 no 5 pp 395ndash401 2003

[10] E Deniz H Orhan andHM Srivastava ldquoSome sufficient con-ditions for univalence of certain families of integral operatorsinvolving generalized Bessel functionsrdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 883ndash917 2011

[11] G M Zaslavsky ldquoFractional kinetic equation for Hamiltonianchaosrdquo Physica D vol 76 no 1ndash3 pp 110ndash122 1994

[12] A I Saichev andGM Zaslavsky ldquoFractional kinetic equationssolutions and applicationsrdquo Chaos vol 7 no 4 pp 753ndash7641997

[13] H J Haubold and A M Mathai ldquoThe fractional kinetic equa-tion and thermonuclear functionsrdquo Astrophysics and Space Sci-ence vol 273 no 1ndash4 pp 53ndash63 2000

[14] R K Saxena A M Mathai and H J Haubold ldquoOn fractionalkinetic equationsrdquo Astrophysics and Space Science vol 282 no1 pp 281ndash287 2002

[15] R K Saxena A M Mathai and H J Haubold ldquoOn generalizedfractional kinetic equationsrdquo Physica A vol 344 no 3-4 pp657ndash664 2004

[16] R K Saxena AMMathai andH J Haubold ldquoSolution of gen-eralized fractional reaction-diffusion equationsrdquo Astrophysicsand Space Science vol 305 no 3 pp 305ndash313 2006

[17] R K Saxena and S L Kalla ldquoOn the solutions of certain frac-tional kinetic equationsrdquo Applied Mathematics and Computa-tion vol 199 no 2 pp 504ndash511 2008

[18] V B L Chaurasia and S C Pandey ldquoOn the new computablesolution of the generalized fractional kinetic equations involv-ing the generalized function for the fractional calculus and

Mathematical Problems in Engineering 7

related functionsrdquo Astrophysics and Space Science vol 317 no3-4 pp 213ndash219 2008

[19] V G Gupta B Sharma and F B M Belgacem ldquoOn the solu-tions of generalized fractional kinetic equationsrdquoAppliedMath-ematical Sciences vol 5 no 19 pp 899ndash910 2011

[20] A Chouhan and S Sarswat ldquoOn solution of generalized Kineticequation of fractional orderrdquo International journal ofMathemat-ical Sciences and Applications vol 2 no 2 pp 813ndash818 2012

[21] A Chouhan S D Purohit and S Saraswat ldquoAn alternativemethod for solving generalized differential equations of frac-tional orderrdquo Kragujevac Journal of Mathematics vol 37 no 2pp 299ndash306 2013

[22] A Gupta and C L Parihar ldquoOn solutions of generalized kineticequations of fractional orderrdquo Boletim da Sociedade Paranaensede Matematica vol 32 no 1 pp 181ndash189 2014

[23] R K Saxena and D Kumar ldquoSolution of fractional kineticequation associated with aleph functionrdquo Submitted

[24] M R Spiegel Theory and Problems of Laplace TransformsSchaums Outline Series McGraw-Hill New York NY USA1965

[25] G Mittag-Leffler ldquoSur la representation analytique drsquounebranche uniforme drsquoune fonction monogenerdquo Acta Mathemat-ica vol 29 no 1 pp 101ndash181 1905

[26] A Erdelyi W Magnus F Oberhettinger and F G TricomiTables of Integral Transforms vol 1 McGraw-Hill New YorkNY USA 1954

[27] HM Srivastava andR K Saxena ldquoOperators of fractional inte-gration and their applicationsrdquo Applied Mathematics and Com-putation vol 118 no 1 pp 1ndash52 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Taking Laplace inverse of (42) we arrive at

119871minus1119873 (119901)

= 1198730

infin

sum

119896=0

(minus119888)119896(119889

]2)2119896+119897

Γ (2119896] + ]119897 + 1)119896Γ (119897 + 119896 + (119887 + 1) 2)

times 119871minus1

infin

sum

119903=0

(minus1)119903119886]119903119901minus(2119896]+]119897+]119903+1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times

infin

sum

119903=0

(minus1)119903119886]119903 119905

]119903

Γ (] (119903 + 2119896 + 119897) + 1)

119873 (119905)

= 1198730

infin

sum

119896=0

(minus119888)119896Γ (2119896] + ]119897 + 1)

119896Γ (119897 + 119896 + (119887 + 1) 2)

(

119889]119905]

2

)

2119896+119897

times 119864](2119896+119897)]+1 (minus119886]119905])

(43)

This completes the proof of Theorem 9

Remark 10 The special cases forTheorem 9 can be developedon similar lines to that of Corollaries 6ndash8 but we do not statehere due to lack of space

3 Conclusion

In this paper we have studied a new fractional generalizationof the standard kinetic equation and derived solutions for itIt is not difficult to obtain several further analogous fractionalkinetic equations and their solutions as those exhibited hereby main results Moreover by the use of close relationshipsof the generalized Bessel function of the first kind 120596119901(119911)

with many special functions we can easily construct variousknown and new fractional kinetic equations

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for the very carefulreading and the valuable suggestions

References

[1] A Baricz Generalized Bessel Functions of the First Kind vol1994 of Lecture Notes in Mathematics Springer Berlin Ger-many 2010

[2] A Baricz ldquoGeometric properties of generalized Bessel func-tionsrdquo Publicationes Mathematicae Debrecen vol 73 no 1-2 pp155ndash178 2008

[3] J Choi P Agarwal S Mathur and S D Purohit ldquoCertain newintegral formulas involving the generalized Bessel functionsrdquoBulletin of the KoreanMathematical Society vol 51 no 4 ArticleID 9951003 2014

[4] P Malik S R Mondal and A Swaminathan ldquoFractional inte-gration of generalized bessel function of the first kindrdquo in Pro-ceedings of the ASME International Design Engineering TechnicalConferences and Computers and Information in EngineeringConference (IDETCCIE rsquo11) pp 409ndash418 Washington DCUSA August 2011

[5] S RMondal andK SNisar ldquoMarichev-Saigo-Maeda fractionalintegration operators involving generalized Bessel functionsrdquoMathematical Problems in Engineering vol 2014 Article ID274093 11 pages 2014

[6] D Baleanu P Agarwal and S D Purohit ldquoCertain fractionalintegral formulas involving the product of generalized Besselfunctionsrdquo The Scientific World Journal vol 2013 Article ID567132 9 pages 2013

[7] G N Watson A Treatise on the Theory of Bessel FunctionsCambridge University Press The Macmillan Cambridge UK1944

[8] J K Prajapat ldquoCertain geometric properties of normalizedBessel functionsrdquo Applied Mathematics Letters vol 24 no 12pp 2133ndash2139 2011

[9] L Galue ldquoA generalized Bessel functionrdquo Integral Transformsand Special Functions vol 14 no 5 pp 395ndash401 2003

[10] E Deniz H Orhan andHM Srivastava ldquoSome sufficient con-ditions for univalence of certain families of integral operatorsinvolving generalized Bessel functionsrdquo Taiwanese Journal ofMathematics vol 15 no 2 pp 883ndash917 2011

[11] G M Zaslavsky ldquoFractional kinetic equation for Hamiltonianchaosrdquo Physica D vol 76 no 1ndash3 pp 110ndash122 1994

[12] A I Saichev andGM Zaslavsky ldquoFractional kinetic equationssolutions and applicationsrdquo Chaos vol 7 no 4 pp 753ndash7641997

[13] H J Haubold and A M Mathai ldquoThe fractional kinetic equa-tion and thermonuclear functionsrdquo Astrophysics and Space Sci-ence vol 273 no 1ndash4 pp 53ndash63 2000

[14] R K Saxena A M Mathai and H J Haubold ldquoOn fractionalkinetic equationsrdquo Astrophysics and Space Science vol 282 no1 pp 281ndash287 2002

[15] R K Saxena A M Mathai and H J Haubold ldquoOn generalizedfractional kinetic equationsrdquo Physica A vol 344 no 3-4 pp657ndash664 2004

[16] R K Saxena AMMathai andH J Haubold ldquoSolution of gen-eralized fractional reaction-diffusion equationsrdquo Astrophysicsand Space Science vol 305 no 3 pp 305ndash313 2006

[17] R K Saxena and S L Kalla ldquoOn the solutions of certain frac-tional kinetic equationsrdquo Applied Mathematics and Computa-tion vol 199 no 2 pp 504ndash511 2008

[18] V B L Chaurasia and S C Pandey ldquoOn the new computablesolution of the generalized fractional kinetic equations involv-ing the generalized function for the fractional calculus and

Mathematical Problems in Engineering 7

related functionsrdquo Astrophysics and Space Science vol 317 no3-4 pp 213ndash219 2008

[19] V G Gupta B Sharma and F B M Belgacem ldquoOn the solu-tions of generalized fractional kinetic equationsrdquoAppliedMath-ematical Sciences vol 5 no 19 pp 899ndash910 2011

[20] A Chouhan and S Sarswat ldquoOn solution of generalized Kineticequation of fractional orderrdquo International journal ofMathemat-ical Sciences and Applications vol 2 no 2 pp 813ndash818 2012

[21] A Chouhan S D Purohit and S Saraswat ldquoAn alternativemethod for solving generalized differential equations of frac-tional orderrdquo Kragujevac Journal of Mathematics vol 37 no 2pp 299ndash306 2013

[22] A Gupta and C L Parihar ldquoOn solutions of generalized kineticequations of fractional orderrdquo Boletim da Sociedade Paranaensede Matematica vol 32 no 1 pp 181ndash189 2014

[23] R K Saxena and D Kumar ldquoSolution of fractional kineticequation associated with aleph functionrdquo Submitted

[24] M R Spiegel Theory and Problems of Laplace TransformsSchaums Outline Series McGraw-Hill New York NY USA1965

[25] G Mittag-Leffler ldquoSur la representation analytique drsquounebranche uniforme drsquoune fonction monogenerdquo Acta Mathemat-ica vol 29 no 1 pp 101ndash181 1905

[26] A Erdelyi W Magnus F Oberhettinger and F G TricomiTables of Integral Transforms vol 1 McGraw-Hill New YorkNY USA 1954

[27] HM Srivastava andR K Saxena ldquoOperators of fractional inte-gration and their applicationsrdquo Applied Mathematics and Com-putation vol 118 no 1 pp 1ndash52 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

related functionsrdquo Astrophysics and Space Science vol 317 no3-4 pp 213ndash219 2008

[19] V G Gupta B Sharma and F B M Belgacem ldquoOn the solu-tions of generalized fractional kinetic equationsrdquoAppliedMath-ematical Sciences vol 5 no 19 pp 899ndash910 2011

[20] A Chouhan and S Sarswat ldquoOn solution of generalized Kineticequation of fractional orderrdquo International journal ofMathemat-ical Sciences and Applications vol 2 no 2 pp 813ndash818 2012

[21] A Chouhan S D Purohit and S Saraswat ldquoAn alternativemethod for solving generalized differential equations of frac-tional orderrdquo Kragujevac Journal of Mathematics vol 37 no 2pp 299ndash306 2013

[22] A Gupta and C L Parihar ldquoOn solutions of generalized kineticequations of fractional orderrdquo Boletim da Sociedade Paranaensede Matematica vol 32 no 1 pp 181ndash189 2014

[23] R K Saxena and D Kumar ldquoSolution of fractional kineticequation associated with aleph functionrdquo Submitted

[24] M R Spiegel Theory and Problems of Laplace TransformsSchaums Outline Series McGraw-Hill New York NY USA1965

[25] G Mittag-Leffler ldquoSur la representation analytique drsquounebranche uniforme drsquoune fonction monogenerdquo Acta Mathemat-ica vol 29 no 1 pp 101ndash181 1905

[26] A Erdelyi W Magnus F Oberhettinger and F G TricomiTables of Integral Transforms vol 1 McGraw-Hill New YorkNY USA 1954

[27] HM Srivastava andR K Saxena ldquoOperators of fractional inte-gration and their applicationsrdquo Applied Mathematics and Com-putation vol 118 no 1 pp 1ndash52 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of