Certain fractional kinetic equations involving the

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Alexandria Engineering Journal (2016) xxx, xxx–xxx

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Alexandria University

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ORIGINAL ARTICLE

Certain fractional kinetic equations involving the product of generalized k-Bessel function Praveen Agarwal a,*, Mehar Chand b, Gurmej Singh c,d a

Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India Department of Mathematics, Fateh College for Women, Bathinda 151103, India c Department of Mathematics, Mata Sahib Kaur Girls College, Talwandi Sabo, Bathinda 151103, India d Department of Mathematics, Singhania University, Pacheri Bari, Jhunjhunu, India b

Received 4 June 2016; revised 18 July 2016; accepted 24 July 2016

KEYWORDS k-Pochhammer symbol; k-gamma function; k-Bessel function; Generalized hypergeometric function; Fractional kinetic equations

Abstract We develop a new and further generalized form of the fractional kinetic equation involving the product of the generalized k-Bessel function. The manifold generality of the generalized k-Bessel function is discussed in terms of the solution of the fractional kinetic equation in the present paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

c c Ck ðcÞ ¼ kk1 C k

1. Introduction and preliminaries Recently, papers on unified integrals involving special functions attract the attention of many researchers due to various applications (see [1,2]). Also, Diaz and Pariguan [3] introduced the k-Pochhammer symbol and k-gamma function defined as follows: ( Ck ðcþnkÞ ðk 2 R; c 2 C n f0gÞ Ck ðcÞ ðcÞn;k :¼ cðc þ kÞ . . . ðc þ ðn  1ÞkÞ ðn 2 N; c 2 CÞ ð1:1Þ They gave the relation with the classical Euler’s gamma function (see [4]) as * Corresponding author. E-mail addresses: [email protected] (P. Agarwal), [email protected] (M. Chand), gurmejsandhu11@gmail. com (G. Singh). Peer review under responsibility of Faculty of Engineering, Alexandria University.

ð1:2Þ

When k ¼ 1, (1.1) reduces to the classical Pochhammer symbol and Euler’s gamma function, respectively (see [5]). Recently, Romero et al. [4] (see, also [6]) introduced the k-Bessel function of the first kind for k; c; m 2 C; k 2 R and RðkÞ > 0; RðmÞ > 0 as follows: ðcÞ;ðkÞ

Jk;l ðzÞ ¼

1 X n¼0

ðcÞn;k ð1Þn  z n Ck ðkn þ l þ 1Þ ðn!Þ2 2

ð1:3Þ

The Fox-Wright function p wq ðzÞ with p numerator and q denominators, such that ai ; bj 2 Cði ¼ 1; . . . ; p; j ¼ 1; . . . ; qÞ is defined by the following (see, for detail [24]):  # " 1 Qp X ðai ; ai Þ1;p  Cðai þ ai nÞ zn Qqi¼1 ð1:4Þ w ðzÞ ¼ w z ¼  p q p q ðbj ; bj Þ1;q  j¼1 Cðbj þ bJ nÞ n! n¼0 under the condition

http://dx.doi.org/10.1016/j.aej.2016.07.025 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: P. Agarwal et al., Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.025

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P. Agarwal et al.

Figure 1

q p X X bj  ai > 1 j¼1

Solution of (2.2) for NðtÞ.

Figure 2

ð1:5Þ

i¼1

Particularly, when ai ¼ bj ¼ 1ði ¼ 1; . . . ; p; j ¼ 1; . . . ; qÞ, it immediately reduces to the generalized hypergeometric function p Fq ðp; q 2 N0 Þ (see, for details [17]):  # Q "   p ðai ; 1Þ1;p  a1 ; . . . ; ap ; Cðai Þ z z ¼ Qi¼1 p wq ðzÞ ¼ p wq p Fq q ðbj ; 1Þ1;q  b1 ; . . . ; bq ; j¼1 Cðbj Þ ð1:6Þ In terms of the k-Pochhammer symbol ðcÞn;k defined by (1.1), more generalized form of k-Bessel function xc;k k;m;b;c ðzÞ is given as follows: zlþ2n 1 X ð1Þn cn ðcÞn;k 2 ðzÞ ¼ ð1:7Þ xc;k k;l;b;c 2 bþ1 C ðkn þ l þ Þ ðn!Þ k 2 n¼0 where k; c; l; c; b 2 C and RðkÞ > 0; RðlÞ > 0. In the case of product form the above Eq. (1.7) can be written as zli þ2n r r X 1 Y Y ð1Þn cni ðci Þn;ki 2 xckii;k;lii ;bi ;ci ðzÞ ¼ ð1:8Þ bi þ1 C ðk n þ l þ Þ ðn!Þ2 k i i i i¼1 i¼1 n¼0 2 where ki ; ci ; li ; ci ; bi 2 C and Rðki Þ > 0; Rðli Þ > 0. If we choose bi ¼ ci ¼ 1 then generalized k-Bessel function reduced into the following form:  n z2 n r r  l X 1 Y Y ð1Þ ðc Þ i 4 z i n;ki ci ;ki xki ;li ;1;1 ðzÞ ¼ 2 n¼0 Cki ðki n þ li þ 1Þ ðn!Þ2 i¼1 i¼1

r  l Y z i ðci Þ;ðki Þ z2 ð1:9Þ ¼ Jki ;li 2 2 i¼1 where ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0. If we choose bi ¼ 1; ci ¼ 1 then generalized k-Bessel function reduced to the k-Wright function [23] associated with the following relation:

Solution of (2.2) for NðtÞ.

 n z2 n r r  l X 1 Y Y ð1Þ ðc Þ i 4 z i n;ki ci ;ki xki ;li ;1;1 ðzÞ ¼ 2 2 C ðk n þ l Þ ki i i ðn!Þ i¼1 i¼1 n¼0 2

r  l Y z i ci z ¼ Wki ;ki ;li 2 2 i¼1

ð1:10Þ

where ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0. The importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in physics, dynamical systems, control systems and engineering, to create the mathematical model of many physical phenomena. Especially, the kinetic equations describe the continuity of motion of substance. The extension and generalization of fractional kinetic equations involving many fractional operators were found [7–18]. In view of the effectiveness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a further generalized form of the fractional kinetic equation involving generalized k-Bessel function. The fractional differential equation between rate of change of the reaction was established by Haubold and Mathai [9], and the destruction rate and the production rate are given as follows: dN ¼ dðNt Þ þ pðNt Þ dt

ð1:11Þ

where N ¼ NðtÞ the rate of reaction, d ¼ dðNÞ the rate of destruction, p ¼ pðNÞ the rate of production and Nt denotes the function defined by Nt ðt Þ ¼ Nðt  t Þ; t > 0. In the special case of (1.11) for spatial fluctuations and inhomogeneities in NðtÞ the quantities are neglected, that is the equation dN ¼ ci Ni ðtÞ dt

ð1:12Þ

with the initial condition that Ni ðt ¼ 0Þ ¼ N0 is the number density of the species iat time t ¼ 0 and ci > 0. If we remove the index iand integrate the standard kinetic Eq. (1.12), we have

Please cite this article in press as: P. Agarwal et al., Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.025

Kinetic equations involving the product of k-Bessel function NðtÞ  N0 ¼ c0 D1 t NðtÞ

ð1:13Þ

is the special case of the Riemann-Liouville intewhere 0 D1 t gral operator 0 Dm t defined as Z t 1 m ðt  sÞm1 fðsÞds; ðt > 0; RðmÞ > 0Þ 0 Dt fðtÞ ¼ CðmÞ 0 ð1:14Þ The fractional generalization of the standard kinetic Eq. (1.13) is given by Haubold and Mathai [9] as follows: NðtÞ  N0 ¼ cm 0 Dm t NðtÞ 1 X k¼0

ð1:16Þ

ðRðvÞ > 0Þ;

ð1:17Þ

where NðtÞ denotes the number density of a given species at time t; N0 ¼ Nð0Þ is the number density of that species at time t ¼ 0; c is a constant and f 2 Lð0; 1Þ. By applying the Laplace transform to (1.17),

¼ N0

FðpÞ 1 þ cm pm 1 X

!

ðcm Þn pmn FðpÞ;

n¼0



 

c n 2 N0 ;   < 1 p ð1:18Þ

where the Laplace transform [19] is given by Z 1 ept fðtÞdt; ðRðpÞ > 0Þ FðpÞ ¼ LfNðtÞ; pg ¼

r Y

xckii;k;lii ;bi ;ci ðdm tm Þ  dm 0 Dm t NðtÞ

ð2:4Þ

is given by the following formula: NðtÞ ¼ N0

i¼1 n¼0

ð1Þn cni ðci Þn;ki

1



Cki ðki n þ li þ bi 2þ1Þ ðn!Þ2

dm tm 2

li þ2n

 Cðmðl1 þ    þ lr þ 2nÞ þ 1ÞEm;mðl1 þþlr þ2nÞþ1 ðdm tm Þ; ð2:5Þ Qr

where i¼1 xckii;k;lii ;bi ;ci ðtÞ is the product of r times k-Bessel function and li þ 2n þ 1 > 0; fi ¼ 1; . . . ; rg; Ea;b ð:Þ is the generalized Mittag-Leffler function [20]. Theorem 3. If d > 0; m > 0; ki ; ci ; li ; ci ; bi 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation NðtÞ  N0

r Y xckii;k;lii ;bi ;ci ðtÞ ¼ dm 0 Dm t NðtÞ

NðtÞ ¼ N0

In this section, we will investigate the solution of the generalized fractional kinetic equations by considering generalized kBessel function. The results are as follows. Theorem 1. If a > 0; d > 0; m > 0; ki ; ci ; li ; ci ; bi 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation xckii;k;lii ;bi ;ci ðdm tm Þ  am 0 Dm t NðtÞ

r X 1 Y

ð2:6Þ

is given by the following formula:

2. Solution of generalized fractional kinetic equations

ð2:1Þ

r X 1 Y i¼1 n¼0

ð1Þn cni ðci Þn;ki Cki ðki n þ li þ

bi þ1 Þ 2

1  t li þ2n ðn!Þ2 2

 Cðl1 þ    þ lr þ 2n þ 1ÞEm; l1 þþlr þ2nþ1 ðdm tm Þ; ð2:7Þ Qr

where i¼1 xckii;k;lii ;bi ;ci ðtÞ is the product of r times k-Bessel function and li þ 2n þ 1 > 0; fi ¼ 1; . . . ; rg; Ea;b ð:Þ is the generalized Mittag-Leffler function [20]. Proof. The Laplace transform of Riemann-Liouville fractional integral operator is given by [21,22] m L 0 Dm t fðtÞ; p ¼ p FðpÞ

ð2:8Þ

where FðpÞ is defined in (1.19). Now, applying the Laplace transform to the both sides of (2.1) gives ( ) r Y ci ;ki m m LfNðtÞ; pg ¼ N0 L xki ;li ;bi ;ci ðd t Þ; p i¼1

 a L 0 Dm t NðtÞ; p m

i¼1

is given by

Z

m m li þ2n dt NðtÞ ¼ N0 2 bi þ1 2 C ðk n þ l þ Þ ðn!Þ i i i¼1 n¼0 ki 2 r X 1 Y ð1Þn cni ðci Þn;ki

ð2:3Þ

Theorem 2. If d > 0; m > 0; ki ; ci ; li ; ci ; bi 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation

ð1:19Þ

The objective of this paper was to derive the solution of the fractional kinetic equation involving generalized k-Bessel function. The results obtained in terms of Mittag-Leffler function are rather general in nature and can easily construct various known and new fractional kinetic equations.

r Y

n¼0

ðxÞn Cðan þ bÞ

i¼1

0

NðtÞ ¼ N0

1 X

NðtÞ ¼ N0

Further, [13] considered the following fractional kinetic equation:

LfNðtÞ; pg ¼ N0

Ea;b ðxÞ ¼

i¼1

ð1Þk ðctÞmk Cðmk þ 1Þ

NðtÞ  N0 fðtÞ ¼ cm 0 Dm t NðtÞ;

Q where ri¼1 xckii;k;lii ;bi ;ci ðtÞ is the product of r times k-Bessel function is given by (1.8) and li þ 2n þ 1 > 0; fi ¼ 1; . . . ; rg; Ea;b ð:Þ is the generalized Mittag-Leffler function [20], which is defined as

ð1:15Þ

and obtained the solution of (1.15) is as follows: NðtÞ ¼ N0

3

1

NðpÞ ¼ N0

1

ept

0 m m

 Cðmðl1 þ  þ lr þ 2nÞ þ 1ÞEm;mðl1 þþlr þ2nÞþ1 ða t Þ;

ð2:2Þ

ð2:9Þ

m m li þ2n ! dt dt 2 bi þ1 2 i¼1 n¼0 Cki ðki n þ li þ 2 Þ ðn!Þ

r X 1 Y ð1Þn cni ðci Þn;ki

1

 am pm NðpÞ ð2:10Þ

Please cite this article in press as: P. Agarwal et al., Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.025

4

P. Agarwal et al.

m li þ2n r X 1 Y ð1Þn cni ðci Þn;ki 1 d m m NðpÞ þ a p NðpÞ ¼ N0 2 bi þ1 2 C ðk n þ l þ Þ ðn!Þ i i i¼1 n¼0 ki 2 Z 1  ept tmðl1 þþlr þ2nÞ dt ð2:11Þ 0

¼ N0

r X 1 Y i¼1 n¼0

m li þ2n d 2 bi þ1 Cki ðki n þ li þ 2 Þ ðn!Þ 2 ð1Þn cni ðci Þn;ki

1

2

ð2:12Þ

 Cðmðl1 þ    þ lr þ 2nÞ þ 1Þ ( ) 1 h  m il X p ðmðl1 þþlr þ2nÞþ1Þ   p a l¼0 Taking Laplace inverse of (2.13), and by using t ; ðRðmÞ > 0Þ CðmÞ

r X 1 Y

ð2:14Þ

i¼1 n¼0

ð1Þn cni ðci Þn;ki Cki ðki n þ li þ

bi þ1 Þ 2

m li þ2n d ðn!Þ 2 1

ð2:15Þ

l¼0

ð3:2Þ

NðtÞ ¼ N0

r m m li Y dt

2

i¼1

NðtÞ ¼ N0

r X 1 Y

i¼1 n¼0

!  dm 0 Dm t NðtÞ

m m li þ2n ð1Þn ðci Þn;ki 1 dt Cki ðki n þ li þ 1Þ ðn!Þ2 2

r  li Y t

2

ðc Þ;ðki Þ

Jki i;li

2

t ¼ dm 0 Dm t NðtÞ 2

ð3:5Þ

is given by the following formula: NðtÞ ¼ N0

r X 1 Y i¼1 n¼0

ð1Þn ðci Þn;ki 1  t li þ2n Cki ðki n þ li þ 1Þ ðn!Þ2 2

 Cðl1 þ    þ lr þ 2n þ 1ÞEm;l1 þþlr þ2nþ1 ðdm tm Þ: ð3:6Þ

1

 Cðmðl1 þ    þ lr þ 2nÞ þ 1Þ ( ) 1 X ðam tm Þl l  ð1Þ Cðmðl1 þ    þ lr þ 2n þ lÞ þ 1Þ l¼0

ð3:3Þ

Corollary 3. If d > 0; m > 0; ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation

i¼1

m m li þ2n dt Cki ðki n þ li þ bi 2þ1Þ ðn!Þ2 2 ð1Þn cni ðci Þn;ki

ðdm tm Þ2 2

ð3:4Þ

ð1Þn cni ðci Þn;ki

 Cðmðl1 þ    þ lr þ 2nÞ þ 1Þ ( ) 1 X tmðl1 þþlr þ2nþlÞ l ml ð2:16Þ ð1Þ a  Cðmðl1 þ    þ lr þ 2n þ lÞ þ 1Þ l¼0 r X 1 Y

ðc Þ;ðk Þ Jki i;li i

 Cðmðl1 þ    þ lr þ 2nÞ þ 1ÞEm;mðl1 þþlr þ2nÞþ1 ðdm tm Þ:

NðtÞ  N0

m li þ2n 1 d NðtÞ ¼ N0 2 bi þ1 2 C ðk n þ l þ Þ ðn!Þ k i i i i¼1 n¼0 2 r X 1 Y

m m li þ2n ð1Þn ðci Þn;ki 1 dt Cki ðki n þ li þ 1Þ ðn!Þ2 2

2

 Cðmðl1 þ    þ lr þ 2nÞ þ 1ÞL1 ( ) 1 X l ml ðmðl1 þþlr þ2nþlÞþ1Þ  ð1Þ a p

ð3:1Þ

 Cðmðl1 þ    þ lr þ 2nÞ þ 1ÞEm;mðl1 þþlr þ2nÞþ1 ðam tm Þ:

i¼1 n¼0 r X 1 Y

 am 0 Dm t NðtÞ

is given by the following formula:

we have L1 fNðpÞg ¼ N0

ðdm tm Þ2 2

Corollary 2. If d > 0; m > 0; ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation

ð2:13Þ

m1

ðc Þ;ðki Þ

Jki i;li

is given by the following formula:

i¼1 n¼0

m li þ2n r X 1 Y ð1Þn cni ðci Þn;ki 1 d NðpÞ ¼ N0 2 bi þ1 2 C ðk n þ l þ Þ ðn!Þ ki i i i¼1 n¼0 2

¼ N0

r m m li Y dt i¼1

NðtÞ ¼ N0

Cðmðl1 þ    þ lr þ 2nÞ þ 1Þ  pðmðl1 þþlr þ2nÞþ1Þ

L1 fpm ; tg ¼

NðtÞ ¼ N0

!

When we apply the case given in (1.9), the results in Eqs. (2.2), (2.5) and (2.7) reduced to the following the form. ð2:17Þ

Finally, summing up the above series with the help of Definition (2.3), we arrive at the right-hand side of (2.2). The proof of theorem (2) and (3) would run parallel to those of the theorem (1). So we would like to skip their proof here. h

Corollary 4. If a > 0; d > 0; m > 0; ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation NðtÞ ¼ N0

r m m li Y dt i¼1

2

Wckii ;ki ;li

ðdm tm Þ2 2

!  am 0 Dm t NðtÞ

ð3:7Þ

is given by the following formula: 3. Special cases NðtÞ ¼ N0 When we apply the case given in (1.9), the results in Eqs. (2.2), (2.5) and (2.7) reduced to the following the form.

m m li þ2n r X 1 Y ð1Þn ðci Þn;ki 1 dt 2 C ðk n þ l Þ 2 ðn!Þ k i i i i¼1 n¼0

 Cðmðl1 þ    þ lr þ 2nÞ þ 1ÞEm;mðl1 þþlr þ2nÞþ1 ðam tm Þ: ð3:8Þ

Corollary 1. If a > 0; d > 0; m > 0; ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation

Corollary 5. If d > 0; m > 0; ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation

Please cite this article in press as: P. Agarwal et al., Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.025

Kinetic equations involving the product of k-Bessel function

NðtÞ ¼ N0

Figure 3

Solution of (2.2) for NðtÞ.

Figure 4

Solution of (2.5) for NðtÞ.

r m m li Y dt

2

i¼1

Wckii ;ki ;li

ðdm tm Þ2 2

 dm 0 Dm t NðtÞ

NðtÞ ¼ N0

Corollary 6. If d > 0; m > 0; ki ; ci ; li 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation

2

Solution of (2.7) for NðtÞ.

2

t  dm 0 Dm t NðtÞ 2

r X 1 Y ð1Þn ðci Þn;ki 1  t li þ2n Cki ðki n þ li Þ ðn!Þ2 2 i¼1 n¼0

 Cðl1 þ    þ lr þ 2n þ 1ÞEm;l1 þþlr þ2nþ1 ðdm tm Þ: ð3:10Þ

i¼1

Figure 6

is given by the following formula:

 Cðmðl1 þ    þ lr þ 2nÞ þ 1ÞEm;mðl1 þþlr þ2nÞþ1 ðdm tm Þ:

NðtÞ ¼ N0

Solution of (2.5) for NðtÞ.

ð3:9Þ

m m li þ2n r X 1 Y ð1Þn ðci Þn;ki 1 dt NðtÞ ¼ N0 2 C ðk n þ l Þ 2 k i ðn!Þ i i i¼1 n¼0

Wckii ;ki ;li

Figure 5

!

is given by the following formula:

r  li Y t

5

ð3:11Þ

ð3:12Þ By applying the results in Eqs. (1.1) and (1.2), after little simplification the results in Eq. (2.2), (2.5) and (2.7) reduced to the following form. Corollary 7. If a > 0; d > 0; m > 0; ki ; ci ; li ; ci ; bi 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation

Please cite this article in press as: P. Agarwal et al., Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.025

6

P. Agarwal et al. Corollary 8. If d > 0; m > 0; ki ; ci ; li ; ci ; bi 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation NðtÞ ¼ N0

1l =k ðb þ1Þ=2ki r Y k i i i i

Cðci =ki Þ

i¼1



1 w2

 ðci =ki ; 1Þ; m m  d t  dm 0 Dm t NðtÞ ðli =ki þ ðbi þ 1Þ=ki ; ki =ki Þ; ð1;1Þ; ð3:15Þ is given by the following formula: 1l =k ðb þ1Þ=2ki X r 1 Y k i i i

k =k 1 n

ðci ki i i Þ bi þ1 Cðci =ki Þ i¼1 n¼0 Cðli =ki þ ki n=ki þ 2ki Þ

l þ2n Cðmðl1 þ    þ lr þ 2nÞ þ 1Þ dm tm i  2 2 ðn!Þ

NðtÞ ¼ N0

i

 Em; mðl1 þþlr þ2nÞþ1 ðdm tm Þ: Figure 7

NðtÞ ¼ N0

1l =k ðb þ1Þ=2ki r Y k i i i i

i¼1

 

Cðci =ki Þ

1 w2

 ðci =ki ;1Þ; dm tm  am 0 Dm t NðtÞ ðli =ki þ ðbi þ 1Þ=ki ; ki =ki Þ; ð1;1Þ; ð3:13Þ

1l =k ðb þ1Þ=2ki X r 1 Y k i i i

Corollary 9. If d > 0; m > 0; ki ; ci ; li ; ci ; bi 2 C; ki 2 N and Rðki Þ > 0; Rðli Þ > 0 (where i ¼ 1; . . . ; r) then the solution of the equation NðtÞ  N0  

is given by the following formula: n

ðci kiki =ki 1 Þ bi þ1 Cðci =ki Þ i¼1 n¼0 Cðli =ki þ ki n=ki þ 2kiÞ m m lþ2n Cðmðl1 þ lr þ 2nÞ þ 1Þ d t  Em;mðl1 þlr þ2nÞþ1 ðam tm Þ: 2 ðn!Þ2

NðtÞ ¼ N0

ð3:16Þ

Solution of (2.7) for NðtÞ.

i

Figure 8

ð3:14Þ

1l =k ðb þ1Þ=2ki r Y k i i i i

i¼1

Cðci =ki Þ

1 w2

 ðci =ki ; 1Þ; t ¼ dm 0 Dm t NðtÞ ðli =ki þ ðbi þ 1Þ=ki ; ki =ki Þ; ð1; 1Þ; ð3:17Þ

is given by the following formula:

Solution of (2.7) for NðtÞ.

Please cite this article in press as: P. Agarwal et al., Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.025

Kinetic equations involving the product of k-Bessel function NðtÞ ¼ N0

1l =k ðb þ1Þ=2ki X r 1 Y k i i i i

i¼1

 

Cðci =ki Þ

n¼0

n

ðci kki i =ki 1 Þ i þ1 Cðli =ki þ ki n=ki þ b2k Þ i

References

Cðl1 þ    þ lr þ 2n þ 1Þ  t li þ2n 2

7

ðn!Þ2 Em; l1 þþlr þ2nþ1 ðdm tm Þ: ð3:18Þ

4. Graphical interpretation In this section we plot the graphs of our solutions of the kinetic equation, which are established in Eqs. (2.2), (2.5) and (2.7). In each graph, we gave five solutions of the results on the basis of assigning different values to the parameters, which are denoted as NðtÞ ¼ NðN0 ; b1 ; b2 ; c1 ; c2 ; k1 ; k2 ; k1 ; k2 ; l1 ; l2 ; c1 ; c2 ; d; m; tÞ for r ¼ 2 and different values of the parameters i.e. fractional kinetic equation involving product of two generalized k-Bessel functions, where values of the parameters are given as N0 ¼ c1 ¼ c2 ¼ k1 ¼ k2 ¼ 2; b1 ¼ b2 ¼ k1 ¼ k2 ¼ 3; l1 ¼ l2 ¼ c1 ¼ c2 ¼ d ¼ a ¼ 1; m ¼ 0:1; 0:2; 0:3; 0:4; 0:5 for solutions of the Eq. (2.2). Using these values of the parameters, we plot two graphs of the Eq. (2.2) in Figs. 1–3 for different time intervals t ¼ 0 : 1; t ¼ 0 : 2 and t ¼ 0 : 5. Similarly we plot the graphs of the solution given in Eq. (2.5), which are in Figs. 4 and 5 for different time intervals and values of the parameters as shown in Figs. 4 and 5. The graphs of Eq. (2.7) also have the five solutions on the basis of different values of the parameters, which are denoted as NðtÞ ¼ NðN0 ; b1 ; b2 ; c1 ; c2 ; k1 ; k2 ; k1 ; k2 ; l1 ; l2 ; c1 ; c2 ; d; m; a; tÞ, where values of the parameters are given as N0 ¼ c1 ¼ c2 ¼ k1 ¼ k2 ¼ 2; b1 ¼ b2 ¼ k1 ¼ k2 ¼ 3; l1 ¼ l2 ¼ c1 ¼ c2 ¼ d ¼ 1; m ¼ 0:1 : 0:1 : 0:5 for solutions of the Eq. (2.7) plotted in Figs. 6–8 respectively. It is clear from these figures that NðtÞ > 0 for t > 0 and NðtÞ is monotonic increasing function for t 2 ð0; 1Þ. As shown in Fig. 8, we observe that NðtÞ ! 1 as t ! 1. In our study, we choose first 50 terms of Mittag-Leffler function and first 50 terms of our solutions to plot the graphs. NðtÞ ¼ 0, when t ¼ 0 and NðtÞ ! 1 when t ! 1 for all values of the parameters. Reader can choose any value of the parameters and select any interval for t to plot the graphs and further more investigation. 5. Conclusion In this work we give a new fractional generalization of the standard kinetic equation and derived solution for the same. From the close relationship of the generalized k-Bessel function with many special functions, we can easily construct various known and new fractional kinetic equations. Also from the graphical interpretation we came to the conclusion that the solutions of all the three Eqs. (2.1), (2.4) and (2.6) for all positive values of the parameters NðtÞ are Non-negative and Nðt1 Þ > Nðt2 Þ for t1 > t2 .

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Please cite this article in press as: P. Agarwal et al., Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.025