Solution of Generalized Fractional Kinetic Equation in Terms of Special Functions

This paper is devoted to introduce an analytical method for solving generalized fractional kinetic equation in terms of the generalized 4 K function, the generalized M – Series and the generalized Mittag – Leffler type function, this method is depending on Laplace transform operator technique.


Introduction and Preliminaries
Fractional kinetic equation have gained importance due to their occurrence in science and engineering, the generalized fractional kinetic equation in term of Mittag -Leffler function studied by sexcena, Mathai and Haubold [16], they introduced the solution of the generalized fractional kinetic equation associated with the generalized Mittag -Leffler function.For more result one can refer to the work of Saichev and Zaslavsky [11], Sexcena et al [15,17] Zaslavsky [19] and Sexcena and Kalla [14].
Chaurasia and Kumar derived the solution of generalized fractional kinetic equation involving the M -Series [ ], the result obtained is quite general in nature and capable of yielding a very large number of results.
On the other hand Gupta and Parihar [6] investigated the solution of generalized fractional kinetic equation in term of the 4 K -function and give more precise result of the work of Chaurasia and Pandey [2] and Sharma [18].
In this paper we will derive the solution of the generalized fractional kinetic equation in terms of the generalized Mittag-Leffler function introduced by Salim and Faraj [12], the generalized M -Series and the generalized 4 K -function introduced by the authors in a different papers [4,5] as ,...., !,....,

During this paper one can use the following:
 Generalized Riemman -Liouville operator of fractional calculus [8,10]  Def of Laplace transform and convolution theorem [10]  also from [8] we have The following two relation is needed

Solving General Fractional Kinetic Equation
On integrating the standard kinetic equation Haubold and Mathai [7] derived that Taking Laplace transform operator for both side we get taking Laplace inverse, we get

Lemma 2.1:
The fractional kinetic equation We will using the illustrated technique above to solve different type of the generalized fractional kinetic equation which contain our special functions in the kernel.

Theorem 2.2:
The general fractional kinetic equation: have the solution

Proof:
Beginning with (2.7) and taking Laplace transform for both sides , ,m,n ; ; ; Hence by applying (1.9) Finally the solution becomes

Remark:
The result obtained coincides with the result obtained by the authors [see 5], for the solution by using alternating methodto see that we have from (2.9) (2.10)

Theorem 2.3:
The solution of the general fractional kinetic equation (2.12)

Proof:
Taking Laplace transform for both side of (2.11) ; ; Another form of the solution can be written as in which given by the authors in another paper [ 5 ].

Theorem 2.4:
The general fractional Kinetic equation has the solution given by

 
Again by applying Laplace inverse to both sides of the last equation we get which ends the proof

Corollary 2.5:
The general fractional kinetic equation has the solution given by

Proof:
Beginning with (2.16) and taking Laplace transform to both sides,

;
; ; and now taking Laplace inverse of (2.18), and again applying (1.10) we get One can rewrite the last equation as has the solution given by