SOLUTIONS OF GENERALIZED FRACTIONAL KINETIC EQUATIONS VIA SUMUDU TRANSFORMS INVOLVING BESSEL-STRUVE KERNEL FUNCTION

In this paper, we pursue and investigate the solutions for fractional kinetic equations, involving Bessel-Struve function by means of their Sumudu transforms. In the process, one Important special case is then revealed, and analyzed. The results obtained in terms of Bessel-Struve function are rather general in nature and can easily construct various known and new fractional kinetic equations.

The Sumudu transform over the set functions is defined by The Sumudu transform of S α (x), using (1.2) and (1.3), is given by Interchanging the order of integration and summation gives, Denoting the left hand side by G(u), we have The main aim of this work is to establish the generalized fractional kinetic equation involving S α (x).Here, we use the Sumudu transform methodology to obtain the results.

Generalized Fractional Kinetic Equations
The fractional differential equation between rate of change of reaction was established in [13] , the destruction rate and the production rate as follows where N = N (t) the rate of reaction ,d = d (N) the rate of destruction, p = p (N) the rate of production and N t denote the function defined by The special case of (2.1), for spatial fluctuations or in homogeneities in N (t) the quantity are neglected, that is the equation with t N i (t = 0) = N 0 is the number of density of species i at time t = 0 and c i > 0. If we reject the index i and integrate the standard kinetic equation (2.2), we have is the special case of the Riemann-Liouville integral operator 0 D −υ t defined as The fractional generalization of the standard kinetic equation (2.3) given in [13] as: and obtained the solution of (2.4) as follows Further, Saxena and Kalla [25] considered the following fractional kinetic equation: where N(t) denotes the number density of a given species at time t, N 0 = N (0) is the number density of that species at time t = 0, c is a constant and f ∈ L(0, ∞).By applying the Laplace transform to (2.6), (see [25] ) where the Laplace transform ( [26]) is defined by The Mittag-Leffler functions E ρ (z) (see [14]) and E ρ,λ (x) [29] is defined respectively as The details about fractional kinetic equations and solutions, one can refer to [11,[17][18][19][20][21][22][23][24][25]30] The solution of the generalized fractional kinetic equations involving (1.2) is given in this section.
Theorem 3. If d > 0, υ > 0, α, µ, t ∈ C, a = d and (α) > −1, then for the solution of the equation there holds the formula Proof.Theorem 3 can easily derive from Theorem 2, so the details are omitted.
Special case: 2), we have In view of (3.8) and Theorem 1, 2 and 3, we have following corollaries respectively Corollary 3.1.If d > 0, υ > 0, α, µ, t ∈ C then for the solution of the equation there holds the formula

Conclusion
Solutions of generalized fractional kinetic equation in terms of the Bessel-Struve kernel function is given in this study.The results obtained here are in compact forms appropriate for numerical computation and are rather general in nature and can easily construct various known and new fractional kinetic equations.