ON IMAGE FORMULAE LEADING TO FRACTIONAL KINETIC EQUATIONS SOLUTIONS VIA SUMUDU GENERALIZED K-BESSEL FUNCTIONS

Recently, representation formulae and monotonicity properties of generalized k-Bessel functions, Wk v,c., were established and studied by SR Mondal [24]. In this paper, we pursue and investigate some of their image formulae. We then extract solutions for fractional kinetic equations, involving Wk v,c, by means of their Sumudu transforms. In the process, Important special cases are then revealed, and analyzed.

Here (η) n,k is the k−Pochhammer symbol defined by (see [13]) while Γ k (z) denotes the k−gamma function defined by (see [13]) For k = 1, Γ k (z) reduces to Γ (z) and have the following relations, and The well known Beta function [29] defined by The generalized hypergeometric function p F q (a 1 , . . ., a p ; c 1 , . . ., c q ; x), is given by the power series [29] p F q (a 1 , . . ., a p ; c 1 , . . ., c q ; z) where c i , (i = 1, 2. • • • , q) can not be zero or a negative integer.Here p or q or both are allowed to be zero.The series (1.7) is absolutely convergent for all finite z if p ≤ q and for |z| < 1 if p = q + 1.When p > q + 1, then the series diverge for z = 0 and the series does not terminate.The generalized Wright hypergeometric function p ψ q (z) is given by the series [42] p ψ q (z) = p ψ q where a i , b j ∈ C, and real γ i , η j ∈ R (i = 1, 2, . . ., p; j = 1, 2, . . ., q).The asymptotic behavior of this function for large values of argument of z ∈ C were studied in [15,21] and under the condition (1.9) The more properties of the Wright function are investigated in [20][21][22]42,43].The Mittag-Leffler function E ρ (z) (see, [23]) and E ρ,η (x) (see, [41]) respectively defined by Recently, SR Mondal [24] gives the new generalization of k-Bessel function W k ν,c and is defined by where k > 0, ν > −1 and c ∈ R.

Image formula of W
The fractional integrals of a function f (z) of order η [32] are given by and The fractional derivatives of a function f (z) of order η [32] are given by and Now, we give some image formulas of (1.12) using (2.1)-(2.4).
Proof.Let the left hand side (LHS) of (2.5) is denoted by L 1 and using the (1.12), we get Using (2.1), we have Interchanging the summation and integration and then evaluating the inner integral by substituting t = xu, we get In view of (1.4) and (1.6) we arrived the required result.
Corollary 2.1.If we set c = 1 in Theorem 1, then we get the fractional integration of k−Bessel function which is equation ( 12) of [16].
Using (2.2) and using the (1.12), we get Interchanging the summation and integration and then evaluating the inner integral by substituting t = x u , we get In view of (1.4) and (1.6) we arrived the required result.
In view (2.3) and (2.4) , we have the following left and right handed fractional differentiation as follows: Proof.Using the definition of (1.12) and (2.3), we can easily find the required result.So the details are omitted.Proof.Using the definition of (1.12) and (2.4), we can easily find the desired result.So the details are omitted.
As mentioned in [19],the destruction rate and the production rate as follows, where If spatial fluctuation and inhomogeneities in the quantity Q(t) are neglected, then (3.1) reduced into which is the number density of species i at time t = 0 and c i > 0 is given by the initial condition Q i (t = 0) = Q 0 .Now after integrating and decline the index i, (3.2) reduced into where 0 D t −1 is the Riemann-Liouville fractional integral operator.
Haubold and Mathai [19] gives a generalized form of the fractional kinetic equation (3.2) as follows where The solution of equation (3.4) is true for The use of Laplace transform [37] to (3.4) gives is given by the following formula where is the generalized Mittag-Leffler function [41] Proof.The Sumudu transform of Riemann-Lioville fractional integral operators is given by where G(u) is defined in (1.17).Now applying Sumudu transform both sides of (3.9) and applying the definition of k-Bessel function given in (1.12), we have where By rearranging terms we get, Applying inverse Sumudu transform of (3.14), and by using we have which gives, . which is the desired result.
have the solution: is given by is given by the following formula is given by Proof.The proofs of theorem 7 would run parallel to those of theorem 5.
is given by the formula

Conclusion
In this paper, we establish some fractional and integral representations of generalized k− Bessel function.Also, we give the solution of fractional kinetic equation involving k-Bessel function with the help of Sumudu transform.This paper conclude with the remark that, the results given in this paper are general and can lead to yield many fractional integrals (derivatives) involving the Bessel, generalized Bessel and trigonometric functions by the suitable specializations of arbitrary parameters in the theorems and corollaries.