Certain Image Formulae and Fractional Kinetic Equations of Generalized k -Bessel Functions via the Sumudu Transform

The monotonicity and the representation formulae of generalized k -Bessel functions, W k ν, c , studied by SR Mondal. This paper establishes the image formulae and then extract the solutions for fractional kinetic equations, involving W k ν, c utilizing their Sumudu transforms. Some signiﬁcant particular cases are then deduced and analyzed

The Sumudu transform was appeared in [41,42] and then numerous authors studied its properties (See [1][2][3][4][5][6][7][8][9]). Here, we recall the definition and then derive the Sumudu transform of k-Bessel respectively in (12) and (13): is given by and Now, using the relation It is observed that Indicating the left side of (15) by H (u), then In the next section, we investigate the fractional integration of (11).
Proof Let the LHS of (21) is denoted by L 1 and using (11), we get Using (17), we have Interchanging the summation and integration and then finding the inner integral by substituting t = xu, we get In view of (4) and (6) we reach the required result.
Proof Let the LHS of (23) is denoted by L 2 , Using (18) and using the (11), we get Interchanging the summation and integration and then evaluating the inner integral by substituting t = x u , we get The proof is done in view of (4) and (6).
In view (19) and (20) , we have the following left and right handed fractional differentiation: Proof The definition of (11) and (19) easily gives required result.
Proof Again the proof is follow from the definition of (11) and (20).
As mentioned in [18],the destruction and the production rate respectively, where t is t (t * ) = (t − t * ) , t * > 0. From (t) , the spatial fluctuation and inhomogeneities are neglected, then (29) reduced into At time t = 0, the number density of species is i and ς i > 0 with the initial condition where 0 D t −1 is the operator of Riemann-Liouville (R-L). In [18] gives a generalized form of the equation (30) by where The solution of (32) is true for The use of the Laplace transform [40] to (32) gives where is given by where E ν,ν(2r + μ k ) (−d ν t ν ) is the Wiman function [43] Proof Let us begin with the Sumudu transform (ST) of the R-L operator as where H (u) is given in (16). Now applying the ST on (37) and applying (11), we have * (u) = S [ (t) ; u] where The rearrangement of terms gives, Applying the inverse ST on (42), and by using we have which gives, Corollary 5 Setting k = 1 in (38) gives the GKFE containing the Bessel function as: If c ∈ R and d > 0, μ > −1, ν > 0, ∈ C then have the solution: Theorem 6 If a = d, a > 0, μ > −1, d > 0, t ∈ C and c, k ∈ R, then is given by Proof The proof of Theorem 6 can be run in parallel of Theorem 5. So the details are not given.
Theorem 7 If ν > 0, μ > −1, d > 0, t ∈ C and c, k ∈ R, then is given by Proof We skip the proof because it works as same as the theorems 7 and 5.