Solutions of Fractional Kinetic Equations using the $(p,q;l)$-Extended τ -Gauss Hypergeometric Function

Abstract

The main objective of this paper is to use the newly proposed $(p,q;l)$-extended beta function to introduce the $(p,q;l)$-extended $τ$-Gauss hypergeometric and the $(p,q;l)$-extended $τ$-confluent hypergeometric functions with some of their properties, such as the Laplace-type and the Euler-type integral formulas. Another is to apply them to fractional kinetic equations that appear in astrophysics and physics using the Laplace transform method.

Application of the Pathway-Type Transform to a New Form of a Fractional Kinetic Equation Involving the Generalized Incomplete Wright Hypergeometric Functions

“…Because of the usefulness and great importance of the kinetic equation in some astrophysical issues, fractional kinetic equations have been investigated to describe the various phenomena governed by anomalous reactions in dynamical systems [6][7][8][9]. Several authors have recently presented solutions to various families of fractional kinetic equations involving special functions using the Laplace transform, Sumudu transform, Prabhakar-type operators, Hadamard fractional integrals, and pathway-type transform based on these principles (see, for example, [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]).…”

“…Many generalizations and solutions of the fractional-order kinetic equation have recently been developed, utilizing a variety of fractional integral transforms including the fractional Laplace transform [12][13][14][15][16], fractional Sumudu transform [17][18][19], Hadamard fractional integrals [20][21][22], fractional pathway transform [23,24] and Prabhakar-type operators [25], which have been extensively studied. In particular, Khan et al [14] presented solutions for fractional kinetic equations associated with the (p, q)-extended τhypergeometric and confluent hypergeometric functions using the Laplace transform, while Hidan et al [15] discussed a technique for the Laplace transformation of solutions of fractional kinetic equations involving extended (k, t)-Gauss hypergeometric matrix functions.…”

“…In particular, Khan et al [14] presented solutions for fractional kinetic equations associated with the (p, q)-extended τhypergeometric and confluent hypergeometric functions using the Laplace transform, while Hidan et al [15] discussed a technique for the Laplace transformation of solutions of fractional kinetic equations involving extended (k, t)-Gauss hypergeometric matrix functions. In addition, Abubakar [16] derived solutions for fractional kinetic equations using the (p, q; l)-extended τ-Gauss hypergeometric function. Gaining insight from the last recently mentioned manuscripts, this paper provides an in-depth exploration of fractional kinetic equations and their solutions by using the generalized incomplete Wright hypergeometric function and pathway-type transform technique.…”

Novel Kinds of Fractional λ–Kinetic Equations Involving the Generalized Degenerate Hypergeometric Functions and Their Solutions Using the Pathway-Type Integral

“…Remark 3. The results already established in [6][7][8][9][15][16][17] can be easily obtained as special cases by applying the functions in Remark 2 to Theorems 2-6, which gives us a more comprehensive understanding of the topic and allows for further exploration.…”

“…For instance, Alqarni et al [15] introduced solutions involving the generalized incomplete Wright hypergeometric functions. Meanwhile, Khan et al [16] considered solutions for KEFO associated with the (p, q)−extended τ−hypergeometric and confluent hypergeometric functions, and Abubakar [17] focused on a solution for KEFO using the (p, q; l)-extended τ-Gauss hypergeometric function. In a similar vein, Fuli He et al [18] derived the solution of KEFO in terms of the Hadamard product of (p, k)−hypergeometric functions.…”