Computable extensions of generalized fractional kinetic equations in astrophysics

Abstract

Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomers and physicists to pay more attention to available mathematical tools that can be widely used in solving several problems of astrophysics/physics. In view of the great importance and usefulness of kinetic equations in certain astrophysical problems, the authors derive a generalized fractional kinetic equation involving the Lorenzo-Hartley function, a generalized function for fractional calculus. The fractional kinetic equation discussed here can be used to investigate a wide class of known (and possibly also new) fractional kinetic equations, hitherto scattered in the literature. A compact and easily computable solution is established in terms of the Lorenzo-Hartley function. Special cases, involving the generalized Mittag-Leffler function and the R-function, are considered. The obtained results imply the known results more precisely.

A generalization of truncated M-fractional derivative and applications to fractional differential equations

“…Example 3. Now we give the truncated M-series fractional derivatives of some well-known functions by using the result (8). Let n ∈ R and α ∈ (0, 1].…”

“…For t > 0, α ∈ (n, n + 1] and for (n + 1) times differentiable function f , it is easy to show that i D α;n M f (t) = Kt n+1−α f (n+1) (t). by using (13), (8) and induction on n.…”

“…Proof. Using the definition of M-series fractional integral (14), (8) and applying fundamental theorem of calculus for integer-order derivatives, we get…”

Fractional calculus and application of generalized Struve function

“…Recently, many papers investigated the solutions of generalized fractional kinetic equations (GFKE) involving various types of special functions. For instance, the solutions of GFKE involving M-series (Chaurasia and Kumar 2010), generalized Bessel function of the first kind (Kumar et al. 2015), Aleph function (Choi and Kumar 2015) and the generalized Struve function of the first kind (Nisar et al.…”

New Generalization of Fractional Kinetic Equation using Aleph-Function of Two Variables

“…Further, a number of research workers have also studied the generalization of kinetic equation in term of Mittag-Leffler functions. Recently, Chaurasia and Kumar [20] investigated the solution of the fractional kinetic equation associated with the generalized M-series.…”