In view of the usefulness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a new and further generalized form of the fractional kinetic equation involving the G-function, a generalized function for the fractional calculus. This new generalization can be used for the computation of the change of chemical composition in stars like the Sun. The MellinBarnes contour integral representation of the G-function is also established. The manifold generality of the G-function is discussed in terms of the solution of the above fractional kinetic equation. A compact and easily computable solution is established. Special cases, involving the generalized Mittag-leffler function and the R-function, are considered. The obtained results imply more precisely the known results.Keywords Generalized function for the fractional calculus and fractional kinetic equations: general · Fractional kinetic equations and generalized functions: individual (the G-function and its relationships with other special functions, the Mellin-Barnes contour integral representation of the G-function, generalized fractional kinetic equations)

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.

Abstract. We discuss and derive the analytical solution for the generalized time-fractional telegraph equation. These problems are solved by taking the Laplace and Fourier transforms in variable t and x respectively. Here we use Green function also to derive the solution of the given differential equation.Mathematics subject classification (2010): Primary 26A33, 33C20, 33E12; secondary 47B38, 47G10.

We obtain the solution of a unified fractional Schrödinger equation. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the Mittag-Leffler function. The result obtained here is quite general in nature and capable of yielding a very large number of results (new and known) hitherto scattered in the literature. Most of results obtained are in a form suitable for numerical computation.

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The Laplace transform and its inverse are fundamental and powerful tools in solving boundary value problems occurring in the diverse fields of engineering. Here we will establish some useful formulas giving the inverse Laplace transform of various products of algebraic powers and $ \overline{H} $-function, involving one and more variables, which are unified and likely to have applications in several different areas.

Abstract. The object of the present paper is to consider a unified and extended form of certain families of elliptic-type integrals, which have been discussed in number of earlier works on the subject due to their importance and applications in problems arising in radiation physics and nuclear technology. The results obtained are of general character and include the investigations carried out by several authors. We obtain asymptotic formulas for the unified elliptic-type integrals. IntroductionElliptic integrals occur in a number of physical problems [1-2], and frequently in the form of multiple integrals. For example, the problems dealing with the computation of the radiations field off axis from certain uniform circular disc radiating according to an arbitrary angular distribution law [3], when treated with Legendre polynomials expansion method, give rise to Epstein and Hubbell [4,21] family of elliptic-type integrals:Elliptic integrals (1.1) have been studied and generalized by many authors notably by Kalla [5,6]

The aim of this paper is to establish a solution of a certain class of convolution integral equation of Fredholm type whose kernel involve certain product of special function by using Riemann-Liouville and Weyl fractional integral operators. Some interesting particular cases are also considered.