A General Solution of a Space-Time Fractional Anomalous Diffusion Problem Using the Series of Bilateral Eigen-Functions

Abstract

In the present paper, we consider an anomalous diffusion problem in two dimensional space involving Caputo time and Riesz-Feller fractional derivatives and then solve it by using a series involving bilateral eigen-functions. Also, we obtain a numerical approximation formula of this problem and discuss some of its particular cases.

Obtaining Voigt Functions Via Quadrature Formula for the Fractional in Time Diffusion and Wave Problem

“…Besides establishing some interesting integral and series representations of special functions, the results given in [13] and [14] may provide a new way of solution of a space-time fractional anomalous diffusion problem using the series of bilateral eigenfunctions and series solution for initial value problems of time fractional generalized anomalous diffusion equations as on the lines of [11,12] and [13].…”

“…authors ( [11,14]). The computation of anomalous diffusion problems in the form of integral equations can be found in ( [5,12] and [13]). For the theory and analysis of the fractional differential equations, we refer the work of the researchers including authors (e.g., [2, 6-8, 18, 21] and [23]).…”

Multiple Fractional Diffusions via Multivariable H-Function

“…In Case IV), the fundamental solution of anomalous diffusion problem is obtained. On putting r = 1, the special cases are checked by the results in one dimensional in space-time fractional derivatives of previous work of many researchers in the literature for example ( [4], [6], [7], [8], [10], [12]).…”

“…In the similar manner, by Theorem 3.1, we also obtain the Green function solution of the Eqn. (2.1) of the case for r = 2 , 0 < α < 1, 0 < β 1 < 1, 1 < β 2 < 2; of the anomalous diffusion problem due to Kumar, Pathan and Srivastava [6]. For further directions of the researches in this field, we omit them.…”

“…Recently, series and analytic solutions of fractional in time and space -fractional anomalous diffusion problems are obtained and studied by many authors ( [6], [7], [8], [13]). In [10] the solution of fractional diffusion is found in terms of Fox's H-function (see also [5], [12, p.2], [17]).…”