Abstract
The present study aimed at solving the stochastic generalized fractional diffusion equation (SGFDE) by means of the random finite difference method (FDM). Moreover, the conditions of mean square convergence of the numerical solution are studied and numerical examples are presented to demonstrate the validity and accuracy of the method.
Highlights
Many time-dependent processes in science have elements of randomness
The R-L fractional derivative is usually discussed in pure mathematical problems, while the Caputo fractional derivative is always employed for depicting the real-world models, since the initial and boundary conditions required are of classical style
A stochastic difference scheme Lnk unk = Gkn approximating stochastic partial differential equations (SPDEs) Lv = G is consistent in mean square at time t = (n + 1)∆t, if for any differentiable function Φ = Φ(x, t), we have in mean square
Summary
Many time-dependent processes in science have elements of randomness. most of the problems in epidemiology and financial mathematics take stochastic effects into account and generally lead to stochastic differential equations (SDEs) [1]. Some of the main numerical methods for solving stochastic partial differential equations (SPDEs), like finite difference and finite element schemes, have been considered [3,4,5] (e.g., [6,7,8]), based on a finite difference scheme in both space and time. The generalized fractional diffusion equations can be considered with random parameters imposed by environmental factors on the problem. Addressing such equations with random terms is closer to actual problem modeling. This paper is organized as follows: In Section 2, important preliminaries are discussed, and the new generalized fractional derivative (GFD) is introduced.
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