The couple of Hermite-based approach and Crank-Nicolson scheme to approximate the solution of two dimensional stochastic diffusion-wave equation of fractional order

Abstract

The main aim of this study is presenting a semi-discretization scheme to find the numerical solution of two dimensional (2D) stochastic time fractional diffusion-wave equation, which obtains from classical 2D diffusion-wave equation by replacing integer time derivative with Caputo fractional time derivative of order α (1 < α ≤ 2) and inserting some stochastic factors. In this scheme, first Crank-Nicolson and linear spline techniques are used to discrete mentioned problem in the time direction and then Hermite-based approach is applied to obtain the approximate solution in each time step. It is not required any discretization in the spatial directions and therefore this approach is an efficient tool to solve various problems which have been defined on irregular domains. Finally, to confirm this claim that obtained numerical results are accurate and in reliable agreement with the theoretical discussion, some test problems are included in the numerical example section. The values of maximum error, error associated with norm 2, and RMS–error are reported to demonstrate accuracy and reliability of the proposed method. Also, the domain of last example is considered in a long range of the time interval to study the stability of our method on time variable.