Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

Abstract

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.