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

Abstract

In 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 (Figure presented.) 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 (Figure presented.). 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. © 2023 Wiley-VCH GmbH.