An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction–diffusion-wave equation

Abstract

The aim of this paper is to develop an efficient numerical treatment for the two-dimensional fractional nonlinear reaction–diffusion-wave equation with the time-fractional derivative of order α (1<α<2). For this purpose, we employ the alternating direction implicit (ADI) method based on the Crank–Nicolson scheme for the time stepping, while we apply the Legendre–Galerkin spectral method for the space discretization. The stability and convergence analysis are rigorously set up. In addition, the proposed method is extended to solve the time-fractional Klein–Gordon and sine-Gordon models. Numerical experiments are included, which verifies the theoretical predictions.