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

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.