An efficient spectral-Galerkin method for fractional reaction-diffusion equations in unbounded domains

Highlights: • Since fractional derivatives are usually nonlocal, here we study the unbounded domain directly, different from other previous works where domain truncation is usually used. • The biorthogonal mapped Chebyshev functions are used as the basis for our spectral-Galerkin method, which lead to diagonal stiffness matrix, and thus very efficient even in higher dimensions. • The ETDRK4 time stepping method is used for the resulted semidiscrete problem which is 4th-order accurate in time. This ensures that numerical treatment of the fractional reaction-diffusion equation is both more efficient and accurate. In this work, we apply a fast and accurate numerical method for solving fractional reaction-diffusion equations in unbounded domains. By using the Fourier-like spectral approach in space, this method can effectively handle the fractional Laplace operator, leading to a fully diagonal representation of the fractional Laplacian. To fully discretize the underlying nonlinear reaction-diffusion systems, we propose to use an accurate time marching scheme based on ETDRK4. Numerical examples are presented to illustrate the effectiveness of the proposed method.