LAUSR

The fractional reaction–diffusion equation has profound physical background and theoretical connotation, and its numerical methods are of great scientific significance and practical value. In this paper, a class of explicit–implicit (E–I) difference method and implicit–explicit (I–E) difference method are proposed for one-dimensional time fractional reaction–diffusion equation. The methods are constructed by combining classical explicit scheme and classical implicit scheme. The existence, uniqueness and convergence of solutions are given for E–I and I–E schemes. Theoretical analysis and numerical experiments show that the E–I and I–E schemes are unconditionally stable, with second-order spatial accuracy and $$2-\alpha $$2-α order temporal accuracy. Compared with the classical implicit difference method with the similar accuracy, the computation time of the presented methods is improved by nearly 41$$\%$$%. It is shown that the E–I difference method and I–E difference method are effective for solving the time fractional reaction–diffusion equation.