An Effective Numerical Simulation Algorithm for the Time‐Space Fractional Reaction–Diffusion–Wave Equation With ψ‐Caputo and Riesz Derivatives

ABSTRACT This paper constructs an effective numerical simulation algorithm for the time‐space fractional reaction–diffusion–wave equation, where the temporal derivative is in the sense of ‐Caputo and the spatial derivative in the sense of Riesz. The H2N2 formula is applied to approximating the ‐Caputo derivative in time and the finite element method is utilized for the spatial discretization. Numerical stability and convergence of the proposed fully discrete scheme are rigorously proved. Numerical implementation for the proposed simulation algorithm is shown in more detail. And the numerical examples are presented to support the theoretical analysis.