A Linearized Numerical Scheme for Two-Dimensional Semi-Linear Time–Space Fractional Diffusion-Wave Equation with Weakly Singular Solution

In this paper, an efficient finite difference scheme is investigated for the two-dimensional semi-linear time–space fractional diffusion-wave equations that have solutions with weak singularity in time–direction at the initial time. First, the original equation is transformed into a system of two equations with the Caputo derivatives of the same order between 0 and 1 by using the symmetric fractional-order reduction method. Then, the Caputo derivative is discretized by the L1 scheme on the nonuniform meshes, and the Riesz derivatives are discretized by the second-order weighted and shifted Grünwald scheme. Newton’s method is used to linearize the nonlinear term. For the resulting linearized finite difference scheme, in the discrete [Formula: see text]-norm, the unconditional stability is analyzed and the optimal error estimate is derived. It is demonstrated that the proposed scheme can be efficiently solved by using the conjugate gradient method, and some numerical experiments are given to verify the theoretical results.