A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients

In this article, we first propose an unconditionally stable implicit difference scheme for solving generalized time–space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the L1-type formula for the generalized Caputo fractional derivative in time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula in spatial discretization, respectively. Theoretical results and numerical tests are conducted to verify the (2 − γ)-order and 2-order of temporal and spatial convergence with γ ∈ (0, 1) the order of Caputo fractional derivative, respectively. The fast sum-of-exponential approximation of the generalized Caputo fractional derivative and Toeplitz-like coefficient matrices are also developed to accelerate the proposed implicit difference scheme. Numerical experiments show the effectiveness of the proposed numerical scheme and its good potential for large-scale simulation of GTSFDEs.