A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients
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In this paper, 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.
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