3 research outputs found
A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients
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
-type formula for the generalized Caputo fractional derivative in time
discretization and the second-order weighted and shifted Gr\"{u}nwald
difference (WSGD) formula in spatial discretization, respectively. Theoretical
results and numerical tests are conducted to verify the -order
and 2-order of temporal and spatial convergence with 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.Comment: 23 pages, 10 tables, 1 figure. Make several corrections again and
have been submitted to a journal at Sept. 20, 2019. Version 2: Make some
necessary corrections and symbols, 13 Jan. 2020. Revised manuscript has been
resubmitted to journa