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Finite difference/spectral approximations for the two-dimensional time Caputo-Fabrizio fractional diffusion equation
The main contribution of this work is to construct and analyze stable and
high order schemes to efficiently solve the two-dimensional time
Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite
difference method in time and spectral methods in space, the proposed scheme is
unconditionally stable and has the global truncation error
, where , and are the time step size,
polynomial degree and regularity in the space variable of the exact solution,
respectively. It should be noted that the global truncation error
is well established in [ Li, Lv and Xu, {\em
Numer. Methods Partial Differ. Equ}. (2019)]. Finally, some numerical
experiments are carried out to verify the theoretical analysis. To the best of
our knowledge, this is the first proof for the stability of the third-order
scheme for the Caputo-Fabrizio fractional operator.Comment: At first sight, fractional derivatives defined using non-singular
kernels may appear very attractive. Thus, it is unsurprising that these
simpler operators have become quite popular since their appearance about five
years ago. But these operators with non-singular kernels have serious
shortcomings that strongly discourage their use, see [Fract. Calc. Appl.
Anal., 23, 610-634, 2020