A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS(2021)

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Abstract
In this paper, we derive a fourth order approximation for thegeneralizedfractional derivative that is characterized by a scale functionz(t)and a weight functionw(t). Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.
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Key words
diffusion,fourth order approximation,generalized fractional derivative,generalized weighted and shifted Grunwald-Letnikov difference method,numerical solution
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