Vabishchevich P.  

Numerical solution of problems for a space-fractional diffusion equation

Nowadays, non-local applied mathematical models based on the use of fractional derivatives in time and space
are actively discussed. Many models involve both sub-diffusion (fractional in time) and supper-diffusion (fractional in space) operators. Supper-diffusion problems are treated as evolutionary problems with a fractional power of an elliptic operator.

We have proposed [1] a computational algorithm for solving an equation for fractional powers of elliptic operators on the basis of a transition to a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes are applied. The computational algorithm is simple for practical use, robust, and applicable to solving a wide class of problems. This computational algorithm for solving equations with fractional powers of operators is promising when considering transient problems. 

This work was supported by the Russian Foundation for Basic Research (projects
14-01-00785, 15-01-00026).

1. Vabishchevich, P.N.: Numerically solving an equation for fractional powers of elliptic operators. Journal of Computational Physics 282(1), 289–302 (2015).

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