Abstract
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two- level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.
Highlights
Non-local applied mathematical models based on the use of fractional derivatives in time and space are actively discussed in the literature [1, 18]
Super-diffusion problems are treated as evolutionary problems with a fractional power of an elliptic operator
More general problems are associated with fractional powers of an elliptic operator
Summary
Non-local applied mathematical models based on the use of fractional derivatives in time and space are actively discussed in the literature [1, 18]. The simplest variant is associated with the explicit construction of the solution using the known eigenvalues and eigenfunctions of the elliptic operator with diagonalization of the corresponding matrix [5,14,15] All these approaches demonstrates too high computational complexity for multidimensional problems. We have proposed [31] a computational algorithm for solving an equation with fractional powers of elliptic operators on the basis of a transition to a pseudo-parabolic equation. A small number of pseudo-time steps is required to reach a steady solution This computational algorithm for solving equations with fractional powers of operators is promising when considering transient problems. The computational algorithm for solving the equation with a fractional power of an operator based on the Cauchy problem for a pseudo-parabolic equation is proposed. At the end of the work the main results of our study are summarized
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