Numerical Solving Unsteady Space-Fractional Problems with the Square Root of an Elliptic Operator

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.

Keyword : fractional partial differential equations, elliptic operator, square root of an operator, two-level difference scheme, regularized scheme, convection-diffusion problem