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Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator

  • Petr N. Vabishchevich EMAIL logo

Abstract

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Padé-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

Funding statement: The publication was financially supported by the Ministry of Education and Science of the Russian Federation (the Agreement # 02.a03.21.0008).

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Received: 2017-4-17
Revised: 2017-7-14
Accepted: 2017-7-18
Published Online: 2017-8-17
Published in Print: 2018-1-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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