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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 18, Issue 1


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

Petr N. Vabishchevich
  • Corresponding author
  • Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya,115191 Moscow; and Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow,Russia
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Published Online: 2017-08-17 | DOI: https://doi.org/10.1515/cmam-2017-0028


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.

Keywords: Elliptic Operator; Fractional Power of an Operator; Finite Element Approximation; Two-Level Schemes; Stability Of Difference Schemes

MSC 2010: 26A33; 35R11; 65F60; 65M06


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About the article

Received: 2017-04-17

Revised: 2017-07-14

Accepted: 2017-07-18

Published Online: 2017-08-17

Published in Print: 2018-01-01

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

Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 1, Pages 111–128, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0028.

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