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Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators

  • Petr N. Vabishchevich ORCID logo EMAIL logo
Published/Copyright: October 20, 2021

Abstract

Many non-local processes are modeled using mathematical models that include fractional powers of elliptic operators. The approximate solution of stationary problems with fractional powers of operators is most often based on rational approximations introduced in various versions for a fractional power of the self-adjoint positive operator. The purpose of this work is to use such approximations for the approximate solution of nonstationary problems. We consider Cauchy problems for the first and second order differential-operator equations in finite-dimensional Hilbert spaces. Estimates for the proximity of an approximate solution to an exact one are obtained when specifying the absolute and relative errors of the approximation of the fractional power of the operator. We construct splitting schemes based on the additive representation with a rational approximation of the operator’s fractional power. The stability and accuracy of factorized two-level additive operator-difference schemes for the first order evolution equation and three-level schemes for a second order equation are established.

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

Award Identifier / Grant number: 20-01-00207

Funding statement: This work was supported by the research grant 20-01-00207 of Russian Foundation for Basic Research.

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Received: 2021-04-06
Revised: 2021-08-13
Accepted: 2021-09-27
Published Online: 2021-10-20
Published in Print: 2022-04-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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