Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

DOI10.1016/J.CAM.2022.114096arXiv2105.10432WikidataQ114201908 ScholiaQ114201908MaRDI QIDQ6368230

Publication date: 21 May 2021

Abstract: Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. In this paper, the possibility of using approximation by exponential products is noted. The solution of an equation with an operator function is based on the transition to standard stationary or evolutionary problems. General approaches are illustrated by a problem with a fractional power of the operator. The first class of methods is based on the integral representation of the operator function under rational approximation, approximation by exponential sums, and approximation by exponential products. The second class of methods is associated with solving an auxiliary Cauchy problem for some evolutionary equation.

Mathematics Subject Classification ID

Fractional derivatives and integrals (26A33) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) A priori estimates in context of PDEs (35B45) Numerical quadrature and cubature formulas (65D32) Fractional partial differential equations (35R11) Numerical computation of matrix exponential and similar matrix functions (65F60)

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