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

TL;DR

This paper reviews numerical methods for solving equations involving operator functions, especially fractional powers of operators, highlighting approaches like rational approximation, exponential sums, and auxiliary evolutionary problems.

Contribution

It introduces the use of exponential product approximation and compares different classes of methods for operator function equations.

Findings

Exponential product approximation offers a viable alternative for operator functions.

Methods based on integral representations and auxiliary problems are effective.

The paper illustrates approaches with fractional power operator problems.

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.