Splitting Schemes for Non-Stationary Problems with a Rational Approximation for Fractional Powers of the Operator

TL;DR

This paper develops stable splitting schemes with weights for rational approximations of fractional powers of operators, enabling efficient numerical solutions of non-stationary problems involving fractional operators.

Contribution

It introduces novel splitting schemes with weights for rational approximations of fractional powers, improving stability and efficiency in time-dependent fractional operator problems.

Findings

Successfully applied to a 2D non-stationary fractional Laplace problem.

Demonstrated stability and accuracy of the proposed schemes.

Showed potential for use in various fractional operator problems.

Abstract

Problems of the numerical solution of the Cauchy problem for a first-order differential-operator equation are discussed. A fundamental feature of the problem under study is that the equation includes a fractional power of the self-adjoint positive operator. In computational practice, rational approximations of the fractional power operator are widely used in various versions. The purpose of this work is to construct special approximations in time when the transition to a new level in time provided a set of standard problems for the operator and not for the fractional power operator. Stable splitting schemes with weights parameters are proposed for the additive representation of rational approximation for a fractional power operator. Possibilities of using similar time approximations for other problems are noted. The numerical solution of a two-dimensional non-stationary problem with a fractional power of the Laplace operator is also presented.