A Singularly Perturbed Boundary Value Problems with Fractional Powers of Elliptic Operators

TL;DR

This paper addresses the numerical solution of singularly perturbed boundary value problems involving fractional powers of elliptic operators, using time-dependent pseudo-parabolic equations and weighted schemes, with results demonstrated on a 2D model.

Contribution

It introduces a numerical approach for fractional elliptic boundary problems with small perturbation parameters using pseudo-parabolic equations and standard two-level schemes.

Findings

Numerical solutions effectively handle small perturbation parameters.

The method is validated on a 2D boundary value problem.

Results demonstrate stability and accuracy of the approach.

Abstract

A boundary value problem for a fractional power $0 < \varepsilon < 1$ of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when $\varepsilon \rightarrow 0$. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. The numerical results are presented for a model two-dimen\-sional boundary value problem with a fractional power of an elliptic operator. Our work focuses on the solution of the boundary value problem with $0 < \varepsilon \ll 1$.