Approximate representation of the solutions of fractional elliptical BVP through the solution of parabolic IVP

TL;DR

This paper presents a method to approximate solutions of fractional elliptic boundary value problems by transforming them into parabolic initial value problems, using quadrature formulas and computational experiments for validation.

Contribution

It introduces an integral representation approach for fractional elliptic BVPs via parabolic IVPs and constructs quadrature formulas to improve solution accuracy.

Findings

Quadrature formulas effectively approximate fractional elliptic solutions.

Key parameters significantly influence approximation accuracy.

Computational experiments validate the method's effectiveness.

Abstract

Boundary value problem for a fractional power of an elliptic operator is considered. An integral representation by means of a standard solution problem for parabolic equations is used to solve such problems. Quadrature generalized Gauss-Laguerre formulas are constructed. We examine the effect of key parameters on the accuracy of the approximate solution: the number of nodes of the quadrature and fractional power of the operator. Computational experiments were performed to model two-dimensional problem with a fractional power of an elliptic operator.