# Approximation of a fractional power of an elliptic operator

## TL;DR

This paper introduces new integral representations for fractional powers of elliptic operators, enabling more efficient numerical approximations and solutions for boundary value problems involving fractional diffusion.

## Contribution

It proposes novel integral representations and quadrature-based approximations for fractional elliptic operators, improving computational efficiency and accuracy.

## Key findings

- New integral representations for fractional elliptic operators.
- Numerical experiments demonstrate improved approximation accuracy.
- Successful application to a 2D fractional diffusion boundary value problem.

## Abstract

Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present paper, new and more convenient integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two-dimensional boundary value problem for fractional diffusion.