# Numerical solution of time-dependent problems with fractional power elliptic operator

## TL;DR

This paper develops numerical methods for solving unsteady space-fractional elliptic equations using finite element spatial discretization and Pade-type approximations for fractional powers, with demonstrated effectiveness through 2D tests.

## Contribution

It introduces a novel approach combining finite element methods with Pade-type approximations for time-dependent fractional elliptic problems.

## Key findings

- Effective finite element discretization in space.
- Successful implementation of Pade-type approximations for fractional powers.
- Numerical experiments confirm method accuracy for 2D problems.

## Abstract

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Pade-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.