# Numerical Approximation of Space-time Fractional Parabolic Equations

## TL;DR

This paper introduces a novel numerical scheme for space-time fractional parabolic equations, utilizing Dunford-Taylor integrals, finite element methods, and sinc quadrature to achieve optimal space discretization and efficient fully discrete solutions.

## Contribution

It develops a new numerical approach combining finite element approximation with Dunford-Taylor integrals and sinc quadrature for space-time fractional parabolic equations, avoiding traditional time-stepping.

## Key findings

- Optimal order space error up to logarithm of 1/h
- Fully discrete method free of time stepping
- Effective approximation of convolution via sinc quadrature

## Abstract

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e., an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator $E(t)$ for the initial value problem can be written as a Dunford-Taylor integral involving the Mittag-Leffler function $e_{\alpha,1}$ and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here $\alpha$ denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator $W(t)$ and the forcing function $F(t)$. We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford-Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of $1/h$. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying an exponentially convergent sinc quadrature technique to approximate the Dunford-Taylor integral of the discrete operator and is free of any time stepping. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo midpoint quadrature. This involves the average of $W_h(s)$, (the semi-discrete approximation to $W(s)$) over the quadrature interval. This average can also be written as a Dunford-Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford-Taylor integrals for computing the averages.