Approximating fractional derivative of the Gaussian function and Dawson's integral

TL;DR

This paper introduces a novel method for approximating fractional derivatives of the Gaussian function and Dawson's integral using a moment problem approach, providing error bounds and extending to derivatives with respect to the order.

Contribution

The paper presents an alternative to discretization-based methods by expressing fractional derivatives as weighted sums, enabling extensions to PDE optimization problems.

Findings

Provides error bounds for the approximation

Expresses derivatives as weighted sums of Gaussian and Dawson's integral

Extends to partial derivatives with respect to the fractional order

Abstract

A new method for approximating fractional derivatives of the Gaussian function and Dawson's integral are presented. Unlike previous approaches, which are dominantly based on some discretization of Riemann-Liouville integral using polynomial or discrete Fourier basis, we take an alternative approach which is based on expressing the Riemann-Liouville definition of the fractional integral for the semi-infinite axis in terms of a moment problem. As a result, fractional derivatives of the Gaussian function and Dawson's integral are expressed as a weighted sum of complex scaled Gaussian and Dawson's integral. Error bounds for the approximation are provided. Another distinct feature of the proposed method compared to the previous approaches, it can be extended to approximate partial derivative with respect to the order of the fractional derivative which may be used in PDE constraint optimization problems.