Approximating fractional derivative of the Gaussian function and Dawson's integral
This work provides a novel computational tool for fractional calculus, potentially benefiting researchers in PDE-constrained optimization and related fields.
The paper presents a new method for approximating fractional derivatives of the Gaussian function and Dawson's integral using a moment problem formulation, achieving accurate results with provided error bounds. The method also extends to approximating partial derivatives with respect to the fractional order.
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.