Approximating fractional derivative of Faddeeva function, Gaussian function, and Dawson's integral

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