Numerical Computation of Fractional Derivatives of Complex-Valued Analytic Functions

Abstract

Highly accurate numerical approximations of analytic Caputo fractional derivatives are dif-ficult to compute due to the upper bound singularity in its integral definition. However, it has been recently demonstrated that Caputo fractional derivatives of analytic functions can be numerically evaluated to double-precision accuracy by utilizing only function values in a grid. This is done by considering a modified Trapezoidal Rule (TR) and placing equispaced finite difference (FD) correction stencils at both endpoints. In terms of complex-valued analytic functions f ( z ), these fractional derivatives are multi-valued. In this paper, we provide several test functions for this numerical method of evaluating Caputo fractional derivatives. We produce figures of the principal branch of the functions’ approximated fractional derivatives, and include error plots of these approximations.