Numerical evaluation of two and three parameter Mittag-Leffler functions

TL;DR

This paper introduces an efficient numerical method for evaluating the Mittag-Leffler function and its three-parameter variant using Laplace transform inversion with optimized contours, demonstrating improved accuracy and efficiency.

Contribution

It presents a novel numerical inversion technique with optimal contour selection for accurate computation of Mittag-Leffler functions, including the three-parameter case.

Findings

Method achieves high accuracy in numerical experiments.

Approach reduces computational effort and error propagation.

Effective for both two- and three-parameter Mittag-Leffler functions.

Abstract

The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.