Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and their Approximation for Fractional Calculus

Abstract

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approxi-
 mation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods
 are exact methods of inverse Laplace transforms which allow us a numerical approximation
 using Sinc methods. The inverse Laplace transform converges exponentially and does not use
 Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions
 incorporating one, two, and three parameters. The three parameter Mittag-Leffler function
 represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential
 equations of constant and variable differentiation order.