Numerical calculations accuracy comparison of the Inverse Laplace Transform algorithms for solutions of fractional order differential equations

In the paper we present results of a numerical experiment in which we evaluate and compare some numerical algorithms of the Inverse Laplace Transform for inversion accuracy of some fractional order differential equations solutions. The algorithms represent diverse lines of approach to the subject of the numerical inversion and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for Fourier series convergence acceleration. We used C++ and Python languages and applied arbitrary precision mathematical libraries to address some crucial issues of an numerical implementation. Introductory test set includes Laplace transforms which are considered as difficult to compute as well as some others commonly applied in fractional calculus. In the main part of the evaluation, there is assessed accuracy of the numerical Inverse Laplace Transform of some popular fractional differential equations solutions, e.g., the initial value problem in case of the inhomogeneous Bagley-Torvik equation and composite fractional oscillation equation. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods which use Fourier series expansion with accelerated convergence provide the most accurate inversions. They can be applied to a wide variety of inversion problems.