Abstract
Abstract In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem 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 the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.
Highlights
Fractional calculus (FC) can be interpreted as an extension of the concept of derivative operator from integer order n to arbitrary order ν, where n is an integer and ν can be a real number
Talbot developed a method for the numerical inversion of the Laplace transform, in which the inversion is approximated by the Trapezoidal rule along a special deformed contour
The GNU Multiple Precision Floating-Point Reliable Library (MPFR) [39] is an arbitrary precision package for C language and it is based on GNU Multiple-Precision Library (GMP) [40]
Summary
Fractional calculus (FC) can be interpreted as an extension of the concept of derivative operator from integer order n to arbitrary order ν, where n is an integer and ν can be a real number. NumILPT is an ill-posed inversion problem due to the inclusion of the multiplication of an exponential function of time in the inversion formula, in which algorithmic and finite precision errors can lead to exponential divergence of numerical solutions [14]. For the same purpose there can be used Fourier series expansion, within which the exponential part is approximated with the use of a Padée approximant In this context, the main task within the scope of this paper is present the results of the evaluation of some of these algorithms for the inversion accuracy of Laplace transforms frequently used in FC. The first section includes short information about the Inverse Laplace Transform, its numerical counterpart - the numerical approximation of the Inverse Laplace Transform It list all the methods used in the evaluation and gives briefly insights into their mathematical algorithms as well. The paper end with usual conclusions and extended list of references
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