Numerical Inversion of Laplace Transforms by Relating Them to the Finite Fourier Cosine Transform

Abstract

In this paper the problem of readily determining the inverse Laplace transform numerically by a method which meets the efficiency requirements of automatic digital computation is discussed. Because the result inverse function is given as a Fourier cosine series, the procedure requires only about ten FORTRAN statements. Furthermore, it does not require the use of involved algorithms for the generation of any special functions, but uses only cosines and exponentials. The basis of the method hinges on the fact that in evaluating the inverse Laplace transform integral there exists a freedom in choosing the contour of integration. Given certain restrictions, the contour may be any vertical line in the right-half plane. Specifying a line, the integral to be evaluated is essentially a Fourier integral. However, the method is concerned with determining the proper line, so that when the integral (along this line) is approximated, the error is as small as desired by virtue of having chosen the particular contour.

A reduced‐order model for chemical species transport in a tube with a constant wall concentration

“…By applying a numerical Laplace inversion technique, [25,26] the time domain solutions can be determined for the problem under study. Here, the method of Fourier series [27] is used to obtain the concentration in the time domain as follows:…”

Viscoelastic Plate Analysis Based on Gâteaux Differential

“…In this study, Maximum Degree of Precision (MDOP), Durbin's, and Dubner and Abate's methods are employed. Papers for discussion of Laplace inversion process, see [17][18][19][20].…”

Partitions Approximated by Finite Cosine-Series

“…The Laplace transform of each triangle is easy to derive, and (2) is the sum of these transforms. However, from (1), oo YAn)e-ns = f(e-s); n=0 therefore, (3) Gis)=s-2il-e-s)2fie~s).…”

“…For generating functions encountered in combinatorial theory it is almost always impossible to find an exact inversion for (3). However, the methods of [3,1] can be used to numerically invert these functions, but the question is whether the accuracy of the method is adequate when applied to complicated partition functions.…”

“…In the late 1960s two papers appeared about numerical inversion of Laplace transforms [3,1]. Both papers were aimed at the engineering and probability world, where functions encountered were reasonably simple and 6-digit accuracy was more than adequate.…”