Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation

A method is presented for numerically inverting a Laplace transform that requires, in addition to the transform function itself, only sine, cosine, and exponential functions. The method is conceptually much like the method of Dubner and Abate, which approximates the inverse function by means of a Fourier cosine series. The method presented here, however, differs from theirs in two important respects. First of all, the Fourier series contains additional terms involving the sine function selected such that the error in the approximation is less than that of Dubner and Abate and such that the Fourier series approximates the inverse function on an interval of twice the length of the corresponding interval in Dubner and Abate's method. Second, there is incorporated into the method in this paper a transformation of the approximating series into one that converges very rapidly. In test problems using the method it has routinely been possible to evaluate inverse transforms with considerable accuracy over a wide range of values of the independent variable using a relatively few determinations of the Laplace transform itself.