Numerical Inversion of Laplace Transforms

In this chapter we present some procedures for numerically computing inverse Laplace transforms. Several methods based on different approaches have been proposed during the past fifty years. The numerical computation of an inverse Laplace transform is not readily done. It is a so-called ill-conditioned or ill-posed problem. In Chapter 6 we saw that the inverse Laplace transform is essentially determined by the poles and branch points of the image function, hence by the singularities of the image. Thus, as it was pointed out, asymptotic analysis is a naturally adapted tool that can help to find information about an inverse Laplace transform. It is now clear that if the singularities, i.e. the behavior of the image function near the singularities, essentially determine the corresponding original function, numerical methods basically are ill adapted for the computation of the original. Nevertheless, several methods were proposed. Some methods are working astonishingly well for certain image functions, while giving poor results for other examples. Because there is no method that is the best in all situations, we propose to use several different methods for a given inversion problem. If two or more methods give about the same result, you can be more confident and accept the numerically computed inverse Laplace transform. It is a feature common to the different numerical inversion methods that they work the better, the smoother the original f(t) is. Singularities of F(s) at infinity have, in general, an unfavourable influence on the precision. Image functions which are readily analytically inverted can offer tough problems for a numerical inversion, on the other hand, there are image functions that are very hard or impossible to transform back analytically, but can be numerically inverted very efficiently. Many useful informations about the numerical inversion of Laplace transforms can be found in Peter Valko’s Website: http://pumpjack.tamu.edu/~valko/.