We describe a FORTRAN implementation, and some related problems, of Talbot's method which numerically solves the inversion problem of almost arbitrary Laplace transforms by means of special contour integration.The basic idea is to take into account computer precision to derive a special contour where integration will be carried out.
In recent years many results have been obtained in the field of the numerical inversion of Laplace transforms. Among them, a very accurate and general method is due to Talbot: this method approximates the value of the inverse Laplace transform
f(t)
, for
t
fixed, using the complex values of the Laplace transform
F(s)
sampled on a suitable contour of the complex plane. On the basis of the interest raised by Talbot's method implementation, the author has been induced to investigate more deeply the possibilities of this method and has been able to generalize Talbot's method, to approximate simultaneously several values of
f(t)
using the same sampling values of the Laplace transform. In this way, the only unfavorable aspect of the classical Talbot method, that is, that of recomputing all of the samples of
F(s)
for each
t
, has been eliminated.
Under certain conditions, starting from the Riemann inversion
formula, which gives an explicit representation of the inverse
Laplace transform in the complex form, we derive an integral
equation, of convolution type, whose solution is the inverse
Laplace transform function.
This formula can be used if the Laplace transform has a finite
number of singularities, located everywhere in the complex plane,
and provided that their corresponding residues are known.
It only requires the knowledge of the Laplace transform function
on the real negative axis. Preliminary numerical experiments
illustrating the reliability of the inversion algorithm are
described.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.