Numerical inversion of 2-D Laplace transforms by fast Hartley transform computations

Numerical inversion of 2-D Laplace transforms applied to fractional diffusion equations

“…Even for a restricted class of transforms, we are not able to provide rigorous error criteria. However, we have found a simple empirical error estimate for a large class of transforms which we label F. In fact, these are the error estimates given by (12) and (18). They seem to be valid even though the theoretical basis for (12) and (18) …”

“…Note, for a fixed value of M, the accuracy decreases as t increases. That is, the error estimates (12) and (18) do not apply for transforms in class G.…”

“…Note, the FT routine in the module requires a precision of 3M. Recall the output of FT will have precision of about (0.6)(3M), as given by (12), which is the required input precision for the routine GWR.…”

“…We are aware of seven numerical procedures that have been published for double Laplace transform inversion [20,11,6,18,12,17,2]. All these procedures are multivariate extensions of well-known inversion methods for one-dimensional Laplace transforms.…”

“…The procedures [20,11] are based on a numerical inversion scheme which we call the Zakian method. The procedures [6,18,12] are based on the Fourier series method while [17,2] are based on the Laguerre method. In Section 2, we discuss these various methods and their merits.…”