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

The problem of numerically inverting the double Laplace transform arises, for instance, in solving linear partial differential equations of fractional order on the semi-infinite domain. This work introduces two algorithms for numerical inversion of the double Laplace transform. The algorithms have only one free parameter (the number of terms in the summation) and provide increasing accuracy with increasing value of the parameter. They utilize the public domain one-dimensional routines: FT and GWR. Inherent in the algorithms is multi-precision computing with automatically adjusting the required number of precision digits. By using simple transform pairs and the example of classical diffusion in a force field we show that the procedures can provide extremely high accuracy. Then we derive the operational solution of the fractional diffusion equation subject to reflecting and absorbing boundary conditions at the origin. Finally, we compare the inversion results to the exact solutions given by Metzler and Klafter [Physica A 278 (2000) 107].