Numerical inversion of the Laplace transform and its application to fractional diffusion

A procedure for computing the inverse Laplace transform of real data is obtained by using a Bessel-type quadrature which is given in terms of Laguerre polynomials LN(α)(x) and their zeros. This quadrature yields a very simple matrix expression for the Laplace transform g(s) of a function f(t) which can be inverted for real values of s. We show in this paper that the inherent instability of this inversion formula can be controlled by selecting a proper set of the parameters involved in the procedure instead of using standard regularization methods. We demonstrate how this inversion method is particularly well suited to solve problems of the form L−1[sg(s);t]=f′(t)+f(0)δ(t). As an application of this procedure, numerical solutions of a fractional differential equation modeling subdiffusion are obtained and a mean-square displacement law is numerically found.