Fully numerical Laplace transform methods

The role of the Laplace transform in scientific computing has been predominantly that of a semi-numerical tool. That is, typically only the inverse transform is computed numerically, with all steps leading up to that executed by analytical manipulations or table look-up. Here, we consider fully numerical methods, where both forward and inverse transforms are computed numerically. Because the computation of the inverse transform has been studied extensively, this paper focus mainly on the forward transform. Existing methods for computing the forward transform based on exponential sums are considered along with a new method based on the formulas of Weeks. Numerical examples include a nonlinear integral equation of convolution type, a fractional ordinary differential equation, and a partial differential equation with an inhomogeneous boundary condition.