| In this dissertation, we investigate three numerical methods for inverting the Laplace transform. These methods are all based on the trapezoidal-type approximations to the Bromwich integral. The first method is the direct integration method: It is a straightforward application of the trapezoidal rule to the Bromwich integral, followed by convergence acceleration techniques to sum efficiently the infinite series that arises. We identify and analyze three sources of error associated with this method, namely the discretization, truncation, and conditioning error. An integral representation of the discretization error is derived and the truncation and conditioning error are also estimated. The method contains a free parameter a (in fact, the position of Bromwich line) that can be adjusted to maximize the accuracy. We present both theoretical formulas and algorithmic techniques for selecting the optimal value of a. The second method we investigate owes to Talbot. It is likewise based on the trapezoidal-type approximation of the Bromwich integral, but uses a deformed contour. We derive a formula for the discretization error associated with this method. Based on this, we propose an algorithm for the optimal selection of the free parameters contained in Talbot's method. The third method we believe to be new. It is based on Ooura and Mori's so-called double exponential formulas for integrals of Fourier-type that we have adapted to the Laplace inversion problem. Throughout the thesis, we test our theoretical formulas and practical algorithms on a wide range of transforms, many of which are taken from the engineering literature. |
