The Numerical Inversion of the Laplace Transform: Gaver-stehfest, Piessens, and Regularized Collocation Methods

The Laplace transform ? is an important integral transform with several applications in physics and engineering. It is used in the analysis of time?invariant systems such as electrical circuits, mechanical systems, optical devices, harmonic oscillators, etc. In the case when the Laplace transform is measured, computed or known only on the real positive axis, the problem of reconstructing the original function is extremely ill?posed. In this case, stable inversion formulas do not exist. As a result, the author examined two known numerical inversion algorithms: the Gaver?Stehfest and the Piessen's method, and proposed a regularized collocation inversion method based on Tikhonov regularization. Most of the chapters are devoted to a review of integral equations, inverse and ill-posed problems, and regularization of ill?posed problems. However, the last two chapters focus on the mathematical derivation of the inversion methods, as well as their numerical implementation and results. This specialized book is intended for advanced undergraduate and graduate students in mathematics, engineering, and physics. Mathematicians, scientists, and engineers will also find this book useful.