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This book gives background material on the theory of Laplace transforms, together with a fairly comprehensive list of methods that are available at the current time. Computer programs are included for those methods that perform consistently well on a wide range of Laplace transforms. Operational methods have been used for over a century to solve problems such as ordinary and partial differential equations.
The classical theory of the Laplace Transform can open many new avenues when viewed from a modern, semi-classical point of view. In this book, the author re-examines the Laplace Transform and presents a study of many of the applications to differential equations, differential-difference equations and the renewal equation.
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.