The solution of differential equations using numerical Laplace transforms

Abstract

The Laplace transform provides a technique for the solution of ordinary and partial differential equations for initial-value problems in which the number of independent variables is reduced by one. Ordinary differential equations become algebraic equations and equations such as the one-dimensional wave or diffusion equation become ordinary differential equations. The difficulty associated with the method manifests itself in the inversion which is required after the algebraic or ordinary differential equations have been solved. If the equations have suitable analytic solutions then the inversion may be effected either directly from tables or by using the complex inversion formula. If, however, such solutions are not suitable or if numerical solutions are obtained then inversion can cause serious problems. A numerical procedure developed some 25 years ago provides the opportunity for a straightforward numerical inversion. The method works well for ordinary and partial differential equations. In the latter case the resulting boundary-value problem may be solved by either the finite difference method, the finite element method or the boundary element method. The major advantage is that the method does not suffer from possible stability problems that may occur with the usual finite difference procedures in the time variable. The method is ideally suited to implementation on a spreadsheet.