Chapter Nine - The BEM for Time Dependent Problems

Abstract

In this chapter, the boundary element method (BEM) is developed for solving problems described by the general second order hyperbolic and parabolic partial differential equations with variable coefficients. Though a reciprocal identity can be derived for these equations using time dependent reciprocal theorems, the establishment of the fundamental solution, except for some simple equations, is out of question. Even in cases where a time dependent solution is available, the implementation of the BEM is a tedious task and requires special care. Instead, time dependent problems can be efficiently solved directly in the time domain using the simple known fundamental solution of the Laplace equation. The analog equation method (AEM) is suitable to solve these problems, thus, rendering the BEM an efficient computational tool for solving all problems described by the general second order hyperbolic or parabolic equations and systems of them as well. It also applies to equations involving time fractional derivatives, such as the wave-diffusion equation, which results in the hyperbolic and parabolic equations as special cases. The efficiency of the AEM is illustrated through well corroborated examples.