Chapter 10: Nonhomogeneous Initial Boundary Value Problems

In previous chapters, particularly those exploring solution techniques for the heat and wave equations, some examples and exercises have included nonhomogeneous boundary conditions and ad hoc methods for finding the solutions to these initial boundary value problems were presented. The purpose of this chapter is to present a generally applicable method for solving nonhomogeneous initial boundary value problems. This method will systematically solve nonhomogeneous problems by breaking them into a collection of related initial boundary value problems in which the nonhomogeneous terms are isolated and separated in the various partial differential equations, boundary conditions, or initial conditions. The main result is driven by Duhamel’s Principle which is closely related to a method used to find particular solutions to linear, nonhomogeneous ordinary differential equations called variation of parameters.