Chapter I. Introduction to Boundary-Value Problems

Abstract

This chapter applies the theory of linear ordinary differential equations to certain boundary-value problems for partial differential equations.

Section 1 briefly introduces some notation and defines the three partial differential equations of principal interest—the heat equation, Laplace’s equation, and the wave equation.

Section 2 is a first exposure to solving partial differential equations, working with boundary-value problems for the three equations introduced in Section 1. The settings are ones where the method of “separation of variables” is successful. In each case the equation reduces to an ordinary differential equation in each independent variable, and some analysis is needed to see when the method actually solves a particular boundary-value problem. In simple cases Fourier series can be used. In more complicated cases Sturm’s Theorem, which is stated but not proved in this section, can be helpful.

Section 3 returns to Sturm’s Theorem, giving a proof contingent on the Hilbert–Schmidt Theorem, which itself is proved in Chapter II. The construction within this section finds a Green’s function for the second-order ordinary differential operator under study; the Green’s function defines an integral operator that is essentially an inverse to the second-order differential operator.