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An Introduction to Partial Differential Equations by Jacob Rubinstein, Yehuda Pinchover

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6

Sturm–Liouville problems and eigenfunction expansion

6.1 Introduction

In the preceding chapter we presented several examples of initial boundary value problems that can be solved by the method of separation of variables. In this chapter we shall discuss the theoretical foundation of this method.

We consider two basic initial boundary value problems for which the method of separation of variables is applicable. The first problem is parabolic and concerns heat flow in a nonhomogeneous rod. The corresponding PDE is a generalization of the heat equation. We seek a function u (x, t) that is a solution of the problem

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The second problem is hyperbolic. ...

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