Differential Equations, 3rd Edition

11Partial Differential Equations and Boundary Value Problems

DOI: 10.1201/9781003214526-11

• Boundary value problems

• Ideas from physics

• The wave equation

• The heat equation

• The Laplacian

• The Dirichlet problem

• The Poisson integral

• Sturm–Liouville problems

11.1 Introduction and Historical Remarks

In the middle of the eighteenth century, much attention was given to the problem of determining the mathematical laws governing the motion of a vibrating string with fixed endpoints at 0 and π (Figure 11.1). An elementary analysis of tension shows that, if $y(x,t)$ denotes the ordinate of the string at time t above the point x, then $y(x,t)$ satisfies the wave equation

$$\begin{array}{l}\frac{{\partial }^{2}y}{\partial t^2}=a^2\frac{{\partial }^{2}y}{\partial x^2}.\hfill \end{array}$$

Figure 11.1 The wave equation.

Here a is a parameter that depends on ...