Application: Solution of a Boundary Value Problem

Abstract

The two-point boundary value problem

$$\begin{array}{*{20}c} { - y'' - \lambda y = 0} \\ {y\left( 0 \right) = y\left( 1 \right) = 0} \\\end{array}$$

((1))

occurs, e.g. when we separate the variables in the wave equation

$$\frac{{\partial ^2 z}}{{\partial x^2 }} = \frac{{\partial ^2 z}}{{\partial t^2 }}$$

by assuming that

$$z\left( {x,t} \right) = y\left( x \right)\exp i\sqrt {\lambda t.} $$

If the boundary conditions are

$$\begin{array}{*{20}c} {z\left( {0,t} \right) = z\left( {1,t} \right) = 0,} \\ {z\left( {x,0} \right){\rm{given}}} \\\end{array}$$

the problem can be interpreted in terms of the vibrations of a uniform string, with fixed end-points (x=0, x = 1) and with initial displacement z(x, 0).