Sturm-Liouville boundary value problems

Abstract

In Section 5.5 we described the remarkable result that an arbitrary piecewise differentiable function f(x) could be expanded in either a pure sine series of the form

$$f\left( x \right) = \sum\limits_{n = 1}^\infty{{b_n}\sin \frac{{n\pi x}}{l}} $$

(1)

or a pure cosine series of the form

$$f\left( x \right) = \frac{{{a^0}}}{2}\sum\limits_{n = 1}^\infty{{a_n}\cos \frac{{n\pi x}}{l}} $$

(2)

on the interval 0 < x < l. We were led to the trigonometric functions appearing in the series (1) and (2) by considering the 2 point boundary value problems

$$y'' + \lambda y = 0,y\left( 0 \right) = 0,y\left( l \right) = 0,$$

(3)

and

$$y'' + \lambda y = 0,y'\left( 0 \right) = 0,y'\left( l \right) = 0.$$

(4)

Recall that Equations (3) and (4) have nontrivial solutions

$${y_n}\left( x \right) = c\sin \frac{{n\pi x}}{l}and{\kern 1pt} {y_n}\left( x \right) = c\cos \frac{{n\pi x}}{l},$$

respectively, only if λ = λ _n = n ₂ π ₂/l ₂. These special values of λ were called eigenvalues, and the corresponding solutions were called eigenfunctions.