Sturm-Liouville boundary value problems

  • Martin Braun
Part of the Texts in Applied Mathematics book series (TAM, volume 11)


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
$$f\left( x \right) = \sum\limits_{n = 1}^\infty{{b_n}\sin \frac{{n\pi x}}{l}} $$
or a pure cosine series of the form
$$f\left( x \right) = \frac{{{a^0}}}{2}\sum\limits_{n = 1}^\infty{{a_n}\cos \frac{{n\pi x}}{l}} $$
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
$$y'' + \lambda y = 0,y\left( 0 \right) = 0,y\left( l \right) = 0,$$
$$y'' + \lambda y = 0,y'\left( 0 \right) = 0,y'\left( l \right) = 0.$$
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 = λn = n2π2/l2. These special values of λ were called eigenvalues, and the corresponding solutions were called eigenfunctions.


Nontrivial Solution Product Space Orthogonal Basis Hermitian Operator Laguerre Polynomial 
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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Martin Braun
    • 1
  1. 1.Department of Mathematics, Queens CollegeCity University of New YorkFlushingUSA

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