Lecture 28: Sturm-Liouville Boundary Value Problems

In this lecture we abstract the eigenvalue problems that we have found so useful thus far for solving the PDEs to a general class of boundary value problems that share a common set of properties. The so-called Sturm-Liouville Problems define a class of eigenvalue problems, which include many of the previous problems as special cases. The S − L Problem helps to identify those assumptions that are needed to define an eigenvalue problems with the properties that we require. 30 Boundary value problems and Sturm-Liouville theory: 30.1 Eigenvalue problem summary • We have seen how useful eigenfunctions are in the solution of various PDEs. • The eigenvalue problems we have encountered thus far have been relatively simple (2k+1)π 2L x V: Mixed Boundary Value Problem B: X + λ 2 X = 0 X (0) = 0 = X(L) =⇒ λ k = (2k+1)π 2L , k = 0, 1, 2,. . .