M257-316Notes_Lecture31

M257-316Notes_Lecture31 - Chapter 27 Lecture 31...

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Chapter 27 Lecture 31 Sturm-Liouville Theory 27.1 Boundary value problems and Sturm-Liouville theory: Up till now we have been able to solve partial differential equations by separating variables which typically reduces the PDE to solving an eigenvalue problem and some initial value problem (for the Heat and Wave Equations) or some inhomogeneous boundary value problem (in the case of Laplace’s Equation). The eigenvalue problems thus far have been simple 1. x + λ 2 X = 0 X (0) = 0 = X ( L ) λ n = L n = 1 , 2 , . . . X n = sin nπx L . 2. X + λ 2 X = 0 X (0) = 0 = X ( L ) λ n = L n = 0 , 1 , . . . X n = cos nπx L . 3. X + λ 2 X = 0 X ( x + 2 π ) = X ( x ) 2 π periodic BC λ n = L , n = 0 , 1 , . . . X n = sin ( nπx L ) cos ( nπx L ) X 0 = 1 . 157

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Lecture 31 Sturm-Liouville Theory In this section we abstract these problems to a general class of bound- ary value problems which share a common set of properties. The so- called Sturm-Liouville Problems define a class of eigenvalue prob- lems which include many of the above problems as special cases. The S L Problem helps to identify those assumptions that are needed to define an(delete the word an?) eigenvalue problems with the properties
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