M257-316Notes_Lecture34

M257-316Notes_Lecture34 - Chapter 30 Lecture 34 Sturm...

This preview shows pages 1–3. Sign up to view the full content.

Chapter 30 Lecture 34 — Sturm Liouville Theory 30.1 Properties of SL Problems: 1. Eigenvalue Properties: (a) the eigenvalues λ are all real. (b) there are an infnite number oF eigenvalues λ j with λ 1 < λ 2 <. . j →∞ as j . (c) λ j > 0 provided α 1 α 2 < 0, β 1 β 2 > 0 and q ( x ) > 0. 2. EigenFunction Properties: Corresponding to each eigenvalue λ j there is an eigenFunction φ j ( x ) that is unique up to multiplication by a constant, and which satisFy: (a) φ j ( x ) are real and can be normalized ` R 0 r ( x ) φ 2 j ( x ) dx =1. (b) the eigenFunctions corresponding to di±erent eigenvalues are orthogonal with respect to the weight Function r ( x ): ` Z 0 r ( x ) φ j ( x ) φ k ( x ) dx =0 iF j 6 = k. (30.1) (c) φ j ( x ) has exactly j 1 zeros on 0 <x<` . 171

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture 34 — Sturm Liouville Theory 3. Expansion Property: The eigenfunctions φ j ( x ) form a complete set so that any piecewise smooth function f ( x ) can be expanded as a generalized Fourier Series: f ( x )= X n =1 c n φ n ( x ) (30.2) by orthogonality: c n = ` R 0 r ( x ) f ( x ) φ n ( x ) dx ` R 0 r ( x ) φ 2 n ( x ) dx . (30.3) 30.2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 5

M257-316Notes_Lecture34 - Chapter 30 Lecture 34 Sturm...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online