Chapter 6: Sturm-Liouville Problems

The following sections are included:

• Introduction

• The Vibrating String

  • Fixed Ends

  • One Fixed and One Free Ends

  • Both Ends Free

  • Discussion

• Sturm Separation and Comparison Theorem

  • Examples A

  • Example B

  • Example C

• Sturm-Liouville Problems

  • Fundamental Definition

  • Properties of Eigenvalues and Eigenfunctions

  • Orthogonality of Eigenfunctions

  • Expansion of Functions

  • The Completeness Relation

• Applications

  • The Special Functions

  • Fourier Expansion of f(x) = x(1 − x)

  • Representation of δ(x) in Terms of Cosine Functions

  • Reality of the Eigenvalues

  • A Boundary Value Problem [3]

• Green’s Functions

• Worked Examples for Green Functions

  • y″(x) = –f(x); y(0) = y(L)=0

  • y″(x) = –f(x) : y(0) = 0, y′(L)=0

  • y″(x) + k²y(x) = −f(x) : y(0) = y(L)=0

  • y″(x) + y(x) = x : y(0) = y(1)=0

• Asymptotic Behavior of Solutions to Differential Equations

  • Elimination of First-Derivative Terms

  • The Liouville-Green Transformation

• Worked Examples

  • The Airy Equation

  • The Bessel Equation

  • A General Expansion Procedure

• Comments and References

• Bibliography