1. Sturm-Liouville Theory
ECE 6382
Fall 2016
David R. Jackson
Notes 18
Sturm-Liouville Theory
Notes are from D. R. Wilton, Dept. of ECE

2. Sturm-Liouville Differential Equations

3. Homogeneous Boundary Conditions
Note:
This Includes Dirichlet and
Neumann boundary conditions
as special cases!

4. Sturm-Liouville Form

5. Sturm-Liouville Operator
This is called the self-adjoint form of the differential equation:
or (using u instead of y):
Where L is the (self-adjoint) “Sturm-Liouville” operator:

6. The Adjoint Problem

7. The Adjoint Problem (cont.)

8. The Adjoint Problem (cont.)

9. The Adjoint Operator (cont.)
(see note below)
Note: In our definition of inner product, the order of the terms inside the inner product is not important, but in other definitions it might be.

10. Boundary Conditions

11. Boundary Conditions (cont.)

12. Adjoint in Linear Algebra
Note:
For complex matrices, the adjoint is defined as the conjugate of the transpose.
Conclusion:
A symmetric real matrix is self-adjoint.

13. Eigenvalue Problems
We often have an eigenvalue problem of the form

14. Orthogonality of Eigenfunctions

15. Orthogonality of Eigenfunctions (cont.)
The LHS is:
Hence, for the RHS we have

16. Orthogonality of Eigenfunctions (cont.)
Conclusion:
The eigenvectors corresponding to a self-adjoint operator equation are orthogonal if the eigenvalues are distinct.
Note:
This orthogonality property will be important later when we construct Green's functions.

17. Orthogonality of Eigenfunctions (cont.)
Example
Orthogonality of Bessel functions
What is w(x)?
We need to identify the appropriate DE that y(x) satisfies in Sturm-Liouville form.

18. Orthogonality of Eigenfunctions (cont.)

19. Orthogonality of Eigenfunctions (cont.)
Rearrange to put into Sturm-Liouville form:

20. Orthogonality of Eigenfunctions (cont.)
Hence, we have
Compare with our standard Sturm-Liouville form:
We ca now identify:

21. Orthogonality of Eigenfunctions (cont.)
Hence we have

22. Orthogonality in Linear Algebra

23. Diagonalizing a Matrix
(proof on next slide)

24. Diagonalizing a Matrix (cont.)
Proof
Hence, we have
Note:
The inverse will exist since the columns of the matrix [e] are linearly independent.
so that