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

Sturm-Liouville Differential Equations

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

Sturm-Liouville Form

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:

The Adjoint Problem

The Adjoint Problem (cont.)

The Adjoint Problem (cont.)

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.

Boundary Conditions

Boundary Conditions (cont.)

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.

Eigenvalue Problems We often have an eigenvalue problem of the form

Orthogonality of Eigenfunctions

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

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.

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.

Orthogonality of Eigenfunctions (cont.)

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

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

Orthogonality of Eigenfunctions (cont.) Hence we have

Orthogonality in Linear Algebra

Diagonalizing a Matrix (proof on next slide)

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