Unit 11 of 13
Boundary Problems and Eigenvalue Methods
- 5 lessons
- Videos 62 to 66
Conditions at both ends of an interval can allow no solution, exactly one or infinitely many. This unit studies solvability and builds eigenfunctions, Green's functions and beam modes.
A resonant linear boundary problem has a solution only when the forcing satisfies a compatibility condition, and then it has infinitely many. Dirichlet, Neumann, Robin and periodic conditions give different eigenvalues and eigenfunctions, sometimes with a constant zero mode. A regular Sturm-Liouville problem with separated boundary conditions has real simple eigenvalues and eigenfunctions orthogonal under a weight. A Green's function gives the solution as an integral against the forcing, and the Fredholm alternative covers the resonant case. Beams lead to fourth-order eigenvalue problems.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Solvability and Eigenfunctions
Boundary-Value Problems and Solvability
Why a linear boundary-value problem has no solution, one or infinitely many, and how the boundary map, the kernel of the homogeneous problem, resonance and a compatibility condition decide unique solvability.
Video 62
Eigenvalues and Eigenfunctions
How boundary conditions select the eigenvalues and eigenfunctions of an interval problem, how Dirichlet, Neumann, Robin and periodic conditions change them, when a zero mode appears, and how eigenfunctions are normalized.
Video 63
Sturm-Liouville Theory and Green's Functions
Regular Sturm-Liouville Problems
How self-adjoint form, a positive leading coefficient and weight, and separated boundary conditions define a regular Sturm-Liouville problem with simple real eigenvalues and weighted orthogonal eigenfunctions.
Video 64
Green's Functions and the Fredholm Alternative
How a Green's function solves a boundary-value problem as an integral kernel, why its slope jumps at the source, and how the Fredholm alternative and its compatibility condition handle an eigenvalue obstruction.
Video 65
Fourth-Order Problems and Beams
Higher-Order Eigenvalue Problems and Beam Models
How the Euler-Bernoulli beam leads to a fourth-order eigenvalue problem, how clamped, pinned and free ends become conditions on the mode shape, and how beam mode shapes and natural frequencies follow.
Video 66