Math 4337: Homework 05a: Sturm-Liouville Theory I
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Math 4337: Homework 05a:
Sturm-Liouville Theory I
Problem 1. Each of the following differential operators are written in standard form. Put them into into Sturm-Liouville form and identify the weight function.
(a) c [u] = 
(b) c [u] = 
- 4u
(c) c [u] = r2 
+ r 
(d) c [u] = 
+ 
- 
u
(e) c [u] = 
+ 
- 
u
Problem 2. Consider the following Sturm-Liouville boundary-value problem:
r 
╱r 
← = -1
u (1) = 0
u (2) = 1.
(a) Find the solution directly by repeated manipulation and integration.
(b) Split the problem into boundary and interior parts, and find the boundary solution.
(c) Put the interior problem into Sturm-Lioville form, and identify the weight function.
(d) Find the eigenvalues and eigenfunctions for the interior problem.
(e) Transform the interior problem, solve, invert, and form the final solution.
(f) Plot 1-, 3-, and 5-term expansions of the solution, compared to the exact solution.
Problem 3. Consider the following Sturm-Liouville boundary-value problem:
uzz + 2uz - 3u = -1
u (0) = 1
u (1) = 1.
(a) Find the solution directly using the method of Undetermined Coefficients.
(b) Split the problem into boundary and interior parts, and find the boundary solution.
(c) Put the interior problem into Sturm-Lioville form, and identify the weight function.
(d) Find the eigenvalues and eigenfunctions for the interior problem.
(e) Transform the interior problem, solve, invert, and form the final solution.
(f) Plot 1-, 3-, and 5-term expansions of the solution, compared to the exact solution.
2022-12-12