Preparing for Third Exam: Sturm-Liouville Problems and Phase Portraits - Prof. Kenneth How, Study notes of Mathematics

Download Preparing for Third Exam: Sturm-Liouville Problems and Phase Portraits - Prof. Kenneth How and more Study notes Mathematics in PDF only on Docsity! Preparing for the Third Exam The test covers all the material in sections §36.2 through §38.5 of the lecture notes. This includes the construction and use of phase portraits for nonlinear systems, simple boundary-value problems, and the material on Sturm-Liouville problems up to what we discussed last Friday. Also review the assigned homework from Mon. 3/9 through Fri. 4/3. Most of the problems will be taken from the material on Sturm-Liouville problems. In particular: ♦ As usual, know the basic notation and terminology. I will not ask for definitions, but if, for example, I ask a question involving “critical points” or “Sturm-Liouville problems”, I expect you to know what these things are well enough to answer the question. ♦ Be able to find critical points, compute Jacobians, find eigenvalues and eigenvectors, etc., just as you had to do on the last test, but do not expect to do many of these computations. I am more likely to give you a nonlinear system describing some sort of “competing species” model with the critical points already found and with the appropriate eigenvalues and eigenvectors given at each critical point. You will then have to draw a rough phase portrait of the trajectories, and tell me a little about what this phase portrait is telling you about the eventual outcome of the system. ♦ Keep in mind that the material in chapter 37 of the notes was mainly to prepare you for chapter 38. You may have to solve one or more boundary-value problems, but that may be part of finding all the solutions to some Sturm-Liouville problem. ♦ I may give you an orthogonal set of vectors e fb b b" # $ß ß and have you express another vector v b Av A in terms of the s. I may also have you compute for some matrix 5 for which the s are eigenvectors. (That is, I may give you something similar tob5 exercises 38.1 — 38.3.) Expect to have to rewrite a given differential equation with in Strum-Liouville form.-♦ Expect to have to solve a simple Sturm-Liouville problem.♦ ♦ Remember that deriving the preliminary Green's formula was an exercise (exercise 38.10). It may be a problem on the test, as well ♦ Understand what an “inner product for functions over an interval with a given weight function” is. I will probably give you such an inner product and have you compute some inner products and norms. I will also probably try to come up with a clever, yet simple, problem to do regarding “orthogonality”.