Stephen P. Shipman
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Department of Mathematics
Louisiana State University
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Ordinary Differential Equations

Math 7320
Louisiana State University
Spring Semester, 2019

Prof. Stephen Shipman


Place: Lockett 232
Time: 10:30-11:50 Tuesday and Thursday

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@lsu.edu
Office Hours: By appointment


Course Description

The field of ODEs is pervasive in science and mathematics and can be fascinating and beautiful, as opposed to the contrary stereotype. We will study both linear and nonlinear ODEs. These are genuinely different areas of mathematics, as the questions that are investigated are very different in nature. In the linear theory, one studies problems of spectral theory of differential operators, such as Schrödinger operators with different kinds of potentials on different domains. This invokes delicate properties of entire and meromorphic functions, moment problems, asymptotic analysis, and of course linear algebra. New problems in inverse spectral theory continue to be interesting--where we seek to characterize the differential operators that possess certain spectral data. I also want to include some more specialized topics, such as linear systems governed by indefinite quadratic forms--this is a part of linear algebra that seems to be new to most people.

The overarching concepts for the nonlinear theory are flows of vector fields and dynamical systems. Upon that basis, one studies diverse phenomena such as bifurcations, separation of time scales, bursting (such as in neurobiology), hysteresis, stability, control systems, chaos, strange attractors, and the Noether theorem on conservation laws corresponding to continuous symmetries. We will introduce these notions and study some of them in depth.

A semester course cannot do any justice to all of these topics. My goal is twofold: (1) to present the foundational rigorous theory of ODEs, and (2) to introduce a broad variety of topics in ODEs that highlight what makes the field interesting.

Textbook

Supporting material will come from my notes and various selections from the literature.

Prerequisite

The prerequisites are real and complex analysis.

Literature

There is a bibliography of relevant works below.

Assignments and Evaluation

Routine problems will be worth 70% of the grade. Assigned problem sets are listed below.

A set of problems at the end of the semester will take the place of the final exam. These will be worth 30% of the grade.

Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.

Lecture Notes

General ideas of flows; Linear equations with constant coefficients
Normal Modes of Linear Equations
Maxwell ODEs for electromagnetics in layered media
Integral equations of Volterra type
Schrödinger ODO 1 (updated)
Schrödinger ODO 2
Existence and Uniqueness
1D and 2D systems and bifurcations
Lagrangian Mechanics

Mathematica file of ODE examples

Problem Sets

Problem Set 1: Due January 29
Problem Set 2: Due February 5
Problem Set 3: Due February 26
Problem Set 4: Due March 21
Problem Set 5: Due April 23 (Four problems)

Final Exam

Final Exam: Due Friday, May 3, 5:00 PM.

No collaboration or communication with any human being is allowed regarding any of the problems on the final exam, except that you may ask me questions for clarification. You must cite all references that you utilize in devising your solution.

Grading scale

A+: at least 95% A: at least 90% A-: at least 88%
B+: at least 85% B: at least 80% B-: at least 78%
C+: at least 75% C: at least 70% C-: at least 68%
D+: at least 65% D: at least 60% D-: at least 50%
F: less than 50%

Bibliography

  1. Vladimir I. Arnold, Ordinary Differential Equations, Springer Verlag, 1984 (Russian), 1992 (English translation by Roger Cooke).
  2. B. Malcolm Brown, Michael S. P. Eastham, and Karl Michael Schmidt, Periodic Differential Operators, Birkhäuser 2013.
  3. Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.
  4. P. Deift and E. Trubowitz, Inverse Scattering on the Line, Communications on Pure and Applied Mathematics XXXII (1979) 121-251.
  5. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Volume 19.
  6. Gerald B. Folland, Introduction to Partial Differential Equations, Second Edition, Princeton University Press, 1995. Excerpt: Chapter 0.
  7. G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science Publishers, 2001. (See Theorem 5.1.1.)
  8. I. Gohberg, P. Lancaster, and L. Rodman, Indefinite Linear Algebra and Applications, Birkhäuser 2005.
  9. Eugene M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of excitability and Bursting, MIT Press, Cambridge, MA, 2007.
  10. Erich Müller-Pfeiffer, Spectral Theory of Ordinary Differential Operators, Ellis Horwood Ltd., 1981.
  11. F. G. Tricomi, Integral Equations, Interscience Publishers, 1957.
  12. Stephen P. Shipman and Aaron T. Welters, Resonant electromagnetic scattering in anisotropic layered media, Journal of Mathematical Physics, Vol. 54, Issue 10 (2013) 103511:1-40.
  13. Stephen P. Shipman and Aaron T. Welters, Pathological scattering by a defect in a slow-light periodic layered medium, J. Math. Phys. 57 (2016) 022902.
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