MTH 480, Spring, 2005

MTH 480: SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS, Spring, 2005

LECTURE: MWF 1500 - 1550 Bexell Hall 328 CRN 35155
Instructor: R.E. Showalter Kidder 368 show@math.oregonstate.edu

Variational methods and related concepts from classical and modern applied mathematics are introduced in the discrete setting. Models of conduction and vibration lead to large systems of linear equations and ordinary differential equations. These lead us to consider diagonalization of matrices, eigen-value problems and minimization, asymptotics of eigenvalues, separation of variables. Nonlinear topics include a selection from phase portraits, linearization and stability of equilibria, conservative systems, reversible systems, limit cycles and the Poincare-Bendixson Theorem.
Prerequisite: MTH 256 and MTH 341 or instructor approval required. MTH 480 and MTH 481 cannot both be taken for credit.
Final Exam: Tuesday, June 7, 1400.
Textbook: Notes will be available from this page.
We shall supplement and illustrate material with MATLAB exercises from the book
Ordinary Differential Equations using MATLAB by John C. Polking and David Arnold.
Class Notes:
Discrete Models of Diffusion and Vibration.pdf
Negative Poisson Ratio
Schedule:
1. Organizational meeting, introductory remarks. (3/28)
Discrete Models of Diffusion and Vibration.pdf
2. Section 1: Heat Conduction in an Interval. pp. 1-2.
3. pp. 3-4. Exercise 1: Due Friday, (4/8).
4. pp. 5-6. (4/4)
5. pp. 7-8.
6. Section 2: Vibrations in an Interval. pp. 8-9.
7. pp. 10-11. Transverse vibrations. (4/11)
8. pp. 11-13. Examples. (No class today)
9. pp. 13-14. Friction; The rotating String. (No class today)
10.pp. 15-16. Longitudinal vibrations. (4/18)
11.pp. 17-18. Viscoelasticity, Poisson Ratio , and Transverse inertia.
Exercise 9, p. 17 (updated notes): Due Monday, (4/25).
Linear Systems.pdf
12.pp. 1-2. Vector spaces, linear functions. Read Chapter 1 of the MatLab manual.
13.pp. 2-3. Coordinates, matrices, change of bases, similar matrices. (4/25)
14. p. 4. Scalar product space.
15. p. 5. Orthonormal bases, unitary operators, unitary matrices.
16. p. 6. Eigenvalues, eigenvectors. (5/2)
17. p. 7. Representation of Symmetric operators/matrices.
18. TEST #1. One problem from Discrete Models. Selection from Exercises 1 - 4 of Linear Systems.
19.pp. 9-10 Linear Systems of Ordinary Differential Equations. Read Chapter 2 of the MatLab manual. (5/9)
20. Second order equations and systems.
21. The trig functions from ODE.
22. pp. 28 - 33 of Chapter 3 of the MatLab manual. Existence of solutions. (5/16)
23. Uniqueness: monotonicity, Lipschitz conditions.
24. Uniqueness and continuous dependence estimates.
25. Test #2 due on Wednesday, 6/1. (5/23)
26. No class today.
27. pp. 93 - 98 of Chapter 7 of the MatLab manual. Existence of solutions.
Memorial Day (5/30)
28. Limit cycles (6/1)
29. Here are graphs of the solution to the discrete diffusion equation (N=20) with initial value u(x,0) = sin(pi x) and boundary conditions u(0,t) = u(1,t) = 0.
Here are graphs of the solution with initial value u(x,0) = 1 and boundary conditions u(0,t) = u(1,t) =0.