MTH 621: Differential and Integral Equations of Mathematical Physics (Fall, 2010)

LECTURE:	MWF 1300 -- 1350	Weniger 201	CRN 12694
Instructor:	R.E. Showalter	Kidder 286	show@math.oregonstate.edu

Office Hours Tuesday 1200 - 1350 (TBA) Dec 7.

Partial differential equations arise as basic models of flow, diffusion, dispersion, and vibration. Topics include first and second order partial differential equations and classification, particularly the wave, diffusion, and potential equations, their origins in applications and properties of solutions, characteristics, maximum principles, Green's functions, eigenvalue problems, and Fourier expansion methods.

Prerequisite: 6 credits of senior-level analysis. Must be taken in order. May be repeated up to 6 credits.

Essentially complete notes for the course will be provided online during the term.
Supplementary course material for all three terms MTH 621-622-623 is available in the (optional)
Reference: L. C. Evans, Partial Differential Equations: Second Edition

Final Exam Problems: Thursday, December 9, 1200.

Schedule:
1. The Transport Equation.pdf Read Evans: Introduction pp. 1-12 sometime (and again). (9/27)
2. Ordinary Differential Equations Read pp. 1-2 of Introduction.pdf.
3. Cauchy-Picard Theorem Read pp. 3-4 in Introduction.pdf
4. Quasi-linear first order equations Read pp. 6-8 of Introduction.pdf. (10/04)
5. Read pp. 9-11 of Introduction.pdf. See Evans 3.1, 3.2 (pp. 91-115) for the fully-nonlinear case.
6. Second Order PDE Read pp. 12-14 of Introduction.pdf.
7. Characteristics & Classification Read pp. 14-16 of Introduction.pdf. (10/11)
8. Characteristics & Discontinuities Read pp. 16-19 of Introduction.pdf.
9. Classification in R^n Read pp. 19-22 of Introduction.pdf.

10. The Cauchy problem Read pp. 1-6 of The Wave Equation.pdf. (10/18)
11. The Divergence Theorem. Read pp. 30-33 in The Wave Equation.
12. The Wave Equation Read pp. 16-17, 7.
13. The Semi-linear equation. Read pp. 7-10 in The Wave Equation. (10/25)
14. The Goursat Problem Read Evans: Wave equation pp. 65-85.
15. The Initial-Boundary-Value Problem Read pp. 11-13 in The Wave Equation.
16. Huygen's Principle Read pp. 17-24 of The Wave Equation. (11/01)
17. Energy Integrals Read pp. 25-27 of The Wave Equation.
18. Read pp. 28-29 of The Wave Equation.

19. Read pp. 1-3 in The Diffusion Equation.pdf.(11/08)
20. Lp estimates Read p. 4 in Diffusion Equation.pdf and Energy Methods in Evans: pp. 44, 62-63.
21. Maximum principle Read pp. 5-6 in Diffusion Equation.pdf
22. Fundamental solution Read pp. 7-8 in Diffusion Equation.pdf and Evans pp. 45-46. (11/15)
23. Initial-value problem Read pp. 9-10 in Diffusion Equation.pdf and Evans pp. 47-48.
24. Duhamel formula: The Nonhomogeneous Equation. Read pp. 10-12 and Evans pp. 49-51.
25. Quarter-plane and Cylinder IBVPs. Read pp. 15-17. (11/22)
26. Holiday.
27. IBVP on Cylinder: Fourier expansion (11/29)
28. Duhamel formula: The Wave Equation
29. IBVP on Cylinder: Fourier expansion
Problems