MTH 621: Differential and Integral Equations of Mathematical Physics (Fall, 2014)

LECTURE:	MWF 1300 - 1350	Kidder 280	CRN 12279
Instructor:	R.E. Showalter	Kidder 286	show@math.oregonstate.edu

Office Hours: Monday 1500-1530, Wednesday 1400-1430.

Partial differential equations arise as basic models of flow, diffusion, dispersion, and vibration. Topics included in this first quarter include first and second order partial differential equations and their classification into types (wave, diffusion, and potential equations), their origins in applications, and properties of solutions. Classical methods (of calculus) will be used to describe characteristics, maximum principles, Green's functions, eigenvalue problems, and Fourier expansions. After a brief discussion of ordinary differential equations, we will study initial-boundary-value problems for the (first order) transport equation, the wave equation, the diffusion equation, and some of their nonlinear extensions.

Prerequisite: 6 credits of senior-level analysis.
Final Exam: Thursday, December 11, 1200-1400.
Textbook : R. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations.
Class NOTES :
Introduction.pdf
The Wave Equation.pdf
The Diffusion Equation.pdf

Schedule:
1. Ordinary Differential Equations Read pp. 1-2 of Introduction.pdf. (9/29)
2. Read pp. 2-3 of Introduction.pdf.
3. Read pp. 3-4 of Introduction.pdf.
4. The Transport Equation.pdf Text: Section 1-4, pp. 8-9. (10/06)
5. Quasi-linear first order equations Read Text: 2-1 pp. 19-23 and pp. 8-9 of Introduction.pdf.
6. Read Text: 2-2 pp. 24-26 and pp. 10-11 of Introduction.pdf.
Exercises # 1, 3, 5 on page 12. Due 10/17.
7. Burgers' equation . (10/13)
8. Second Order PDE Read Text: pp. 40-41 and pp. 13-15 of Introduction.pdf.
9. Characteristics & Classification Read Text: pp. 42-45 and pp. 16-18 of Introduction.pdf.
Read Characteristics & Discontinuities pp. 16-19 of Introduction.pdf. and Classification in R^n pp. 19-22 of Introduction.pdf.
10. Read pp. 30-33 Stokes Theorem in The Wave Equation.pdf. (10/20)
11. Read pp. 1-6 The Cauchy Problem in The Wave Equation.pdf and Text: pp. 122-124.
12. Read pp. 14-16 Riemann's Representation in The Wave Equation.pdf and Text: pp. 129-133.
Exercises # 2.2, 2.3, 2.4 on page 6 of Wave Eqn. Due 10/31.
13. The Initial-Value Problem Read Text 1-2 and 4-1 and pp. 16-17 in The Wave Equation.pdf. (10/27)
14. See p. 2 of Duhamel formula
15. Spherical mean Read Text 10-4 and pp. 18-20 in The Wave Equation.pdf.
16. Huygen's principle Read pp. 22-24 of The Wave Equation.pdf. (11/03)
17. Energy Integrals Read pp. 25-28 of The Wave Equation.pdf and Text p.102 and 10-3.
Exercises # 6.2, 6.3, 6.4 on pp. 21-22 of Wave Eqn. Due 11/14.
18. The Heat Equation Read Text 1-3 and pp. 1-3 in The Diffusion Equation.pdf.
19. Lp estimates and Maximum principle Read pp. 4-6 in Diffusion Equation.pdf (11/10)
20. Fundamental solution Read (lightly) p. 7 in Diffusion Equation.pdf and Text: pp. 166-167.
21. Initial-value problem Read pp. 8-9 in Diffusion Equation.pdf and Text: pp. 168-171.
22. Duhamel formula: Read pp. 10-12 in Diffusion Equation.pdf. (11/17)
23. Green's function Read pp. 12-15.
24. IBVP on Quarter-plane & Cylinder. Read pp. 15-17 and Text: 171-172, 174-175.
25. Eigenfunction expansion method and Section 5-1 of Text.(11/24)
26. No class.
27. Rotating string Read Text: 7-1. (12/01)
28. Eigenvalue problems, Green's function & Heat equation Read Text: 7-2.
29. Wave Equation revisited. Read Text: 4-2, 4-3.

Final Problem Set