MTH 623: Differential and Integral Equations of Mathematical Physics, Spring, 2006

For information on the corresponding sequence of courses for the academic year 2005-2006, see MTH 621,-2,-3.



LECTURE: MWF 1400 - 1450 Kidder 236 CRN 33165
Instructor: R.E. Showalter Kidder 368 show@math.oregonstate.edu

The third term will begin with a discussion of the Laplace - Poisson equation and then develop corresponding models of flow, transport, and vibration in higher dimension. We obtain useful representations and discuss Hilbert space techniques for the corresponding boundary-value problems and the associated initial-boundary-value problems. Properties of solutions of the various types of equations will be developed and compared.
Prerequisite: 6 credits of senior-level analysis.
Final Exam: Thursday, June 15, 1800.
Textbook: R. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations.
Class Notes and Schedule:
1. Introduction (& Review); an elementary example. Text: pp. 219-222. (4/3/06)
2. Sturm-Liouville BVPs; construction of Green's function. Text: pp. 223-231.
3. The Divergence Theorem ; Conservation laws. Text: pp. 5-10.
The Potential Equation.pdf , The Potential Equation.ps
4. Text: 295-299. Notes: p. 1. Poisson equation and invariance. (4/10)
5. Text: 306-308. Notes: p. 2. Integral identities and uniqueness.
6. Text: 300-305. Solutions on special regions.
7. Poisson's formula for the circle. (4/17)
8. Text: 309-310. Notes: p. 3. Fundamental Integral Representation.
9. Text: 318. Notes: p. 5-6. Mean Value Theorems, subharmonic functions.
10. Text: 314-320. Notes: p. 7. Maximum principle. (4/24)
11. Text: 310-323. Notes: p. 8 - 9. Green's function; The Half-Space.
12. Notes: pp. 10-11. The Sphere.
13. Poisson's formula, reflection, and Weierstrass' Theorem. Notes: pp. 12-13. (5/1)
14. Exercises
15. Harnack's inequality, Monotone convergence theorem. Notes: pp. 14-15.
16. The Dirichlet Problem, Perron's method. Notes: pp. 15-16. (5/8)
17. Barriers, regular boundary, and Green's function. pp. 16-17.
. . . Exercises
The Dirichlet Principle
18. Read Chapter 11, Section 1, of Textbook. Stretched membrane, minimization principle.
19. Chapter 11, Section 4, of Textbook. L^2(G) and Generalized derivatives. Exercises (5/15)
20. Sobolev space, H^1(G), and Boundary Trace for the Half-space.
21. Kernel of trace operator.See Chapter II, Section 3 of HSM.
22. Boundary Trace for H^1(G). See Chapter III, Section 1 of HSM. (5/22)
23. The Dirichlet Problem. Homework: Pick one of the two preceding Exercises (either 3 and 4 or 5 and 6); due Friday, June 2.
24. No Class Friday, 5/26
No Class Monday : Memorial Day. (5/29)
25. Mixed, Neumann, Robin Boundary-value problems.
26. Periodic Boundary-value problems.
27. Advection-Diffusion Boundary-value problems. (6/5)