MTH 622: Differential and Integral Equations of Mathematical Physics (Winter 2015)


LECTURE: MWF 1300 - 1350 Weniger 287 CRN 32080
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
Office Hours: Mondays 1400-1540 and by appointment.
The topic of the first half of this second term is a classical treatment of the elliptic Laplace - Poisson equation. We obtain useful representations and properties of solutions and then characterize the corresponding boundary-value problems by a minimization principle. This provides the transition to Hilbert space methods for boundary-value problems.
Prerequisite: 6 credits of senior-level analysis.
Final Exam: Monday, March 16, 1400.
Textbooks : R. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations.
RES, Hilbert Space Methods for PDEs.

The Potential Equation.pdf,
1. The Divergence Theorem ; Heat conduction. Text 1-3: pp.5-10 & 8-1: pp.295-299. (01/05)
2. Poisson equation, BVPs and Uniqueness. Text: 8-3: pp.306-308. Notes: pp. 1-3.
3. Fundamental Integral Representation. Text: 309-310. Notes: pp. 3-6. Exercise # 4 on pp. 4-5 of Notes. Due Friday 1/16.
4. Text: 314-320. Notes: pp. 6-7. Subharmonic Functions, Mean Value & Maximum principle. (01/12)
5. Text: 311-313. Notes: pp. 8-10. Green's function on Half-Space, Quadrant.
6. Text: 317-318. Notes: pp. 11-12. Green's function on Sphere: Poisson's formula.
7. MLK: No class (01/19)
8. Mean-value property, Reflection, Weierstrass. Notes: p. 13.
9. Harnack inequality, Monotone convergence theorem. Notes: pp. 14-15. Exercises # 4 p. 15 and #6 p. 12 of Notes. Due Friday 01/30.
10. Text: 329-339. The Dirichlet Problem: Perron's method. Notes: pp. 16-17. (01/26)

Variational Method in Hilbert Space.pdf
11. Section 1: Preview. Notes: pp. 1-3. Text: pp. 238-241, 254-255.
12. derivatives. pp. 4 - 6.
13. anti-derivatives. pp. 7 - 8. (02/02)
14. Section 2: Hilbert space, Sobolev spaces. pp. 9 - 10.
15. Continuity, dual space. pp. 11 - 12. Text: pp. 242-247. Exercises # 1 and #2 p.8 of VMHS. Due Friday 02/13.
16. Minimization Principle. pp. 12 - 13. (02/09)
17. Projection, Riesz operator. pp. 14 - 15.
18. Review, Exercise #2.
19. Examples. pp. 15 - 18. (02/16)
20. Examples of Variational inequalities.
21. Section 3: Approximation of solutions pp. 19 - 20. Text: 11-5.
22. Approximation & Interpolation, pp. 21 - 22. (02/23).
23. solution error estimates, pp. 23 - 24.
24. Fourier series, pp, 24 - 25.
25. Eigenvalue Problem, pp. 26 - 28. (03/02)
26. Expansion theorem, pp. 29 - 31.
27. Examples, p. 32; Text: 3-3.
28. Interpolation of subspaces, Fredholm alternative. pp. 33 - 34. (03/09)
29. Diffusion equation, pseudo-parabolic equation. pp. 35 - 36; Text: 6-5.
30. Wave equation, strong damping, inertia. pp. 36 - 38; Text: 6-4.
Final Exercises