MTH 623: Differential and Integral Equations of Mathematical Physics


LECTURE: MWF 1300 - 1350 Weniger 285 CRN 52465
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
We shall begin with a general discussion of Fourier series, orthonormal bases and eigenvalue problems. These are illustrated with examples from heat conduction and vibration in one spatial dimension, and these lead to the resolution by separation of variables (Faedo-Galerkin method) of corresponding initial-boundary-value problems. Then we extend these to the variational theory for weak solutions of elliptic boundary-value problems in Sobolev space. The associated eigenvalue problems are developed together with representation of Green's functions and the solution of the initial-boundary-value problems for heat and wave equations in higher dimension. Finally, we develop the semigroup theory and its application to the resolution of initial-boundary-value problems of flow, diffusion, and vibration.
Prerequisite: 6 credits of senior-level analysis.
Final Exam: Wednesday, June 8, 1800.
Textbook: R. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations.
Class Notes :
VMH : Variational Method in Hilbert Space.pdf,
HSM : Hilbert Space Methods for PDE,
SS : Stokes System.
Schedule:
Expansion in Eigenfunctions.
1. VMH (Section 4): pp. 24 - 25, Fourier Series. Text: Section 7-6, pp. 254-256, Section 11-3, pp. 448-453. (03/28)
2. VMH: pp. 26 - 27, Eigenvector expansions.
3. VMH: pp. 27 - 28.
4. VMH: pp. 29, Boundary-Value Problems. (04/04)
5. VMH: pp. 30-31, Compact Resolvent Operator. Exercises due April 15.
6. VFM: pp. 31 - 33, The Expansion Theorem for BVPs. Text: Fourier Series, Sections 3-1 to 3-3: pp. 47-72.
7. VFM: pp. 34 - 35, The Sturm-Liouville Eigenvalue Problem. Text: Section 7-1, pp. 219-222. (04/11)
8. VFM: pp. 35 - 36, Wave Equation. Text: Section 7-10, pp. 290-293. See Duhamel formulae.
9. VFM: p. 37, Diffusion Equation. Exercises 14 (p. 36) & 16 (p. 37) due April 22. (See Text: p. 217.)
Elliptic Boundary-Value Problems.
10. HSM Chapter 2: pp. 31 - 40, distributions. (04/18)
11. HSM pp. 40 - 42, Sobolev space, H^m(G).
12. HSM pp. 43 - 48, localization, Boundary Trace.
13. HSM Chapter 3: pp. 59 - 61, Text: Section 11-4,5, pp. 458-464. (04/25)
14. HSM pp. 62 - 65, general Green's operator.
15. HSM pp. 66 - 68, 2nd order Elliptic BVPs.
16. HSM pp. 69 - 71, Neumann, Robin, & Interface BVPs. Exercise 3.2 on page 90 of HSM due May 13. (05/02)
17. HSM pp. 72 - 76, coercivity, Garding's inequality.
18. HSM pp. 86 - 89, eigenfunction expansions.
19. Periodic BVP. (05/09)
20. SS pp. 1 - 3, Stokes System, weak solution. Text: p. 490.
21. Review.
22. Stress & Strain. (05/16)
23. Momentum equation, and SS pp. 4 - 5, strong solution.
24. SS pp. 5 - 6, normal trace.
25. SS pp. 7 - 8, mixed problem. (05/23)
Semi-groups of Operators.
26. HSM Chapter 4: pp. 95 - 100, Cauchy problems to Semigroups.
27. HSM pp. 100 - 102, Semigroups to Generators.
28. No class. (05/30)
29. HSM pp. 102 - 105, Generators to Cauchy problems.
30. "There and Back Again".