MTH 622, Winter, 2006

MTH 622: Differential and Integral Equations of Mathematical Physics, Winter, 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 280 CRN 23185
Instructor: R.E. Showalter Kidder 368 show@math.oregonstate.edu

This second term is an introduction to partial differential equations in two variables, including the first-order transport equation, the second-order wave equation and the diffusion equation in one spatial dimension. Each of these equations is introduced as a model of basic convection, diffusion or vibration processes, and their role in applications is emphasized. The discussion is aimed at the classification of equations, properties of the solutions of each type, and construction and representation of solutions of initial-value problems in the half-plane and of the simplest initial-boundary-value problem in the quarter plane. We also develop the expansion theory for compact self-adjoint operators in Hilbert space and its application in the representation of solutions to initial-boundary-value problems.
Prerequisite: 6 credits of senior-level analysis.
Final Exam: Tuesday, March 21, 1400.
Textbook: R. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations.
Class Notes and Schedule:
Flow and Transport.pdf
1. The transport equation. Notes: pp. 1-3. Text: Section 1-4. (1/9)
2. The porous medium equation. Notes: pp. 4-5. Text: Section 1-5.
3. Longitudinal vibrations. Notes: pp. 9-11. Text: Section 1-2.
MLK Day Observed (1/16)
4. Classification. Text: Section 2-6, pp. 40-46.
5. Characteristics and The Three Types.
6. Examples. See Sections 4 and 6 of PDE1.pdf(1/23)
7. Wave Equation: the initial-value problem. Section 4-1, pp. 89-92. Problems: 6,7,8.
8. The (quarter-plane) Initial-Boundary-Value problem.
9. A little reflection .... (1/30)
10. I-B-V problem on a bounded interval, Section 4-2, pp. 94-99.
11. The non-homogeneous equation, Section 4-3, pp. 101-106.
12. Diffusion Equation: I-B-V problem on a bounded interval, Section 5-1, pp. 144-152. (2/6)
13. No class today.
14. The Green's function, Section 5-3, pp. 159-165.
15. The Initial-Value Problem: Fundamental Solution, Section 5-4, pp. 166-167. (2/13)
16. pp. 168-171.
Diffusion Equation.pdf
17. Diffusion Equation, pp. 7-12, Discontinuous data and the non-homogeneous equation.
18. pp. 1-4, Energy estimates, comparison, maximum principle for IBVP. (2/20)
19. pp. 5-6, Extensions to IVP, Quarter-plane problem. Text: pp. 171-173.
20. Quarter-plane non-homogeneous boundary-values. Text: pp. 174-175.
Variational Method in Hilbert Space.pdf ,
Section 4: Expansion in Eigenfunctions.
21. Notes: pp. 24 - 25, Fourier Series. Text: Section 7-6, pp. 254-256. (2/27)
22. Notes: pp. 26 - 27, Eigenvector expansions.
23. Notes: pp. 28 - 29, Boundary-Value Problems.
24. Notes: pp. 30 - 31, The `Strong' Operator'. (3/6)
Caution: I have added material starting at p. 28.
25. Notes: pp. 32 - 33, The Expansion Theorem for BVPs.
26. Notes: pp. 34 - 36, The Non-Homogeneous Eigenvalue Problem; Wave Equation.
27. Notes: pp. 37, Diffusion Equation; Strongly-Damped (viscous) Wave Equation. (3/13)
28. Text: p. 217, A pseudo-parabolic example: Barenblatt's equation.
29. Notes: p. 37, Wave equation with inertia: Sobolev's equation.