MTH 621, Fall, 2005

MTH 621: Differential and Integral Equations of Mathematical Physics, Fall, 2005

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

LECTURE:	MWF 1400 - 1450	Milam 234	CRN 13420
Instructor:	R.E. Showalter	Kidder 368	show@math.oregonstate.edu

We shall begin with a discussion of models of heat conduction and vibration and the corresponding initial-boundary-value problems. Then we develop fundamental variational principles in Hilbert Space and Fourier expansion theory for boundary-value problems. These lead naturally to generalized solutions in function spaces and provide tools to establish well-posedness of boundary-value problems and to obtain useful numerical approximation schemes of finite-element type. All these concepts are introduced in one spatial dimension where the technical prerequisites are minimal.

Prerequisite: 6 credits of senior-level analysis.
Final Exam: Monday, March 15, 1800.
Textbook: R. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations.
Class Notes and Schedule:
1: Discrete Models of Diffusion and Vibration.pdf
1. Section 1: Heat conduction in an interval, pp. 1-3. (9/26/05)
NOTE correction to equation 2.b, a misplaced parenthesis.
2. Boundary Conditions; Section 1.3: Dirichlet type.
3. Sections 2.1-2.4: Vibrations in an Interval.

2: Linear Systems.pdf ............. Corrections on p. 4 10/06.
4. Sections 1.1: Vector spaces,subspace, basis, linear map; scalar product space. (10/03)
5. Section 1.3: geometry and examples of scalar product space, p. 4.
Due Friday, 10/14: Chap1,p.9, Exc 5, p.14, Exc 8; Chap2, p.2, Exc 1 (last part only).

3: Continuous Models of Diffusion and Vibration.pdf ............... Exercises updated 10/19.
6. Section 1: Heat conduction. Text: Read Chapter 1, Sections 1-1, 1-3.
7. Section 2: Eigenfunction Expansion Method, pp. 4 - 6. (10/10)
8. Fourier series. Text: Read Chapter 3, Section 3-1.
9. Examples, non-homogeneous boundary conditions, p. 7.
10. Non-homogeneous equation, pp. 8 - 9, 17. Text: Read Chapter 5, Sections 5-1. (10/17)
Due Friday, 10/28: Chap3,p.9, Exc 5, and p.10, Exc 6.

Here are graphs of solutions to the diffusion equation with initial value u(x,0) = 1 and boundary conditions u(0,t) = 0 and at x = 1
of Dirichlet type, and of Neumann type, respectively.
Here are graphs of solutions with initial value u(x,0) = sin(pi x) and boundary conditions u(0,t) = 0 and at x = 1
of Dirichlet type, and of Neumann type, respectively.

4: Variational Method in Hilbert Space.pdf
11. Section 1: Function spaces; the weak formulation. pp. 1 - 3.
12. functions and functionals. p. 4.
13. derivatives. pp. 5 - 6. (10/24)
14. anti-derivatives. p. 7.
15. weak solution = strong solution. p. 8.
16. Section 2: Hilbert space, Sobolev spaces. pp. 8 - 9. (10/31)
17. Estimates; continuity. pp. 10 - 11.
18. Dual space; the Minimization Principle. pp. 11 - 12.
19. The Variational Equation. pp. 13 - 14. (11/07)
20. Orthogonal projection. pp. 14 - 15.
Due Friday, 11/18 : Chap4, Exc 1, p.8, and p.18, Exc 9(a).
21. No class today.
22. Review and questions. Examples. pp. 15 - 16. (11/14)
23. Section 3: Approximation of solutions. (Raleigh-Ritz-Galerkin) pp. 19 - 20.
24. The discrete model again. pp. 20 - 21.
25. pp. 22 - 23. (11/21)
26. Review (at most).
Thanksgiving Break
27. Transport equation: the IVP and Duhamel Principle. Text: pp. 8-9. (11/28)
28. Transport equation: the IBVP for quarter plane.
29. An hyper - sensitive problem!
Final Exercise