MTH 621-3: DIFFERENTIAL AND INTEGRAL EQUATIONS - Winter 2009
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General information
Assignments

In the schedule below [GL *] means a section from Guenther/Lee text, [S *] means sections from Strauss' text. Material marked with E means additional reading is available there.
Assignments and schedule
  1. 1/5: Introduction
  2. 1/7-14: Functionals and variational calculus: linear and nonlinear functionals; finding stationary points; Euler-Lagrange conditions; Hamilton-Ostrogradski principle; examples from applications. [S 14.3, GL 11.1-2]
  3. Assignment 1 due 1/16: turn in solutions to 2-4 for credit. If you wish, you can turn in solution to 1 (I will grade it but no credit will be given).
  4. 1/14, 16, 21 : Linear continuous functionals; distributions. weak derivatives, adjoint differential operators, weak and generalized solutions to differential equations. [S 12.1, GL 11.3-4]
    Also, for review of important information that will be needed shortly, please read Notes on linear systems (and Hilbert spaces) by R.E. Showalter
  5. Assignment 2 due Wed. 1/28: turn in 1-2, 4. You can also turn in 5 but no credit will be given.
  6. 1/21-: Hilbert spaces and examples.
  7. 1/23-25 Sobolev spaces. Space H^1 and H^1_0. Handout in class. Read [GL 11.3-4] and notes on Hilbert space methods for PDEs by R.E. Showalter
  8. 1/28: minimization principles for functionals for whcih classical solutions cannot be found
  9. 1/30: conditions of equivalence of classical and variational formulations
  10. 2/2: Riesz theorem, Greens' function as Riesz representer for \delta
  11. Assignment 3 due Fri. 2/13: turn in 1-2, and use remaining problems for practice.
  12. 2/4: bilinear forms and properties
  13. 2/6: Lax-Milgram Theorem
  14. 2/9: examples of variational BVP with different boundary conditions and different forms
  15. 2/11: nonhomogeneous Boundary value problems and variational formulation
  16. 2/13: Review for exam.
    Ritz-Galerkin method. [S 11.1-2] Rayleigh-Ritz quotient and minimization principles.
    These help to compute the best Poincare-Friedrichs constant which provides a bridge between Fourier series methods and variational methods.
  17. 2/16: Exam.
  18. 2/18: Green's function in constructing solutions to Au=f. Fourier series representation of Green's function in 1D. Properties of Green's function. Connection to solving Au=f with spd A. [GL 7.2 * has more general context of Sturm-Liouville problems]
  19. 2/20: Extra HW (worksheet on Fourier series) due.
  20. 2/20: Fourier series on (0,1): review using worksheet. \\ Fundamental solutions to Laplace equation in 2D and 3D. [S 6.1, GL 8.1-2]
  21. Assignment 4 due Wed. 3/4: turn in 2-4 and at least one part out of 5. Use 1 for practice/enrichment.
  22. 2/23: Dirichlet problem on a circle using separation of variables. [S 6.4, GL 8.2]
  23. 2/25: BVP on a square/rectangle using separation of variables and linearity [GL 8.2pbms, S 6.2].
  24. 2/26: worksheet on harmonic functions, fundamental solutions, and normal derivatives
  25. 3/2: Poisson's formula (2D) and consequences: averaging property and maximum/minimum principle of harmonic functions (weak and strong form). [S 7.1-7.3, GL 8.4]
  26. 3/4-6: scaling of fundamental solutions in 3D (use 2D as exercise). Representation formulas using Laplace operator for i) smooth functions, ii) harmonic functions.
  27. 3/9: Use method of images to find Green's function for half-space and sphere.[S 7.4, GL 8.3]
  28. 3/11: Wrap-up of Greens' functions, their properties and representation formulas. Additional reading: [*GL 8.6]
  29. 3/13: Review for Final Exam.