MTH 622: DIFFERENTIAL AND INTEGRAL EQUATIONS - Winter 2016
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General information
Assignments
Assignments and schedule
  1. 1/4/15: INCLEMENT WEATHER day: class cancelled.
  2. 1/6/15: Introduction and review of solving the inhomogeneous diffusion equation [Read 5.1]
    HW 1. Part A (do not turn in, use as practice problems): 5.1-3, 5.1-5. 5.1-12. Provide formal solution to the pbm in 5.2-7 (ignore the steady-state part). Carry out details of Corollary 5-2-3.
    Part B due Friday 1/15/15.
  3. 1/8/15: Maximum principle on (0,L)x(0,T) [read 5.2].
  4. 1/11/15: Green's function on (0,L)x(0,T) and how to relax assumptions on f(). [Read 5.3].
  5. 1/13/15: Green's function for inhomogeneous IBVP.
  6. 1/15/15: Solving heat equation on R. The heat/diffusion kernel aka findamental solution.
  7. 1/20/15: Representation formula for solutions to diffusion eqns on R. Examples. Fourier transforms.
  8. 1/22/15: Use Fourier transform to solve the diffusion eqn on R. Analogues between representation formulas on r and (0,L).
    Solving heat equation on RxR_+. Properties of the solution. [Read 5.4]
  9. HW 2. Part A (do not turn in, use as practice problems): 5.3.5-6. Prove the properties of solutions on R discussed in class. Solve 5-4-1 in analogy what we did for the IBVP for the diffusion equation. Extra: solve 5-7-3.
    Part B due Wednesday 1//27.
  10. 1/25/15: Overview of solving other higher order pbms (beam, damped wave eqn) via SOV. Diffusion eqn with variable coefficients and general boundary conditions [Read Chapter 6-1,2,3,5].
  11. 1/27/15: Solving the general Sturm-Liouville pbm, and finding its Green's function.
    Practice problems: 7-2-4, 7-2-6, 7-2-7.
  12. 1/29/15: Review for Midterm.
  13. 2/1/15: Midterm. One index card note allowed.
  14. 2/3/15: Problem solving.
  15. 2/5/15: Laplace equation and more general elliptic equations in 2d, 3d. [Read 8-1-2-3]. Solution via SOV on a rectangle.
  16. 2/8/15: Solution to Laplace eqn on a circle.
  17. 2/10/15: Fundamental solution to Laplace eqn & Green's function.
  18. 2/12/15: Maximum/minimum principle. Uniqueness for Dirichlet problem.
    HW 3. Part A (do not turn in): 8-1. Pbms 1-4; 8-2. Pbms 1-3; 8-3. Pbms 1,5,7. 8-4. Pbms 2, 11, 12.
    Part B: solve one of 8-2, Pbms 1-3. Solve 8-2-4, 8-3-9. Due Monday 2/22.
  19. 2/15/15: [Make-up class in STAG 210 at 12:00-1:00]. Introduction to variational approaches for elliptic PDEs. Functionals (linear, continuous).
    Read Chapter 11-1, 3,4,5.
    Also, read notes from Variational Methods for Hilbert spaces by Prof. R. Showalter, Sections 1-2.
  20. 2/15/15: First variation of a functional. Euler-Lagrange equations.
  21. 2/17/15: [Make-up class in GLK 104 at 12:00-1:00]. Solving problems with Euler-Lagrange equations.
  22. 2/17/15: Spaces D(Omega), D'(Omega). Distributions: singular and regular, and distributional derivatives.
  23. 2/19/15: Sobolev spaces H^k; weak derivatives. Solving \del u=f via examples.
  24. 2/22/15: Characterization of the space H^1. Poincare-Friedrichs inequality.
  25. 2/24/15: Riesz representation theorem, and examples of applications.
  26. 2/26/16: Outline of the proof of Riesz theorem. Equivalence of of weak formulation of Dirichlet BVP and of variational formulation: minimization of a functional.
  27. 2/29/16: No class today; attend Math Colloquium instead. Extra credit: summarize the type and applications of the PDEs discussed in the talk.
  28. 3/2/16: Examples of variational/weak formulations. Variable coefficients, other boundary conditions.
  29. 3/4/16:
    3/7-9: no class today.
  30. 3/11/16: Review.
    HW 4. Part A (do not turn in): 11-1-2, 3. 11-1-14, 15. 8-4. Pbms 2, 11, 12.
    Part B due Friday 3/11. SOLVE ONLY THREE PROBLEMS.