MTH 621-3: DIFFERENTIAL AND INTEGRAL EQUATIONS - Fall 2011
Links
General information
Assignments
Assignments, [reading material], and schedule
  1. 9/26/11: General information. Classification of ODEs, PDEs, IEs. Examples.
  2. 9/28/11: (Review) Well-posedness of ODEs [use any rigorous text on ODEs]. Local and global. Examples of well- and ill-posed IVP and BVP.
  3. 9/30/11: First order quasilinear PDEs [Guenter/Lee 2.1-2]
  4. 10/3/11: Examples of quasilinear problems. What if your initial data is not smooth ?

    HW assignment 1 due 10/12.

    • Solve Assignment 1.
    • Challenge: You can substitute any one of the problems 1-3 from the above list by solving pbm 2-1.1 or 2-1.2 (pages 23-24) or 2-2.1 or 2-2.2 (page 27) from the textbook.
    • Extra: go to the page with PDE Coffee table book, by Trefethen et al, to see examples of various PDEs. Pick three examples and classify them. [You can turn in your solutions if you want]. Read about their qualitative nature.

  5. 10/5/11: Modeling of Transport(=advection). Physical examples where the flux and velocity depend on position or on the unknown solution. Need for weak solutions.
  6. 10/7/11: Burger's equation. Existence/uniqueness for quasilinear first order PDEs.
  7. 10/10/11: Second order PDEs. Classification of canonical types. [GLee, 2-6].
  8. 10/12/11: Examples of classification. Change of variables formula.

    HW assignment 2 due 10/21.

    • Solve Assignment 2.
      Note: in problem 2-6.5b, you can skip the mixed derivative term. Or, solve the original problem for (a bit of) extra credit.

  9. 10/14/11: Change of variable to reduce a second order PDE to a canonical form.
  10. 10/17/11: Modeling leading to second order PDEs. [Read: GLee, Chapter I.]
  11. 10/19/11: continue.
  12. 10/21/11: Solving the wave equation in one space dimension. D'Alembert's solution. [Read: GLee, 1-2, 1-6., GLee, 4-1].
  13. 10/24/11: Solving the wave equation on an interval: separation of variables.

    HW assignment 3 due 11/2.

  14. 10/26/11: continue separation of variables
  15. 10/28/11: Convolutions. Construction of solutions to non-homogeneous first and secord order evolution problems using source operators.
  16. 10/31/11: Solution to the non-homogeneous wave equation. Introduction to Fourier series via best approximations in inner product spaces.
  17. 11/2/11: Review.
  18. 11/4/11: Midterm.
  19. 11/7/11: Inner product spaces, L^2, and approximation in finite dimensional subsets and closed convex subsets of normed spaces. Handout on Fourier series.
  20. 11/9/11: Examples of best approximations: how to compute Fourier coefficients. Orthonormal system in a Hilbert space. Bessel's inequality.[Read GLee Chapter 3-1, 3-2, 3-3].
  21. 11/11//11: Orthonormal basis in a Hilbert space and Parseval's identity. Examples of Fourier series.

    HW assignment 4 due 11/23.

    • Solve Assignment 4.
      Note: you can use the code in fourier.m to get a feeling for how the functions should behave.
      Other useful problems related to this HW: 3-1.4, 3-1.11, 3-2.2. Also, read the sequence 3-1.8-10 to understand about even and odd extensions.

  22. 11/14/11: Examples of Fourier series. Pointwise and uniform convergence and L^2 convergence. [Handouts: worksheets].
  23. 11/16/11: No class. (work on the next assignment).
  24. 11/18/11: Even and odd extensions and expansions on (0,pi).
  25. 11/21/11: Differentiability of formal (Fourier series) solutions. Existence result for the wave equation on (0,L).
  26. 11/23/11: Uniqueness of solutions to the wave equation on R and (0,L).
    Start diffusion equation: derive formal solutions via separation of variables. [GLee, 5-1].

    HW assignment 5 (do not turn in, use as the basis for self-study and exam preparation)

    • Solve 5-1.[1-5,10,14] and TBA
    • 11/28/11: Analytical properties of the solution by separation of variables. [Glee, 5-1, Thm 5-1-1].
    • 11/30/11: HW review. Applications of diffusion equation.
    • 12/2/11: Source and Green's function in solving diffusion equation. Preparation for Final Exam.