MTH 621-3: DIFFERENTIAL AND INTEGRAL EQUATIONS - Winter 2012
Links
General information
Assignments
Assignments, [reading material], and schedule
  1. 1/9/12: General information; class overview
  2. 1/11/12: Green's function/source operator for diffusion equation on (0,L). Inhomogeneous data. Maximum (minimum) principle and consequences. [GLee 5.2-3].
    Exercises: derive solution for inhomogeneous data directly and compare with the solution obtained using source function
  3. 1/13/12: Diffusion equation on R.
    Exercises: prove properties stated in class
    For another derivation of the solution read [GLee, 5.4]
  4. 1/18/12: Diffusion equation on R cd.: heat kernel (aka fundamental solution, source function, Green's function) and its porperties
    Exercises: find solutions for the initial data as given in class
  5. 1/20/12: class cancelled (campus closed due to inclement weather)
  6. 1/23/12: Functionals. Origins of elliptic BVP: equlibrium problems, Euler-Lagrange equations [Glee, 11-1]
  7. 1/25/12: cd
    Exercises: [Glee, 11-1.(2-5)]
    1/27/12: NO CLASS (a make-up class will be scheduled)
    Read [Glee, 8.1]
    Exercises: [Glee, 8-1.1-4]
    Assignment 1 due 2/3/12 in class.
    Potential equation [Glee, Chap. 8] or these [ Notes on Potential Equation by Ralph Showalter.
  8. 1/30/12: Divergence theorem and Green's identities. Fundamental solution in R^2 R^3.
  9. 1/30/12: 16:00-17:30 Make-up class , in Kidd 350: Potential equation [Glee, Chap. 8] Integral representation for any smooth function using fundamental solution. Green's function for Dirichlet problem. Poisson's formula for the ball and circle.
  10. Properties of harmonic functions: infinite smoothness and mean value property. Uniqueness of solutions to Poisson's equation.
  11. 2/1/12: Maximum principle for Laplace equation.
    Assignment 2 due 2/10/12 in class.
  12. 2/3/12: Distributions: motivation, definition of D(\Omega), D'(\Omega). Approximation by smooth functions, convergence.
    Read: [GLee, Chap.11.3-4], [Showalter, Variational methods in Hilbert spaces
  13. 2/6/12: Convergence of distributions. Distributional derivatives.
  14. 2/8/12: Weak solutions to differential equations. How weak ?
  15. 2/10/12: Construction of the "anti-derivative" of a distribution
  16. 2/13/12: How to solve \del u = f in the sense of distributions: examples
    Exercises: as given in class.
  17. 2/15/12: Review for exam.
    Sobolev spaces. Characterization of H^1 in 1D. Poincare-Friedrichs inequality.
  18. 2/17/12: Riesz representation theorem and examples. Weak/variational formulation of -u''=f.
  19. 2/20/12: MIDTERM (2 hours)
  20. 2/20/12: MIDTERM (cd)
    Assignment 3 due in class on Friday 2/24.
  21. 2/22/12: Examples of Riesz representers (from constructive proof). Alternative proof of Riesz representation theorem using minimization principle.
  22. 2/24/12: Variational formulation of BVP.
  23. 2/27/12: Bilinear forms and properties
  24. 2/29/12: Characterize variational solution to (Dirichlet) BVP
  25. 3/2/12: Neumann problem in variational form
    Corrections due to HW #3

    3/5/12: NO CLASS
    3/7/12: NO CLASS
    Assignment 4 due in class on Monday 3/12.
  26. 3/9/12: Non-homogeneous Neumann and Robin boundary conditions.
  27. 3/12/12: E/U for non-symmetric variational problem: Lax-Milgram Thm.
  28. 3/14/12: Spectral properties of self-adjoint compact operators
  29. 3/16/12: Wrap-up