MTH 621-3: DIFFERENTIAL AND INTEGRAL EQUATIONS - Fall 2015
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Assignments
Assignments and Schedule
  1. 9/25/15: Introduction and overview. Classification of DE. Review notion of existence/uniqueness for ODE: recall condition of Lipschitz continuity.
  2. 9/28/15: [HW on handout in class on examples of existence/uniqueness. Do not turn in]. [HW on handout in class: practice changing variables]. Idea behind the method of characteristics using directional derivatives. [Read: [use any rigorous text on ODEs].]
  3. 9/30/15: general PDEs of first order. Method of characteristics for quaslilinear PDEs in 2D: more formally. Examples.
  4. 10/2/15: Continue method of characteristics. What to do if the initial data is nonsmooth ?

    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. 10/5/2015: Generalize the transport model. Two examples of non-linear conservation laws. Solving a problem with smooth initial data and a nonlinear flux function: shock development. Riemann problem.
  6. 10/7/2015: Continue Riemann problem.
  7. 10/9/2015: Generalized (weak) solutions to transport model.
    Start classification of second order PDEs. [Section 2.6]. Canonical form of PDEs. "Factoring the second order differential opoerator".

    HW Assignment 2 due 10/19.

  8. 10/12/2015: Classification (of quadratic curves in the plane and) of second order PDEs into: elliptic, hyperbolic, parabolic. [Section 2.6]
    General solution to a hyperbolic equation using the method of characteristics.
  9. 10/14/2015: Solving the IVP for the wave equation using MOC. [Section 4.1]. More general second order PDEs that can be or cannot be classified. Solving a more general IVP for the nonhomogeneous wave equation. Domain of influence and domain of dependence.
  10. 10/16/2015: Recap the big ideas so far: changing variables to reduce the order, using linearity, and transforming a PDE to a canonical form.
    Solving the wave equation as a system of first order PDEs.
    Models leading to the wave equation: elastic bar as the limit of "balls and springs". [Read Chapter 1].
  11. 10/19/2015: continue modeling: Vibrating spring. Telegraph equation.
  12. 10/21/2015:
    Extra office hours on Thursday, 10/22, time 10:00-11:30am and 3:00-4:00pm..
  13. 10/23/2015: Midterm.
  14. 10/26/2015: Separation of variables for an IBVP for the wave equation.
  15. 10/28/2015: cd separation of variables. Fourier series. [Read Chapter 3, 3-1, 3-2, 3-3]
  16. 10/30/2015: Examples of how to compute Fourier series. Handout on analytical properties.
    Assignment 2.5 (do not turn in): Practice your Fourier series skills. Use 3-1.1, 3-4. Be familiar with ways to derive the formulas as in 3-1.5-10, 13-16. (I will assume in class you have these skills).
  17. 11/2/2015: Review: convergence of (functional) sequences/series and properties of limits. Inner product spaces.
  18. 11/4/2015: Approximation of functions in C(-pi,-pi]) and L^2([-pi,pi]). Utility fourier.m
  19. 11/6/2015: Best approximation in inner product spaces. Orthogonal and orthonormal system and basis. Bessel's inequality and Parseval's identity.
  20. 11/9/2015: Examples of approximations. Sketch of the proof that L^2 is well approximated by trigonometric polynomials. Riemann-Lebesque Lemma, and sketch of Dirichlet Theorem.
    Assignment 3 due Wednesday, 11/18. In pbm 3-2-3, you only need to work out s=1/2 or s=1/4. (Both, s=1/2, and s=1/4 are for extra credit.)
    11/11/2015: No class (Veteran's Day)
  21. 11/13/2015: Use red and green class handouts: predict the convergence/uniform convergence/order of terms for the F-series of various functions defined on (-L,L), and considered on (0,L). [Theory from class handout on F-series]. Density of trigonometric polynomials.
  22. 11/16/2015: What smoothness of initial data and what compatibility conditions are needed for the existence of (classical) solutions to the wave equation on (0,L): use class handout on F-series and reverse engineering. [Also, check Thm 4-4-1].
  23. 11/18/2015: Existence and uniqueness of solutions to the wave equation. Solving inhomogeneous wave equation with inhomogeneous boundary conditions [Read 4.3]
  24. 11/20/2015: The operator "A=-d2/dx2", with Dirichlet bc is self-adjoint and positive definite. Contrast the solutions to "u''+Au=f" with those to "u'+Au=f".
  25. 11/23/2015: Diffusion equation via separation of variables. Analysis of convergence of the formal F-series solution. [Chapter 5] [Read 5.1].
    Assignment 4 due Friday, 12/4.
  26. 11/25/2015: Modeling leading to the diffusion equation. [Read 1.3-1.4]
  27. 11/30/2015: Continue modeling.
  28. 12/2/2015: Green's function for the operator A, stationary pbm Au=f, and for the evolution euqation u'+Au=f.[Read 5.3]
  29. 12/4/2015: Review.
    Final exam: Friday Dec. 11, at 7:30am, in GLK 115.
    Office hours Finals sweek: Wed 12/9 4:00-6:00 and Thu 12/10, 12:00-2:00.