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Assignments, [reading material], and schedule |
- 1/9/12: General information; class overview
- 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
- 1/13/12: Diffusion equation on R.
Exercises: prove properties stated in class
For another derivation of the solution read [GLee, 5.4]
- 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
- 1/20/12: class cancelled (campus closed due to inclement weather)
- 1/23/12: Functionals. Origins of elliptic BVP: equlibrium problems,
Euler-Lagrange equations [Glee, 11-1]
- 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.
- 1/30/12:
Divergence theorem and Green's identities.
Fundamental solution in R^2 R^3.
- 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.
- Properties of harmonic functions: infinite smoothness and
mean value property. Uniqueness of solutions to Poisson's equation.
- 2/1/12: Maximum principle for Laplace equation.
Assignment 2 due 2/10/12 in class.
- 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
- 2/6/12: Convergence of distributions. Distributional derivatives.
- 2/8/12: Weak solutions to differential equations. How weak ?
- 2/10/12: Construction of the "anti-derivative" of a distribution
- 2/13/12: How to solve \del u = f in the sense of distributions: examples
Exercises: as given in class.
- 2/15/12: Review for exam.
Sobolev spaces. Characterization
of H^1 in 1D. Poincare-Friedrichs inequality.
- 2/17/12: Riesz representation theorem and examples.
Weak/variational formulation of -u''=f.
- 2/20/12: MIDTERM (2 hours)
- 2/20/12: MIDTERM (cd)
Assignment 3 due in class on Friday 2/24.
- 2/22/12: Examples of Riesz representers (from constructive
proof). Alternative proof of Riesz representation theorem using
minimization principle.
- 2/24/12: Variational formulation of BVP.
- 2/27/12: Bilinear forms and properties
- 2/29/12: Characterize variational solution to (Dirichlet) BVP
- 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.
- 3/9/12: Non-homogeneous Neumann and Robin boundary conditions.
- 3/12/12: E/U for non-symmetric variational problem: Lax-Milgram
Thm.
- 3/14/12: Spectral properties of self-adjoint compact operators
- 3/16/12: Wrap-up
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