|
MATH 456-556 Applied
Partial Differential Equations, Spring 2009 |
|
|
|
|
|
Lectures:
|
|
Text: Partial Differential Equations by Walter A. Strauss
Problem sets:
Seven homework problems will be assigned on Fridays each two weeks and will be due
on Jan 23, Feb 6, Feb 20, Mar 6, Mar 27, Apr 10, Apr 24 until
Exams: Three midsemester class exams will be given: February 16, March 30 and May 3. On May 6 the Practice test will be offered and solved in class (not grated). The date of two-hour final exam will be announced later. Laptops and calculators will not be allowed during the exams.
Grading: The homeworks will count 30 percent of the final grade. Each of midsemester test will count 15 percent of the final grade. The final exam will count 25 percent of the final grade.
Grades "W" and "I": The grade "W" will be awarded to any student who requests this grade before March 7. The grade of "I" can be awarded only in an exceptional case to a student who has a valid reason for not completing the course in time, who has not completed a small portion of the course, and who has shown a passing performance in class.
APPROXIMATE SYLLABUS
|
Monday |
Wednesday |
Friday |
|
January |
||
|
|
14 Partial Differential Equations. Examples. Linear and nonlinear equations. First order equations. |
16 Second-order linear equations with constant coefficients. Classification. |
|
19 Martin Luther King Day |
21 Wave equation on the line. Initial and boundary conditions. Diffusion equation on the line. |
23 Scalar and field gradient. Vector field divergence. |
|
26 Wave equation in space. Initial and boundary conditions. Diffusion equation in space. Laplace and Poisson equations. |
28 Solution of the wave equation on the line. Dalambert’s formula. Wave equation with forcing. |
30 Causality. Energy conservation. Uniqueness. Characteristics. |
|
|
|
|
|
February |
||
|
2 Wave equation on the half-line. Reflections. Multiple reflections. |
4 Solution of the diffusion equation on the line. Self-similar solutions. Well-posed and ill-posed problems. |
6 Diffusion equation on the half-line (heat transport). |
|
9 Separation of variables. The Dirichlet condition. |
11 Separation of variables. The Neumann condition. |
13 Separation of variables. The Robin condition. |
|
16 First Class Test |
18 Sine Fourier series. Examples. |
20 Cosine Fourier series. Examples. |
|
23 General Fourier series.
Examples. |
25 Rate of convergences and smoothness. Even, odd and periodic continuation. Gibbs phenomenon. |
27 Distributions. δ-function and its derivatives. |
|
|
|
|
|
March |
||
|
2 Even, odd and periodic continuation of δ-function. Completeness of Fourier harmonics. Parseval’s identity. |
4 Fourier Transforms. Examples. |
6 Fourier Transforms (continuation). |
|
9 Harmonic functions. Principle of maximum. Solution of |
11 |
13 Poisson formula. Circle mean. |
|
Spring Break |
||
|
23 Circles, Wedges, Annuly. |
25 Potential flow of ideal fluid. Vortices. |
27 Vibration of drumhead. Diffusion equation in the circular domain. Bessel equation. |
|
30 Second Class Test |
|
|
|
April |
||
|
|
1 Bessel functions. |
3 Bessel functions (continuation). |
|
6 Diffusion and wave equations in the cylinder. |
8 Green’s first and second identity. |
10 Green function. Solution of Poisson equation in space. Using of the Fourier Transform. |
|
13 Green function in half-space and in sphere. |
15 Wave in spaces. Energy and Causality. |
17 Green function for the wave equation. |
|
20 Diffusion equation in space. |
22 Schrödinger equation in space. |
24 Hydrogen atom. |
|
27 Eugenvalues and Eugenfunctions of the symmetric differential operator. Orthogonality. |
29 Oscillator in quantum mechanics. |
|
|
May |
||
|
|
|
1 Asymptotics of the Eugenvalues and Eugenfunctions. WKB approximation. Bohr’s quantization rules. |
|
4 Third Class Test |
6 Practice
test solution |
|
|
|
|
|