Problem 2 was more subtle than I realized when I assigned it. Also, the correct answer (about uniqueness) is different from what one reads in some textbooks. My understanding of this is greater today than it was last week, and I've summarized my current thinking in these notes. I found this problem quite interesting, and if you have any further thoughts on it yourself, I'd be glad to hear them.
Here is the final Grade Sheet.
These are the best grades I have ever given in a class,
and you deserve them! Well done!
Please let me know immediately if you see
any errors or omissions.
I wish you a really great summer!
Photos of blackboards are maintained at the location I announced in class. (Ask me if you've forgotten where.)
KdV, solitons: kuaii, kdv.mws
Tidal bores: Alaska,
China image
description,
bore.mws.
Exact solution of Burgers with diffusion: mws
Burgers eqn, rarefaction wave and shock: rarefaction2.mws, p144.mws
HW2 X4: an interesting family of solutions of the heat equation: hw2x4ratio.mws
plot thru m=120
Burgers eqn: burgers1.mws, burgers.mws
Characteristics converging: 3.2_case_1.mws
Initial temperature zero: 2D plot, rocked 3D plot
Characteristics: Ex 3
Envelopes: eikonal example, Clairaut envelope, Example 4
Wave equation:
in R,
in R+,
results summary
p1
p2
p3
Heat equation:
heat ball, function v on p57
fourier modes,
sinusoidally heated half space,
fundamental solution,
summary
Numerical solution of Laplace's equation applet
Theorems for Laplace's equation: summary
Green for half-plane mws
Balls for Ch.2 Thm 7 mws
Mollifying a harmonic function: mws:
original,
blurred.
Standard mollifier: mws
Fundamental solutions of Laplace eqn: mws
Q: Why do we choose the seemingly strange normalization of the fundamental
solution (8)?
A: So that -&Delta u = f, rather than some non-unit constant times f.
Q: Why does the death rate not appear in the rate of aging-in?
A: here
Required text: PDE by Lawrence C. Evans (American Mathematical Society)
We will use Maple for visualizing some concepts and examples,
but no prior experience with Maple is assumed. A brief guide to
its use is given below. The HTML files can be opened with any browser. You need Maple
to open the mws files. Maple is available for Linux, Mac, and Windows for
less than $2 at UBMicro in the UB Commons (open 9am-6pm, 1st week of classes).
Maple as calculator: HTML,
mws.
Maple as analytical tool: HTML,
mws.
Maple as programming language: HTML,
mws.
Maple for pictures: HTML,
mws.
Maple for 3D pictures: HTML,
mws.
