ne dimensional version
Problem Consider a long thin water pipe. Say (x,t) represents the concentration of a solute at
point x and time t. Suppose the ambient solution is moving with velocity v (which could depend
on x and t). Given v, how does evolve with
372 PDE: Midterm 2.
Wed 4/2/2014
This is a closed book test. No calculators or computational aids are allowed.
You have 50 mins. The exam has a total of 4 questions and 20 points.
You may use any result from class or homework PROVIDED it is independent
21-372 PDE: Midterm 2.
Mar 23th , 2012
This is a closed book test. No calculators or computational aids are allowed.
You have 50 mins. The exam has a total of 4 questions and 40 points.
You may use without proof any result that has been proved in class
21-372 PDE: Midterm 1.
April 1st , 2013
This is a closed book test. No calculators or computational aids are allowed.
You have 50 mins. The exam has a total of 3 questions and 20 points.
You may use any result from class or homework PROVIDED it was pro
Assignment 4: Assigned Wed 02/08. Due Wed 02/15
Assignment 5: Assigned Wed 02/15. Due Wed 02/22
1. Sec. 2.3. 3, 5, 6, 7.
1. Sec. 2.4. 6, 18.
2. (a) Let L, T > 0, and a, b be two continuous functions such that a(x, t)
0
(with no assumption on b). Suppose u
Assignment 6: Assigned Wed 02/22. Due Wed 02/29
3. Sec. 4.2. 2, 4.
2
1. (a) Compute the solution of t u 1 xx u = f given u(x, 0) = 0, and f (x, t) = 1
2
if |x| 1, and f (x, t) = 0 otherwise.
(b) For
1
u(x, t).
t t
2
t ux u = 0
max u(x, t)
0, compute lim
Math 372: PDE Homework.
The problem numbers refer to problems from your text book (second edition). I will
often assign problems which are not in the text book. Keep in mind that there is a
rm no late homework policy.
4. Let D be a region in R3 , and c, r
Assignment 10: Assigned Wed 03/28. Due Wed 04/04
4. Let f be a complex valued 2L-periodic function, and cn =
th
be the n Complex Fourier coecient of f . Let SN f =
N 1
1
SN f .
sums and N f = N 0
1. Sec. 5.3. 2, 4.
2. Sec. 5.4. 1, 12, 13 [Assume that the
Assignment 13: Assigned Wed 04/18. Due Wed 04/25
Assignment 14: Assigned Wed 04/25. Due Wed 05/02
1. Sec. 6.4. 4, 12.
1. Sec. 7.1. 6.
2. Let D be a disc with center 0 and radius 1. Let f be some function with
f ()2 d < , and u be the solution of u = 0 in
21-372 PDE: Midterm 1.
Feb 17th , 2012
This is a closed book test. No calculators or computational aids are allowed.
You have 50 mins. The exam has a total of 4 questions and 40 points.
You may use without proof any result that has been proved in class
21-372 PDE: Midterm 1.
Feb 20th , 2012
This is a closed book test. No calculators or computational aids are allowed.
You have 50 mins. The exam has a total of 4 questions and 20 points.
You may use any result from class or homework PROVIDED it was prov
One Dimensional Version
Consider a long thin rod, and let (x,t) be the temperature of the rod at point x and time t. Our
aim is to find an equation determining the evolution of u. For this, we use Fourier's law
Fourier's Law: The heat flux is proportional
One Dimensional Version
Hold a thin string taught at two points. Let u(x,t) be the displacement of the string from it's
mean position at point x and time t. Our goal is to find an equation determining the evolution of
u.
Let T be the magnitude of the tens
Poisson's equation
Consider a conductor with a (possibly non-uniform) stationary (i.e. time independent) charge
density . Then the electric and magnetic fields are constant in time and Maxell's equations give
E=tB=0.
We know (from homework!) that any curl
D'Alembert's Principle
Proposition (D'Alembert's principle). If 2tuc22xu=0 for xR, t>0 with initial data
u(x,0)=(x) and tu(x,0)=(x), then
u(x,t)=(x+ct)+(xct)2+12cx+ctxct.
Lemma. The general solution of the wave equation is of the form u(x,t)=f(x+ct)
+g(xc
General solutions of ODE's usually involved undetermined constants. The examples we have so
far (from the method of characteristics) indicates that general solutions of PDE's involve
undetermined functions. To describe our solution completely, just specif
Assignment 12: Assigned Wed 04/23. Due Wed 04/30
1. (Hopf lemma) Heres an outline to solve the optional challenge from HW12.
(a) Given 0 < R0 < R1 , let A(R0 , R1 ) be the annulus cfw_x R2 | R0 < |x| <
R1 . Let c0 , c1 R with c0 < c1 , and suppose v satis
21-372 PDE: Final.
May 9th , 2013
This is a closed book test. No calculators or computational aids are allowed.
You have 3 hours. The exam has a total of 7 questions and 70 points.
You may use any result from class or homework PROVIDED it was proved in
Math 372: PDE Homework.
(b) If f is holomorphic, let f = lim
0
The problem numbers refer to problems from your text book (second edition). I will
often assign problems which are not in the text book. Keep in mind that there is a
rm no late homework policy
372 PDE: Midterm 1.
Mon 02/10/2014
This is a closed book test. No calculators or computational aids are allowed.
You have 50 mins. The exam has a total of 4 questions and 20 points.
You may use any result from class or homework PROVIDED it is independe
Assignment 13: Assigned Wed 04/17. Due Wed 04/24
Assignment 14: Assigned Wed 04/24. Due Wed 05/01
1. Let D be a disc with center 0 and radius 1. Let f be some function with
f ()2 d < , and u be the solution of u = 0 in D with u = f on D.
Dene g to be the