University of Alberta
Math 337 Q1, Introduction to Partial Dierential Equations, Winter 2011
solution of Assignment # 3,
1. Exercises 4.2: 4.2.1.
(a) Using (4.2.7), compute the sagged equilibrium position uE (x) if Q(x, t) = g. The boundary
conditions are
University of Alberta
Math 337 Q1, Introduction to Partial Dierential Equations, Winter 2011
Solution of Assignment # 1,
Due time: 4:00 pm, Friday January 21 2011
1. Exercise 1.2: 1.2.3.
Derive the heat equation for a rod assuming constant thermal propert
MATH 337
Assignment 1 - Solutions
Section 1.2: 7) set y = x t and U (y, t) = u(x, t) = u(y + t, t), so u(x, t) = U (x t, t).
Then, by the chain rule, ux = Uy and ut = Uy + Ut , so our PDE becomes:
Uy + Ut + Uy 3U
Ut 3U
= t
= t
To solve this, we multiply b
University of Alberta
Math 337 Q1, Introduction to Partial Dierential Equations, Winter 2011
Solution of exercises 4.4.2 (c), 5.4.2,5.4.5 and 5.5.8,
Exercise 4.4: 4.4.2 (c).
Solution. Let u(x, t) = (x)h(t). Two ODEs are
h (t) = h(t)
and
T0
(x) + (x) + (
University of Alberta
Math 337 Q1, Introduction to Partial Dierential Equations, Winter 2011
Solution of Assignment # 5,
Due time: 4:00 pm, Friday April 1 2011
Exercises 8.2: 8.2.1 (a)+(b):
Consider the heat equation with time-independent sources and boun
Page 1
1. Given the following initial boundary value problem
(a)
(b)
an 8221,
a—«ggﬁ, 0<33<L, t>0,
“(3370) = f($l a
Bu Bu
Use separation of variables to reduce the PDE into a time dependent ODE
and a boundary value problem (space dependent ODE). Explain a
EXERCISES 1.4
1.4.1. Determine the equilibrium temperature distribution for a. one-dimensional rod with
constant thermal properties with the following sources and boundary conditions:
(b) Q = 0, ”(0) = T, u(L) = 0
Bu
(c) Q = 0, 5(0) = 0, u(L) = T
(e) 7% =
Math 337
Solutions to Midterm Examination
Question 1. Let f (x) = x2 ,
0 x .
(a) Find the Fourier cosine series for f on the interval [0, ].
(b) For which values of x R does this Fourier series converge?
(c) By choosing appropriate values of x with 0 x ,
University of Alberta
Department of Mathematical and Statistical Sciences
Solution of Midterm Exam Math 337 Q1 Instructor: Zhichun Zhai Winter 2011
Name(print):
Instructions
1. This is a close-book exam.
2. Only pens, pencils and erasers are allowed.
3. A
University of Alberta
Math 337 Q1, Introduction to Partial Dierential Equations, Winter 2011
Assignment # 4,
Due time: 4:00 pm, Friday March 18 2011
Exercises 7.3: 7.3.1 (a)+(b)+(c)
Consider the heat equation in a two-dimensional rectangular region 0 < x
1
Math 337, Summer 2010
Assignment 5
Dr. T Hillen, University of Alberta
Exercise 0.1.
Consider Laplaces equation
1
r r
u
r
r
+
1 2u
=0
r2 2
in a semi-circular disk of radius a centered at the origin with boundary conditions
u(r, 0) = 0,
0 < r a,
u(r, )
MATH 337
Assignment 8 - Solutions
1) We use the method of characteristics. The characteristic equations are
t (s) = s, t(0) = 0
x (s) = 2t, x(0) = a
z (s) = z + et , z(0) = a
The solution to the rst equation is t = s. Substituting into the second ODE yiel
1
Math 337, Summer 2010
Assignment 4
Dr. T Hillen, University of Alberta
Exercise 0.1.
The neutron ux u in a sphere of uranium obeys the dierential equation
1 d
2 du
r
+ (k 1)A u = 0
3 r2 dr
dr
for 0 < r < a, where is the eective distance traveled by a n
1
Math 337, Summer 2010
Assignment 3
Dr. T Hillen, University of Alberta
Exercise 0.1.
Find the values of 2 for which the boundary value problem
d2 u
+ 2 u = 0,
dx2
0<x<
2
u(0) = 0
2
u(t) dt = 0
0
has nontrivial solutions.
Solution to Exercise 0.1: We con
Problem 1:
1. Compute the Fourier series of the 27rperiodic function f given by
1 if 0 < a: < 7r/2,
f(.7:) = 0 if 7r/2 S 3 7T,
1 if 7r/2<$<0.
2. For which values of a: does the Fourier series for f converge?
3. Sketch the graph of the function and of