Suggested Problem Set 3
February 10, 2015
Theres nothing to hand in, these are obviously optional, and an opportunity for those of
you wishing to work on more problems. I suggest also reviewing previous recommended required problems from throughout the co

1.2.2 Solve the equation 3uy + uxy = 0 (Hint: Let v = uy ) 1.2.5 Solve the equation 1.5.1 Consider the problem
(1 x2 ) ux + uy = 0 with the condition u(0, y ) = y .
d2 u +u=0 dx2 u(0) = 0 u(L) = 0,
and consisting of an ODE and a pair of boundary condition

Suggested Problem Set 4
February 26, 2015
Theres nothing to hand in, these are obviously optional, and an opportunity for those of
you wishing to work on more problems. I suggest also reviewing previous recommended required problems from throughout the co

Suggested Problem Set 5
March 17, 2015
Theres nothing to hand in, these are obviously optional, and an opportunity for those of
you wishing to work on more problems. I suggest also reviewing previous recommended required problems from throughout the cours

Recommended Problem Set 1 Solutions
January 22, 2015
Look over the solutions carefully and make sure you feel comfortable obtaining the answers.
1. Classify and solve xux yuy = 0.
Ans: Its first-order, linear, non-constant coefficient and homogeneous. Can

Suggested Problem Set 2
January 22, 2015
To help prepare for the upcoming midterm, you may wish to try out some of the following
exercises. You may find some of these are quite challenging. Theres nothing to hand in, these
are obviously optional, and an o

MATH 18.152 COURSE NOTES - CLASS MEETING # 1
18.152 Introduction to PDEs, Fall 2011
Professor: Jared Speck
Class Meeting # 1: Introduction to PDEs
1. What is a PDE?
We will be studying functions u = u(x1 , x2 , , xn ) and their partial derivatives. Here x

Suggested Problem Set 1
January 14, 2015
These optional problems are not to be handed in but are excellent practice for the exams
and quizzes and the remainder of the material we will cover in the course. It can sometimes
be nice to find people to work on

FACULTY OF APPLIED SCIENCE AND ENGINEERING
University of Toronto
APM346F
Partial Dierential Equations
Midterm Exam, October 6, 2014
Examiner: N. Hoell
Duration: 50 minutes
NO AIDS ALLOWED.
Total: 100 marks
Family Name:
(Please Print)
Given Name(s):
(Pleas

FACULTY OF APPLIED SCIENCE AND ENGINEERING
University of Toronto
APM346F
Partial Dierential Equations
Midterm Exam, November 12, 2014
Examiner: N. Hoell
Duration: 50 minutes
NO AIDS ALLOWED.
Total: 100 marks
Family Name:
(Please Print)
Given Name(s):
(Ple

MATH 18.152 COURSE NOTES - CLASS MEETING # 3
18.152 Introduction to PDEs, Fall 2011
Professor: Jared Speck
Class Meeting # 3: The Heat Equation: Uniqueness
1. Uniqueness
The results from the previous lecture produced one solution to the Dirichlet problem

Problem 1 (not from the textbook). Solve the initial value problem:
ux + 2 x uy = 2 x u,
u(0, y ) = y 2
by two dierent methods (using parametrization of the characteristic curve such as (s) = (x(s), y (s) and such as (x) = (x, y (x). Hint - see pdf notes

Problem 1
Solve the initial value problem for the inhomogeneous wave equation:
utt = uxx + cos(2 x) sin(t), ux (0, t) = ux (, t) = 0, u(x, 0) = 0,
where is a real valued parameter.
0 < x < ,
ut (x, 0) = 0,
Problem 2
By using Fourier Integral Formula deriv

Problem 1 Solve the initial value problem for the nite interval:
utt uxx = 0, ux (0, t) = ux (, t) = 0, u(x, 0) = 1,
Problem 2 Solve the initial value problem on the line:
0 < x < ,
ut (x, 0) = 1.
4 u + 4 ut + utt 4 uxx = 0, u(x, 0) = x, ut (x, 0) = 1.
1

Problem 1 (not from the textbook).
Solve the initial value problem:
2 uxx − 3 uxy + uyy = ex−y ,
u(x, o) = ex ,
uy (x, 0) = x
from scratch by two dierent methods: using method of characteristics
and using Green's formula.
(Hint: see notes)
1