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University of Toronto
,s FACULTY OF APPLIED SCIENCE AND ENGINEERING
FINAL EXAMINATIONS, DECEMBER 1999
APM 384H1F Partial Differential Equations
Year III, Program III 5a, 5bm(c), 5env, 5p
Examiner: Professor R.A. Ross
Duration: 2% hours
Exam Type C
All q

£073 RcTep CC; PV
. University of Toronto I
FACULTY OFAPPLIED SCIENCE AND ENGINEERING
Final Examination, December 1998
APM 384E
Year III, Program III 5a, Sbm, 5env, 5p
Examiner: Prof. R.A. Ross
Duration : 2% hours
Exam Type C
All questions have EQUAL valu

APM 384 HlF
Final Exam, 12/16/09 (WED)
Name:
ID:
Instruction: You must justifyyour answer for full credit by showing all the necessary
steps in a readable manner. If you do not have enough room, use the backside. If- you use
any information from the formu

FACULTY OF APPLIED SCIENCE AND ENGINEERING
University of Toronto
APM384F
Partial Differential Equations
Final Exam December 10, 2012
Examiner: Nicholas Hoell
Duration: 150 minutes
NO AIDS ALLOWED. Total: 185 marks
Family Name:
(Please Print)
Given Name(

FACULTY OF APPLIED SCIENCE AND ENGINEERING
University of Toronto
APM384F
Partial Dierential Equations
Midterm Exam, October 24, 2013
Examiner: J. Ortmann
Duration: 1 hour 45 minutes
NO AIDS EXCEPT RULERS ALLOWED.
Total: 100 marks
Family name (surname):
Fi

FACULTY OF APPLIED SCIENCE AND ENGINEERING
University of Toronto
APM384F
Partial Differential Equations-
Final Exam, December 6, l
/ Y Q
I gal whoa
Examiner: J. Ortmann
Duration: 2 hour 30 minutesT ix 7 ,I : y
NO AIDS EXCEPT RULERS ALLOWED. Total: 140 ma

Topics for the Midterm
APM 384 Partial Dierential Equations
Autumn 2014
Here is a list of concepts you should be familiar with for the midterm, which will
take place during the lecture slot on Friday, October 24.
1. Denitions and basic classication of die

Suggested Exercises for the Midterm
APM 384: PDEs
Autumn 2014
1
Questions
Below are some suggested problems you may want to attempt in preparation for the
midterm. There are many excellent problems in Haberman for which you can nd
solutions in the back of

Problem Sheet 2
APM 384
September 18, 2014
1. Find the general solution to the PDE
ux + uy = y.
(1)
subject to the boundary condition u(0, y) = ey .
So we have a(x, y) = b(x, y) = 1, c(x, y) = 0, g(x, y) = y and x0 = 0. Thus
our characteristic curves are

Problem Sheet 3
APM 384
Autumn 2014
1. Using the method of separation of variables or otherwise, solve the BVP
2u 2u
+
=0
x2 y 2
u(0, y) = u(x, 0) = u(x, H) = 0
u(L, y) = g2 (y).
if 0 < x < L and 0 < y < H
(1)
(2)
(3)
2. Using the method of separation of

Problem Sheet 4
APM 384
October 16, 2014
1. [10 marks] Suppose that u : (0, ) R3 R satises the heat equation in
three dimensions, i.e.
u
(t, x) = u (t, x) .
t
(1)
(a) Show that for any R the function u : (0, ) R3 R dened by
u (t, x) = u (2 t, x) also sati

Problem Sheet 7
APM 384
Autumn 2014
On this sheet, questions 1 - 4 and 8 are assessed and have equal weight. Solutions
are due before the lecture on Friday 28 November.
1. Consider the boundary value problem
u(x, y) = f (x, y)
u
(x, 0) = 0.
y
if y > 0
(a)

Problem Sheet 1: Solutions
APM 384
Autumn 2014
Throughout this problem sheet let V1 , V2 , V3 be vector spaces.
1. Let T1 , T2 : V1 V2 be linear operators and 1 , 2 R. Show that the following are also linear operators:
(a) 1 T1
(b) T1 + T2
(c) 1 T1 + 2 T2

Problem Sheet 6
APM 384
Autumn 2014
On this sheet, all questions are assessed and have equal weight. Solutions are due
in the lecture on Monday 17 November.
1. Consider non-uniform heat equation
c(x)(x)
u
u
=
K0 (x)
t
x
x
(1)
on the interval [0, L], subje

[4)
UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
FINAL EXAMINATION. DECEMBER 2002
Third Year A Engineering Science
APM 384 HiF - PARTIAL DIFFERENTIAL EQUATIONS
Exam Type: C
Examiner NA. Derzko
QueStions have equal value. Total marks -

/
University of Toronto
FACULTY OF APPLIED SCIENCE AND ENGINEERING 1
FINAL EXAMINATION, DECEMBER 1996 5;
Third Year Program III-5(a), 5bme, 5env, 5p
APM 384 F - Partial Differential Equations
Exam Type: C 1'
Examiner: R. Ross
Instructions: All questi

OCTOBER 2009 MIDTERM TEST 1 APM346H1 Partial Dierential Equations (closed book test, no calculators) Instructors: M. Chugunova, V. Ivrii Marker: Ioannis Anapolitanos Duration: 1.5 hours Total Marks: 100 Question 1 (5 marks ). Write the general form of the

Innite Domain Problems ( < x, y < +): Problem 0
Find the region in xy plane where the equation
(1 x)uxx + 2 y uxy + (1 + x)uyy = 0
is elliptic, hyperbolic, parabolic.
Problem 1
Solve by any method:
uyy uyx 2 uxx = 0,
u(x, 2x) = (x),
uy (x, 2x) = (x)
Probl

APM384
Questions:
1) Existence?
[2) Uniqueness?]
3) Properties
1
Introduction
Definition 1.1
The order of a differential equation is the highest derivative that appears.
2
Eg.
u u
+ =0
2
x t
(second order)
3
u u
=
3
x t
(3rd order)
Definition 1.2
A DE

Problem Sheet 7
APM 384
Autumn 2014
On this sheet, questions 1 - 4 and 8 are assessed and have equal weight. Solutions
are due before the lecture on Friday 28 November.
1. Consider the boundary value problem
u(x, y) = f (x, y)
u
(x, 0) = 0.
y
if y > 0
(a)

Problem Sheet 6
APM 384
Autumn 2014
On this sheet, all questions are assessed and have equal weight. Solutions are due
in the lecture on Monday 17 November.
1. Consider non-uniform heat equation
u
u
=
K0 (x)
c(x)(x)
t
x
x
(1)
on the interval [0, L], subje

Problem Sheet 5
APM 384
Autumn 2014
On this sheet, all questions are assessed. Exercises 1-2 are worth 5 points, exercises
3-4 are worth 10 points. Solutions are due in the lecture on Monday 10 November.
1. Consider the partial differential equation
(x)