Math2T03: Mid-Term Test 2 Instructor: Dr. D. Pelinovsky Date: March 20 2009, 11:30-12:20
NAME: STUDENT ID NUMBER: Instructions: This test paper is printed on both sides of the paper. It includes four questions on six pages. The last page can be used for r
Math2T03: Mid-Term Test 1 Instructor: Dr. D. Pelinovsky Date: February 6 2009, 11:30-12:20
NAME: STUDENT ID NUMBER: Instructions: This test paper is printed on both sides of the paper. It includes four questions on six pages. The last page can be used for
LAST (family) NAME:
Test # 1
FIRST (given) NAME:
Math 3I03
ID # :
October 17, 2013
Instructor:
Dr. J.-P. Gabardo
Test duration: 50 min.
Instructions: You must use permanent ink. Tests submitted in pencil will not be considered later for remarking. This ex
Math 3I03: Assignment # 3
(Due Thursday October 29, 2015 by 4 pm)
Problem 1. Consider the following heat equation on a circular rod (which
includes a heat source and heat loss through the lateral side):
u
2u
u + sin2 (x),
=
t
x2
< x < ,
t > 0,
u(, t) =
Math 3I03: Assignment # 4
(Due Monday November 16, 2015 by 4 pm)
Problem 1. [20] Use the method of eigenfunctions expansion to solve the
non-homogeneous heat equation
u
2u
+ 2 cos2 x cos t,
=
t
x2
0 < x < ,
t > 0,
subject to the boundary conditions
ux (0,
Math 3I03: Assignment # 2: Solutions
(Due Thursday October 8, 2014 by 4 pm)
Total of Marks= 50
Problem 1.
(a) [15 ] Solve the heat equation
u
2u
= k 2,
t
x
L < x < L,
t > 0,
subject to the periodic boundary conditions
u(L, t) = u(L, t)
and
u
u
(L, t) =
(L
Math 3I03: Assignment # 5: Solutions
(Due Monday November 30, 2015 by 4 pm)
Problem 1. [10] Consider the wave equation on the real line
2u
2u
= c2 2 , < x < , t > 0,
2
t
x
with initial conditions
u
(x, 0) = g(x),
t
u(x, 0) = f (x),
< x < .
Suppose that b
(1)
Find eigenvalues (n ) and eigenfunctions (n )of the following ordinary dierential equation
( (x)
d2
+ = 0
dx2
subject to
(0) = 0, (L) = 0.
(2)
Find eigenvalues and eigenfunctions of the same equation above with boundary conditions,
d
d
dx (0) = 0, dx
Math 3I03: Assignment # 1: Solutions
(Due Thursday September 24, 2015 by 4 pm)
Total of Marks= 50
Problem 1. [20] Use the method of separation of variables and the superposition principle to solve the heat equation
u
2u
= k 2,
t
x
0 < x < ,
t > 0,
subject
Math 3I03: Assignment # 2
(Due Thursday October 8, 2014 by 4 pm)
Problem 1.
(a) [15 ] Solve the heat equation
u
2u
= k 2,
t
x
L < x < L,
t > 0,
subject to the periodic boundary conditions
u(L, t) = u(L, t)
and
u
u
(L, t) =
(L, t),
x
x
t > 0,
and satisfyin
Math 3I03: Assignment # 5
(Due Monday November 30, 2015 by 4 pm)
Problem 1. [10] Consider the wave equation on the real line
2u
2u
= c2 2 , < x < , t > 0,
2
t
x
with initial conditions
u(x, 0) = f (x),
u
(x, 0) = g(x),
t
< x < .
Suppose that both f (x) a
Math 3I03: Assignment # 3
(Due Thursday October 29, 2015 by 4 pm)
Problem 1. Consider the following heat equation on a circular rod (which
includes a heat source and heat loss through the lateral side):
u
2u
u + sin2 (x),
=
t
x2
< x < ,
t > 0,
u(, t) =
Math 3I03: Assignment # 1
(Due Thursday September 24, 2015 by 4 pm)
Problem 1. [20] Use the method of separation of variables and the superposition principle to solve the heat equation
2u
u
= k 2,
t
x
0 < x < ,
t > 0,
subject to the boundary conditions
u(
Math 3I03: Assignment # 4
(Due Monday November 16, 2015 by 4 pm)
Problem 1. [20] Use the method of eigenfunctions expansion to solve the
non-homogeneous heat equation
u
2u
+ 2 cos2 x cos t,
=
t
x2
0 < x < ,
subject to the boundary conditions
ux (0, t) = u
h n f n l f k h ngg
Cqzz"qdqdHdr
sddPd2zer"idmmddHzqerPeelepFmR8x
g g kp s g kp l g n g g kp r p g n h v n
p j h r pg l ~ k g p p g r g g gn n x g kp gng k f r pg l ~ | k pgp k
d$edSYdqyrdPHdPqd811qruPdqud2Pudqcfw_z"j
h gp l x w v p gp kp s r p n l g k
Spring 2010
Math 322
Dr. Hermi
Exam # 1Version A
Partial credit is possible, but you must show all work. You may
use a calculator, however all integrals have to be either worked out
directly, or performed using the attached table.
Name:
TA:
I hereby testi
MATH 3I03 ASSIGNMENT 6 (DUE ON 19TH OF NOVEMBER)
Problem 1 (12.2.1 in the textbook) Show that the wave equation can be considered as
the following system of two coupled rst-order partial dierential equations:
u
u
c
= w,
t
x
w
w
+c
= 0.
t
x
Problem 2 Solve
ASSIGNMENT 2 (DUE ON THE 10TH OF OCTOBER)
Problem 1
Explicitly show that there are no negative eigenvalues for
d2
=
dx2
subject to
d
dx (0)
= 0 and
d
dx (L)
= 0.
Problem 2
Solve for Bn in the following equation
Bn sin
f (x) =
n=1
nx
L
on 0 < x < L.
Supp
Solve the heat equation
u
2u
=k 2
t
x
on 0 < x < L
subject to boundary conditions
u(0, t) = 0, u(L, t) = 0
and an initial condition
u(x, 0) = f (x).
1
Set u(x, t) = (x)G(t) for the heat equation
u
t
d2
2
= k u and get
x2
d
[(x)G(t)] = k 2 [(x)G(t)]
dt
dx
MATH3I03 ASSIGNMENT 5 (DUE ON NOVEMBER 12TH)
Modeling of a Vibrating String
Problem 1
Consider the one-dimensional wave equation
2u
2u
= c2 2 in 0 x L
t2
x
where u(x, t) describes the displacement in y-direction at position x and time t, and c is a
consta
MATH 3I03 ASSIGNMENT 7 (DUE ON 28TH OF NOVEMBER)
Problem 1 (Lecture 24)
Consider the wave equation for a semi-innite vibrating string (0 x < ),
2u
2u
= c2 2 ,
t2
x
subject to initial conditions
1
0
u(x, 0) =
3<x<4
otherwise,
u
(x, 0) = 0
t
and the boundar
(a) Solve Laplaces equation
2u 2u
+ 2 =0
x2
y
inside a rectangle 0 x L, 0 y H, with the following boundary conditions:
u
u
x (0, y) = 0, x (L, y) = 0, u(x, 0) = 0, u(x, H) = f (x)
SOLUTION:
We look for solution in the form
u(x, y) = h(x)(y).
Then, substit
Evaluate
L
sin
0
nx
mx
sin
dx
L
L
for n > 0, m > 0.
1
Using the trigonometric identity
1
sin a sin b = [cos(a b) cos(a + b)],
2
we can write
mx
nx
sin
=
L
L
1
=
cos
2
Then we integrate
sin
1
nx mx
nx mx
cos
cos
+
2
L
L
L
L
x
x
(n m) cos
(n + m) .
L
L
nx
MATH3I03 ASSIGNMENT 3 (DUE ON 22TH OF OCTOBER)
Problem 1. 2.5.1. (f) in the textbook
Solve the Laplaces equation inside a rectangle 0 x L, 0 y H, with the following
boundary conditions:
u(0, y) = f (y), u(L, y) = 0,
u
u
(x, 0) = 0,
(x, H) = 0.
y
y
Problem
MATH322 (SOLUTIONS TO THE SAMPLE EXAM)
1. Give all the possible values of the following:
(a) log(1
3i) =
3 = 2ei(/3+2k) (k = 0, 1, 2, .) so that we have
log(1 3i) = log(2ei(/3+2k) )
= log 2 i + i2k, (k = 0, 1, 2, .).
3
(b) Log(1 3i) =
Write 1
Recall tha
h n f n l f k h ngg
Cq1zz"qdqdHdr
sddPdzer"idmmddHzqerPeelepFmRix
g g kp s g kp l g n g g kp r p g n h v n
p j h r pg l ~ k g p p g r g g gn n x g kp gng k f r pg l ~ | k pgp k
d$edSYdqysdPHdPqdiqsuPdqudPudqcfw_z"j
h gp l x w v p gp kp s r p n l g k j h
4
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
6. ex
e
x
67
=2
solution:
ex
e
x
e
x
= 2
1
= 2
ex
1
= 2
ex ex
ex
x 2
(e )
1 = 2ex
(ex )2 +
2ex
1 = 0
If we take y = ex , then this is a quadratic again
y2
y=
p
2 22
2y
4(1)( 1)
2
Since y = e we have two solutions:
x
Math3I03 Assignment3
Solution
Solution to Problem 1.
(a)
u(x, t) = a0 t +
1
X1
e
n2
n=1
where
n2 t
[an cos(nx) + bn sin(nx)] ,
Z
1
Q(x)dx,
2
Z
1
Q(x) cos(nx)dx,
Z
1
Q(x) sin(nx)dx.
a0 =
an =
bn =
(b)
In the solution u(x, t) above, we have the term a0 t,