Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 3 Solutions
1. Suppose fn : R ! R is a sequence of continuous, nonnegative functions such that
1
x=0
fn (x) !
0
x 6= 0
as n ! +1. In addition, assume
Z
1
fn (x) dx = 1
1
for all n, and there exists a closed, bounded subset
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 7
Due Thursday, August 14, 2003
1. Consider the eigenvalue problem,
w = w
w
+ a(x)w = 0
n
x
x .
(1)
Let cfw_vi be the eigenfunctions for this problem. Let
Yn cfw_w C 2 : w 6 0, hw, vi i = 0 for i = 1, . . . , n 1.
Let
R
J
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Physical Interpretation of Single and Double Layer Potentials
Below is a brief description of a physical interpretation for the single and double layer
potentials.
References
Folland, G. Introduction to Partial Differential Equations, Chap. 3.
McOwen, R
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 6
Due Thursday, August 7, 2003
1. Consider the Neumann problem,
u = f
u
=g
x
x
Assume the compatibility condition holds. That is,
Z
Z
f (x) dx =
g(x) dS(x).
Just as the Greens function allowed us to find a representation
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 5
Due Thursday, July 31, 2003
1. Let = (0, k)(0, l). Use separation of variables to solve the following boundaryvalue
problem for Laplaces equation on a square,
(x, y)
u = 0
u(0, y) = 0, ux (k, y) = (y)
0<y<l
uy (x, 0)
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Comments on RayleighRitz Approximation and Minimax
Principle
Let cfw_w1 , . . . , wn be n linearly independent trial functions. Let A = (ajk ), B = (bjk )
where ajk = hwj , wk i and bjk = hwj , wk i. Consider the equation
det(A B) = 0.
Assume 1 , . . .
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 1
Due Thursday, July 3, 2003
1. Consider the eigenvalue problem
X 00 = X
X satisfies symmetric B.C.s.
Suppose
0<x<l
x = 0, l.
x=b
f (x)f 0 (x)x=a 0
for all realvalued functions f (x) which satisfy the boundary conditions
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 4 Solutions
1. Let be an open, bounded set in Rn . Let C(), g C(), and suppose
u C 2 ( [0, ) is a solution of
(x, t) (0, )
ut (x, t) = ku(x, t)
u(x, t) = g(x)
(x, t) [0, )
(1)
u(x, 0) = (x)
x .
Also suppose that v C 2 ()
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 2 Solutions
1. (a) Compute the Fourier transform of xf in terms of fb.
Answer:
Z
1
c
xf () =
eix xf (x) dx
2
Z
1 d ix
1
=
e
f (x) dx
2 i d
Z
1 d
1
ix
e
f (x) dx
=
i d
2
Z
d
1
ix
=i
e
f (x) dx
d
2
d
= i fb().
d
c (
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 3 Solutions
1. Suppose fn : R R is a sequence of continuous, nonnegative functions such that
x=0
fn (x)
0
x 6= 0
as n +. In addition, assume
Z
fn (x) dx = 1
for all n, and there exists a closed, bounded subset K R such th
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 5 Solutions
1. Let = (0, k)(0, l). Use separation of variables to solve the following boundaryvalue
problem for Laplaces equation on a square,
(x, y)
u = 0
u(0, y) = 0, ux (k, y) = (y)
0<y<l
uy (x, 0) = 0, u(x, l) = 0
0
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 1 Solutions
1. Consider the eigenvalue problem
X 00 = X
X satisfies symmetric B.C.s.
Suppose
0<x<l
x = 0, l.
x=b
f (x)f 0 (x)x=a 0
for all realvalued functions f (x) which satisfy the boundary conditions. Show there
are
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 7 Solutions
1. Consider the eigenvalue problem,
w = w
w
+ a(x)w = 0
n
x
x .
(1)
Let cfw_vi be the eigenfunctions for this problem. Let
Yn cfw_w C 2 : w 6 0, hw, vi i = 0 for i = 1, . . . , n 1.
Let
R
J(w)
w2 dx +
R
R
a
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 6 Solutions
1. Consider the Neumann problem,
u = f
u
=g
x
x
Assume the compatibility condition holds. That is,
Z
Z
f (x) dx =
g(x) dS(x).
Just as the Greens function allowed us to find a representation formula for soluti
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 2
Due Thursday, July 10, 2003
1. (a) Compute the Fourier transform of xf in terms of fb.
2
(b) Compute the Fourier transform of xetx .
2. Use the Fourier transform to show that the solution of the inhomogeneous heat equati
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 4
Due Tuesday, July 22, 2003
1. Let be an open, bounded set in Rn . Let C(), g C(), and suppose
u C 2 ( [0, ) is a solution of
(x, t) (0, )
ut (x, t) = ku(x, t)
u(x, t) = g(x)
(x, t) [0, )
(1)
u(x, 0) = (x)
x .
Also suppo
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 4 Solutions
1. Let be an open, bounded set in Rn . Let 2 C(), g 2 C(@), and suppose
u 2 C 2 ( [0, 1) is a solution of
8
(x, t) 2 (0, 1)
< ut (x, t) = ku(x, t)
u(x, t) = g(x)
(x, t) 2 @ [0, 1)
(1)
:
u(x, 0) = (x)
x 2 .
Also
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 5 Solutions
1. Let = (0, k)(0, l). Use separation of variables to solve the following boundaryvalue
problem for Laplaces equation on a square,
8
(x, y) 2
< u = 0
u(0, y) = 0, ux (k, y) = (y)
0<y<l
:
uy (x, 0) = 0, u(x, l
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 2 Solutions
b
1. (a) Compute the Fourier transform of xf in terms of f .
Answer:
Z 1
c () = p1
xf
eix xf (x) dx
2 1
Z 1
1
1 d ix
=p
e
f (x) dx
2 1 i d
Z 1
1 d
1
ix
p
=
e
f (x) dx
i d
2 1
Z 1
d
1
ix
p
=i
e
f (x) dx
d
2 1
d
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 1 Solutions
1. Consider the eigenvalue problem
X 00 = X
X satises symmetric B.C.s.
Suppose
0<x<l
x = 0, l.
x=b
f (x)f 0 (x)x=a 0
for all realvalued functions f (x) which satisfy the boundary conditions. Show there
are no
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 6 Solutions
1. Consider the Neumann problem,
u = f
@u
=g
@
x2
x 2 @
Assume the compatibility condition holds. That is,
Z
Z
f (x) dx =
g(x) dS(x).
@
Just as the Greens function allowed us to nd a representation formula for
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
2
2.1
Heat Equation
Derivation
Ref: Strauss, Section 1.3.
Below we provide two derivations of the heat equation,
ut kuxx = 0
k > 0.
(2.1)
This equation is also known as the diusion equation.
2.1.1
Diusion
Consider a liquid in which a dye is being diused t
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
3
Laplaces Equation
We now turn to studying Laplaces equation
u = 0
and its inhomogeneous version, Poissons equation,
u = f.
We say a function u satisfying Laplaces equation is a harmonic function.
3.1
The Fundamental Solution
Consider Laplaces equation i
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
6
Eigenvalues of the Laplacian
In this section, we consider the following general eigenvalue problem for the Laplacian,
v = v
x2
v satises symmetric BCs
x 2 @.
To say that the boundary conditions are symmetric for an open, bounded set in Rn
means that
hu,
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
4
Greens Functions
In this section, we are interested in solving the following problem. Let be an open, bounded
subset of Rn . Consider
u = f
x 2 Rn
(4.1)
u=g
x 2 @.
4.1
Motivation for Greens Functions
Suppose we can solve the problem,
y G(x, y) = x
G(x,
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
5
Potential Theory
Reference: Introduction to Partial Dierential Equations by G. Folland, 1995, Chap. 3.
5.1
Problems of Interest.
In what follows, we consider an open, bounded subset of Rn with C 2 boundary. We let
c = Rn (the open complement of ). We ar
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 7 Solutions
1. Consider the eigenvalue problem,
w = w
@w
+ a(x)w = 0
@n
x2
x 2 @.
(1)
Let cfw_vi be the eigenfunctions for this problem. Let
Yn cfw_w 2 C 2 : w 6 0, hw, vi i = 0 for i = 1, . . . , n 1.
Let
J(w)
R
rw2 d
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
7
Calculus of Variations
Ref: Evans, Sections 8.1, 8.2, 8.4
7.1
Motivation
The calculus of variations is a technique in which a partial dierential equation can be
reformulated as a minimization problem. In the previous section, we saw an example of this
t
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
Math 220B  Summer 2003
Homework 3
Due Thursday, July 17, 2003
1. Suppose fn : R R is a sequence of continuous, nonnegative functions such that
x=0
fn (x)
0
x 6= 0
as n +. In addition, assume
Z
fn (x) dx = 1
for all n, and there exists a closed, bounded
Partial Differential Equations of Applied Mathematics
MATH 220B

Winter 2013
5
Potential Theory
Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5.1
Problems of Interest.
In what follows, we consider an open, bounded subset of Rn with C 2 boundary. We let
c = Rn (the open complement of ). We