2/19/2016
MATH 4557 - MIDTERM 1 - SOLUTIONS
1. (50) Let v(x, t) be a function given by the following formula,
S(x y, t)(y) dy
v(x, t) =
1
x2
S(x, t) = exp
t
t
with
.
where (x) is a piecewise continuous function on the whole line R.
(a) (20) Show that v(x

1/20/2016
MATH 4557 - SOLUTIONS FOR 1.2 (7) AND 1.5 (2)
1. Section 1.2, 7:
2
(a) Solve the equation yux + xuy = 0 with u(0, y) = ey .
(b) In which region of the xy-plane is the solution uniquely determined?
Solution:
(a) Consider a curve C := cfw_x(t), y(

MATH 4557
PARTIAL DIFFERENTIAL EQUATIONS SPRING 2015
MIDTERM I
WEDNESDAY, FEBRUARY 18, 2015
3:00 - 3:55 PM, RAMSEYER 115
Solutions
No books, notes or electronic devices.
Show your work.
1. (10 points) What is the general solution of the ordinary different

MATH 4557
PARTIAL DIFFERENTIAL EQUATIONS SPRING 2015
MIDTERM II
WEDNESDAY, APRIL 1, 2015
3:00 - 3:55 PM, RAMSEYER 115
Solutions
No books, notes or electronic devices.
Show your work.
1. (10 points) Given the formula for the solution of the diffusion equat

2/17/2016
MATH 4557 - SAMPLE MIDTERM 1 - SOLUTIONS
1. (50) Let v(x, t) be a function given by the following formula,
S(x y, t)(y) dy
v(x, t) =
1
x2
S(x, t) = exp
t
t
with
.
where (x) is a piecewise continuous function on the whole line R.
(a) (20) Show t

MATH 110
SOLUTIONS TO HW #1
Section 1.1, Problem 3
Recall that an equation is linear homogeneous if you can add two solutions and
get another. A good way to check for this is simply to see wether or not the
equation involves any multiplications or non-lin

MATH 110
SOLUTIONS TO HW #2
Section 1.4, Problem 3
With insulating boundary conditions, one has that
total heat content:
H(t) =
dH
dt
= 0 where H(t) is the
u(x, t) dx .
It is a fact that in this kind of situation, the function u(x, t) will become constant

MATH 110
SOLUTIONS TO HW #4
Typed Problem # 1
Recall from the general discussion in class that if u(x, t) solves the Dirichlet BC
system:
utt = c2 uxx ,
06x6`,
u(0, t) = u(`, t) = 0 ,
X
n
u(x, 0) =
An sin( x) ,
`
16n
X
n
ut (x, 0) =
Bn sin( x) ,
`
16n
the

MATH 110
SOLUTIONS TO HW #6
Section 5.3, Problem 2
Well solve each part of this problem separately:
a) We are trying to show that 1, x = 0. That is:
1
1 x dx = 0 .
1
This is a simple computation.
b) Next, we are trying to nd p2 (x) = a+bx+cx2 such that 1,

MATH 110
SOLUTIONS TO HW #3
Section 2.3, Problem 4
For this problem it is necessary to use the strong maximum principle which is
mentioned p. 41. This simply says that if u is not constant, then one can replace
in the maximum principle by < in the interio

MATH 110
SOLUTIONS TO HW #5
Typed Problem # 1
The coefficients of the sin series are given by the formula:
Z 1
f (x) sin(nx) dx ,
An = 2
0
where f (x) is given by the formula on the worksheet. Integrating by parts one time,
we have that:
Z 1
2
An =
f 0 (x

MATH 110
SOLUTIONS TO HW #7
Section 6.1, Problem 6
The way we will solve this is the following. Notice that since the boundary
conditions are rotationally symmetric, the solution must be also. Therefore, we
assume that the solution is a function u(r) whic

2/15/2016
MATH 4557 - SOLUTIONS OF 3.1 (4) AND 3.2 (2)
1. Section 3.1, 4: Consider the diffusion equation with a Robin boundary condition,
PDE :
ut = kuxx ,
IC :
0 < x < ,
for
u(x, 0) = x,
0<x<
for
ux (0, t) 2u(0, t) = 0,
BC :
0<t<
for
0<t<
Let f (x) = x

3/6/2016
SOLUTIONS OF SELECTED PROBLEMS IN CHAPTERS 4 AND 5
1. Section 4.1, 6: Separate the variables for the equation tut = uxx + 2u with the boundary
conditions u(0, t) = u(, t) = 0. Show that there are an innite number of solutions that
satisfy the ini

3/6/2016
MATH 4557 - SAMPLE QUIZ 3
Consider the initial-boundary value problem for the diffusion equation,
0<x<1
PDE: ut = uxx ,
for
0<t
(
ux (0, t) = 0,
BCs:
for 0 t,
ux (1, t) u(1, t) = 0,
IC: u(x, 0) = (x),
for 0 < x < 1.
(a) (10) Step 1: Using the met

4/1/2016
MATH 4557 - QUIZ 4 - SOLUTIONS
Consider the Laplace equation in the polar coordinates,
2 u 1 u
1 2u
+
= 0,
+
r2
r r
r2 2
for D :=
0 < r < 1,
0<
with the boundary condition given by
1. u = 0
on cfw_ = 0, = and 0 < r < 1,
u
+ au = sin + b sin(3) o

MATH 4557
PARTIAL DIFFERENTIAL EQUATIONS
MIDTERM II SAMPLE QUESTIONS
SPRING 2015
No books, notes or electronic devices.
Show your work.
A complete exam is 100 points.
1. (10 points) Write down (do not derive) the formula for the solution of the initial-va

MATH 4557
PARTIAL DIFFERENTIAL EQUATIONS SPRING 2015
FINAL EXAMINATION SAMPLE QUESTIONS
No books, notes or electronic devices.
Show your work.
A complete exam is 200 points.
The final will be comprehensive; roughly 35% will be material covered by midterm

MATH 4557
PARTIAL DIFFERENTIAL EQUATIONS
MIDTERM I SAMPLE QUESTIONS
SPRING 2015
No books, notes or electronic devices.
Show your work.
A complete exam is 100 points.
1. (15 points) Find the general solution of the ordinary differential equation y 00 +3y 0

2/5/2016
MATH 4557 - QUIZ 2 - SOLUTIONS
1. (10) Using the maximum principle, prove the comparison principle for the diusion equation: If u and
v are two solutions, and if u v for t = 0, for x = 0, and for x = , then u v for 0 x , 0 t < .
Solution: Let D b

1/22/2016
MATH 4557 - QUIZ 1 - SOLUTIONS
1. (20) Consider the rst order equation with the boundary condition on x = 0,
2yux + uy = 0
2
u(0, y) = ey .
with
(a) (10) Find the solution.
(b) (5) In which region of the xy-plane is the solution uniquely determi

3/28/2016
SOLUTIONS OF SELECTED PROBLEMS IN CHAPTER 6
1. Section 6.1, 11: Show that there is no solution of
u
u = f in D,
= g on D,
n
in three dimensions, unless
ZZZ
ZZ
f dV =
g dS.
D
D
Also show the analogue in one and two dimensions.
Solution: Let F~ be

9/4/2015
MATH 4557 - QUIZ 1 - SOLUTIONS
1. (20) Consider the rst order equation, yux xuy = 0.
(a) Find the general solution (including an arbitrary function of single variable).
(b) If u(0, y) = f (y), what condition on f (y) is necessary for the existenc

2/2/2016
MATH 4557 - CHAPTER 2 - SOLUTIONS
1. Section 2.1, 7: The general solution of the wave equation utt = c2 uxx on the whole real line
is given by
u(x, t) =
1
1
[(x + ct) + (x ct)] +
2
2c
x+ct
(s) ds,
xct
where (x) = u(x, 0) and (x) = ut (x, 0). If b

2/2/2016
MATH 4557 - QUIZ 2 - SAMPLE
Consider the wave equation on the real line,
PDE:
utt = 4uxx ,
< x <
0<t
for
u(x, 0) = 0,
ICs:
for
2
ut (x, 0) = 8xex ,
< x < .
(a) (10) Show that the following form of u(x, t) satises the PDE,
u(x, t) = f (x + 2t)

11/6/2015
MATH 4557 - QUIZ 4
Consider the Laplace equation in the polar coordinates,
2 u 1 u
1 2u
+
+ 2 2 = 0,
2
r
r r
r
for
D :=
0 < r < 1,
0 < 2
(a) (20) Using the method of separation of variables, u(r, ) = R(r)(), nd the general solution which is bo

10/23/2015
MATH 4557 - QUIZ 3 - SOLUTIONS
Consider the initial-boundary value problem for the diusion equation,
PDE:
ut = uxx ,
0<x<1
0<t
ux (0, t) = 0,
BCs:
IC:
for
for
ux (1, t) u(1, t) = 0,
u(x, 0) = (x),
for
0 t,
0 < x < 1.
(a) (10) Step 1: Using the

9/19/2015
MATH 4557 - QUIZ 2 - SOLUTIONS
Consider the wave equation on the real line,
PDE:
utt = 4uxx ,
< x <
0<t
for
u(x, 0) = 0,
ICs:
for
2
ut (x, 0) = 8xex ,
< x < .
(a) (10) Show that the following form of u(x, t) satises the PDE,
u(x, t) = f (x +

3/28/2016
MATH 4557 - SAMPLE QUIZ 4
Consider the Laplace equation in the polar coordinates,
2 u 1 u
1 2u
0 < r < 1,
+ 2 2 = 0,
+
for D :=
2
0 < 2
r
r r r
(a) (20) Using the method of separation of variables, u(r, ) = R(r)(), find the
general solution wh