Math 523/Spring 2013, Assignment 1
Solutions
1. (10 pts) Show that the Laplace operator is invariant under orthogonal change of coordinates and
2
2
translations. In other words, let x = x1 + . . . + xn as usual and let y = U x + a be a change of variables
Math 523 Spring 2013 Exam 2
1. (15 pts)
Consider the Cauchy problem
u2 uzz + u2 ux
xy
u(x, y, 0)
uz (x, y, 0)
=
0,
= x y,
= sin x.
Find uxz , uyz , uzz , uxyz , uxzz on the plane z = 0 as functions of x, y .
Answer:
uxz (x, y, 0) = cos x
uzz (x, y, 0) = (
Math 523 Spring 2013 Exam 3
1.
(15 pts) Let D be the semi-disk x2 + y 2 < 1, y > 0 Solve
u = 0
in D,
u|D1 = (2x 2) 1 x2 ,
u|D2 = 0,
where D1 = cfw_x2 + y 2 = 1, y 0 is the upper part of the boundary, and D2 = cfw_y = 0 is the lower part.
If you use polar
Math 523 Spring 2013 Test 1
1. (15 pts) Consider the Cauchy problem
uy = xuux ,
u(x, 0) = x.
Find ux , uy , uxx , uxy , uyy on the line y = 0 as a function of x.
Answer:
ux (x, 0) = 1, from the initial condition.
uy (x, 0) = x2 , from the PDE and the abov
Math 523
Qualifying Examination
January 3, 2007
Name.
I. D. no. .
Problem Score Max. pts.
1
20
2
20
3
20
4
20
5
20
Total
100
Problem 1. 1) Let Sn1 = cfw_ Rn | | | = 1 be the unit sphere centered at the origin. Prove
that the function u(x, t) = ei t (x), w
Name:
Math 523 Qualifying Exam August 2013
Note: Some of the subquestions (a), (b), etc., in problems 1 and 2 are independent of each other and
can be answered even if the previous ones are not answered. You have to justify each answer on this exam.
1. (2
Math 523 Spring 2013 Assignment 5: solutions to some of the
problems
p.110, #4
(a) We are looking for a solution, in polar coordinates, of the form
rn (an cos(n) + bn sin(n) .
u = a0 /2 +
n=1
To satisfy the boundary condition, we need to satisfy
r u + u =
Math 523
Qualifying Examination
August 18, 2009
Prof. N. Garofalo
Name.
I. D. no. .
Problem Score Max. pts.
1
20
2
20
3
20
4
30
5
40
Total
130
Problem 1. Let be a continuous function on Rn with compact support. Prove that if is
spherically symmetric, i.e.
Math 523
Qualifying Examination
August , 2008
Prof. N. Garofalo
Name.
I. D. no. .
Problem Score Max. pts.
1
20
2
20
3
20
4
20
5
20
Total
100
Problem 1. 1) Let Rn be an open set and consider a sequence cfw_fk kN , fk C 2 (), of
harmonic functions in such t
Math 523
Qualifying Examination
August 9, 2010
Prof. N. Garofalo
Name.
I. D. no. .
Problem Score Max. pts.
1
20
2
20
3
30
4
30
5
30
Total
130
Problem 1. Let E be a regular hexagon centered at the origin of the plane R2 . Let f be the
harmonic function in
Qualifying exam in Partial Dierential Equations
August, 2011
Name.Total: 160 points
I solved problems (give four numbers here).
Solve any FOUR of the following ve problems. You HAVE to specify which
problems you solved. Only four will be graded.
The total
Math 523
Qualifying Examination
August 7, 2012
Prof. N. Garofalo
Name.
I. D. no. .
Problem Score Max. pts.
1
20
2
30
3
20
4
30
5
20
Total
120
Problem 1. Let be a continuous function on Rn with compact support, F C (Rn (0, )
with compact support. Write an
Math 523
Qualifying Examination
January 6, 2013
Prof. N. Garofalo
Name.
I. D. no. .
Problem Score Max. pts.
1
30
2
40
3
30
4
40
5
30
Total
170
Problem 1. Prove that the solution to the Cauchy problem
u utt = 0,
in R3 (0, ),
|x|u(x, 0) = sin |x|,
ut (x, 0)
Math 523
Qualifying Examination
January, 2012
Prof. N. Garofalo
Name.
I. D. no. .
Problem Score Max. pts.
1
20
2
30
3
20
4
30
5
20
Total
120
Problem 1. Let be a continuous function on Rn with compact support, F C (Rn (0, )
with compact support, a Rn \ cfw
MA 523 Qualifying Exam, Spring 2011. Name:
There are ve questions, each worth 20 points.
1. Let I (x) be the indicator function for B1 (0) = cfw_x R3 : |x| < 1. Let u be such that
for x R3 ,
u = I (x)
u(x) 0
as
|x| .
a) Represent u using the fundamental s
MA523 Qualifying Exam, January 2008
P. Bauman and P. Stefanov
Each problem is worth 20 points. Everywhere in this exam, be a bounded domain (an open connected
set) in Rn with smooth boundary.
1.
Let u be a harmonic function in , continuous in .
(a) Show t
Qualifying Exam - Math 523 - January 2010 - P. Bauman
Instructions: Show your reasoning in all problems.
1. Consider the following Cauchy problem:
xux yuy = u 1,
u(x, x) = 1 + x3 .
a) (8 pts.) At what values x0 is there a unique C 1 solution in a neighbor
Math 523 Spring 2013 Exam 3
1.
(15 pts) Let D be the semi-disk x2 + y 2 < 1, y > 0 Solve
u = 0
in D,
u|D1 = (2x 2) 1 x2 ,
u|D2 = 0,
where D1 = cfw_x2 + y 2 = 1, y 0 is the upper part of the boundary, and D2 = cfw_y = 0 is the lower part.
If you use polar
Name:
Instructor: P. Stefanov
Math 523/Spring 2013, Final Exam , Answers
1. (15 pts)
(a) Find the solution of the Cauchy problem
uyy = uxx + u,
u(x, 0) = ex ,
uy (x, 0) = 0
in the form of a power series expansion with respect to y , i.e., u =
an (x)y n .
MATH 523/FALL 2013, ASSIGNMENT 2
Name:
1. (10 pts). (1999) Solve the initial value problem
= u3
1
u(x, 0) = ,
x
3ux + uy
Answer:
x > 0.
The characteristic system is
dx
dy
dz
= 3,
= 1,
= z3
dt
dt
dt
with initial conditions at t = 0 given by (x, y, z ) = (s
Math 523 Spring 2013 Assignment 3
Name:
1. (14 pts)
(1989) Given the equation
xuxx + 2yuxy + uyy + ex ux + 2u = 0.
(a) Write down the equation for the characteristic curves.
(b) Describe the regions in the plane where the equation is hyperbolic, parabolic
Math 523 Spring 2013 Assignment 4: solutions to some of the
problems
Name:
p. 72, #13
(a) Use the denition. Since L = L, we need to show that
F1 , L = (0),
C0 (R2 ).
We have
F1 , L =
H ()H ( ) (, ) d d =
(, ) d d
0
0
Integrate twice by part to get
F1 ,