MATH 220: MIDTERM
OCTOBER 29, 2009
This is a closed book, closed notes, no calculators exam.
There are 5 problems. Solve all of them. Write your solutions to problems 1 and 2 in
blue book #1, and your solutions to problems 3, 4 and 5 in blue book #2. With
MATH 220: INTRODUCTION TO PDE
ANDRAS VASY
Partial dierential equations are ubiquitous in applications of mathematics, and
indeed they play an important role in pure mathematics as well. In this course we
study these equations, and we start by describing g
MATH 220: INNER PRODUCT SPACES, SYMMETRIC
OPERATORS, ORTHOGONALITY
When discussing separation of variables, we noted that at the last step we need to
express the inhomogeneous initial or boundary data as a superposition of functions
arising in the process
MATH 220: THE FOURIER TRANSFORM BASIC
PROPERTIES AND THE INVERSION FORMULA
The Fourier transform is the basic and most powerful tool in studying constant
coecient PDE on Rn . It is based on the following simple observation: for Rn ,
the functions
v (x) =
MATH 220: FINAL EXAM DECEMBER 11, 2009 SOLUTIONS
This is a closed book, closed notes, no calculators exam.
There are 7 problems. Solve all of them. Total score: 200 points.
Problem 1.
(i) (20 points) For |y 1| small, solve
xux + yuy = 1, u(x, 1) = x2 .
Sk
MATH 220: FIRST ORDER SCALAR SEMILINEAR EQUATIONS
First order scalar semilinear equations have the form
(1)
a(x, y )ux + b(x, y )uy = c(x, y, u);
here we assume that a, b, c are at least C 1 , given real valued functions. Let V be
the vector eld on R2 giv
MATH 220: MIDTERM SOLUTIONS
OCTOBER 29, 2009
This is a closed book, closed notes, no calculators exam.
There are 5 problems. Solve all of them. Write your solutions to problems 1
and 2 in blue book #1, and your solutions to problems 3, 4 and 5 in blue boo
MATH 220: MIDTERM SOLUTIONS
NOVEMBER 1, 2012
This is a closed book, closed notes, no computers/calculators exam.
There are 5 problems. Solve Problems 1-3 and one of Problems 4 and 5. Write
your solutions to problems 1 and 2 in blue book #1, and your solut
MATH 220: DISTRIBUTIONS AND WEAK DERIVATIVES
ANDRAS VASY
The basic motivation for distribution theory is sometimes a PDE does not have
a classical solution, i.e. there are mth order PDEs which do not have C m solutions
(with whatever conditions we want to
MATH 220: PROPERTIES OF SOLUTIONS OF SECOND ORDER
PDE
ANDRAS VASY
We have solved the initial value problem for the wave equation
2
2
(t c2 x )u = 0, u(x, 0) = (x), ut (x, 0) = (x),
namely we showed that the solution is
u(x, t) =
1
1
(x + ct) + (x ct) +
2
MATH 220: FIRST ORDER SCALAR QUASILINEAR
EQUATIONS
We now consider the quasilinear equations; these have the form
(1)
a(x, y, u) ux + b(x, y, u) uy = c(x, y, u),
1
with a, b, c at least C , given real valued functions. There is an immediate dierence
betwe
MATH 220: FINAL EXAM DECEMBER 11, 2009
This is a closed book, closed notes, no calculators exam. There are 7 problems. Solve all
of them. Write your solutions to problems 1, 2 and 3 in blue book(s) #1, and your solutions
to problems 4, 5, 6 and 7 in blue
MATH 220: CONVERGENCE OF THE FOURIER SERIES
We now discuss convergence of the Fourier series on compact intervals I . Convergence depends on the notion of convergence we use, such as
(i) L2 : uj u in L2 if uj u L2 0 as j .
(ii) uniform, or C 0 : uj u unif