Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 1.
1. Solve the equation
u u
+
= sin x,
x y
in two dimensions (x, y) R2 , with the initial condition u(x, 0) = f (x).
What would happen if we prescribe the data u(x, x) = f (x) along the line
x = y?
2. Let u Rn be a fixed vector in Rn , and c R
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
MATH 220 SOLUTIONS TO HOMEWORK 1
VITALY KATSNELSON
1.) We set (s) = (x(s), y(s) with (0) = (x0 , 0), and z(s) = u(s). We want to
solve the ODEs
z(s)
= x(s)u
= y(s)
=1
x (s) + y(s)u
y (s) = sin(x(s), x(s)
subject to the initial conditions. A solution is gi
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 2.
Problem 1. Consider the Poisson equation in a bounded domain U :
u = f (x) in U .
(1)
However, instead of prescribing the value of u at the boundary we prescribe
its normal derivative:
u
(2)
= g on U .
(i) Show that if u(x) solves the proble
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 5.
Problem 1. (i) Solve the following initial boundary value problem:
utt uxx = 0, t > 0, x > 0,
u(t, 0) = 0 for all t > 0,
u(0, x) = f (x), for x 0,
ut (0, x) = g(x) for x 0.
Hint: extend the functions f and g as odd functions over the whole r
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
MATH 220 SOLUTIONS TO HOMEWORK 2
VITALY KATSNELSON
1.) (i)Since we have a linear PDE and the derivative of a constant is 0, then u + C
also satisfies the PDE.
(ii)Using Greens theorem we have
Z
Z
Z
f dx = udx =
U
U
U
u
dS =
Z
gdS
U
(iii) First observe t
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
MATH 220 SOLUTIONS TO HOMEWORK 3
VITALY KATSNELSON
We will prove the general statement in ndimensions for the operator L =
P1.)
aij (x)xi xj . First, suppose u has a local max at x0 U and that Lu(x0 ) > 0. Let
1 , . . . , n be the positive eigenvalues of
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework #6
1. (i) Let (t, x) solve the wave equation
1
tt xx = 0.
c2
Show that u = t and p = x satisfy the 2 2 system
1
ut + px = 0
c2
pt + ux = 0.
(ii) Let A be a symmetric positive definite n n matrix, D a symmetric
matrix, and let v = (v1 , . . . , vn
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 1.
1. Solve the equation
u u
+
= sin(x y),
x y
in two dimensions (x, y) R2 , with the initial condition u(x, 0) = f (x).
What would happen if we prescribe the data u(x, x) = f (x) along the line
x = y?
2. Let u Rn be a fixed vector in Rn , and
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 4.
Problem 1. Let a(x) be a smooth function such that 1 < a(x) < 10 for
all x R, and let b R. Let u(t, x) be a bounded solution (you may assume
it exists and is unique) of the initial value problem
u
u
2u
a(x) 2 + b
= 0, t > 0, x Rn
t
x
x
u(0,
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework #6
1. Let u(t, x) solve the initial value problem for the wave equation in one
dimension:
utt uxx = 0,
with the initial data u(0, x) = g(x), ut (0, x) = f (x). Suppose that g and f
have compact support. Define the kinetic and potential energy as
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 2.
1. Let u(x) be the solution of
u = f (x) in B(0, R)
u = 0, on B(0, R).
here B(0, R) = cfw_x R is the ball of radius R in Rn , n 3, centered at
x = 0, and B(0, R) is its boundary (that is, the sphere of radius R centered
at the origin). Mod
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 1.
1. Solve the equation
u u
+
= cos(x y),
x y
in two dimensions (x, y) R2 , with the initial condition u(x, 0) = f (x).
What would happen if we prescribe the data u(x, x) = f (x) along the line
x = y?
2. Let u Rn be a fixed vector in Rn , and
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 3.
Problem 1. Let u(t, x) satisfy the heat equation
u
u = 0, t > 0, x R3
t
in three dimensions, with the initial condition u(0, x) = f (x), where f (x) is
a continuous function that vanishes outside of the ball cfw_x 1. Define
Z
u(t, x)dt.
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
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Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 4.
Problem 1. Let u(t, x) satisfy the heat equation
u
u = 0, t > 0, x R3
t
in three dimensions, with the initial condition u(0, x) = f (x), where f (x) is
a continuous function that vanishes outside of the ball cfw_x 1. Define
Z
u(t, x)dt.
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Lecture notes for Math 220
Lenya Ryzhik
December 3, 2013
Essentially nothing found here is original except for a few mistakes and misprints here
and there. These lecture notes are based on material from the following books: L. Evans
Partial Differential E
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
Homework # 3.
Problem 1. Consider an equation of the form
(1)
a
2u
2u
2u
+
b
+
c
= 0,
x2
xy
y 2
posed in a bounded domain R2 . Here a, b, c R are constants. Show
that if the matrix
a b/2
A=
b/2 c
is positive definite then no solution u(x, y) of such an eq
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), with a, b, c at least C 1 , given real valued functions. There is an immediate dierence betwe
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: SOLVING PDES
We now return to solving PDE using duality arguments and energy estimates. Before getting into details, we note that the ideal kind of wellposedness result we would like is the following. We are given a PDE (including various addit
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
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: SEPARATION OF VARIABLES
Separation of variables is a method to solve certain PDEs which have a warped product structure. The general idea is the following: suppose we have a linear PDE Lu = 0 on a space Mx Ny . We look for solutions u(x, y ) = X
MATH 220: DUHAMELS PRINCIPLE
Although we have solved only the homogeneous heat equation on Rn , the same method employed there also solves the inhomogeneous PDE. As an application of these methods, lets solve the heat equation on (0, )t Rn : x (1)
ut k u
MATH 220: PDES AND BOUNDARIES
We have used the Fourier transform and other tools (factoring the PDE) to solve PDEs on Rn . We now study how we can use these results to solve problems on the half space, or indeed on intervals, cubes, etc. As you have shown
MATH 220: THE FOURIER TRANSFORM TEMPERED DISTRIBUTIONS
Beforehand we constructed distributions by taking the set Cc (Rn ) as the set of very nice functions, and dened distributions as continuous linear maps u : Cc (Rn ) C (or into reals). While this was
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: 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: SECOND ORDER CONSTANT COEFFICIENT PDE
ANDRAS VASY
We consider second order constant coecient scalar linear PDEs on Rn . These have the form Lu = f,
n n
L=
i,j =1
aij xi xj +
i=1
bi xi + c
where aij , bi and c are (complex) constants, and f is gi
Partial Differential Equations of Applied Mathematics
CME 303

Spring 2014
MATH 220 SOLUTIONS TO HOMEWORK 4
VITALY KATSNELSON
1.) (i) First we compute
Z Z
Z
1 xy2 /(4t)
1
e
f (y)dydt f L1
dt < .
3/2
3/2
t
t
1
1
We also know that u(t, x) is bounded and continuous so
is welldefined.
R1
0
u(t, x)dt < . Hence, v
(ii)The boun