Solution Set 9
1. Check the validity of the maximum principle for the harmonic function f (x, y ) =
(1 x2 y 2 )/(1 2x + x2 + y 2) in the closed disk D = cfw_x2 + y 2 1.
Answer
The function can be writ
Solution Set 8
1. Solve the diusion equation with constant dissipation:
ut kuxx + bu = 0 for < x < u(x, 0) = (x),
where b > 0 is a constant. [Hint: Make the change of variables u(x, t) =
ebt v(x, t).]
Solution Set 7
1. Consider waves in a resistant medium that satisfy the problem
utt = c2 uxx rut for 0 < x < L
u(0, t) = u(L, t) = 0
u(x, 0) = (x)
ut (x, 0) = (x)
where r is a constant, 0 < r < 2c/L.
Solution Set 5
1. a. Solve the IVP
utt = c2 uxx < x < ,
u(x, 0) = 0,
ut (x, 0) =
1 for x < 1
0 for x > 1.
1
1
2
), u(x, ), and u(x, ).
2c
c
c
2
c. Compute the total energy at t = .
c
Answer
b. Ske
Solution Set 2
1. With the general transformation = (x, y ), = (x, y ), show that the pde
Auxx + 2Buxy + Cuyy + Dux + Euy + F u + G = 0
can be changed to the form
aw + bw + cw + dw + ew + f w + g = 0,
Problem Set 4
1. (Problem of 3.6)
Prove the dierentiation and multiplication by xn formulas for the Fourier
transform.
Answer
It suces to illustrate the steps with the n = 1 cases.
(a) As lim f (x) =
Solution Set 3
1. Write the Fourier series for f (x) = ex on [, ].
Answer
The Fourier series is
1
2 sinh( )(1)n
sinh( ) +
[cos(nx) n sin(nx)].
(1 + n2 )
n=1
2. For each point of the interval [3, 3],
Solution Set 1
1. Solve the characteristic equation, nd the general solution of ux uy + yu = 0
Answer
The characteristic equation is dy/dx = 1, so the characteristics are
straight lines y = x + k. Use
Solution Set 11
1. In the cube (0, a)3, a substance is diusing whose molecules multiply at a
rate proportional to the concentration. It therefore satises the PDE ut =
ku + u, where is a constant. Assu
Solution Set 10
1. Generalize the energy method to prove uniqueness for the diusion equation
with Dirichlet boundary conditions in three dimensions.
Answer
Suppose that u is a solution of the diusion
Fist
order
t
Linear
L
PDE
linear
,
Lu=g
:
differential operator
L=acx*Ix
L
EI
's
H
doesn't
tbcx
,y)z
dependent
U*tuy=o
UtuUy=o
on
U
:C L=+Fy
(
)u=0
Ixtusg
,
Luo
second
Uxxtuyyto
order of

equation
an
2
chapter
wave
equations
:
Diffusions
c2u=o

of string
vibration
Hangout along
vector
tension
:
u*
:
physical example
T
and
Waves
:
the
,
.
hyperbolic
equation
string )
)
tEtm
aDMs#t
Newton 's
law
T
2.3
Diffusion equation
.
(
Heat
ni
)
MH
:
.
+MH=
moves
concentration
lower
proportional
concentration
the
to
from regions
nx
u
,txdx=kux
Hu
?
S
=
is
dye )
regions of
to
motion
of
rate
substance (
:
c
diffusion equation
U+=kU
o
Ydfximnm
?TIE!eite
t
was
,
in
oexel
bonded domain
on
(
rectangular
%
,otT )
Exel
o

.org.in#of
a
in
space
value
of
time

initially
Root
M=
;
then
,
not
)
Lto
)
Let
the
)
n
EI
h
,
"+u=o
UCOKUCL )=O
general solution
acosotbsino
u6)=
UCL
)
basin
=
when
=o
NUDIO
L
SmL=o
or
a
=
L=O
b=o
either
0
Ht
bshx
ucxk
.
.
Ltnt
there
n=,1
at
=
is
when
L=nT
There
,
solution
trivial
b
EI
Problem Set 9
1. Check the validity of the maximum principle for the harmonic function f (x, y ) =
(1 x2 y 2 )/(1 2x + x2 + y 2 ) in the closed disk D = cfw_x2 + y 2 1.
2. (a) Find the harmonic functi
Problem Set 11
1. In the cube (0, a)3, a substance is diusing whose molecules multiply at a rate proportional to
the concentration. It therefore satises the PDE:
ut = k u + u, where is a constant.
Ass
Problem Set 8
1. Solve the diusion equation with constant dissipation:
ut kuxx + bu = 0 for < x < u(x, 0) = (x),
where b > 0 is a constant. [Hint: Make the change of variables u(x, t) = ebt v (x, t).]
GREENS IDENTITIES AND GREENS
FUNCTIONS
Greens rst identity
First, recall the following theorem.
Theorem:
(Divergence Theorem) Let D be a bounded solid region
with a piecewise C 1 boundary surface D. L
Uniform convergence of Fourier series
A smooth function on an interval [a, b] may be represented by
a full, sine, or cosine Fourier series, and pointwise convergence
can be achieved, except possibly a
THE DIFFUSION EQUATION
Diusion in one spatial dimension
The initial value problem on R
Consider the initial value problem (IVP)
ut = kuxx
( < x < , 0 < t < )
u(x, 0) = (x).
where k is a positive const
THE WAVE EQUATION
The wave equation on R
Consider the equation
utt c2uxx = 0 for < x < +
where t is usually interpreted as time. This is a second order
linear pde of the hyperbolic type. The character
FOURIER SERIES
Fourier series are important for solving pdes. We discuss the
basic properties of this kind of series here. The application
will be discussed later.
Expanding a function in terms of sin