LINEAR SECOND-ORDER EQUATIONS
Classication
In two independent variables x and y , the general form is
Auxx + 2Buxy + Cuyy + Dux + Euy + F u + G = 0.
The coecients are continuous functions of (x, y ) in .
We will use the 1-1 transformation
= (x, y ), = (x
Problem 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. Write down the series expansion of the
solution.
2. Use
Problem Set 10
1. Generalize the energy method to prove uniqueness for the diusion equation
with Dirichlet boundary conditions in three dimensions.
2. Derive the representation formula
u(x0 ) =
1
2
u(x)
D
u
(ln |x x0 |)
ln |x x0 | ds
n
n
for harmonic fun
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).]
2. Solve the diusion equation with advection:
ut kuxx
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.
Assume that u = 0 on all six sides. What is the condition
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 function in the semi-innite strip cfw_0 x , 0 y <
that satis
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 the transformation
= x, = x + y.
The transformed dier
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], determine the sum of the Fourier series
of the function
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) = 0,
x
F [f ] =
ix
f (x)e
ix
dx = f (x)e
(b) F [xf ] =
+
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,
2
2
with a = Ax + 2Bx y + Cy , b = Axx + B (xy + y x )
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. Sketch u(x,
a. As shown in the left panel of the gure belo
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. Write down the series expansion of the
solution.
Answer
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).]
Answer
Let u = ebt v, then ut = bebt v + ebt vt and ux
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 written as
1 (x2 + y 2 )
f=
(x 1)2 + y 2
which is > 0 for x
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. Assume that u = 0 on all six sides. What is
the condition o
Problem Set 6
1. (Problem 2 of 4.6)
Solve the IVP:
utt 4uxx = 2xt
<x<
u(x, 0) = sin x,
ut (x, 0) = 2x
2. Let f (x, t) C 1 (R, [0, ) and let u(x, t) = (1/2c)
f , where (x,t) is
(x,t)
the triangle of dependence of (x, t). Verify directly by dierentiation th
Problem Set 4
1. (Problem of 3.6)
Prove the dierentiation and multiplication by xn formulas for the Fourier
transform.
2. (Problem 11 of 3.6)
Find the Fourier transform of f (x) = e|x|.
3. (Problem 26 of 3.6)
Prove Parsevals theorem. Let f be real-valued,
Problem Set 3
1. (Problem 2 of 3.2)
Write the Fourier series for f (x) = ex on [, ].
2. (Problem 4(a) of 3.3)
For each point of the interval [3, 3], determine the sum of the Fourier series
of the function
2x for 3 < x 2
f (x) =
0 for 2 < x < 1
2
x for 1
Fourier transform
Suppose that f is piecewise smooth on each interval [L, L]
and |f (x)|dx converges (i.e. absolutely integrable). By
substituting the expressions for an and bn into the Fourier
series, one gets
as cos( ) cos(x) + sin( ) sin(x) = cos( ( x)
Pointwise convergence of Fourier series
We discuss here under what conditions a Fourier series converges, and converges to what.
All the terms of a Fourier series are periodic of period 2 .
For the convenience of derivation, we are going to extend f
dened
INTRODUCTION
Partial dierential equations originated from physical applications. Though this course emphasizes analysis, a little bit
of physical interpretations are going to be useful.
Particle vs function
Let x be the vertical axis pointing upward. Cons
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 sine and cosine
Supposed that f (x) is a function dened on
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 characteristic equations
are
dx/dt = c.
The characteristic coord
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 constant (called diusion coecient).
The solution can be writ
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 at the boundary points. If the
smooth function satises t
Wave equation in a nite interval
The Dirichlet condition
Consider the IBVP with homogeneous Dirichlet conditions
pde: utt c2 uxx = 0
for 0 < x < L,
IC:
u(x, 0) = (x) ut(x, 0) = (x),
BC:
u(0, t) = u(L, t) = 0.
The method used here is to build up the genera
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. Let n be the
unit outward normal vector on D. Let f be a
Diusion on the half-line
The Dirichlet problem
Consider the initial boundary value problem (IBVP) on the
half line (0, ):
vt kvxx = 0
v (x, 0) = (x)
v (0, t) = 0.
The solution will be obtained by the reection method with
odd extensions.
For the extended p
LAPLACE EQUATION
If a diusion or wave problem is stationary (time independent), the pde reduces to the Laplace equation
u = u = 0,
an archetype of second order elliptic pde.
= 2 is call the Laplacian or del-square operator. In
two dimensions, the expres