Partial Differential Equation

Partial Differential Equation

E: easy, M: moderate, D: difficult.

1. (E, 10 points) Let u( x, y) be a nonconstant harmonic function in the disk x2

+

^y2

< R

2. Define for each 0

<

r

<

R,

Prove that M(r) is a monotone increasing function in the interval (0, R).

2. (E, 15 points) Let u(r, e) be a harmonic function in the disk D

= {(r,e)lo:::::

r

<

R,-1r

<

e::::: 1r},

such that u is continuous in the closed disk jj and satisfies u(R e) = {sin²

2e, lei:::::

1r/2,

, 0, 1r/2

< lei:::::

1r.

(a) Evaluate u(O,O).

(b) Show that 0

<

u(r, e)

<

1 holds at each point (r, e) in the disk.

3. (M, 20 points) Use the energy method to prove uniqueness for the problem Utt - c^2u xx

+

hu

=

F(x, t), - ⁰⁰

<

x

<

^00, t

>

O.

lim u(x, t)

=

lim ux(x, t)

=

lim Ut(x, t)

=

0, t _~ O.

x~±oo x~±oo x~±oo

2u;

+

hu^2)dx

f:

^(u;

⁺

^c

^<

^{00, t}

^~

^0,

u(x,O) = j(x), Ut(x,O) = g(x), - 00

<

x

<

00,

where c and h are positive constants.

4. (M, 15 points) Consider the Cauchy problem

Utt - 2uxx

=

0, - 00

<

x

<

00, t > 0, u(x,O)

=

j(x), Ut(x,O)

=

g(x), - ⁰⁰

<

x

<

^00.

Fix T

>

O. Please prove that the above problem in the domain -00

<

x

<

^00, 0 ::::: t ::::: T is

well-posed for j E C2(R), 9 E Cl(R).

1

5. (M, 10 points) Let DR

== R

² \

B

_R be the exterior of the disk with radius

R

centered at the origin.

Find the Green function (for the Laplace operator) of DR'

6. (M, 20 points) Solve the following heat problem:

Ut -

ku

x x

A

cos

at, a

^{< x < 1,}

^t >

0,

ux(O, t)

=

u

x (1,

t) -

0,

t

~ 0,

21rX,

u(x, 0) - 1

+

^COS

a

^{S; x S; 1.}

7. (M, 10 points)

(a) Find the eigenfunction expansion of the function on [0,2]

j(x) = {x, as x

S; 1, 1, 1 S; x S 2 with respect to the (classical Fourier) orthonormal system

{

~}

^{U {costucx }}

::"~l

^{U {sinnnx }}

::"~l

(b) Does the series you obtain in (a) converge to j ?

2