1.  (12 marks) The solution to the following PDE with boundary and initial conditions:

vt  = v北北     t > 0, 0 < x < 2

v(0, t) = 0

v北 (2, t) = 0

v(x, 0) = f (x)

is given by

v(x, t) = n Kn exp ╱ 一  (n 一 )2 π 2 t、sin ╱  (n 一2)πx、

where

Kn  =   ← 2 f (x) sin ╱  (n 一2)πx、dx.

Using this information, find the solution to the following PDE with boundary and initial conditions:

ut  = u北北     t > 0, 0 < x < 2

u(0, t) = 5

u北 (2, t) = 一2

u(x, 0) = g(x)

Explain your reasoning.

2.  (7 marks) Consider the following Sturm-Liouville problem:

d2y         dy

dx2          dx

(a) Use an integrating factor to show that the differential equation can be

rewritten as

 ╱e4x、 + λe4xy = 0

(b)  Show that the Sturm-Liouville problem has no non-negative eigenvalues.

3.  (26 marks) Consider the following PDE with boundary conditions: uxx + uyy  = 0   0 < x < 1,   0 < y < 1

with

ux (0, y) = 0,   ux (1, y) = 0,   u(x, 0) = 0,   u(x, 1) = f (x).

Use separation of variables to find a series solution to this problem.  Give a formula for the coefficients.

4.  (5 marks) Suppose that

utt  = u北北 ,

t > 0,   一& < x < &

with ut (x, 0) = 0 and u(x, 0) given by the following graph:

1

1

- 1

-1

Sketch the solution to this problem at t = 1 and at t = 2.