*2.4.1. Solve the heat equation du/dt = k (d/dx)^2 u, 0 < x < L, t > 0, subject to
u_x(0.t)=0,  t>0
u_x(L,t)=0, t>0.

(a) u(x,0) = 0  on x < L/2 and u(x,0) = 1 for x>L/2

(b) u(x,0)=6+4 cos (3 Pi x/L)

(c) u(x, 0) = -2 sin(Pi x/L)

(d) u(x, 0) = -3 cos(8 Pi x/L)



*2.4.2. Solve du/dt = k (d/dx)^2 u, 0 < x < L, t > 0,
with u_x(0,t) = 0,
u(L, t) = 0,
u(x,0) = f(x).

For this problem you may assume that no solutions of the heat equation
exponentially grow in time. You may also guess appropriate orthogonality
conditions for the eigenfunctions.

*2.4.3. Solve the eigenvalue problem
X'' + lambda X = 0 
subject to
X(0) = X(2 Pi) and X'(0) = X'(2 Pi).

2.4.4. Explicitly show that there are no negative eigenvalues for
X'' + lambda X = 0 subject to X'(0)=0 and X'(L) = 0.

2.4.5. This problem presents an alternative derivation of the heat equation for a
thin wire. The equation for a circular wire of finite thickness is the two-dimensional
heat equation (in polar coordinates). Show that this reduces
to du/dt = k (d/dx)^2 u if the temperature does not depend on r and if the wire is very
thin.

2.4.6. Determine the equilibrium temperature distribution for the thin circular
ring derived in H2.4 part 2:

(a) Directly from the equilibrium problem (see H1.4).

(b) By computing the limit as t approaches infinity of the time-dependent problem.


2.4.7. Solve Laplace's equation inside a circle of radius a,

(1/r)(d/dr)(r du/dr)) + (1/r^2)(d^2u/d theta^2 )= 0,

subject to the boundary condition u(a,theta) = f(theta).