due 11:59pm Sunday, April 18.

Consider the functions *u*(*x*, *t*) = sin(*α**x*)*e*^{βt}.

(a) For which values of *α* does *u*(*x*, *t*) satisfy the boundary conditions
*u*(0, *t*) = 0 and *u*(*L*, *t*) = 0?

(b) For each of the acceptable values of *α* found in (a), for what value of *β* does
*u*(*x*, *t*) satisfy the diffusion (heat) equation *u*_{t} = *D**u*_{xx}?

Show that the IBVP
*u*_{t} = *D**u*_{xx},
*u*(*x*, 0) = *f*(*x*)),
*u*(*a*, *t*) = *l*(*t*) and *u*(*b*, *t*) = *r*(*t*),
where *l* and *r* are differentiable functions,
can be recast as an IBVP for a function
*v*(*x*, *t*)
with *homogeneous* Dirichlet boundary conditions
(i.e., *v*(*a*, *t*) = 0 and *v*(*b*, *t*) = 0)
by defining
*v*(*x*, *t*)
as the difference between
*u*(*x*, *t*)
and the function
*z*(*x*, *t*)
where, at each
*t*,
*z*(*x*, *t*)
is the linear interpolant between
*l*(*t*)
and
*r*(*t*).
Conclude your answer by writing down the IBVP satisfied by
*v*(*x*, *t*).

Heating a bar at both ends (Forward Difference Method)

Modify the code I showed you in class for the heat equation to approximately solve and visualize the solution of the problem

where
*L* = 1,
*D* = 4,
*l*(*t*) = 50*t* while
*t* ≤ *t**
and *l*(*t*) = *l*(*t**) for
*t* > *t**, and
*t** = 0.02,
using 51 spatial grid points (h=.02).

Things I'll be looking for:

- display of actual code that generates the results that are presented (run code to check, if in doubt)
- correct boundary and initial conditions in code
- a plot of the approximate solutions in some fashion
- an illustration of what happens when the time-step k is a little too large.