# Homework #8

due 11:59pm Sunday, April 18.

# 1

Consider the functions u(x, t) = sin(αx)eβt.

(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 ut = Duxx?

# 2

Show that the IBVP ut = Duxx, 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).

# 3

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

ut = Duxx, 0 < t < 0.05, 0 < x < L
u(x, 0) = 0,
u(0, t) = l(t), u(L, t) = 1.5*l(t)

where L = 1, D = 4, l(t) = 50t 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.