m-3150-001 Midterm exam 2

Andrej Cherkaev

Your name, ........................................................... Solve any three problems:

1. Find the displacement u(x, t) of a string of a length 5, if its initial

displacement u(x, 0) is

u(x, 0) = sin (\frac{πx}{5}) + \frac{1}{3} sin (\frac{3πx}{5}) ,

its initial speed is zero. and the spring constant c is c = 2.

2. Initially, a rod has zero temperature u(x, 0) = 0. Its left end (x = 0) is kept at zero temperature u(0, t) = 0 and its right end (x = 2) – at the temperature 100, u(2, t) = 100. The diffusivity constant c is equal to one, c = 1. Find the temperature inside the rod at t = 2.

3. The left end of a rod is thermo-insulated, and the right end is kept at zero temperature.

\frac{∂u(x, t)}{∂x}|_{x=0} = 0, u(3, t) = 0, ∀t ∈ [0, ∞).

Find a general solution to the heat problem, assuming that the initial temperature is given. u(x, 0) = f(x) and the diffusivity constant c is equal to one, c = 1.

4. Separate variables in the the two-dimensional heat equation (proceed similarly to the two-dimensional wave equation analysis). Show the scheme of separation, the resulting differential differential equations, general soultion, do not compute coefficients.

\frac{∂u(x, y, t)}{∂t} = c^{2} (\frac{∂^{2}u(x, y, t)}{∂x^{2}} + \frac{∂^{2}u(x, y, t)}{∂y^{2}})

if 0 ≤ x ≤ a, 0 ≤ y ≤ b, u(0, y, t) = (u(a, y, t) = 0, u(x, 0t) = (u(x, b, t) = 0 and u(x, y, 0) = f(x, y)