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m-3150-001 Midterm exam 2

Andrej Cherkaev

Your name ...........................................................

1. Find the displacement u(x, t) of a string of a length 5, x ∈ [0, 5] if its initial displacement u(x, 0) and the initial speed speed are

u(x, 0) = 0, \frac{∂(u(x, t)}{∂t}_{t=0} sin (πx)

and the spring constant c is c = 2.

3. The left end of a rod of the length three (x ∈ [0, 3]) is thermoinsulated, 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.