Zach

A metal bar of length 1 lies along the unit interval. Its temperature distribution is given by:

$$g(x) = 5x^2 - 1x .$$

At time t = 0, its left end is set to temperature -2 and its right end to -1. Sketch the temperature distribution at times:

t = 0, t = 0.01, t = 0.1, t = 10.

Heat flow with a constant internal heat source is governed by:

$$u_t = 1u_{xx} + 2, u(0,t) = -3, u(1,t) = 5.$$

When at the steady state temperature distribution, $u_t = 0$ therefore:

$$0 = u_{xx} + 2$$

This can be rearranged to give $u_{xx} = -2$. Next this function was integrated with respect to $x$ twice giving $$u(x) = -1x^2 + \alpha*x + \beta$$.

Plugging in the conditions $u(0,t) = -3$ and $u(1,t) = 5$ allowed for finding of $\alpha = 9$ and $\beta = -3$.

Therefore, the steady state temperature distribution is:

$$u(x) = -1x^2 + 9x - 3$$

The displacement from the equilibrium of the midpoint of the string at time t=1.3 seconds into the vibration is approximately -0.351.