flappy_bird

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

$$g(x) = 4x^2 - 3x$$

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

$$t=0, \: t=0.01, \: t=0.1, \text{ and } t=10$$

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

$$u_t = 4u_{xx} + 5$$

Find the steady-state temperature distribution $u(x,t)$ if the boundary conditions are:

$$u( 0, t) = -2; u( 1, t) = 4$$

Steady-state implies that nothing changes with time, therefore:

$$\frac{du}{dt} = 0$$

Plugging the result into the model and rearranging the terms:

$$u_{xx} = - \frac{5}{4}$$

Integrate twice:

$$u_x = - \frac{5}{4} x + \phi (t)$$

$$u = - \frac{5}{8} x^2 + \phi (t)x + \psi (t)$$

Use boundary conditions to determine the unknown functions $\phi (t)$ and $\psi (t)$

$$u(0, t) = \psi (t) = -2$$

$$u(1, t) = -\frac{5}{8} + \phi (t) + (-2) = 4 \rightarrow \phi(t) = \frac{53}{8}$$

Therefore, the steady state temperature distribution is:

$$\boxed{u(x,t) = - 0.625 x^2 + 9.6 x -2}$$

The vertical displacement of the midpoint from equilibrium at time t = 1.6 [seconds] is approximately equal to -0.16 [unit lengths(?)].