flappy_bird

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

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:

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

Bird face just looks ominous (。_。)

My matrix was:

  [0, 1, 3, 1, 2],

  [2, 0, 1, 3, 1],

  [2, 4, 0, 1, 3],

  [2, 3, 1, 0, 1],

  [4, 2, 2, 2, 0]

My ranking was:

Team 3: rating = 0.5273129974036033
Team 5: rating = 0.5215157378625016
Team 1: rating = 0.4172591286356548
Team 4: rating = 0.3713873702087703
Team 2: rating = 0.3713873702087703

I tried the following:

a = 0

b = 2

n =25

1) The 24x24 matrix that approximates the 2nd derivative is

$$U_{xx}=\begin{pmatrix} -312.5 & 156.25 & 0 & \ldots & 0 & 0 & 0\\ 156.25 & -312.5 & 156.25 & \ldots & 0 & 0 & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & 0 &\ldots & 156.25 & -312.5 & 156.25\\ 0 & 0 & 0 & \ldots & 0 & 156.25 & -312.5 \end{pmatrix}$$

2) The eigenvalue of smallest magnitude and its corresponding eigenvector are:

$$\lambda = -2.4642$$

$$\vec u = \begin{pmatrix} -0.0354\\ -0.0703\\-0.1041\\-0.1362\\-0.1662\\-0.1936\\-0.2179\\-0.2388\\-0.2559\\-0.2689\\-0.2778\\-0.2822\\-0.2822\\-0.2778\\-0.2689\\-0.2559\\-0.2388\\-0.2179\\-0.1936\\-0.1662\\-0.1362\\-0.1041\\-0.0703\\-0.0354\end{pmatrix}$$

3) The eigenfunction for this problem was: $$k* sin(\sqrt{-\lambda}*x)$$.

If we drop k, we get the following plot:

But if we let $k = -0.09\pi$, we get the following:

Now, I don't remember talking about using the approximation to find $k$ in class. @mark Could you explain why this was necessary here?

Given the boundary and initial conditions, $u(t=1s) \approx 0.92$ [temperature units] at the midpoint of the insulated boundary.

The steady-state temperature of the bar at the lower right corner is approximately $0.909$ [temperature units]

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:

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