joshb5643

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

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

At time $t = 0$ its left end is set to temperature 0 and its right end to 3. Sketch the temperature distribution at times: $t = 0,\ t = 0.01,\ t = 0.1,\ t = 10$ .

My matrix: [

[0,4,1,1,4],

 [1,0,1,2,3],

  [1,3,0,1,4],

  [2,3,1,0,1],

  [1,1,1,4,0]

]

Eigenranking:

Team 1: rating = 0.5312098699110835
Team 3: rating = 0.4855376577979557
Team 4: rating = 0.408991251670405
Team 5: rating = 0.39800103471102516
Team 2: rating = 0.39546250061814175

Team 1 is the best.

At $t = 1s$, the temperature near the midpoint of the insulating edge of the triangle is $0.29975$.

The temperature at the lower right corner of the bar is $0.89869$.

Heat flow with a constant internal heat source is governed by

$$u_t = 3u_{xx} + 4\ ; \ u(0,t) = 1, \ u(1,t) = 2$$

Find the steady state temperature distribution.

Steady state: $u_t = 3u_{xx} + 4 = 0 \ \longrightarrow \ u_{xx} = -\frac{4}{3}$

$$u = \int\int -\frac{4}{3}\ dx = -\frac{2}{3}x^2 + c_{1}x + c_2$$

Initial conditions:

$u(0,t) = 1 \ \longrightarrow \ c_2 = 1$

$u(1,t) = 2 \ \longrightarrow \ -\frac{2}{3} + c_1 + 1 = 2 \ \longrightarrow \ c_1 = \frac{5}{3}$

So, $\boxed{u(x,t) = -\frac{2}{3}x^2 + x + \frac{5}{3}}$