math_boss

My matrix was matrix = [

  [0,2,4,1,1],

  [1,0,3,2,1],

  [3,4,0,1,2],

  [1,3,3,0,4],

  [4,3,1,4,0]

]

The eigen ranking was:

Team 5: rating = 0.5351649756702808
Team 4: rating = 0.516996248452163
Team 3: rating = 0.43493231817629663
Team 1: rating = 0.3710347855055461
Team 2: rating = 0.3456593618764917

Team 5 was the best team.

Given the conditions of my problem with $\kappa=0$ and $f=0$ the temperature at the lower right hand corner of the bar at steady state is 0.75700.

The temperature of the midpoint at the insulated edge at t=1s is 0.16587 as shown in the image below.

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

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

Since this is a steady state temperature distribution, $u(t)=0$ since $u(x,t)$ does not change with time. Thus,

$$1u_{xx}+3=0.$$

Solving for $u_{xx}$ yields $u_{xx}=-3$. Integrating this twice with provide the result,

$$u(x,t)=-\frac{3}{2}x^2+Ax+B,$$

where A and B are two constants. Solving for A and B using the initial conditions of $u(0,t)=1, u(1,t)=2$ produces the steady state distribution of

$$u(x,t)=-\frac{3}{2}x^2+\frac{5}{2}x+1.$$

The displacement of the midpoint from equilibrium at time t=1.9s is -0.028.