leland

At $t=1s$, the temperature near the midpoint of the insulated edge of the triangle is approximately $.15480$.

The temperature at the lower right corner is approximately 0.806 at a steady state.

My matrix was:

matrix = [

  [0,3,3,3,1],

  [2,0,3,1,3],

  [3,3,0,3,1],

  [4,1,1,0,4],

  [1,1,3,4,0]

]

and the eigen ranking was:

Team 3: rating = 0.4627375562956863
Team 1: rating = 0.46273755629568625
Team 4: rating = 0.46088857575916814
Team 5: rating = 0.4269598459305646
Team 2: rating = 0.4207551766581456

Team 3 is barely ahead of team 1 by rounding. But otherwise these two teams are dead even for the best team.

A metal bar of length 1 lies along the unit interval. Its temperature distribution is given by $$g(x)=6x^2-1x$$.

At time $t=0$, its left end is set to temperature -2 and its right end to 2. Sketch the temperature distribution at times $$t=0,t=0.01,t=0.1, \text{ and } t=10.$$