dmajor

My matrix was:

matrix = [

  [0,2,1,2,4],

  [4,0,4,4,2],

  [4,2,0,1,2],

  [3,2,4,0,4],

  [4,1,4,4,0]

]

The ranking for my league is:

Team 2: rating = 0.5165517842331686
Team 4: rating = 0.48670829470439997
Team 5: rating = 0.47767125643216296
Team 1: rating = 0.3784150823149017
Team 3: rating = 0.3534423380504801

Team 2 is the best.

The temperature near the midpoint of the insulated edge of the triangle at $t=1s$ is about $0.17972$

The temperature in the lower right hand corner of the bar at steady state is roughly 0.91361.

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

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

At time $t=0$ its left end is set to temperature -1 and its right end to -3. 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

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

Find the steady state temperature distribution.

Steady state occurs when $u_t=0$. Solving the above equation for $u_{xx}$ given $u_t=0$, yields $u_{xx}=-3/2$. Integrating this twice gives $u(x)=\frac{-3}{4}x^2+cx+d.$ Using the boundary condition to solve for c and d gives you the steady state equation of $$u(x)=\frac{-3}{4}x^2+\frac{31}{4}x-4$$