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A random vibration problem
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Random heat evolution problem
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Eigenranking
My matrix is:
matrix = [
[0,4,2,1,4],
[4,0,2,3,1],
[3,1,0,3,2],
[1,1,1,0,4],
[4,4,1,2,0]
]
Team 1: rating = 0.5079982820145703 Team 5: rating = 0.5032938392103756 Team 2: rating = 0.4485785148338271 Team 3: rating = 0.4101302319490532 Team 4: rating = 0.34525869395863085
That means Team 1 was the best but doesn't look like it is by too much.
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Modeling 2D Heat Flow
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Modeling a steady state heat distribution in 2D
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Steady state heat flow with source
We will look at heat flow with a constant internal heat source. This will be governed by, $$u_t=4{u_xx}+8.$$
Such that we have the initial conditions, $u(0,t)=-3$ and $u(1,t)=3.$
Given that we are looking for a steady-state distribution, we can set $u_t=0,$ such that $u(x,t)$ does not change with respect to time, $t.$
Therefore, we have the equation $$0=4{u_xx}+8.$$
Simplifying we get:
$$4{u_xx}=-8$$
$${u_xx}=-2.$$
Integrating twice we get
$${u_x}=-2x+{c_1}$$
$$u=-x^{2}+x{c_1}+{c_2}.$$
Using our initial conditions, we will solve for $c_1$ and $c_2$:
We get ${c_1}=7$ and ${c_2}=-3.$
Thus, our steady-state distribution function is:
$$u=-x^2+7x-3.$$