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Eigenranking
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Modeling a steady state heat distribution in 2D
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Modeling 2D Heat Flow
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Random heat evolution problem
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Steady state heat flow with source
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, $\boxed{u(x,t) = -\frac{2}{3}x^2 + x + \frac{5}{3}}$