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Steady state heat flow with source
My heat flow equation is
$$ u_t=4u_{xx}+4$$
And my boundary conditions are
$$u(0,t)=-2$$ and $$u(1,t)=5$$ when $u_t=0$, solving for $u_{xx}$ gives us $u_{xx}=-1$.
Integrating twice, we get
$$u(x) = -x^2+ax+b$$
By using our initial conditions we get
$$u(x)=-x^2+(17/2)x-(7/2)$$ which is our steady state temperature distribution
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Random heat evolution problem
A metal bar of length 1 lies along the unit interval. Its temperature distribution is given by $$g(x)=4x^2−1x.$$ At time $t=0$, its left end is set to temperature 2 and its right end to 0. Here is my (tentative) sketch the temperature distribution at times $t=0$, $t=0.01$, $t=0.1$, and $t=10$.
