Consider heat conduction in a $1$-D material. The temperature distribution $u$ after sufficiently long time can be modeled by $$-(a(x)u_x)_x = f(x), \qquad x \in [0,1]$$ where $a$ is heat conductivity, $f$ denotes internal heat sources.
Now i want to understand how this equation is modeled? Can anyone explain it plz? Also what is the difference between this equation and general heat equation?
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$\begingroup$You can start from the derivation of the general heat equation in one dimension. For example, you can look at the . You just need to account for the fact that the heat conductivity is position dependent. You should get: $$\partial_tu-\partial_x(a(x)\partial_xu)=f(x)$$ Now all you need to do is assume that the system reaches an equilibrium when $t\rightarrow\infty$. If the system is in equilibrium, it does not change with time, so $\partial_tu=0$
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