= Slab conduction with exponentially decaying heat generation
{title2=$\beta=\mu a$}
= Exponential slab heating
{synonym}
A one-dimensional <thermal conduction> model with constant <thermal conductivity>, source $q'''(x)=q_0e^{-\mu x}$ and prescribed face <temperatures> $T(0)=T(a)=T_0$. Its <steady state> solution is
$$
T-T_0=\frac{q_0}{k\mu^2}\left[1-e^{-\mu x}-(1-e^{-\mu a})\frac{x}{a}\right].
$$
Positive internal heating makes the profile concave. Its maximum lies toward the more strongly heated face, at $x_m=\mu^{-1}\ln[\mu a/(1-e^{-\mu a})]$. The uniform-source limit is $T-T_0=q_0x(a-x)/(2k)$.
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