= Surface temperature under constant heat flux on a half-line
{title2=$u(0,t)=\frac{2Q\sqrt{\kappa t}}{k\sqrt\pi}$}
For <thermal diffusivity> $\kappa>0$ and <thermal conductivity> $k>0$, a semi-infinite conductor satisfies $u_t=\kappa u_{xx}$ for $x>0$. Take zero initial <temperature> and constant incoming <heat flux> $-ku_x(0,t)=Q$. Select the physical solution by boundedness at spatial infinity together with sufficient growth control for the time <Laplace transform>.
Writing the time transform as $U$ gives $U_{xx}=(p/\kappa)U$. The decaying branch is $U=A(p)e^{-x\sqrt{p/\kappa}}$, where $\operatorname{Re}\sqrt p>0$. The transformed flux condition gives $A=Q\sqrt\kappa/(kp^{3/2})$. Since $\mathcal L[2\sqrt t]=\sqrt\pi p^{-3/2}$, inversion gives the displayed surface <temperature>. More generally the full profile is
$$
u(x,t)=\frac Qk\left[2\sqrt{\frac{\kappa t}{\pi}}e^{-x^2/(4\kappa t)}-x\operatorname{erfc}\!\left(\frac{x}{2\sqrt{\kappa t}}\right)\right].
$$
Differentiation verifies the <heat equation> and the applied flux; for every $x>0$ it tends to zero at initial time and decays at spatial infinity. Its surface <temperature> grows as $\sqrt t$, because the depth significantly heated by <diffusion> has the same scale.
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