 Surface temperature under constant heat flux on a half-line

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 $-k{u}_x(0,t)=Q$. 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 $\mathrm{Re}\sqrt{p}>0$. The transformed flux condition gives $A=Q\sqrt{\kappa }/(k{p}^{3/2})$. Since $\mathcal{L}[2\sqrt{t}]=\sqrt{\pi }{p}^{-3/2}$, inversion gives the displayed surface temperature. More generally the full profile isDifferentiation 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.