(Solved): Error function or its complement appears in the the solution of the following problem in the theory ...

the solution of the following problem in the theory of heat conduction. A semi-infinite slab occupying the half-space $x?0$ is initially heated to temperature $T_0$. It then cools off by radiating heat through its surface at $x=0$ into the surrounding medium which is at zero temperature. The quantity of greatest interest is the surface temperature $T(x=0,t)$ of the object at subsequent time $t$. The solution to the heat diffusion equation, subject to the initial condition $T(x,0)=T_0$ and boundary conditions ${\left(k\frac{?T}{?x}??T\right)?}_{x=0}=0,{T?}_{x??}=T_0$ is given by $T(0,t)=T_0{e}^{\left({?}^{2}Dt/k^2\right)}[1?\mathrm{erf}(?\sqrt{Dt}/k)]$ Here $D\left[{\mathrm{m}}^2/\mathrm{s}\right]$ is heat diffusion constant the slab has thermal conductivity $k[\mathrm{W}/\mathrm{m}?\mathrm{K}]$ and heat transfer coefficient $?\left[\mathrm{W}/{\mathrm{m}}^2?\mathrm{K}\right]$. Find an expression for the surface temperature in terms of elementary functions in the (i) short time and (ii) long time limits. Compute at least two terms in each case and be sure to define short and long times when these approximations can be used meaningfully.