$4\mathrm{.}3.$ Transient Conduction with a Constant$-$Flux Boundary Condition Suppose that a solid of indefinite thickness is subjected to a constant heat flux q$\_0$ at x$=0.$ rginning at t$=0.$ It occupies the space x$>0$ and its initial temperature is uniform at T$\_\backslash$infty $.($a$)$ State the equations that govern T$($x$,$ t$),$ and use order$-$of$-$magnitude reasoning to infer how the temperature at x$=0$ varies with time. What variable transformations does this suggest when seeking a similarity solution? $($b$)$ Use the similarity method to determine T$($x$,$ t$).[$Hint: The solution involves an integral of the complementary error function, denoted as ierfc and defined as

ierfc$($x$)=\_$x$^\backslash$infty erfc$($s$)$ d s $.$

The properties of this and other integrals of erfc are described in Abramowitz and Stegun $(1970,$ p$.299).$ This integral error function can be evaluated from erfc$($x$)$ as

$.$ierfc$($x$)=-$x erfc$($x$)+$e$^-$x$^2/(\backslash$pi $)]$

$($c$)$ How does T$(0,$ t$)$ from the similarity solution in part $($b$)$ compare with the order$-$ofmagnitude estimate in part $($a$)?$