Q$8.$ A solid confined in a half$-$space of $x>0$ is initially at the melting temperature $T_m^{**}.$ For times $t>0,$ the boundary surface at $x=0$ is subjected to a heat flux in the form of

$-k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}}{|}_x|=0=H=$ constant

Please use the integral method of solution and a $2$nd$-$order polynomial approximation for the temperature to obtain an expression for the location of the solid$-$liquid interface as a function of times $t.$