A semi-infinite cylinder of metal lies along the positive x-axis with its sides and the face at $\displaystyle x = 0$ insulated. The initial temperature distribution is given by $\displaystyle u(x,0) = u_0 e^{\frac{-x}{L}}$ where $\displaystyle u_0$ and $\displaystyle L$ are positive constants. Show that the temperature in the cylinder is given for $\displaystyle t > 0$ and $\displaystyle x > 0$ by

$\displaystyle u(x,t) = \frac{u_0}{2} e^{\frac{c^2 t}{L^2}} $ $\displaystyle \Biggl( \ e^{-\frac{x}{L}} \Biggl[ 1 + \mbox{erf}$ $\displaystyle \ \ \Biggl( \frac{x - \frac{2c^2t}{L}}{\sqrt{(4c^2t)}} \Biggr) \Biggr]$ $\displaystyle + \ e^{\frac{x}{L}} $ $\displaystyle \Biggl[1 - \mbox{erf}$ $\displaystyle \ \ \Biggl( \frac{x - \frac{2c^2t}{L}}{\sqrt{(4c^2t)}} \Biggr) \Biggr] \Biggr)$