Question

Solution (2) is the solution related in diffusion especially found in carburizing or decarburizing steel.

Can you give me an idea to derive (2) equation from (1) using Laplace transform?

Thank you.

$c(x, t) = \frac{c'}{2} * \left [ 1 + erf\left ( \frac{x}{2\sqrt{Dt}} \right ) \right ]$ ????(1)

$erf(z)= \frac{2}{\sqrt{\pi }}\int_{0}^{z}exp(-u^2)du$ ????(1a)

$u=\frac{x-\alpha }{2\sqrt{Dt}}$    ????(1b)

$c(x,t)-c_o=\frac{c'-c_o}{2}\left [ 1-erf(\frac{x}{2\sqrt{Dt}}) \right ]$    ????(2)

when

$\\c=c_o~~~for ~x> 0, at~t=0 \\c=c'~~~for ~x< 0, at ~t=0$

Answer 1

first substitute the values in equation(2)

after substitution , solve integral equation

using the limits c=c0 and c=c' by choosing boundary conditions.