Chapter 11, Problem 19P

The one-dimensional diffusion equation is given by:

$\frac{\partial u}{\partial t}=m\frac{{\partial }^{2}u}{\partial x^2}$

Show that the following are solutions to the diffusion equation.

(a) $u=A\frac{1}{\sqrt{t}}\exp \left(\frac{-x^2}{4mt}\right)+B$, where A and B are constants

(b) u=A exp(-ax)cos(ax-2ma²t+B)+C, where A, B, C, and a are constants.