Chapter 14.3, Problem 81E

The diffusion equation

$\frac{\partial c}{\partial t}=D\frac{{\partial }^{2}c}{\partial x^2}$

where D is a positive constant, describes the diffusion of heat through a solid, or the concentration of a pollutant at time t at a distance x from the source of the pollution, or the invasion of alien species into a new habitat. Verify that the function

$c(x,t)=\frac{1}{\sqrt{4\pi Dt}}{e}^{-x^2/(4Dt)}$

is a solution of the diffusion equation.