Time-Dependent density and heat equation
In nonstationary case, we can further simplify the diffusion equation by eliminating the term ∂p∕∂x. Set
and choose α = -sgn(x) and β = -1/2.
Then (8.7) becomes the standard heat-equation form
This is a specially simple and well-known parabolic partial differential equation. It is called the heat equation because it arose as a model of the temperature distribution in one-dimensional heat-conducting media in steady heat flow (conduction or diffusion). The book by Sommerfeld (1949) discusses this and other physics examples. As mentioned in the introductory section, the option-pricing equation by Black and Scholes is a slightly more complicated example of this equation, to which it may be reduced by suitable transformation as in Willmot etal. (1993, Sec. 5.4).
The solution is of the form
rrom the conservation of the probability
mass we impose
The constant c is givvn by
where 
8.9