The Diffusion Equation

The diffusion equation for a concentration $c(X,T)$ of some chemical (given by number of particles/unit length) is

$$\frac{\partial c}{\partial T} = D \frac{\partial^2 c}{\partial X^2}.$$ (1)

Suppose that our domain is the box $0 \le X \le L$.

Let's define new variables $x = X/L$ and $t = T/\tau$. Then,

$$\frac{\partial}{\partial X} = \frac{1}{L} \frac{\partial}{\partial x},$$

$$\frac{\partial}{\partial T} = \frac{1}{\tau} \frac{\partial}{\partial t},$$

so equation (1) becomes

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

Choosing $\tau = L^2/D$, we obtain the equation for the concentration $c(x,t)$

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

on the domain $0 \le x \le 1$.

To make this equation well-posed, we need to specify the boundary conditions at $x=0$ and $x=1$. The two main cases of interest are

• Constant concentration boundary conditions: $c(0,t) = c(1,t) = c_0$. Such boundary conditions might be due to the contact of the edges of our domain with an ``infinite'' reservoir with essentially constant concentration $c_0$.
• No-flux boundary conditions: $\left. \frac{\partial c}{\partial x} \right\vert _{x=0} = \left. \frac{\partial c}{\partial x} \right\vert _{x=1} = 0$. Such boundary conditions correspond to having no flux out of the domain.