$$\newcommand{\D}{\displaystyle} \renewcommand{\eqref}{Eq.~(\ref{#1})}$$

# 7.1 Introduction

A one-dimensional diffusion equation takes the canonical form: $$\begin{equation} \tag{7.1} \frac{\partial u}{\partial t}=\alpha\frac{\partial^2u}{\partial x^2} \end{equation}$$

where $$t$$ is an evolutionary variable, which might be both a time-coordinate and a spatial coordinate. Some classical diffusion problems are listed below:

• Heat conduction
$$\begin{equation*} \frac{\partial T}{\partial t}=\alpha \frac{\partial^2T}{\partial x^2} \end{equation*}$$
• Unsteady boundary layers (Stokes' problem):
$$\begin{equation*} \frac{\partial u}{\partial t}=\nu \frac{\partial^2u}{\partial y^2} \end{equation*}$$
• Linearized boundary layer equation with $$x$$ as an evolutionary variable:
$$\begin{equation*} \frac{\partial u}{\partial x}=\frac{\nu}{U_0} \frac{\partial^2u}{\partial y^2} \end{equation*}$$
• Flow in porous media:
$$\begin{equation*} \frac{\partial u}{\partial t}=c \frac{\partial^2u}{\partial x^2} \end{equation*}$$

Our model equation (7.1) may be classified according to (5.19): $$\begin{equation} A \frac{\partial^2\phi }{\partial x^2} +B \frac{\partial^2 \phi }{\partial x \partial y} +C\frac{\partial^2 \phi }{\partial y^2} +f=0 \tag{7.2} \end{equation}$$ $$\begin{equation} A\cdot (dy)^2-B\cdot dy \cdot dx+C\cdot (dx)^2=0 \tag{7.3} \end{equation}$$ $$\begin{equation} \lambda_{1,2} = \frac{B\pm \sqrt{B^2-4AC}}{2A} \tag{7.4} \end{equation}$$

$$B=C=0$$ and $$A=1$$ which by substitution in (5.34) and (5.36) yield: $$\begin{equation*} dt =0 ,\ B^2-4AC=0 \end{equation*}$$

And we find that (7.1) is a parabolic PDE with the characteristics given by $$t=$$ constant. By dividing $$dt$$ with $$dx$$ we get: $$\begin{equation} \tag{7.5} \frac{dt}{dx}=0\to \frac{dx}{dt}=\infty \end{equation}$$

which corresponds to an infinite propagation speed along the characteristic curve $$t=$$ constant.