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:
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.