# time-dependent diffusion equation - why is it parabolic?

engineer here, looking for some help!

Studying the classification of PDEs I am confused about the following, probably trivial, problem:

The time-dependent diffusion equation is

$$\frac{\partial \phi}{\partial t} - \alpha \left(\frac{\partial² \phi}{\partial x²} + \frac{\partial² \phi}{\partial y²}\right) = 0$$

and is considered to be a parabolic PDE.

Is it correct that there are 3 independent variables, $$x, y \text{ and } t$$? If so, how do I apply the common rules for classifying PDEs using the comparison of the discriminant $$B² - AC$$ with $$0$$?

Comparing with the general case of such a PDE (as in wikipedia), I'd get $$A=-\alpha$$, $$B=0$$ and $$C=-\alpha$$ again, which appears to be incorrect. But where is my mistake?

Thanks a lot!

https://en.wikipedia.org/wiki/Partial_differential_equation

You are right. There are 3 independent variables here. So, you cannot apply the discriminant criterion $$B^2 - AC$$, as that holds only for 2 independent variables.
Thus, you must apply the method given in the Wikipedia page for a general PDE. In this case, taking $$x_1 = x, x_2 = y, x_3 = t$$, the coefficient matrix of the PDE is - $$\begin{pmatrix} -\alpha & 0 & 0 \\ 0 & -\alpha & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$$
As this has eigenvalues $$\lambda = -\alpha, -\alpha, 0$$, this is a parabolic differential equation.