where does the pde's mathematical classification names come from?

Question

PDEs are classified into hyperbolic, parabolic and elliptic. where do these names come from? Do they have anything to do their geometric shapes?

Answer 1

It's primarily due to the similarity to the corresponding algebraic equations. For example, let $A \in \mathbb{R}^{n\times n}$ be a symmetric positive definite matrix. We say that the PDEs: $$ \begin{cases} \sum_{i,j=1}^n A_{ij} \partial_i \partial_j u =f &\text{is elliptic} \\ \partial_t u - \sum_{i,j=1}^n A_{ij} \partial_i \partial_j u =f&\text{is parabolic} \\ \partial_{t}^2 u - \sum_{i,j=1}^n A_{ij} \partial_i \partial_j u =f&\text{is hyperbolic}. \end{cases} $$ Similarly, we say that the algebraic equations for $\xi \in \mathbb{R}^n$ and $\tau \in \mathbb{R}$ $$ \begin{cases} \sum_{i,j=1}^n A_{ij} \xi_i \xi_j = C &\text{is elliptic} \\ \tau - \sum_{i,j=1}^n A_{ij} \xi_i \xi_j =C &\text{is parabolic} \\ \tau^2 - \sum_{i,j=1}^n A_{ij} \xi_i \xi_j =C &\text{is hyperbolic}. \end{cases} $$ In the latter case we say this because the solution sets are the ellipses, parabolas, and hyperbolas.

There is a deeper connection between these ideas when one uses the Fourier transform to study the above PDEs because on the Fourier side the differential operators become polynomials.