This article considers the classification of linear, constant-coefficient, second order PDEs.
Second-order PDE with two variables can be written as $A u_{xx} + 2B u_{xy} + C u_{yy} + D u_x + E u_y + F =0$. The coefficient matrix of the second order terms is $T = \begin{pmatrix} A & B \\ B & C \end{pmatrix}$, with eigenvalues $\lambda_{1,2} = ((A+C)\pm \sqrt{(A-C)^2+4B^2})/2$. Denote discriminant $\Delta = B^2 - AC$, then $\Delta = -\lambda_1 \lambda_2$.
Table: Classification of Bivariate Second-order PDEs
| Type | Criteria | Alt Criteria | Standard Form | Alt Standard Form |
|---|---|---|---|---|
| Hyperbolic | $\Delta > 0$ | $(α,β,γ) = (1,1,0)$ | $u_{xy} + \dots =0$ | $u_{xx} - u_{yy} + \dots =0$ |
| Elliptic | $\Delta < 0$ | $(α,β,γ) = (2,0,0)$ | $u_{xx} + u_{yy} + \dots =0$ | |
| Parabolic | $\Delta = 0$ | $(α,β,γ) = (1,0,1)$ | $u_{xx} + \dots =0$ | $u_{yy} + \dots =0$ |
Second-order PDE with n independent variables can be written as: (all coefficients are real) $$\sum_{i,j=1}^{n} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{k=1}^{n} b_k \frac{\partial u}{\partial x_k} + cu = f$$ The major part of the PDE is: $$\sum_{i,j=1}^{n} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} = \left( \frac{\partial}{\partial x_1}, \cdots , \frac{\partial}{\partial x_n} \right) \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{pmatrix} \begin{pmatrix} \frac{\partial}{\partial x_1} \\ \vdots \\ \frac{\partial}{\partial x_n} \end{pmatrix}$$
The coefficient matrix $T$ is a real symmetric matrix, and can be standardized into a diagonal matrix with entries -1, 0, or 1. Denote the multiplicity (here the geometric multiplicity equals the algebraic ones) of positive, negative and zero (real) eigenvalues as $\alpha, \beta, \gamma$ respectively, then $\alpha + \beta + \gamma = n$. Noticing that cases $(\alpha, \beta, \gamma)$ and $(\beta, \alpha, \gamma)$ are essentially the same, we ignore situations when $\alpha$ and $\beta$ switches.
Table: Classification of Second-order PDEs
| Type | Criteria | Example |
|---|---|---|
| elliptic | $(\alpha, \beta, \gamma) = (n,0,0)$ | Laplace equation $u_{xx} + u_{yy} + u_{zz} =0$ |
| parabolic | $(\alpha, \beta, \gamma) = (n-1,0,1)$ | heat equation $u_{t} = a^2 (u_{xx} + u_{yy} + u_{zz})$ |
| hyperbolic | $(\alpha, \beta, \gamma) = (n-1,1,0)$ | wave equation $u_{tt} = a^2 (u_{xx} + u_{yy} + u_{zz})$ |
| ultrahyperbolic | $\alpha, \beta \geq 2, \gamma = 0$ | |
| elliptic-parabolic | $\beta=0, \gamma \geq 2$ | |
| hyperbolic-parabolic | $\alpha \beta \gamma \neq 0$ |