Below we consider the classification of linear, constant-coefficient, 2nd-order PDEs.

Two independent variables

General form of 2nd-order PDEs with two independent variables: $$A u_{xx} + 2B u_{xy} + C u_{yy} + D u_x + E u_y + F =0$$

The coefficient matrix of the 2nd-order terms is $T = \begin{pmatrix} A & B \\ B & C \end{pmatrix}$ , with eigenvalues $\lambda_{1,2} = \frac{(A+C)\pm \sqrt{(A-C)^2+4B^2}}{2}$. Denote discriminant $\Delta = B^2 -AC$, then $-\lambda_1 \lambda_2$.

Classification:

  1. Hyperbolic type: $\Delta >0$ (or $(\alpha,\beta,\gamma) = (1,1,0)$, see notation in section "$n$ independent variables" );
  2. Elliptic type: $\Delta <0$ (or $(\alpha,\beta,\gamma) = (2,0,0)$ );
  3. Parabolic type: $\Delta =0$ (or $(\alpha,\beta,\gamma) = (1,0,1)$.

Standard form:

  1. Hyperbolic PDE:
    1. 1st standard form: $u_{xy} + \dots =0$
    2. 2nd standard form: $u_{xx} - u_{yy} + \dots =0$
  2. Elliptic PDE: $u_{xx} + u_{yy} + \dots =0$
  3. Parabolic PDE: $u_{xx} + \dots =0$ or $u_{yy} + \dots =0$

$n$ independent variables

General form

General form of 2nd-order PDEs with $n$ independent variables: $$\sum_{i,j=1}^{n} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{k=1}^{n} b_k \frac{\partial u}{\partial x_k} + cu = f$$ , where all the coefficients are real.

The major part of the PDE is $$\sum_{i,j=1}^{n} a_{ij} \frac{\partial^2 u}{\partial x_i \partial x_j} = \left( \frac{\partial}{\partial x_1}, \cdots , \frac{\partial}{\partial x_n} \right) \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{pmatrix} \begin{pmatrix} \frac{\partial}{\partial x_1} \\ \vdots \\ \frac{\partial}{\partial x_n} \end{pmatrix} $$

Denote the coefficient matrix as $T$, then $T$ is a real symmetric matrix. 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.

Classification

  1. Elliptic type: $(\alpha, \beta, \gamma) = (n,0,0)$
  2. Parabolic type: $(\alpha, \beta, \gamma) = (n-1,0,1)$
  3. Hyperbolic type: $(\alpha, \beta, \gamma) = (n-1,1,0)$
  4. Ultrahyperbolic type: $\alpha, \beta \geq 2, \gamma = 0$
  5. Elliptic-Parabolic type: $\beta=0, \gamma \geq 2$
  6. Hyperbolic-Parabolic type: $\alpha \beta \gamma \neq 0$

Examples:

  1. Hyperbolic type - Wave equation $$u_{tt} = a^2 (u_{xx} + u_{yy} + u_{zz}) \quad : \quad (\alpha, \beta, \gamma) = (3,1,0)$$
  2. Elliptic type - Laplace equation $$u_{xx} + u_{yy} + u_{zz} =0 \quad : \quad (\alpha, \beta, \gamma) = (3,0,0)$$
  3. Parabolic type - Heat equation $$u_{t} = a^2 (u_{xx} + u_{yy} + u_{zz}) \quad : \quad (\alpha, \beta, \gamma) = (3,0,1)$$

Standard form

Coefficient matrix can be standardized into a diagonal matrix with entries $0, \pm 1$.

🏷 Category=Dynamical System