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