Classification of PDEs

A general 2${}^{\text{nd}}$ order PDE

$$\mathcal{L}\psi =b$$

Where $\mathcal{L}\psi$ is given by

$$\mathcal{L}\psi =\sum _{i,j=1}^4{A}_{ij}(x_k)\frac{{\partial }^2\psi }{\partial x_i\partial x_j}+\sum _{i=1}^4{B}_i(x_k)\frac{\partial \psi }{\partial x_i}+C(x_k)$$

$b$ is a source term

1. Elliptic: ${\lambda }_{ij}$ do not vanish and have the same sign.

   $$A_{ij}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

2. Parabolic: One Eigen Value Vanishes.

   $$A_{ij}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

3. Hyperbolic: Eigen Values don’t vanish and 1 of the eigen values have a different sign.

   $$A_{ij}=\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

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References

1. https://books.physics.oregonstate.edu/LinAlg/pdeclass.html