Classification of PDEs
Why do you want to classify solutions?
If we can classify a PDE that we are trying to solve, according to the scheme given below, it will help us extend qualitative knowledge that we have about the nature of solutions of similar PDEs to the current case. Most importantly, the types of initial conditions that are needed to ensure that our solution is unique vary according to the classification–see PDE Theorems. In physics situations, the classification is usually obvious: if there are two time derivatives, the equation is hyperbolic and we will need two initial conditions on the entire spatial region to make the solution unique; if there is only a single time derivative, the equation is parabolic and we will need only a single initial condition; if the equation has no time derivatives, the equation is elliptic and the solutions are qualitatively different from the previous two cases. You should look at the section on Important PDEs in Physics and classify each one.
If you want to see the formal mathematical notation for the classification (optional), read on:
Form of the equation
The most general 2nd order, linear, PDE can be written: $${\cal L}\psi = b$$ where $${\cal L}\psi=\sum_{i,j=1}^4 A_{ij}(x_k) {\partial^2\psi\over\partial x_i\partial x_j} +\sum_{i=1}^4 B_i(x_k) {\partial \psi\over\partial x_i} + C(x_k)\psi$$
Optional Mathematical Details of the Classification Scheme
The PDE is classified according to the signs of the eigenvalues $\lambda_i(x_k)$ of the matrix of functions $A_{ij}(x_k)$.
- Elliptic: $\lambda_i(x_k)$ are nowhere vanishing. All have the same sign.
Ex: Poisson, Laplace, Helmholtz
$$\nabla^2={\partial^2 \over \partial x^2} + {\partial^2 \over \partial y^2}+ {\partial^2 \over \partial z^2}$$
$$A_{ij}=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$$
- Parabolic: One eigenvalue vanishes everywhere (usually time dependence), the others are nowhere vanishing and have the same sign.
Ex: Diffusion, Schroedinger
$$A_{ij}=\begin{pmatrix}0&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$
- Hyperbolic: All eigenvalues are nowhere vanishing. One sign differs from the others.
Ex: Wave, Klein-Gordon
$$A_{ij}=\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$