Second-order partial differential equation

A second-order partial differential equation contains derivatives of the unknown function of order at most two and at least one second derivative. Its principal symbol determines its characteristic directions and elliptic, parabolic or hyperbolic type.

In two independent variables, a scalar second-order partial differential equation with a definite principal symbol. It has no real characteristic direction. Laplace's equation is the basic example.

In two independent variables, a scalar second-order partial differential equation with a nonzero rank-one principal symbol. Its two characteristic directions coincide. The heat equation is the basic example.

For $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+\cdots =0$, the sign of $\Delta =B^2-AC$ distinguishes a hyperbolic partial differential equation when $\Delta >0$, an elliptic partial differential equation when $\Delta <0$, and a parabolic partial differential equation when $\Delta =0$ and the principal symbol has rank one. If $A=B=C=0$, the second-order principal symbol has rank zero and the differential order drops; this fully degenerate case must be checked separately for compatibility with the lower-order terms and forcing.