Classification of second-order PDEs in n independent variables
The above discussion is readily extended to more than two dimensions or independent variables. Given coordinates x = (x_{1}, x_{2},...,x_{n}) consider the second-order PDE
where the coefficient matrix [Ay] is constant. Then we have the following classification determined by the set of eigenvalues of this matrix:
- 1. If the eigenvalues are all positive or all negative, then the PDE is elliptic.
- 2. If the eigenvalues are all positive or all negative, except for one that is zero, the PDE is parabolic.
- 3. If one eigenvalue is positive and the rest negative, or one negative and the rest positive, then the PDE is hyperbolic.
- 4. If more than one eigenvalue is positive and more than one eigenvalue is negative, the PDE is ultrahyperbolic.
It turns out that the fundamental PDEs used in physics to encode the properties of time as we know it, such as causality, are all hyperbolic.
Boundary conditions
It is not sufficient to have equations of motion that can propagate information throughout spacetime: we need boundary conditions, that is, the information that is propagated from one region of spacetime to another. There are three important types of boundary condition that lead to different effects, depending on the kind of PDE involved.
Cauchy boundary conditions are such that the value of the field ф and its normal derivative are specified on part of the boundary of a region. In spacetime physics, such a boundary will usually be a so-called spacelike hypersurface. Cauchy boundary conditions are the field theoretic analogues of initial position and velocity that are required to solve the second-order equations of motion in Newtonian mechanics.
Dirichlet boundary conditions appear to be more limited, being a specification of the value of the field over a complete boundary of a region. These conditions are the analogue of the specification of initial and final positions in the Calculus of Variations.
Neumann boundary conditions are a variant of the Dirichlet conditions: the normal derivative of the field is specified over the boundary of the region concerned.
The following table shows how the various boundary conditions match up with the various classes of PDE [Arfken, 1985].
Table 12.1 Boundary conditions versus PDE classification.
Boundary condition |
Elliptic |
Parabolic |
Hyperbolic |
Cauchy open surface |
unstable |
too restrictive |
unique, stable solution |
Cauchy closed surface |
too restrictive |
too restrictive |
too restrictive |
Dirichlet open surface |
insufficient |
unique, stable in one direction |
insufficient |
Dirichlet closed surface |
unique, stable |
too restrictive |
solution not |
solution |
unique |
||
Neumann open surface |
insufficient |
unique, stable in one solution |
insufficient |
Neumann closed surface |
unique, stable |
too restrictive |
solution not |
solution |
unique |