- The wave equations (3.6) and (3.8)
are special, and very, very simple, examples of a broader
class of equations, which are called the
*hyperbolic*equations. They are called*hyperbolic*because the equation:

*x*^{2}-*t*^{2}=*a*^{2}

describes a hyperbola. Hyperbolic equations almost always have some sort of waves as their solutions, but in general those waves may have very complex and nonlinear characteristics. - In absence of charges and currents a static, i.e., time
independent solutions to wave equations (3.6) and
(3.8) satisfy:

This is called the*Laplace*equation and is one of the simplest field equations. - The Laplace equation describes many other systems, for example
heat transfer.
- Heat always flows from a hotter to a cooler point,
i.e.,
*against*temperature gradient

(3.15)

where**Q**is the heat flux and*T*is the temperature field. - The divergence of
**Q**, i.e., describes*production*of heat in space, as describes*production*of electric field by charge density. - If heat is brought to a given point, then
temperature at that point must increase so that

(3.16)

(recall that convergence of field lines implies negative divergence ) Assuming that*c*_{p}is a constant (we have already assumed this about*h*) we get:

where . - This is not a wave equation.
This is a
*diffusion*equation. - A heat diffusion equation is a very simple example
of a broader class of equations, which are called
*parabolic*equations. They are called*parabolic*because the equation:

*x*^{2}-*t*=*a*

describes a parabola. - A static, i.e., time independent solution of
the diffusion equation must satisfy the Laplace
equation:

(3.18)

- Heat always flows from a hotter to a cooler point,
i.e.,
- The Laplace equation is a very simple example of a broader
class of equations, which are called
*elliptic*equations. They are called*elliptic*because

*x*^{2}+*y*^{2}=*a*^{2}

describes a circle, which is a special case of an ellipse. - Real life equations usually have both hyperbolic and diffusive terms. That is, their solutions have waves, but those waves eventually diffuse.
- Real life problems are usually described by
*many*coupled equations, with various characteristics - the more realistic a model, the more complex its behaviour. - In general:
- Elliptic equations describe static situations, e.g., static electromagnetic fields, static temperature distribution, static distribution of tensions in a geode.
- Parabolic equations describe diffusive systems, e.g., diffusion of heat, diffusion of impurities in semiconductors, diffusion of electrons across a pn-junction.
- Hyperbolic equations describe highly dynamic, vibrating systems, e.g., propagation of sound, propagation of electromagnetic waves, propagation of tectonic waves.

- Schrödinger equation looks like a parabolic equation at
first glance, but
there is an
*i*in front of . That*i*is responsible for the presence of waves (dispersive for unbound states). - Elliptic, parabolic, and hyperbolic equations are specific
cases of
*partial differential equations*. Supercomputers are machines designed specifically for finding numerical solutions to partial differential equations. It is seldom necessary to use supercomputers for anything else. Some examples of non-PDE problems that are run on supercomputers: