Elliptic Equations -- Laplace's equation

We shall deal with elliptic equations later when we come to consider matrix methods. For the moment it suffices to note that, apart from the formal distinction, there is a very practical distinction to be made between elliptic equations on the one hand and hyperbolic and parabolic on the other hand. Generally speaking elliptic equations have boundary conditions which are specified around a closed boundary. Usually all the derivatives are with respect to spatial variables, such as in Laplace's or Poisson's Equation. Hyperbolic and Parabolic equations, by contrast, have at least one open boundary. The boundary conditions for at least one variable, usually time, are specified at one end and the system is integrated indefinitely. Thus, the wave equation and the diffusion equation contain a time variable and there is a set of initial conditions at a particular time. These properties are, of course, related to the fact that an ellipse is a closed object, whereas hyperbolæ and parabolæ are open.