4.2 Cauchy's Method

The description of the method of characteristics given above relies on the concept of a directional derivative, which is basically a geometric concept and therefore hard to generalize to higher dimensions at least without losing the insight that the geometric interpretion provides. Therefore, Cauchy's method, a more abstract formulation of the method of characteristics that is easily generalized to higher dimensions, will now be presented. Consider again a linear first-order partial differential equation of the form

(4.8)

with an initial condition given on the initial manifold M. As the first step, parameterize the initial manifold by the parameter ?. This parameter does not have to be the arc length, but can be the parameter that gives the simplest and most convenient parameterization of M. We can then formally write the initial condition as

along M. Define now the so-called characteristic equations

These equations are first-order, coupled ordinary differential equations that in the general case are obtained as follows. For the first two equations, take the coefficient functions of Eq. (4.8), i.e., p( t, x) and q( t, x), and convert these to functions of one argument s by setting each free variable, t and x, equal to a function of s. Then set the derivative (with respect to s) of the nth free variable equal to the nth coefficient function. The equations obtained this way are called the base equations