First-order Partial Differential Equation - Two-dimensional Theory

Two-dimensional Theory

The notation is relatively simple in two space dimensions, but the main ideas generalize to higher dimensions. A general first-order partial differential equation has the form

where

A complete integral of this equation is a solution φ(x,y,u) that depends upon two parameters a and b. (There are n parameters required in the n-dimensional case.) An envelope of such solutions is obtained by choosing an arbitrary function w, setting b=w(a), and determining A(x,y,u) by requiring that the total derivative

In that case, a solution is also given by

Each choice of the function w leads to a solution of the PDE. A similar process led to the construction of the light cone as a characteristic surface for the wave equation.

If a complete integral is not available, solutions may still be obtained by solving a system of ordinary equations. To obtain this system, first note that the PDE determines a cone (analogous to the light cone) at each point: if the PDE is linear in the derivatives of u (it is quasi-linear), then the cone degenerates into a line. In the general case, the pairs (p,q) that satisfy the equation determine a family of planes at a given point:

where

The envelope of these planes is a cone, or a line if the PDE is quasi-linear. The condition for an envelope is

where F is evaluated at, and dp and dq are increments of p and q that satisfy F=0. Hence the generator of the cone is a line with direction

This direction corresponds to the light rays for the wave equation. To integrate differential equations along these directions, we require increments for p and q along the ray. This can be obtained by differentiating the PDE:

Therefore the ray direction in space is

The integration of these equations leads to a ray conoid at each point . General solutions of the PDE can then be obtained from envelopes of such conoids.