example of rewriting a differential equation as a Pfaffian system

To show how one may reformulate a differential equationMathworldPlanetmath as Pfaff’s problem for a set of differential formsMathworldPlanetmath, consider the wave equationMathworldPlanetmath


The first step is to rewrite the equation as a system of first-order equations


To translateMathworldPlanetmath these equations into the language of differential forms, we shall use the fact that


from which it follows that


We can do likewise with a or b or c in the place of u; there is no point in repeating the formulas for each of these variables.

Multiplying the differential equations through by the form dtdxdy and using the above identities to eliminate partial derivativesMathworldPlanetmath, we obtain the following system of differential forms:


From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with coordinatesPlanetmathPlanetmath x,y,t,a,b,c,u which solves Pfaff’s problem and can be parameterized by x,y,t corresponds to the graph of a solution to the system of differential equations, and hence to a solution of the wave equation.

Note: These considerations are purely local. The global topology of the seven-dimensional space will depend on the domain on which the original wave equation was formulated and on the boundary conditionsMathworldPlanetmath.

Title example of rewriting a differential equation as a Pfaffian system
Canonical name ExampleOfRewritingADifferentialEquationAsAPfaffianSystem
Date of creation 2013-03-22 14:38:49
Last modified on 2013-03-22 14:38:49
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Example
Classification msc 53B99