## You are here

Homeexample of rewriting a differential equation as a Pfaffian system

## Primary tabs

# example of rewriting a differential equation as a Pfaffian system

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

${\partial^{2}u\over\partial t^{2}}={\partial^{2}u\over\partial x^{2}}+{% \partial^{2}u\over\partial y^{2}}$ |

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

${\partial a\over\partial t}-{\partial b\over\partial x}-{\partial c\over% \partial y}=0$ |

${\partial u\over\partial t}-a=0$ |

${\partial u\over\partial x}-b=0$ |

${\partial u\over\partial y}-c=0$ |

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

$du={\partial u\over\partial t}\,dt+{\partial u\over\partial x}\,dx+{\partial u% \over\partial y}\,dy$ |

from which it follows that

$du\wedge dx\wedge dy={\partial u\over\partial t}\,dt\wedge dx\wedge dy$ |

$du\wedge dy\wedge dt={\partial u\over\partial x}\,dt\wedge dx\wedge dy$ |

$du\wedge dt\wedge dx={\partial u\over\partial y}\,dt\wedge dx\wedge dy$ |

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 $dt\wedge dx\wedge dy$ and using the above identities to eliminate partial derivatives, we obtain the following system of differential forms:

$da\wedge dx\wedge dy-db\wedge dy\wedge dt-dc\wedge dt\wedge dx$ |

$du\wedge dx\wedge dy-a\,dt\wedge dx\wedge dy$ |

$du\wedge dy\wedge dt-b\,dt\wedge dx\wedge dy$ |

$du\wedge dt\wedge dx-c\,dt\wedge dx\wedge dy$ |

From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with coordinates $x,y,t,a,b,c,u$ 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 conditions.

## Mathematics Subject Classification

53B99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections