A partial differential equation (PDE) is an equation involving functions and their Partial Derivatives; for example, the Wave Equation

(1) |

Fortunately, partial differential equations of second-order are often amenable to analytical solution. Such PDEs are of
the form

(2) |

(3) |

If
is a Positive Definite Matrix, i.e., det
, the PDE is said to be Elliptic. Laplace's Equation and Poisson's Equation are examples. Boundary conditions
are used to give the constraint
on
, where

(4) |

If det
, the PDE is said to be
Hyperbolic.
The Wave Equation is an example of a hyperbolic
partial differential equation. Initial-boundary conditions are used to give

(5) |

(6) |

(7) |

(8) |

If det
, the PDE is said to be parabolic. The Heat Conduction Equation equation and other
diffusion equations are examples. Initial-boundary conditions are used to give

(9) |

(10) |

(11) |

**References**

Partial Differential Equations

Arfken, G. ``Partial Differential Equations of Theoretical Physics.'' §8.1 in
*Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 437-440, 1985.

Bateman, H. *Partial Differential Equations of Mathematical Physics.* New York: Dover, 1944.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Partial Differential Equations.'' Ch. 19 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 818-880, 1992.

Sobolev, S. L. *Partial Differential Equations of Mathematical Physics.* New York: Dover, 1989.

Sommerfeld, A. *Partial Differential Equations in Physics.* New York: Academic Press, 1964.

Webster, A. G. *Partial Differential Equations of Mathematical Physics, 2nd corr. ed.* New York: Dover, 1955.

© 1996-9

1999-05-26