## Partial Differential Equation

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

 (1)

In general, partial differential equations are much more difficult to solve analytically than are Ordinary Differential Equations. They may sometimes be solved using a Bäcklund Transformation, Characteristic, Green's Function, Integral Transform, Lax Pair, Separation of Variables, or--when all else fails (which it frequently does)--numerical methods.

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

 (2)

Second-order PDEs are then classified according to the properties of the Matrix
 (3)

as Elliptic, Hyperbolic, or Parabolic.

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)

holds in .

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)

where
 (8)

holds in .

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)

where
 (11)

holds in .

See also Bäcklund Transformation, Boundary Conditions, Characteristic (Partial Differential Equation), Elliptic Partial Differential Equation, Green's Function, Hyperbolic Partial Differential Equation, Integral Transform, Johnson's Equation, Lax Pair, Monge-Ampère Differential Equation, Parabolic Partial Differential Equation, Separation of Variables

References

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.