A partial differential equation (PDE) is an equation involving functions and their partial derivatives; for example, the wave equation

(1)

Some partial differential equations can be solved exactly in Mathematica using DSolve[eqn, y, x1, x2], and numerically using NDSolve[eqns, y, x, xmin, xmax, t, tmin, tmax].

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, characteristics, Green's function, integral transform, Lax pair, separation of variables, or--when all else fails (which it frequently does)--numerical methods such as finite differences.

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

(2)

Linear second-order PDEs are then classified according to the properties of the matrix

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as elliptic, hyperbolic, or parabolic.

If is a positive definite matrix, i.e., , 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 .

The following are examples of important partial differential equations that commonly arise in problems of mathematical physics.

Benjamin-Bona-Mahony equation

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Biharmonic equation

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Boussinesq equation

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Cauchy-Riemann equations

			(15)
			(16)

Chaplygin's equation

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Euler-Darboux equation

(18)

Heat conduction equation

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Helmholtz differential equation

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Klein-Gordon equation

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Korteweg-de Vries-Burgers equation

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Korteweg-de Vries equation

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Krichever-Novikov equation

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where

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Laplace's equation

(26)

Lin-Tsien equation

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Sine-Gordon equation

(28)

Spherical harmonic differential equation

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Tricomi equation

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Wave equation

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Weisstein, E. W. "Books about Partial Differential Equations." http://www.ericweisstein.com/encyclopedias/books/PartialDifferentialEquations.html.

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Weisstein, Eric W. "Partial Differential Equation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PartialDifferentialEquation.html