Partial Differential Equation
A partial differential equation (PDE) is an equation involving functions and their partial derivatives; for example, the wave equation
 |
(1)
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Some partial differential equations can be solved exactly in the Wolfram Language using DSolve[eqn, y, ![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/Inline1.gif" style="height:14px; width:5px" />x1, x2![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/Inline2.gif" style="height:14px; width:5px" />], and numerically using NDSolve[eqns, y, ![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/Inline3.gif" style="height:14px; width:5px" />x, xmin, xmax![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/Inline4.gif" style="height:14px; width:5px" />, ![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/Inline5.gif" style="height:14px; width:5px" />t, tmin, tmax![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/Inline6.gif" style="height:14px; width:5px" />].
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
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Linear second-order PDEs are then classified according to the properties of the matrix
![Z=[A B; B C]](http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/NumberedEquation3.gif) |
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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
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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
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where
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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)
|
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where
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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
Chaplygin's equation
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Euler-Darboux equation
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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
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Lin-Tsien equation
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Sine-Gordon equation
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Spherical harmonic differential equation
![[1/(sintheta)partial/(partialtheta)(sinthetapartial/(partialtheta))+1/(sin^2theta)(partial^2)/(partialphi^2)+l(l+1)]u=0.](http://mathworld.wolfram.com/images/equations/PartialDifferentialEquation/NumberedEquation27.gif) |
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Tricomi equation
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Wave equation
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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.
Conte, R. "Exact Solutions of Nonlinear Partial Differential Equations by Singularity Analysis." 13 Sep 2000. http://arxiv.org/abs/nlin.SI/0009024.
Kamke, E. Differentialgleichungen Lösungsmethoden und Lösungen, Bd. 2: Partielle Differentialgleichungen ester Ordnung für eine gesuchte Function. New York: Chelsea, 1974.
Folland, G. B. Introduction to Partial Differential Equations, 2nd ed. Princeton, NJ: Princeton University Press, 1996.
Kevorkian, J. Partial Differential Equations: Analytical Solution Techniques, 2nd ed. New York: Springer-Verlag, 2000.
Morse, P. M. and Feshbach, H. "Standard Forms for Some of the Partial Differential Equations of Theoretical Physics." Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 271-272, 1953.
Polyanin, A.; Zaitsev, V.; and Moussiaux, A. Handbook of First-Order Partial Differential Equations. New York: Gordon and Breach, 2001.
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.
Taylor, M. E. Partial Differential Equations, Vol. 1: Basic Theory. New York: Springer-Verlag, 1996.
Taylor, M. E. Partial Differential Equations, Vol. 2: Qualitative Studies of Linear Equations. New York: Springer-Verlag, 1996.
Taylor, M. E. Partial Differential Equations, Vol. 3: Nonlinear Equations. New York: Springer-Verlag, 1996.
Trott, M. "The Mathematica Guidebooks Additional Material: Various Time-Dependent PDEs." http://www.mathematicaguidebooks.org/additions.shtml#N_1_06.
Webster, A. G. Partial Differential Equations of Mathematical Physics, 2nd corr. ed. New York: Dover, 1955.
Weisstein, E. W. "Books about Partial Differential Equations." http://www.ericweisstein.com/encyclopedias/books/PartialDifferentialEquations.html.
Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.
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