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 the Wolfram Language 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
(3)
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
(12)
Biharmonic equation
(13)
Boussinesq equation
(14)
Cauchy-Riemann equations
Chaplygin's equation
(17)
Euler-Darboux equation
(18)
Heat conduction equation
(19)
Helmholtz differential equation
(20)
Klein-Gordon equation
(21)
Korteweg-de Vries-Burgers equation
(22)
Korteweg-de Vries equation
(23)
Krichever-Novikov equation
(24)
where
(25)
Laplace's equation
(26)
Lin-Tsien equation
(27)
Sine-Gordon equation
(28)
Spherical harmonic differential
equation
(29)
Tricomi equation
(30)
Wave equation
(31)
See also Bäcklund Transformation ,
Boundary Conditions ,
Characteristic ,
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 Explore
this topic in the MathWorld classroom
Explore with Wolfram|Alpha
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. Referenced
on Wolfram|Alpha Partial Differential Equation
Cite this as:
Weisstein, Eric W. "Partial Differential Equation."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/PartialDifferentialEquation.html
Subject classifications