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Partial Differential Equation

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

(partial^2psi)/(partialx^2)+(partial^2psi)/(partialy^2)+(partial^2psi)/(partialz^2)=1/(v^2)(partial^2psi)/(partialt^2).
(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

Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0.
(2)

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

Z=[A B; B C]
(3)

as elliptic, hyperbolic, or parabolic.

If Z is a positive definite matrix, i.e., det(Z)>0, the PDE is said to be elliptic. Laplace's equation and Poisson's equation are examples. Boundary conditions are used to give the constraint u(x,y)=g(x,y) on partialOmega, where

u_(xx)+u_(yy)=f(u_x,u_y,u,x,y)
(4)

holds in Omega.

If det(Z)<0, 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

u(x,y,t)=g(x,y,t) for x in partialOmega,t>0
(5)
u(x,y,0)=v_0(x,y) in Omega
(6)
u_t(x,y,0)=v_1(x,y) in Omega,
(7)

where

u_(xy)=f(u_x,u_t,x,y)
(8)

holds in Omega.

If det(Z)=0, 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

u(x,t)=g(x,t) for x in partialOmega,t>0
(9)
u(x,0)=v(x) for x in Omega,
(10)

where

u_(xx)=f(u_x,u_y,u,x,y)
(11)

holds in Omega.

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

Benjamin-Bona-Mahony equation

u_t+u_x+uu_x-u_(xxt)=0.
(12)

Biharmonic equation

del ^4phi=0.
(13)

Boussinesq equation

u_(tt)-alpha^2u_(xx)=beta^2u_(xxtt).
(14)

Cauchy-Riemann equations

(partialu)/(partialx)=(partialv)/(partialy)
(15)
(partialv)/(partialx)=-(partialu)/(partialy).
(16)

Chaplygin's equation

u_(xx)+(y^2)/(1-(y^2)/(c^2))u_(yy)+yu_y=0.
(17)

Euler-Darboux equation

u_(xy)+(alphau_x-betau_y)/(x-y)=0.
(18)

Heat conduction equation

(partialT)/(partialt)=kappadel ^2T.
(19)

Helmholtz differential equation

del ^2psi+k^2psi=0.
(20)

Klein-Gordon equation

1/(c^2)(partial^2psi)/(partialt^2)=(partial^2psi)/(partialx^2)-mu^2psi.
(21)

Korteweg-de Vries-Burgers equation

u_t+2uu_x-nuu_(xx)+muu_(xxx)=0.
(22)

Korteweg-de Vries equation

u_t+u_(xxx)-6uu_x=0.
(23)

Krichever-Novikov equation

(u_t)/(u_x)=1/4(u_(xxx))/(u_x)-3/8(u_(xx)^2)/(u_x^2)+3/2(p(u))/(u_x^2),
(24)

where

p(u)=1/4(4u^3-g_2u-g_3).
(25)

Laplace's equation

del ^2psi=0.
(26)

Lin-Tsien equation

2u_(tx)+u_xu_(xx)-u_(yy)=0.
(27)

Sine-Gordon equation

v_(tt)-v_(xx)+sinv=0.
(28)

Spherical harmonic differential equation

[1/(sintheta)partial/(partialtheta)(sinthetapartial/(partialtheta))+1/(sin^2theta)(partial^2)/(partialphi^2)+l(l+1)]u=0.
(29)

Tricomi equation

u_(yy)=yu_(xx).
(30)

Wave equation

del ^2psi=1/(v^2)(partial^2psi)/(partialt^2).
(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

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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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Partial Differential Equation

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

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