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Laplace's Equation

The scalar form of Laplace's equation is the partial differential equation

del ^2psi=0,
(1)

where del ^2 is the Laplacian.

Note that the operator del ^2 is commonly written as Delta by mathematicians (Krantz 1999, p. 16). Laplace's equation is a special case of the Helmholtz differential equation

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

with k=0, or Poisson's equation

del ^2psi=-4pirho
(3)

with rho=0.

The vector Laplace's equation is given by

del ^2F=0.
(4)

A function psi which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima. Because Laplace's equation is linear, the superposition of any two solutions is also a solution.

A solution to Laplace's equation is uniquely determined if (1) the value of the function is specified on all boundaries (Dirichlet boundary conditions) or (2) the normal derivative of the function is specified on all boundaries (Neumann boundary conditions).

Coordinate SystemVariablesSolution Functions
CartesianX(x)Y(y)Z(z)exponential functions, circular functions, hyperbolic functions
circular cylindricalR(r)Theta(theta)Z(z)Bessel functions, exponential functions, circular functions
conicalellipsoidal harmonics, power
confocal ellipsoidalLambda(lambda)M(mu)N(nu)ellipsoidal harmonics of the first kind
elliptic cylindricalU(u)V(v)Z(z)Mathieu function, circular functions
oblate spheroidalLambda(lambda)M(mu)N(nu)Legendre polynomial, circular functions
parabolicBessel functions, circular functions
parabolic cylindricalparabolic cylinder functions, Bessel functions, circular functions
paraboloidalU(u)V(v)Theta(theta)circular functions
prolate spheroidalLambda(lambda)M(mu)N(nu)Legendre polynomial, circular functions
sphericalR(r)Theta(theta)Phi(phi)Legendre polynomial, power, circular functions

Laplace's equation can be solved by separation of variables in all 11 coordinate systems that the Helmholtz differential equation can. The form these solutions take is summarized in the table above. In addition to these 11 coordinate systems, separation can be achieved in two additional coordinate systems by introducing a multiplicative factor. In these coordinate systems, the separated form is

psi=(X_1(u_1)X_2(u_2)X_3(u_3))/(R(u_1,u_2,u_3)),
(5)

and setting

(h_1h_2h_3)/(h_i^2)=g_i(u_(i+1),u_(i+2))f_i(u_i)R^2,
(6)

where h_i are scale factors, gives the Laplace's equation

sum_(i=1)^31/(h_i^2X_i)[1/(f_i)d/(du_i)(f_i(dX_i)/(du_i))]=sum_(i=1)^31/(h_i^2R)[1/(f_i)partial/(partialu_i)(f_i(partialR)/(partialu_i))].
(7)

If the right side is equal to -k_1^2/F(u_1,u_2,u_3), where k_1 is a constant and F is any function, and if

h_1h_2h_3=Sf_1f_2f_3R^2F,
(8)

where S is the Stäckel determinant, then the equation can be solved using the methods of the Helmholtz differential equation. The two systems where this is the case are bispherical and toroidal, bringing the total number of separable systems for Laplace's equation to 13 (Morse and Feshbach 1953, pp. 665-666).

In two-dimensional bipolar coordinates, Laplace's equation is separable, although the Helmholtz differential equation is not.

Zwillinger (1997, p. 128) calls

(a_0x+b_0)y^((n))+(a_1x+b_1)y^((n-1))+...+(a_nx+b_n)y=0
(9)

the Laplace equations.

See also

Boundary Conditions, Ellipsoidal Harmonic of the First Kind, Harmonic Function, Helmholtz Differential Equation, Laplacian, Partial Differential Equation, Poisson's Equation, Separation of Variables, Stäckel Determinant

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972.Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.Eisenhart, L. P. "Separable Systems in Euclidean 3-Space." Physical Review 45, 427-428, 1934.Eisenhart, L. P. "Separable Systems of Stäckel." Ann. Math. 35, 284-305, 1934.Eisenhart, L. P. "Potentials for Which Schroedinger Equations Are Separable." Phys. Rev. 74, 87-89, 1948.Krantz, S. G. "The Laplace Equation." §7.1.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 16 and 89, 1999.Moon, P. and Spencer, D. E. "Recent Investigations of the Separation of Laplace's Equation." Proc. Amer. Math. Soc. 4, 302, 1953.Moon, P. and Spencer, D. E. "Eleven Coordinate Systems." §1 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 1-48, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 125-126 and 271, 1953.Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Brookline, MA: Math. Sci. Press, pp. 306-315, 1950.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 417, 1995.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997.

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Laplace's Equation

Cite this as:

Weisstein, Eric W. "Laplace's Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplacesEquation.html

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