Laplace's equation

Laplace's equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. The solutions of Laplace's equation are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they describe the behavior of electric, gravitational, and fluid potentials.

In three dimensions, the problem is to find twice-differentiable real-valued functions φ of real variables x, y, and z such that

[itex]

{\partial^2 \varphi\over \partial x^2 } + {\partial^2 \varphi\over \partial y^2 } + {\partial^2 \varphi\over \partial z^2 } = 0. [itex]

This is often written as

[itex]\nabla^2 \varphi = 0 [itex]

or

or

[itex]\Delta \varphi = 0,[itex]

Solutions of Laplace's equation are called harmonic functions.

If the right-hand side is specified as a given function f(x, y, z), i.e.

[itex]\Delta \varphi = f[itex]

then the equation is called Poisson's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The partial differential operator [itex]\nabla^2[itex] or [itex]\Delta[itex] (which may be defined in any number of dimensions) is called the Laplace operator or just the Laplacian.

The Dirichlet problem for Laplace's equation consists in finding a solution φ on some domain [itex]D[itex] such that [itex]\phi[itex] on the boundary of [itex]D[itex] is equal to some given function. Since the Laplace operator appears in the heat equation, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain and wait until the temperature in the interior doesn't change anymore; the temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.

The Neumann boundary conditions for Laplace's equation specify not the function [itex]\phi[itex] itself on the boundary of [itex]D[itex], but its normal derivative. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of [itex]D[itex] alone.

The solutions to Laplace's equation which are twice continuously differentiable are called harmonic functions; they are all analytic.

If any two functions are solutions to Laplace's equation, their sum (or any linear combination) is also a solution. This property, called the principle of superposition is very useful, since solutions to complex problems can be constructed by summing simple solutions.

Bibliography

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.de:Laplace-Gleichung

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