NationMaster - Encyclopedia: Laplace's equation

In mathematics, 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. The general theory of solutions to Laplace's equation is known as potential theory. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... A giant Hubble mosaic of the Crab Nebula, a supernova remnant Astronomy is the science of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as auroras and cosmic background radiation). ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... It has been suggested that this article or section be merged with Scalar potential. ... Potential theory may be defined as the study of harmonic functions. ...

1 Definition
2 Boundary conditions
3 Laplace equation in two dimensions
4 Laplace equation in three dimensions
- 4.1 Fundamental solution
- 4.2 Green's function
5 See also
6 External links
7 References

Definition

In three dimensions, the problem is to find twice-differentiable real-valued functions, φ of real variables, x, y, and z, such that In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

${partial^2 varphiover partial x^2 } + {partial^2 varphiover partial y^2 } + {partial^2 varphiover partial z^2 } = 0.$

This is often written as

$nabla^2 varphi = 0 ,$

$operatorname{div},operatorname{grad},varphi = 0,$

where div is the divergence, and grad is the gradient, or In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... For other uses, see Gradient (disambiguation). ...

$Delta varphi = 0,,$

where Δ is the Laplace operator. In mathematics and physics, the Laplace operator or Laplacian, denoted by Î”, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...

Solutions of Laplace's equation are called harmonic functions. In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...

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

$Delta varphi = f,$

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, $nabla^2$ , or $Δ$ , (which may be defined in any number of dimensions) is called the Laplace operator, or just the Laplacian. In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ... In mathematics, an Elliptic operator is a major type of differential operator P defined on spaces of complex-valued functions, or some more general function-like objects, such that the coefficients of the highest-order derivatives satisfy a positivity condition. ... In mathematics and physics, the Laplace operator or Laplacian, denoted by Î”, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...

Boundary conditions

The Dirichlet problem for Laplace's equation consists in finding a solution φ on some domain $D$ such that $varphi$ on the boundary of $D$ 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. In mathematics, Dirichlet problems are a class of partial differential equation (PDE) problems which ask you to solve for the values of a function in a region given the value of the function on the boundary of that region. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...

The Neumann boundary conditions for Laplace's equation specify not the function $varphi$ itself on the boundary of $D$ , 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 $D$ alone. In mathematics, a Neumann boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values the derivative of a solution is to take on the boundary of the domain. ...

Solutions of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogenous differential 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. In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ...

Laplace equation in two dimensions

The Laplace equation in two independent variables has the form

$varphi_{xx} + varphi_{yy} = 0.,$

Analytic functions

The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if z = x + iy, and if In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

$f(z) = u(x,y) + iv(x,y),,$

then the necessary condition that f(z) be analytic is that the Cauchy-Riemann equations be satisfied: In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ...

$u_x = v_y, quad v_x = -u_y.,$

It follows that

$u_{yy} = (-v_x)_y = -(v_y)_x = -(u_x)_x.,$

Therefore u satisfies the Laplace equation. A similar calculation shows that v also satisfies the Laplace equation.

Conversely, given a harmonic function, it is the real part of an analytic function, f(z) (at least locally). If a trial form is

$f(z) = varphi(x,y) + i psi(x,y),,$

then the Cauchy-Riemann equations will be satisfied if we set

$psi_x = -varphi_y, quad psi_y = varphi_x.,$

This relation does not determine ψ, but only its increments:

$d psi = -varphi_y, dx + varphi_x, dy.,$

The Laplace equation for φ implies that the integrability condition for ψ is satisfied:

$psi_{xy} = psi_{yx},,$

and thus ψ may be defined by a line integral. The integrability condition and Stokes' theorem implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the Laplace equation are called conjugate harmonic functions. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if r and θ are polar coordinates and Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

$varphi = log r, ,$

then a corresponding analytic function is

$f(z) = log z = log r + itheta. ,$

However, the angle θ is single-valued only in a region that does not enclose the origin.

The close connection between the Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a power series, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the wave equation, which generally have less regularity. The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ...

There is an intimate connection between power series and Fourier series. If we expand a function f in a power series inside a circle of radius R, this means that The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

$f(z) = sum_{n=0}^infty c_n z^n,,$

with suitably defined coefficients whose real and imaginary parts are given by

$c_n = a_n + i b_n.,$

Therefore

$f(z) = sum_{n=0}^infty left[ a_n r^n cos n theta - b_n r^n sin n thetaright] + i sum_{n=1}^infty left[ a_n r^n sin ntheta + b_n r^n cos n thetaright],,$

which is a Fourier series for f.

Fluid flow

Let the quantities u and v be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The condition that the flow be incompressible is that

$u_x + v_y=0,,$

and the condition that the flow be irrotational is that

$v_x - u_y =0. ,$

If we define the differential of a function ψ by

$d psi = v, dx - u, dy,,$

then the incompressibility condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines. The first derivatives of ψ are given by In fluid dynamics, the stream function is defined for two-dimensional flows. ...

$psi_x = v, quad psi_y=-u, ,$

and the irrotationality condition implies that ψ satisfies the Laplace equation. The harmonic function φ that is conjugate to ψ is called the velocity potential. The Cauchy-Riemann equations imply that A velocity potential is used in fluid dynamics, when a fluid is irrotational. ...

$varphi_x=-u, quad varphi_y=-v. ,$

Thus every analytic function corresponds to a steady incompressible, irrotational fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.

Electrostatics

According to Maxwell's equations, an electric field (u,v) in two space dimensions that is independent of time satisfies In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ...

$nabla times (u,v) = v_x -u_y =0,,$

and

$nabla cdot (u,v) = rho,,$

where ρ is the charge density. The first Maxwell equation is the integrability condition for the differential

$d varphi = -u, dx -v, dy,,$

so the electric potential φ may be constructed to satisfy

$varphi_x = -u, quad varphi_y = -v.,$

The second of Maxwell's equations then implies that

$varphi_{xx} + varphi_{yy} = -rho,,$

which is the Poisson equation. Poissons equation is the partial differential equation: Or alternately: or i. ...

Laplace equation in three dimensions

Fundamental solution

A fundamental solution of Laplace's equation satisfies

$Delta u = u_{xx} + u_{yy} + u_{zz} = -delta(x-x',y-y',z-z'), ,$

where the Dirac delta function $δ$ denotes a unit source concentrated at the point $(x',, y', , z').$ No function has this property, but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see weak solution). The definition of the fundamental solution thus implies that, if the Laplacian of u is integrated over any volume that encloses the source point, then The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives appearing in the equation may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. ...

$iiint_V operatorname{div} nabla u , dV =-1. ,$

The Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance r from the source point. If we choose the volume to be a ball of radius a around the source point, then Gauss' divergence theorem implies that

$-1= iiint_V operatorname{div} nabla u , dV = iint_S u_r dS = 4pi a^2 u_r(a).,$

It follows that

$u_r(r) = -frac{1}{4pi r^2},,$

on a sphere of radius r that is centered around the source point, and hence

$u = frac{1}{4pi r}.,$

A similar argument shows that in two dimensions

$u = frac{-log r}{2pi}. ,$

Green's function

A Green's function (the terminology is not grammatical) is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V. For instance, $G(x,y,z;x',y',z'),$ may satisfy In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ...

$nabla cdot nabla G = -delta(x-x',y-y',z-z') quad hbox{in} quad V, ,$

$G = 0 quad hbox{if} quad (x,y,z) quad hbox{on} quad S. ,$

Now if u is any solution of the Poisson equation in V:

$nabla cdot nabla u = -f, ,$

and u assumes the boundary values g on S, then we may apply Green's formula, (a consequence of the divergence theorem) which states that Greens identities are a set of three identities in vector calculus. ...

$iiint_V left[ G , nabla cdot nabla u - u , nabla cdot nabla G right], dV = iiint_V nabla cdot left[ G nabla u - u nabla G right], dV = iint_S left[ G u_n -u G_n right] , dS. ,$

The notations u_n and G_n denote normal derivatives on S. In view of the conditions satisfied by u and G, this result simplifies to

$u(x',y',z') = iiint_V G f , dV - iint_S G_n g , dS. ,$

Thus the Green's function describes the influence at $(x',y',z'),$ of the data f and g. For the case of the interior of a sphere of radius a, the Green's function may be obtained by means of a reflection (Sommerfeld, 1949): the source point P at distance ρ from the center of the sphere is reflected along its radial line to a point P' that is at a distance

$rho' = frac{a^2}{rho}. ,$

Note that if P is inside the sphere, then P' will be outside the sphere. The Green's function is then given by

$frac{1}{4 pi R} - frac{a}{4 pi rho R'}, ,$

where R denotes the distance to the source point P and R' denotes the distance to the reflected point P'. A consequence of this expression for the Green's function is the Poisson integral formula. Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation inside the sphere is given by In mathematics, the Poisson integral formula gives an explicit solution to the Dirichlet problem for Laplaces equation in a ball in Euclidean space Rn. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$u(P) = frac{1}{4pi} a^3left( 1 - frac{rho^2}{a^2} right) iint frac{g(theta',varphi') sin theta' , dtheta' , dvarphi'}{(a^2 + rho^2 - 2 a rho cos Theta)^{3/2} }, ,$

where

$cos Theta = cos theta cos theta' + sintheta sintheta'cos(theta -theta'). ,$

A simple consequence of this formula is that if u is a harmonic function, then the value of u at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.

External links

Laplace Equation (particular solutions and boundary value problems) at EqWorld: The World of Mathematical Equations.
Example initial-boundary value problems using Laplace's equation from exampleproblems.com.
Eric W. Weisstein, Laplace’s Equation at MathWorld.
Module for Laplace’s Equation by John H. Mathews

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

References

L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
I. G. Petrovsky, Partial Differential Equations, W. B. Saunders Co., Philadelphia, 1967.
A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
A. Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, 1949.

Categories: Elliptic partial differential equations | Harmonic functions | Equations

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