Laplace\'s equation - Citizendia

In mathematics, Laplace's equation is a partial differential equation named after Pierre-Simon Laplace who first studied its properties. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical The general theory of solutions to Laplace's equation is known as potential theory. Potential theory may be defined as the study of Harmonic functions Definition and comments The term "potential theory" arises from the fact that

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, $\scriptstyle\varphi$ of real variables, x, y, and z, such that

${\partial^2 \varphi\over \partial x^2 } + {\partial^2 \varphi\over \partial y^2 } + {\partial^2 \varphi\over \partial z^2 } = 0.$

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

$\Delta \varphi = 0,\,$

where Δ is the Laplace operator. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after

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

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. In Mathematics, Poisson's equation is a Partial differential equation with broad utility in Electrostatics, Mechanical engineering and Theoretical " Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. In Mathematics, an elliptic operator is one of the major types of Differential operator. The partial differential operator, $\scriptstyle\nabla^2$ , or $\scriptstyle\Delta$ , (which may be defined in any number of dimensions) is called the Laplace operator, or just the Laplacian. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after

Boundary conditions

The Dirichlet problem for Laplace's equation consists of finding a solution $\varphi$ on some domain $D$ such that $\varphi$ on the boundary of $D$ is equal to some given function. In Mathematics, a Dirichlet problem is the problem of finding a function which solves a specified Partial differential equation (PDE in the interior of 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 heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in 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. In Mathematics, the Neumann (or second type) boundary condition is a type of Boundary condition, named after Carl Neumann. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of $D$ alone.

Solutions of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. In Mathematics, Mathematical physics and the theory of Stochastic processes a harmonic function is a twice continuously differentiable function This article is about both real and complex analytic functions 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, e. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems g. , solutions to complex problems can be constructed by summing simple solutions.

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. This article is about both real and complex analytic functions That is, if z = x + iy, and if

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

then the necessary condition that f(z) be analytic is that the Cauchy-Riemann equations be satisfied:

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

It follows that

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

Therefore u satisfies the Laplace equation. In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two 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. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from 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

$\varphi = \log r, \,$

then a corresponding analytic function is

$f(z) = \log z = \log r + i\theta. \,$

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 second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves

There is an intimate connection between power series and Fourier series. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions If we expand a function f in a power series inside a circle of radius R, this means that

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

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

$c_n = a_n + i b_n.\,$

Therefore

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

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

$\nabla \times \mathbf{V}=v_x - u_y =0. \,$

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 stream function is defined for two-dimensional flows of various kinds Fluid flow is described in general by a Vector field in three (for steady flows or four (for non-steady flows including time dimensions The first derivatives of ψ are given by

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

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. A velocity potential is used in Fluid dynamics, when a fluid occupies a Simply-connected region and is Irrotational. The Cauchy-Riemann equations imply that

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

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

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

and

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

where ρ is the charge density. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric 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. In Mathematics, Poisson's equation is a Partial differential equation with broad utility in Electrostatics, Mechanical engineering and Theoretical

It is important to note that the Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.

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 Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. In Mathematics, a weak solution (also called a generalized solution) to an ordinary or Partial differential equation is a function 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

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

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

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

It follows that

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

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

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

A similar argument shows that in two dimensions

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

Green's function

A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V. In Mathematics, Green's function is a type of function used to solve inhomogeneous Differential equations subject to boundary conditions For instance, $\scriptstyle G(x,y,z;x',y',z')\,$ may satisfy

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

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

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 identity, (a consequence of the divergence theorem) which states that

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

The notations u_n and G_n denote normal derivatives on S. In Mathematics, Green's identities are a set of three identities in Vector calculus. 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 $\scriptstyle (x',y',z')\,$ 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. In Mathematics, the Poisson integral formula gives an explicit solution to the Dirichlet problem for Laplace's equation in a ball in Euclidean Let ρ, θ, and φ be spherical coordinates for the source point P. In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial 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

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

where

$\cos \Theta = \cos \varphi \cos \varphi' + \sin\varphi \sin\varphi'\cos(\theta -\theta'). \,$

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. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of In the branch of Mathematics called Potential theory, a quadrature domain in two dimensional real Euclidean space is a domain D (an open connected Potential theory may be defined as the study of Harmonic functions Definition and comments The term "potential theory" arises from the fact that In Fluid dynamics, a potential flow is a Velocity field which is described as the Gradient of a scalar function the velocity potential
Laplace Differential Equation on PlanetMath
Example initial-boundary value problems using Laplace's equation from exampleproblems. PlanetMath is a free, collaborative online Mathematics Encyclopedia. com.
Eric W. Weisstein, Laplace’s Equation at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Module for Laplace’s Equation by John H. Mathews
Find out how boundary value problems governed by Laplace's equation may be solved numerically by boundary element method

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.

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