In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

where is the Laplace operator,[note 1] is divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

If the right-hand side is specified as a given function, , we have

This is called Poisson's equation, a generalization of Laplace's equation, Laplace's and Poisson's equation are the simplest examples of elliptic partial differential equations.

The Laplace equation is also a special case of the Helmholtz equation.

The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions,[1] which are important in branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation.[2]

Contents

Forms in different coordinate systemsEdit

In Cartesian coordinates,[3]

 

In cylindrical coordinates,[3]

 

In spherical coordinates, using the  convention,[3]

 

More generally, in curvilinear coordinates,

 

or

 

Boundary conditionsEdit

 
Laplace's equation on an annulus (inner radius r = 2 and outer radius R = 4) with Dirichlet boundary conditions u(r=2) = 0 and u(R=4) = 4 sin(5 θ)

The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain D such that φ 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 according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain 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 φ 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.

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 homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition, is very useful, e.g., solutions to complex problems can be constructed by summing simple solutions.

In two dimensionsEdit

Laplace's equation in two independent variables in rectangular coordinates has the form

 

Analytic functionsEdit

The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if z = x + iy, and if

 

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

 

where ux is the first partial derivative of u with respect to x. It follows that

 

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

 

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

 

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

 

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

 

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

 

then a corresponding analytic function is

 

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.

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

 

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

 

Therefore

 

which is a Fourier series for f. These trigonometric functions can themselves be expanded, using multiple angle formulae.

Fluid flowEdit

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 continuity condition for an incompressible flow is that

 

and the condition that the flow be irrotational is that

 

If we define the differential of a function ψ by

 

then the continuity 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

 

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

 

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.

ElectrostaticsEdit

According to Maxwell's equations, an electric field (u, v) in two space dimensions that is independent of time satisfies

 

and

 

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

 

so the electric potential φ may be constructed to satisfy

 

The second of Maxwell's equations then implies that

 

which is the Poisson equation. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.

In three dimensionsEdit

Fundamental solutionEdit

A fundamental solution of Laplace's equation satisfies

 

where the Dirac delta function δ denotes a unit source concentrated at the point (x′, y′, z′). No function has this property: in fact it is a distribution rather than a function; 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). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a positive operator. 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 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

 

It follows that

 

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

 

Note that, with the opposite sign convention (used in physics), this is the potential generated by a point particle, for an inverse-square law force, arising in the solution of Poisson equation. A similar argument shows that in two dimensions

 

where log(r) denotes the natural logarithm. Note that, with the opposite sign convention, this is the potential generated by a pointlike sink (see point particle), which is the solution of the Euler equations in two-dimensional incompressible flow.

Green's functionEdit

A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V. For instance,

 

may satisfy

 
 

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

 

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

 

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

 

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

 

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

 

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 with Dirichlet boundary values g inside the sphere is given by

  (Zachmanoglou 1986, p. 228)

where

 

is the cosine of the angle between (θ, φ) and (θ′, φ′). 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.

ElectrostaticsEdit

Let   be the electric field,   be the electric charge density, and   be the permittivity of free space. Then Gauss's law for electricity (Maxwell's first equation) in differential form states[4]

 

Now, the electric field can be expressed as the negative gradient of the electric potential  ,

 

if the field is irrotational,  . The irrotationality of   is also known as the electrostatic condition.[4]

Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity,[4]

 

In the particular case of a source-free region,   and Poisson's equation reduces to Laplace's equation for the electric potential.[4]

If the electrostatic potential   is specified on the boundary of a region  , then it is uniquely determined. If   is surrounded by a conducting material with a specified charge density  , and if the total charge   is known, then   is also unique.[5]

A potential that doesn't satisfy Laplace's equation together with the boundary condition is an invalid electrostatic potential.

In the Schwarzschild metricEdit

S. Persides[6] solved the Laplace equation in Schwarzschild spacetime. Using the canonical variables r, θ, φ the solution is

 

where Yl(θ, φ) is a spherical harmonic function, and

 

Here Pl and Ql are Legendre functions of the first and second kind, respectively, while rs is the Schwarzschild radius. The parameter l is an arbitrary non-negative integer.

See alsoEdit

NotesEdit

  1. ^ The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, e.g.  . Its use to represent the Laplacian should not be confused with this use.

ReferencesEdit

  1. ^ Stewart, James. Calculus : Early Transcendentals. 7th ed., Brooks/Cole, Cengage Learning, 2012. Chapter 14: Partial Derivatives. p. 908. ISBN 978-0-538-49790-9.
  2. ^ Zill, Dennis G, and Michael R Cullen. Differential Equations with Boundary-Value Problems. 8th edition / ed., Brooks/Cole, Cengage Learning, 2013. Chapter 12: Boundary-value Problems in Rectangular Coordinates. p. 462. ISBN 978-1-111-82706-9.
  3. ^ a b c Griffiths, David J. Introduction to Electrodynamics. 4th ed., Pearson, 2013. Inner front cover. ISBN 978-1-108-42041-9.
  4. ^ a b c d Griffiths, David J. Introduction to Electrodynamics. Fourth ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4. ISBN 978-1-108-42041-9.
  5. ^ Griffiths, David J. Introduction to Electrodynamics. Fourth ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121. ISBN 978-1-108-42041-9.
  6. ^ Persides, S. (1973). "The Laplace and poisson equations in Schwarzschild's space-time". Journal of Mathematical Analysis and Applications. 43 (3): 571–578. doi:10.1016/0022-247X(73)90277-1.

Further readingEdit

  • Evans, L. C. (1998). Partial Differential Equations. Providence: American Mathematical Society. ISBN 978-0-8218-0772-9.
  • Petrovsky, I. G. (1967). Partial Differential Equations. Philadelphia: W. B. Saunders.
  • Polyanin, A. D. (2002). Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton: Chapman & Hall/CRC Press. ISBN 978-1-58488-299-2.
  • Sommerfeld, A. (1949). Partial Differential Equations in Physics. New York: Academic Press.
  • Zachmanoglou, E. C. (1986). Introduction to Partial Differential Equations with Applications. New York: Dover.

External linksEdit