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A visualisation of a solution to the

heat equation on a two

dimensional plane

Partial differential equationFrom Wikipedia, the free encyclopedia

In mathematics, a partial differential equation(PDE) is a

differential equation that contains unknown multivariable functions

and their partialderivatives. (This is in contrast to ordinary differential

equations, which deal with functions of a single variable and their

derivatives.) PDEs are used to formulate problems involving

functions of several variables, and are either solved by hand, or used

to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena such as

sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or

quantum mechanics.These seemingly distinct physical phenomena

can be formalised similarly in terms of PDEs. Just as ordinary

differential equations often model one-dimensional dynamical

systems, partial differential equations often model multidimensional systems. PDEs find their generalisation

in stochastic partial differential equations.

Contents

1 Introduction

2 Existence and uniqueness

3 Notation

4 Examples

4.1 Heat equation in one spacedimension

4.2 Wave equation in one spatial dimension

4.3 Generalised heat-like equationin one space dimension

4.4 Spherical waves

4.5 Laplace equation in two dimensions

4.5.1 Connection with holomorphic functions

4.5.2 A typical boundary value problem4.6 EulerTricomi equation

4.7 Advection equation

4.8 GinzburgLandau equation

4.9 The Dym equation

4.10 Initial-boundary value problems

4.11 Vibrating string

4.11.1 Vibrating membrane

4.12 Other examples

5 Classification

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5.1 Equations of first order

5.2 Equations of second order

5.3 Systems of first-order equations and characteristic surfaces

5.4 Equations of mixed type

5.5 Infinite-order PDEs in quantum mechanics

6 Analytical methods to solve PDEs

6.1 Separation of variables

6.2 Method of characteristics

6.3 Integral transform

6.4 Change of variables

6.5 Fundamental solution

6.6 Superposition principle

6.7 Methods for non-linear equations

6.8 Lie group method

6.9 Semianalytical methods

7 Numerical methods to solve PDEs

7.1 Finite element method

7.2 Finite difference method

7.3 Finite volume method

8 See also

9 References

10 External links

Introduction

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous

variables. The position of a rigid body is specified by six numbers, but the configuration of a fluid is given

by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The

dynamics for the rigid body take place in a finite-dimensional configuration space the dynamics for thefluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much

harder to solve than ordinary differential equations (ODEs), but here again there will be simple solutions

for linear problems. Classic domains where PDEs are used include acoustics, fluid flow, electrodynamics,

and heat transfer.

A partial differential equation (PDE) for the function is an equation of the form

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IfFis a linear function of uand its derivatives, then the PDE is called linear. Common examples of linear

PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein

Gordon equation, and Poisson's equation.

A relatively simple PDE is

This relation implies that the function u(x,y) is independent ofx. However, the equation gives no

information on the function's dependence on the variabley. Hence the general solution of this equation is

where fis an arbitrary function ofy. The analogous ordinary differential equation is

which has the solution

where cis any constant value. These two examples illustrate that general solutions of ordinary differential

equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A

solution of a PDE is generally not unique additional conditions must generally be specified on the

boundary of the region where the solution is defined. For instance, in the simple example above, the

function f(y)can be determined if uis specified on the linex= 0.

Existence and uniqueness

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very

satisfactory answer with the PicardLindelf theorem, that is far from the case for partial differential

equations. The CauchyKowalevski theorem states that the Cauchy problem for any partial differential

equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique

analytic solution. Although this result might appear to settle the existence and uniqueness of solutions,

there are examples of linear partial differential equations whose coefficients have derivatives of all orders

(which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the

solution of a partial differential equation exists and is unique, it may nevertheless have undesirable

properties. The mathematical study of these questions is usually in the more powerful context of weak

solutions.

An example of pathological behavior is the sequence of Cauchy problems (depending upon n) for the

Laplace equation

with boundary conditions

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where nis an integer. The derivative of uwith respect toyapproaches 0 uniformly inxas nincreases, but

the solution is

This solution approaches infinity if nxis not an integer multiple of for any non-zero value ofy. The

Cauchy problem for the Laplace equation is called ill-posedor not well posed, since the solution does not

depend continuously upon the data of the problem. Such ill-posed problems are not usually satisfactory for

physical applications.

Notation

In PDEs, it is common to denote partial derivatives using subscripts. That is:

Especially in physics, del () is often used for spatial derivatives, and for time derivatives. For

example, the wave equation (described below) can be written as

or

where is the Laplace operator.

Examples

Heat equation in one space dimension

The equation for conduction of heat in one dimension for a homogeneous body has

where u(t,x) is temperature, and is a positive constant that describes the rate of diffusion. The Cauchy

problem for this equation consists in specifying u(0,x)= f(x), where f(x) is an arbitrary function.

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General solutions of the heat equation can be found by the method of separation of variables. Some

examples appear in the heat equation article. They are examples of Fourier series for periodic fand Fourier

transforms for non-periodic f. Using the Fourier transform, a general solution of the heat equation has the

form

whereFis an arbitrary function. To satisfy the initial condition,Fis given by the Fourier transform of f,

that is

If frepresents a very small but intense source of heat, then the preceding integral can be approximated by

the delta distribution, multiplied by the strength of the source. For a source whose strength is normalized to

1, the result is

and the resulting solution of the heat equation is

This is a Gaussian integral. It may be evaluated to obtain

This result corresponds to the normal probability density forxwith mean 0 and variance 2t. The heat

equation and similar diffusion equations are useful tools to study random phenomena.

Wave equation in one spatial dimension

The wave equation is an equation for an unknown function u(t,x) of the form

Here umight describe the displacement of a stretched string from equilibrium, or the difference in air

pressure in a tube, or the magnitude of an electromagnetic field in a tube, and cis a number that

corresponds to the velocity of the wave. The Cauchy problem for this equation consists in prescribing the

initial displacement and velocity of a string or other medium:

where fand gare arbitrary given functions. The solution of this problem is given by d'Alembert's formula:

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This formula implies that the solution at (t,x) depends only upon the data on the segment of the initial line

that is cut out by the characteristic curves

that are drawn backwards from that point. These curves correspond to signals that propagate with velocity

cforward and backward. Conversely, the influence of the data at any given point on the initial line

propagates with the finite velocity c: there is no effect outside a triangle through that point whose sides are

characteristic curves. This behavior is very different from the solution for the heat equation, where the

effect of a point source appears (with small amplitude) instantaneously at every point in space. The

solution given above is also valid if t< 0, and the explicit formula shows that the solution depends

smoothly upon the data: both the forward and backward Cauchy problems for the wave equation are well-

posed.

Generalised heat-like equation in one space dimension

Where heat-like equation means equations of the form:

where is a SturmLiouville operator (However it should be noted this operator may in fact be of the

form

where w(x) is the weighting function with respect to which the eigenfunctions of are orthogonal) in the

xcoordinate. Subject to the boundary conditions:

Then:

If:

where

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Spherical waves

Spherical waves are waves whose amplitude depends only upon the radial distance rfrom a central point

source. For such waves, the three-dimensional wave equation takes the form

This is equivalent to

and hence the quantity rusatisfies the one-dimensional wave equation. Therefore a general solution for

spherical waves has the form

whereFand Gare completely arbitrary functions. Radiation from an antenna corresponds to the case

where Gis identically zero. Thus the wave form transmitted from an antenna has no distortion in time: the

only distorting factor is 1/r. This feature of undistorted propagation of waves is not present if there are two

spatial dimensions.

Laplace equation in two dimensions

The Laplace equation for an unknown function of two variables has the form

Solutions of Laplace's equation are called harmonic functions.

Connection with holomorphic functions

Solutions of the Laplace equation in two dimensions are intimately connected with analytic functions of acomplex variable (a.k.a. holomorphic functions): the real and imaginary parts of any analytic function are

conjugate harmonicfunctions: they both satisfy the Laplace equation, and their gradients are orthogonal.

If f=u+iv, then the CauchyRiemann equations state that

and it follows that

Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at

least locally. Details are given in Laplace equation.

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A typical boundary value problem

A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary

of a domain. For example, we may seek a harmonic function that takes on the values u() on a circle of

radius one. The solution was given by Poisson:

Petrovsky (1967, p. 248) shows how this formula can be obtained by summing a Fourier series for . If r