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Encyclopedia > Partial differential equation

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. Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space, or distributed in space and time. Completely distinct physical problems may have identical mathematical formulations. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ... Partial plot of a function f. ... In an experimental design, the independent variable (argument of a function, also called a predictor variable) is the variable that is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... Sound is a disturbance of mechanical energy that propagates through matter as a longitudinal wave, and therefore is a mechanical wave. ... In physics, heat, symbolized by Q, is defined as transfer of thermal energy [1] Generally, heat is a form of energy transfer associated with the different motions of atoms, molecules and other particles that comprise matter when it is hot and when it is cold. ... Electrostatics (also known as Static Electricity) is the branch of physics that deals with the forces exerted by a static (i. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... This article or section should be merged with Fluid mechanics Fluid dynamics is the study of fluids (liquids and gases) in motion, and the effect of the fluid motion on fluid boundaries, such as solid containers or other fluids. ... Elasticity has meanings in two different fields: In physics and mechanical engineering, the theory of elasticity describes how a solid object moves and deforms in response to external stress. ... Process (lat. ... Space has been an interest for philosophers and scientists for much of human history. ... A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ...

Contents

Introduction

A very simple partial differential equation is

frac{partial u(x,y)}{partial x}=0,

This relation implies that the values u(x,y) are actually independent of x. Hence the general solution of this equation is In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... 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. ...

u(x,y) = f(y),,

where f is an arbitrary function of y. The analogous ordinary differential equation is In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

frac{du}{dx}=0,

which has the solution

u(x) = c,,

where c is any constant value (independent of x). These two examples illustrate that general solutions of ordinary differential equations involve arbitrary constants, but solutions of partial differential equations involve arbitrary functions. A solution of a partial differential equation 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 u is specified on the line x = 0. In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...


Existence and uniqueness

Although the issue of the existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard-Lindelöf theorem, that is far from the case for partial differential equations. There is a general theorem (the Cauchy-Kovalevskaya theorem) that states that the Cauchy problem for any partial differential equation that is analytic in the unknown function and its derivatives has a 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. In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ... In mathematics, the Cauchy-Kovalevskaya theorem is the main existence and uniqueness theorem for analytic partial differential equations. ... Consider a smooth hypersurface having a continuous, non-tangential direction field described by unitary vectors , i. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...


An example of pathological behavior is the sequence of Cauchy problems (depending upon n) for the Laplace equation Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...

frac{part^2 u}{partial x^2} + frac{part^2 u}{partial y^2}=0,,

with initial conditions In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

u(x,0) = 0, ,
frac{partial u}{partial y}(x,0) = frac{sin n x}{n},,

where n is an integer. The derivative of u with respect to y approaches 0 uniformly in x as n increases, but the solution is

u(x,y) = frac{(sinh ny)(sin nx)}{n^2}.,

This solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y. The Cauchy problem for the Laplace equation is called ill-posed or 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. The mathematical term well-posed problem stems from a definition given by Hadamard. ...


Notation and examples

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

u_x = {partial u over partial x}
u_{xy} = {part^2 u over partial y, partial x} = {partial over partial y } left({partial u over partial x}right).

Especially in (mathematical) physics, one often prefers use of the nabla operator (which in cartesian coordinates is written nabla=(part_x,part_y,part_z), for spatial derivatives and a dot dot u, for time derivatives, e.g. to write the wave equation (see below) as In vector calculus, del is a vector differential operator represented by the symbol ∇. This symbol is sometimes called the nabla operator, after the Greek word for a kind of harp with a similar shape (with related words in Aramaic and Hebrew). ... 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. ...

ddot u=c^2triangle u.,(math notation)
ddot u=c^2nabla^2u.,(physics notation)

Heat equation in one space dimension

The equation for conduction of heat in one dimension has the form

u_t = alpha u_{xx} ,

where u(t,x) is temperature, and α 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.


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 f and Fourier Transforms for non-periodic f. Using the Fourier transform, a general solution of the heat equation has the form In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... 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. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

u(t,x) = frac{1}{sqrt{2pi}} int_{-infty}^{infty} F(&# 0; e^{-alpha &# 0;2 t} e^{i &# 0;x} d&# 0; ,

where F is an arbitrary function. In order to satisfy the initial condition, F is given by the (the Fourier transform of f), that is

F(&# 0; = frac{1}{sqrt{2pi}} int_{-infty}^{infty} f(x) e^{-i &# 0;x}, dx. ,

If f represents 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 The Dirac delta function, sometimes 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 δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

F(&# 0; = frac{1}{sqrt{2pi}}, ,

and the resulting solution of the heat equation is

u(t,x) = frac{1}{2pi} int_{-infty}^{infty}e^{-alpha &# 0;2 t} e^{i &# 0;x} d&# 0; ,

This is a Gaussian integral. It may be evaluated to obtain The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

u(t,x) = frac{1}{2sqrt{pi alpha t}} expleft(-frac{x^2}{4 alpha t} right). ,

This result corresponds to a normal probability density for x with mean 0 and variance 2αt. The heat equation and similar diffusion equations are useful tools to study random phenomena. The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...


Wave equation in one spatial dimension

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

u_{tt} = c^2 u_{xx}. ,

Here u might 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 c is 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:

u(0,x) = f(x), ,
u_t(0,x) = g(x), ,

where f and g are arbitrary given functions. The solution of this problem is given by d'Alembert's formula: In mathematics, and specifically partial differential equations, d´Alemberts formula is the general solution to the one-dimensional wave equation: . It is named after the mathematician Jean le Rond dAlembert. ...

u(t,x) = frac{1}{2} left[f(x-ct) + f(x+ct)right] + frac{1}{2c}int_{x-ct}^{x+ct} g(y), dy. ,

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 In mathematics, the method of characteristics is a technique for solving partial differential equations. ...

x - ct = hbox{constant,} quad x + ct = hbox{constant}, ,

that are drawn backwards from that point. These curves correspond to signals that propagate with velocity c forward 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 is negative, 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. Look up point source in Wiktionary, the free dictionary. ...


Spherical waves

Spherical waves are waves whose amplitude depends only upon the radial distance r from a central point source. For such waves, the three-dimensional wave equation takes the form Look up point source in Wiktionary, the free dictionary. ...

u_{tt} = c^2 left[u_{rr} + frac{2}{r} u_r right]. ,

This is equivalent to

(ru)_{tt} = c^2 left[(ru)_{rr} right],,

and hence the quantity ru satisfies the one-dimensional wave equation. Therefore a general solution for spherical waves has the form

u(t,r) = frac{1}{r} left[F(r-ct) + G(r+ct) right],,

where F and G are completely arbitrary functions. Radiation from an antenna corresponds to the case where G is 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 φ has the form In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ...

varphi_{xx} + varphi_{yy} = 0

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 → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...


Connection with analytic functions

Solutions of the Laplace equation are intimately connected with analytic functions of a complex variable: the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are orthogonal. If f=u+iv, then the Cauchy-Riemann equations state that 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,,

and it follows that

u_{xx} + u_{yy} = 0, quad v_{xx} + v_{yy}=0. ,

Conversely, given any harmonic function, it is the real part of an analytic function, at least locally. Details are given in Laplace equation. Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...


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(θ) on a circle of radius one. The solution was given by Poisson: Simeon Poisson. ...

varphi(r,theta) = frac{1}{2pi} int_0^{2pi} frac{1-r^2}{1 +r^2 -2rcos (theta -theta')} u(theta')dtheta'.,

Petrovsky (1967, p. 248) shows how this formula can be obtained by summing a Fourier series for φ. If r<1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if u is continuous but not necessarily differentiable. This behavior is typical for solutions of elliptic partial differential equations: the solutions may be much more smooth than the boundary data. This is in contrast to solutions of the wave equation, and more general hyperbolic partial differential equations, which typically have no more derivatives than the data. 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. ... 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. ... A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ...


Euler-Tricomi equation

The Euler-Tricomi equation is used in the investigation of transonic flow. It is In mathematics, the Euler-Tricomi equation is a nonlinear partial differential equation useful in the study of transonic flow. ... Transonic is an aeronautics term referring to a range of velocities just below and above the speed of sound. ...

u_{xx} , =xu_{yy}

Advection equation

The advection equation describes the transport of a conserved scalar ψ in a velocity field {bold u}=(u,v,w). It is: The advection equation is the partial differential equation that governs the motion of a conserved scalar as it is advected by a known velocity field. ...

psi_t+(upsi)_x+(vpsi)_y+(wpsi)_z , =0.

If the velocity field is solenoidal (that is, nablacdot{bold u}=0), then the equation may be simplified to This article is in need of attention. ...

psi_t+upsi_x+vpsi_y+wpsi_z , =0.

The one dimensional steady flow advection equation ψt + ux = 0 (where u is constant) is commonly referred to as the pigpen problem. If u is not constant and equal to ψ the equation is referred to as Burgers' equation. Burgers equation is a fundamental partial differential equation from fluid mechanics. ...


Ginzburg-Landau equation

The Ginzburg-Landau equation is used in modelling superconductivity. It is In physics, Ginzburg-Landau theory is a mathematical theory used to model superconductivity. ... A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. ...

iu_t+pu_{xx} +q|u|^2u , =igamma u

where p,qinmathbb{C} and gammainmathbb{R} are constants and i is the imaginary unit.


The Dym equation

The Dym equation is named for Harry Dym and occurs in the study of solitons. It is In mathematics, and in particular in the theory of solitons, the Dym equation (also known as DH) is named for Harry Dym. ... Professor Harry Dym is a mathematician at the Weizmann Institute of Science, Israel. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ...

u_t , = u^3u_{xxx}.

Other examples

The Schrödinger equation is a PDE at the heart of non-relativistic quantum mechanics. In the WKB approximation it is the Hamilton-Jacobi equation. In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. ... Fig. ... In physics, the WKB (Wentzel-Kramers-Brillouin) approximation, also known as WKBJ approximation, is the most familiar example of a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be slowly... The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ...


Except for the Dym equation and the Ginzburg-Landau equation, the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier-Stokes equations describing the flow of fluids, and Einstein's field equations of general relativity. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ... Albert Einstein ( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is best known for his theory of relativity and specifically mass-energy equivalence, . He was awarded the 1921 Nobel Prize in Physics for his services to Theoretical Physics, and especially for his discovery of the... For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...


Methods to solve PDEs

The method of separation of variables will yield particular solutions of a linear PDE on very simple domains such as rectangles that may satisfy initial or boundary conditions. Because any superposition of solutions of a linear PDE is again a solution, the particular solutions may then be combined to obtain more general solutions. If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example for use of a Fourier integral. In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ... The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ... 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. ... The Fourier transform, named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...


Initial-boundary value problems

Many problems of Mathematical Physics are formulated as initial-boundary value problems. Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ...


Vibrating string

If the string is stretched between two points where x=0 and x=L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0<x<L and t is unlimited. Since the string is tied down at the ends, u must also satisfy the boundary conditions

u(t,0)=0, quad u(t,L)=0, ,

as well as the initial conditions

u(0,x)=f(x), quad u_t(0,x)=g(x). ,

The method of separation of variables for the wave equation

u_{tt} = c^2 u_{xx}, ,

leads to solutions of the form

u(t,x) = T(t) X(x),,

where

T'' + k^2 c^2 T=0, quad X'' + k^2 X=0,,

where the constant k must be determined. The boundary conditions then imply that X is a multiple of sin kx, and k must have the form

k= frac{npi}{L}, ,

where n is an integer. Each term in the sum corresponds to a mode of vibration of the string. The mode with n=1 is called the fundamental mode, and the frequencies of the other modes are all multiples of this frequency. They form the overtone series of the string, and they are the basis for musical acoustics. The initial conditions may then be satisfied by representing f and g as infinite sums of these modes. Wind instruments typically correspond to vibrations of an air column with one end open and one end closed. The corresponding boundary conditions are

X(0) =0, quad X'(L) = 0.,

The method of separation of variables can also be applied in this case, and it leads to a series of odd overtones.


The general problem of this type is solved in Sturm-Liouville theory. In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...


Vibrating membrane

If a membrane is stretched over a curve C that forms the boundary of a domain D in the plane, its vibrations are governed by the wave equation

frac{1}{c^2} u_{tt} = u_{xx} + u_{yy}, ,

if t>0 and (x,y) is in D. The boundary condition is u(t,x,y) = 0 if (x,y) is on C. The method of separation of variables leads to the form

u(t,x,y) = T(t) v(x,y),,

which in turn must satisfy

frac{1}{c^2}T'' +k^2 T=0, ,
v_{xx} + v_{yy} + k^2 v =0.,

The latter equation is called the Helmholtz Equation. The constant k must be determined in order to allow a non-trivial v to satisfy the boundary condition on C. Such values of k2 are called the eigenvalues of the Laplacian in D, and the associated solutions are the eigenfunctions of the Laplacian in D. The Sturm-Liouville theory may be extended to this elliptic eigenvalue problem (Jost, 2002). The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ...


There are no generally applicable methods to solve non-linear PDEs. Still, existence and uniqueness results (such as the Cauchy-Kovalevskaya theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the Split-step method, exist for specific equations like Non-Linear Schrodinger equation. In mathematics, the Cauchy-Kovalevskaya theorem is the main existence and uniqueness theorem for analytic partial differential equations. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In numerical analysis, the split-step (Fourier) method is a general computational technique used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. ...


Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier-Janet theory is an effective method for obtaining information about many analytic overdetermined systems. This article or section is in need of attention from an expert on the subject. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ... In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. ...


The method of characteristics (Similarity Transformation method) can be used in some very special cases to solve partial differential equations. In mathematics, the method of characteristics is a technique for solving partial differential equations. ...


In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers. Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... A finite difference is a mathematical expression of the form f(x + b) − f(x +a). ... It has been suggested that this article or section be merged with Multigrid method. ... Mathematically, the finite element method (FEM) is used for finding approximate solution of partial differential equations (PDE) as well as of integral equations such as the heat transport equation. ... A BlueGene supercomputer cabinet. ... A supercomputer is a computer that leads the world in terms of processing capacity, particularly speed of calculation, at the time of its introduction. ...


Classification

Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic or elliptic. Others such as the Euler-Tricomi equation have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions, and to smoothness of the solutions. A parabolic partial differential equation is a second-order partial differential equation of the form in which the matrix has the determinant equal to 0. ... A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ... 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, the Euler-Tricomi equation is a nonlinear partial differential equation useful in the study of transonic flow. ...


Equations of second order

Assuming uxy = uyx, the general second-order PDE in two independent variables has the form

Au_{xx} + Bu_{xy} + Cu_{yy} + cdots = 0,

where the coefficients A, B, C etc. may depend upon x and y. This form is analogous to the equation for a conic section:

Ax^2 + Bxy + Cy^2 + cdots = 0.

Just as one classifies conic sections into parabolic, hyperbolic, and elliptic based on the discriminant B2 − 4AC, the same can be done for a second-order PDE at a given point. Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ...

  1. B^2 - 4AC , < 0 : solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler-Tricomi equation is elliptic where x<0.
  2. B^2 - 4AC = 0, : equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler-Tricomi equation has parabolic type on the line where x=0.
  3. B^2 - 4AC , > 0 : hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler-Tricomi equation is hyperbolic where x>0.

If there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form 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. ... A parabolic partial differential equation is a second-order partial differential equation of the form in which the matrix has the determinant equal to 0. ... A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ... 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. ...

L u =sum_{i=1}^nsum_{j=1}^n a_{i,j} frac{part^2 u}{partial x_i partial x_j} quad hbox{ plus lower order terms} =0. ,

The classification depends upon the signature of the eigenvalues of the coefficient matrix.

  1. Elliptic: The eigenvalues are all positive or all negative.
  2. Parabolic : The eigenvalues are all positive or all negative, save one which is zero.
  3. Hyperbolic: There is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
  4. Ultrahyperbolic: There is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).

Systems of first-order equations and characteristic surfaces

The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices Aν are m by m matrices for nu=1, dots,n. The partial differential equation takes the form

Lu = sum_{nu=1}^{n} A_nu frac{partial u}{partial x_nu} + B=0, ,

where the coefficient matrices Aν and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form

varphi(x_1, x_2, ldots, x_n)=0, ,

where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes:

Qleft(frac{partvarphi}{partial x_1}, ldots,frac{partvarphi}{partial x_n}right) =detleft[sum_{nu=1}^nA_nu frac{partial varphi}{partial x_nu}right]=0.,

The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation do not determine the normal derivative of u on S, then the surface is characteristic, and the differential equation restricts the data on S: the differential equation is internal to S.

  1. A first-order system Lu=0 is elliptic if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.
  2. A first-order system is hyperbolic at a point if there is a space-like surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation
Q(lambda &# 0;+ eta) =0, ,

has m real roots λ1, λ2, ..., λm. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form Q(ζ)=0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has m sheets, and the axis ζ = λ ξ runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.


Equations of mixed type

If a PDE has coefficients which are not constant, it is possible that it will not belong to any of these categories but rather be of mixed type. A simple but important example is the Euler-Tricomi equation

u_{xx} , = xu_{yy}

which is called elliptic-hyperbolic because it is elliptic in the region x < 0, hyperbolic in the region x > 0, and degenerate parabolic on the line x = 0.


External links

Wikibooks
Wikibooks has a book on the topic of
Partial Differential Equations
  • Partial Differential Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
  • Partial Differential Equations: Index at EqWorld: The World of Mathematical Equations.
  • Partial Differential Equations: Methods at EqWorld: The World of Mathematical Equations.
  • Example problems with solutions at exampleproblems.com
  • Partial Differential Equations at mathworld.wolfram.com
  • Dispersive PDE Wiki

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ...

References

Wikimedia Commons has media related to:
category:Solutions of PDE
  • R. Courant and D. Hilbert, Methods of Mathematical Physics, vol II. Wiley-Interscience, New York, 1962.
  • L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
  • J. Jost, Partial Differential Equations, Springer-Verlag, New York, 2002.
  • Hans Lewy (1957) An example of a smooth linear partial differential equation without solution. Annals of Mathematics, 2nd Series, 66(1),155-158.
  • I.G. Petrovskii, 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. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. ISBN 1-58488-355-3
  • A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
  • D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
  • Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005. ISBN 978-0-521-84886-2

  Results from FactBites:
 
Partial differential equation - Wikipedia, the free encyclopedia (2676 words)
Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space, or distributed in space and time.
The Schrödinger equation is a PDE at the heart of non-relativistic quantum mechanics.
In the WKB approximation it is the Hamilton-Jacobi equation.
  More results at FactBites »

 
 

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