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Encyclopedia > Differential equation
Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations.
Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations.

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics and other disciplines. Image File history File links Size of this preview: 800 × 571 pixelsFull resolution (1270 × 907 pixel, file size: 85 KB, MIME type: image/png) A simulation using the navier-stokes differential equations of the aiflow into a duct at 0. ... Image File history File links Size of this preview: 800 × 571 pixelsFull resolution (1270 × 907 pixel, file size: 85 KB, MIME type: image/png) A simulation using the navier-stokes differential equations of the aiflow into a duct at 0. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... 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. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... For a non-technical overview of the subject, see Calculus. ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ...

Contents

Introduction

Differential equations arise in many areas of science and technology; whenever a deterministic relationship involving some continuously changing quantities (modeled by functions) and their rates of change (expressed as derivatives) is known or postulated. This is well illustrated by classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's Laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In many cases, this differential equation may be solved, yielding the law of motion. In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...


Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. Only the simplest differential equations admit solutions given by explicit formulas. Many properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...


Directions of study

The study of differential equations is a wide field in both pure and applied mathematics. Pure mathematicians study the types and properties of differential equations, such as whether or not solutions exist, and should they exist, whether they are unique. Applied mathematicians emphasize differential equations from applications, and in addition to existence/uniqueness questions, are also concerned with rigorously justifying methods for approximating solutions. Physicists and engineers are usually more interested in computing approximate solutions to differential equations, and are typically less interested in justifications for whether these approximations really are close to the actual solutions. These solutions are then used to simulate celestial motions, simulate neurons, design bridges, automobiles, aircraft, sewers, etc. [1] Often, differential equations arising in applied disciplines do not have closed form solutions and are solved using numerical methods that work well enough for the purposes of analyzing the original problem. Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of well-known operations. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...


Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolic type. 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. ... In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i. ...


The study of the stability of solutions of differential equations is known as stability theory. In mathematics, stability theory deals with the stability of the solutions of differential equations and dynamical systems. ...


Types of differential equations

Each of those categories is divided into linear and nonlinear subcategories. A differential equation is linear if the dependent variable and all its derivatives appear to the power 1 and there are no products or functions of the dependent variable. Otherwise the differential equation is nonlinear. Thus if u' denotes the first derivative of the function u, then the equation 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. ... 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. ... 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. ... In mathematics, delay differential equations are a type of differential equations. ... A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. ... In the mathematics of probability, a stochastic process is a random function. ... Differential algabraic equations (DAEs) are a generalised form of Differential equation. ...

u' = u

is linear, while the equation

u' = u2

is nonlinear. Solutions of a linear equation in which the unknown function or its derivative or derivatives appear in each term (linear homogeneous equations) may be added together or multiplied by an arbitrary constant in order to obtain additional solutions of that equation, but there is no general way to obtain families of solutions of nonlinear equations, except when they exhibit symmetries; see symmetries and invariants. Linear equations frequently appear as approximations to nonlinear equations, and these approximations are only valid under restricted conditions. Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, an invariant is something that does not change under a set of transformations. ...


Another important characteristic of a differential equation is its order, which is the order of the highest derivative (of a dependent variable) appearing in the equation. For instance, a first-order differential equation contains only first derivatives, like both examples above.


Connection to difference equations

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation. See also: Time scale calculus. In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics, time scale calculus is a unification of the theory of difference equations and standard calculus. ...


Universality of mathematical description

A large number of fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, whose theory was brilliantly developed by Joseph Fourier, is governed by another second order partial differential equation, the heat equation. It turned out that many diffusion processes, while seemingly different, are described by the same equation; Black-Scholes equation in finance is for instance, related to the heat equation. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Chemistry (disambiguation). ... Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, knowledge), also referred to as the biological sciences, is the study of living organisms utilizing the scientific method. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Note: The term model is also given a formal meaning in model theory, a part of axiomatic set theory. ... 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. ... 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. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... diffusion (disambiguation). ... The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...


Famous differential equations

Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ... In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ... Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. ... Nuclear physics is the branch of physics concerned with the nucleus of the atom. ... Heat flow along perfectly insulated wire Conduction is the transfer of heat or electric current from one substance to another by direct contact. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... 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. ... For thermodynamic relations, see Maxwell relations. ... 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. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... 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. ... In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ... In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... 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. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. ... Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. ... The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... Finance studies and addresses the ways in which individuals, businesses, and organizations raise, allocate, and use monetary resources over time, taking into account the risks entailed in their projects. ... 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. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... The shallow water equations are a set of equations that describe the flow below a horizontal pressure surface in a fluid. ...

Notes

  1. ^ Indeed, differential equations permeate most of physical engineering disciplines, and much of the study and practice of these engineering disciplines is in fact the dealing with differential equations masked as a particular problem in that discipline.

See also

Wikibooks
Wikibooks has a book on the topic of
  • Picard-Lindelöf theorem on existence and uniqueness of solutions

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. ... 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. ...

References

  • D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
  • A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.
  • W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
  • E.L. Ince, Ordinary Differential Equations, Dover Publications, 1956

External links


  Results from FactBites:
 
partial differential equation - definition of partial differential equation in Encyclopedia (800 words)
Where ordinary differential equations have solutions that are families with each solution characterized by the values of some parameters, for a PDE it is more helpful to think that the parameters are function data (informally put, this means that the set of solutions is much larger).
Partial differential equations are ubiquitous in science, as they describe phenomena such as fluid flow, gravitational fields, and electromagnetic fields.
In the WKB approximation it is the Hamilton-Jacobi equation.
Differential equation - Wikipedia, the free encyclopedia (571 words)
In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables.
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.
The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates.
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