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Differential Equations

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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.

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.

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.

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.

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.

The study of the stability of solutions of differential equations is known as stability theory.

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 denotes the first derivative of the function , then the equation



is linear, while the equation



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.

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.

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.

Famous differential equations

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

References

External links

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Major fields of mathematics
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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equation is a mathematical statement, in symbols, that two things are the same (or equivalent). Equations are written with an equal sign, as in
.


The equation above is an example of an equality: a proposition which states that two constants are equal.
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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variable (IPA pronunciation: [ˈvæɹiəbl]) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression.
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
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Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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Economics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Greek for oikos (house) and nomos (custom or law), hence "rules of the house(hold).
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In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. Deterministic models thus produce the same output for a given starting condition.
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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.
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Newton's laws of motion are three physical laws which provide relationships between the forces acting on a body and the motion of the body, first compiled by Sir Isaac Newton.
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dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and
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Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics).

One of the earliest mathematical writing is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of ,
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Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction and beauty. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterised
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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.
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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 certain "well-known" functions.
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Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics).

One of the earliest mathematical writing is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of ,
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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
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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.e. to lie in the Lebesgue space . See distributions for an even more general definition.
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In mathematics, stability theory deals with the stability of solutions (or sets of solutions) of differential equations and dynamical systems.

Definition

Let (R, X, Φ) be a real dynamical system with R the real numbers, X
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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.
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In mathematics, a partial differential equation (PDE) is a type of differential equation, i. e. a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables.
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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).
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In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
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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.
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A stochastic process, or sometimes random process, is the opposite of a deterministic process (or deterministic system) in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as is the case, for example, for
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In mathematics, differential algebraic equations (DAEs) are a general form of differential equation, given in implicit form. They can be written



where
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Symmetry in common usage generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection.
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Definition

In mathematics, an invariant is something that does not change under a set of transformations. The property of being an invariant is invariance.
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In mathematics, a recurrence relation is an equation that defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. A difference equation is a specific type of recurrence relation.
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