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8/3/2019 Differential Equation - Wikipedia, The Free Encyclopedia
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Visualization of heat transfer in a pump casing, created bysolving the heat equation. Heat is being generated internally in
the casing and being cooled at the boundary, providing a
steady state temperature distribution.
Diffeenial eaionFrom Wikipedia, the free encyclopedia
A diffeenial eaion is a mathematical equation for an
unknown function of one or several variables that relates the
values of the function itself and its derivatives of various
orders. Differential equations play a prominent role in
engineering, physics, economics, and other disciplines.
Differential equations arise in many areas of science and
technology, specifically whenever a deterministic relation
involving some continuously varying quantities (modeled by
functions) and their rates of change in space and/or time
(expressed as derivatives) is known or postulated. This is
illustrated in classical mechanics, where the motion of a body
is described by its position and velocity as the time value
varies. Newton's laws allow one (given the position, velocity,
acceleration and various forces acting on the body) to express
these variables dynamically as a differential equation for the
unknown position of the body as a function of time. In somecases, this differential equation (called an equation of motion)
may be solved explicitly.
An example of modelling a real world problem using
differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air
resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air
resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This
means that the ball's acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a
function of time involves solving a differential equation.
Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas;
however, some 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.
Conen
1 Directions of study2 Nomenclature
2.1 Classification summary
2.1.1 Ordinary DE classification
2.2 Examples
3 Related concepts
4 Connection to difference equations
5 Universality of mathematical description
6 Exact solutions
7 Notable differential equations
7.1 Physics and engineering
7.2 Biology
7.3 Economics
8 See also
9 References
10 External links
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Directions of stud
T , , , . A
. P
,
. D , ,
, , , . D
- , .. . I,
.
M ( ),
. T ,
, .
T .
omenclatureT
.
A (ODE) (
dependent variable) single . I ,
, , - -:
.
O order
. T first-order second-order differential equations. F , B'
( ) - . I
.
A (PDE) multiple . T
, , , ,
- , . S
mied
tpe.
B linear nonlinear. A linear
1 ( ) nonlinear . T
,
.Homogeneous
.. . T
() ;
constant coefficient linear differential equation.
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There are very few methods of explicitly solving nonlinear differential equations; those that are known typically depend on the
equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended
time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions
for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard
problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier
Stokes existence and smoothness).
Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid
under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum
equation that is valid for small amplitude oscillations (see below).
Classification summar
The mathematical definitions for the various classifications of differential equation can be collected as follows.
Ordinar DE classification
See main article: Ordinar differential equation.
In the table below,
All differential equations are of order n and arbitrary degree d .
F is an implicit function of: an independent variable , a dependent variable (a function of ), and integer derivatives of
(fractional derivatives are in fact possible, but not considered here).
may in general be a vector valued function:
so is an element of the vector space R , an element of a vector space of dimensionm, , where R is
the set of real numbers, is the cartesian product of R with itself m times to form an m-tuple
of real numbers.
This leads to a system of differential equations to be solved for 1, 2,... m.
is characterized by the function mapping .
r( ) is called a source term in , and A( ) is an arbitrary function, both assumed continuous in on defined intervals.
Characteristic Properties Differential equation
Implicit system of
dimension m
Explicit system of
dimension m
Autonomous: No
dependence
Linear: nth derivative
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ca be ie
a a linear
combination f he
he
deiaie
Homogeneous:Sce e
i e
Nice he aig f ced he a f x, , ad he n (n-1) deiaie f
he i, i geea iici.
Eamples
I he fi g f eae, e u be a fci f x, ad c ad ae ca.
Ihgee fi-de iea ca cefficie dia diffeeia eai:
Hgee ecd-de iea dia diffeeia eai:
Hgee ecd-de iea ca cefficie dia diffeeia eai decibig he haic
cia:
Fi-de iea dia diffeeia eai:
Secd-de iea dia diffeeia eai decibig he i f a ed f egh L:
I he e g f eae, he fciu deed aiabe x ad t x ad y.
Hgee fi-de iea aia diffeeia eai:
Hgee ecd-de iea ca cefficie aia diffeeia eai f eiic e, he Laace
eai:
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Third-order nonlinear partial differential equation, the Kortewegde Vries equation:
Related concepts
A delay differential equation (DDE) is an equation for a function of a single variable, usually calledtime, in which
the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and
the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion
equations.
A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in
implicit form.
Connection to difference equations
See also: Time scale calculus
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.
Universalit of mathematical description
Many 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, the theory
of which was 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.
Eact solutions
Some differential equations have solutions which can be written in an exact and closed form. Several important classes are
given here.
In the table below, H ( ), Z ( ), H ( ), Z ( ), or H ( , ), Z ( , ) are any integrable functions of or (or both), and A, B, C , I ,
L, N , M are all constants. In general A, B, C , I , L, are real numbers, but N , M , P and Q may be complex. The differentialequations are in their equivalent and alternative forms which lead to the solution through integration.
Differential equation General solution
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hee
olion ma be implici in and , obained b calclaing aboe inegal hen
ing
back-biing
If DE i eac o ha
hen he olion i:
hee and ae conan fncion fom he inegal ahe han
conan ale, hich ae
e o make he final fncion hold.
If DE i no eac, he fncion H ( , ) o Z ( , ) ma be ed o obain an
inegaing faco, he DE i
hen mliplied hogh b ha faco and he olion poceed he ame a.
If
hen
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8
If
then
If
then
9
where are the solutions of the polynomial of degree :
Note that 3 and 4 are special cases of 7, they are relatively common cases and included for completeness.
Similarly 8 is a special case of 9, but 8 is a relatively common form, particularly in simple physical and engineering problems.
otable differential equations
Phsics and engineering
Newton's Second Law in dynamics (mechanics)
Hamilton's equations in classical mechanics
Radioactive decay in nuclear physics
Newton's law of cooling in thermodynamics
The wave equationMaxwell's equations in electromagnetism
The heat equation in thermodynamics
Laplace's equation, which defines harmonic functions
Poisson's equation
Einstein's field equation in general relativity
The Schrödinger equation in quantum mechanics
The geodesic equation
The NavierStokes equations in fluid dynamics
The CauchyRiemann equations in complex analysis
The PoissonBoltzmann equation in molecular dynamics
The shallow water equations
Universal differential equation
The Lorenz equations whose solutions exhibit chaotic flow.
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Biolog
Verhulst equation biological population growth
von Bertalanffy model biological individual growth
LotkaVolterra equations biological population dynamics
Replicator dynamics may be found in theoretical biology
Hodgkin-Huxley model - neural action potentials
Economics
The BlackScholes PDE
Exogenous growth model
Malthusian growth model
The Vidale-Wolfe advertising model
See also
Complex differential equation
Exact differential equationIntegral equations
Linear differential equation
PicardLindelöf theorem on existence and uniqueness of solutions
Numerical methods
References
D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
A. D. Polyanin and V. F. Zaitsev, Handbook of Eact Solutions for Ordinar Differential Equations (2nd
edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.W. Johnson, A Treatise on Ordinar and Partial Differential Equations
(http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001) , John Wiley and Sons, 1913, in University
of Michigan Historical Math Collection (http://hti.umich.edu/u/umhistmath/)
E.L. Ince, Ordinar Differential Equations, Dover Publications, 1956
E.A. Coddington and N. Levinson, Theor of Ordinar Differential Equations, McGraw-Hill, 1955
P. Blanchard, R.L. Devaney, G.R. Hall, Differential Equations , Thompson, 2006
Calculus, Teach Yourself, P.Abbott and H. Neill, 2003 pages 266-277
Further Elementary Analysis, R.I.Porter, 1978, chapter XIX Differential Equations</ref>
Eternal linksLectures on Differential Equations (http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-
2010/video-lectures/) MIT Open CourseWare Videos
Online Notes / Differential Equations (http://tutorial.math.lamar.edu/classes/de/de.aspx) Paul Dawkins, Lamar
University
Differential Equations (http://www.sosmath.com/diffeq/diffeq.html) , S.O.S. Mathematics
Differential Equation Solver (http://publicliterature.org/tools/differential_equation_solver/) Java applet tool used to
solve differential equations.
Mathematical Assistant on Web (http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode) Symbolic
ODE tool, using Maxima
Exact Solutions of Ordinary Differential Equations (http://eqworld.ipmnet.ru/en/solutions/ode.htm)Collection of ODE and DAE models of physical systems (http://www.hedengren.net/research/models.htm)
MATLAB models
Notes on Diffy Qs: Differential Equations for Engineers (http://www.jirka.org/diffyqs/) An introductory textbook on
differential equations by Jiri Lebl of UIUC
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Categories: Differential equations
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