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Differential equation
*** Shopping-Tip: Differential equation
Image:Differential.png thumb|right|Graph of a differential equation
In
mathematics, a '''differential equation''' is an
equation in which the
derivatives of a
function (mathematics) function appear as variables. Many of the fundamental laws of physics, chemistry, biology and economics can be formulated as differential equations. The mathematical theory of differential equations has developed together with the sciences where the equations originate and where the results find application. Diverse scientific fields often give rise to identical problems in differential equations. In such cases, the mathematical theory can unify otherwise quite distinct scientific fields. A celebrated example is
Joseph Fourier Fourier's theory of the conduction of heat in terms of sums of trigonometric functions
Fourier series, which finds application in the propagation of sound, the propagation of electric and magnetic fields, radio waves, optics, elasticity, spectral analysis of radiation, and other scientific fields.
The '''order''' of a differential equation is that of the highest derivative that it contains. For instance, a first-order differential equation contains only first derivatives.
Types of differential equations
* An
ordinary differential equation (ODE) only contains functions of one independent variable, and derivatives in that variable.
* A
partial differential equation (PDE) contains functions of multiple independent variables and their
partial derivatives.
* A
delay differential equation (DDE) contains functions of one dependent variable, derivatives in that variable, and depends on previous states of the dependent 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.
Each of those categories is divided into linear and nonlinear subcategories. A differential equation is ''linear'' if it involves the unknown function and its derivatives only to the first power; otherwise the differential equation is ''nonlinear''. Thus if
denotes the first derivative of ''u'', 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.
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.
The study of differential equations is a wide field in both
pure mathematics 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, physicists and engineers are usually more interested in how to compute solutions to differential equations. These solutions are then used to simulate celestial motions, design bridges, automobiles, aircraft, sewers, etc. Often, these equations do not have
closed-form expression closed form solutions and are solved using
numerical methods.
Famous differential equations
*
Maxwell's equations in
electromagnetism
*
Einstein's field equation in
general relativity
* The
Schrödinger equation in
quantum mechanics
* The
heat equation in
thermodynamics
* The
wave equation
* The
geodesic#(pseudo-)Riemannian geometry geodesic equation
*
Laplace's equation, which defines
harmonic functions
*
Poisson's equation
* The
Navier-Stokes equations in
fluid dynamics
* The
Lotka-Volterra equation in
population dynamics
* The
Black-Scholes#The Black-Scholes PDE Black-Scholes equation in
finance
* The
Cauchy-Riemann equations in
complex analysis
See also
*
Picard–Lindelöf theorem on existence and uniqueness of solutions
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, [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001 ''A Treatise on Ordinary and Partial Differential Equations''], John Wiley and Sons, 1913, in [http://hti.umich.edu/u/umhistmath/ University of Michigan Historical Math Collection]
* Wikibooks, [http://www.wikibooks.org/wiki/Differential_Equations Differential Equations]
External links
-
MIT Open CourseWare video lectures on differential equations
Category:Differential equations *
{{Mathematics-footer}}
af:Differensiaalvergelyking
bg:Диференциално уравнение
ca:Equació diferencial
cs:Diferenciálnà rovnice
da:Differentialligning
de:Differentialgleichung
es:Ecuación diferencial
ee:Diferentiaalvõrrand
fa:معادلات دیÙ?رانسیل
fr:Équation différentielle
ko:ë¯¸ë¶„ë°©ì •ì‹?
it:Equazione differenziale
he:משוו×?×” ×“×™×¤×¨× ×¦×™×?לית
nl:Differentiaalvergelijking
ja:微分方程�
pl:Równanie różniczkowe
pt:Equação diferencial
ro:Ecuaţie diferenţială
fi:Differentiaaliyhtälö
sv:Differentialekvation
th:สมà¸?ารเชิงà¸à¸™à¸¸à¸žà¸±à¸™à¸˜à¹Œ
tr:Diferansiyel denklemler
zh:微分方程
*** Shopping-Tip: Differential equation
