Revision 2243812 of "Persamaan pembezaan" on mswiki

{{terjemah|en|Differential equation}}
{{distinguish|persamaan beza}}

'''Persamaan pembezaan''' atau '''persamaan kebezaan''' ialah suatu [[persamaan]] [[matematik]] untuk suatu [[fungsi (matematik)|fungsi]] yang tidak dikenali dari satu atau beberapa [[pembolehubah (matematik)|pembolehubah]] yang mengaitkan nilai-nilai fungsi itu sendiri dan [[terbitan (matematik)|terbitannya]] dari pelbagai tertib. Persamaan pembezaan memainkan suatu peranan ketara pada [[kejuruteraan]], [[fizik]], [[ekonomi]], dan disiplin lain.

[[Fail:Airflow-Obstructed-Duct.png|thumb|250px|Pembayangan aliran udara ke dalam sebuah [[model matematik|model]] saluran menggunakan menggunakan [[persamaan Navier-Stokes]], suatu set persamaan pembezaan separa]]

Persamaan kebezaan timbul dalam pelbagai bidang sains dan teknologi: apabila hubungan [[sistem berketentuan (matematik)|berketentuan]] yang melibatkan kuantiti yang berubah-ubah tanpa henti (dimodelkan oleh fungsi) dan kadar perubahannya dalam angkasa dan/atau masa (dinyatakan sebagai terbitan) diketahui atau diharapkan. Ini digambarkan dalam ilmu [[mekanik klasik]], yang mana pergerakan jasad diterangkan melalui kedudukan dan halajunya ketika masa berubah-ubah. [[Hukum Newton]] membolehkan perhubungan kedudukan, halaju, pecutan dan berbagai-bagai daya yang bertindak dalam jasad serta penyataan hubungan ini sebagai persamaan kebezaan kedudukan jasad yang tidak diketahui sebagai fungsi masa. Adakalanya, persamaan kebezaan ini (dipanggil [[persamaan gerakan]]) boleh diselesaikan secara tidak tersirat.

Contoh masalah yang melibatkan persamaan kebezaan adalah menentukan halaju sebiji bola yang jatuh melalui udara, dengan hanya mengambil kira graviti dan rintangan udara. Pecutan bola ke arah tanah ialah pecutan yang terhasil dari graviti tolak nyahpecutan akibat rintangan udara. Graviti adalah malar tetapi rintangan bolanya boleh dimodelkan sebagai berkadar dengan halaju bola. Ertinya, pecutan bola tersebut sebagai terbitan halajunya, bergantung pada halajunya. Pencarian halaju sebagai fungsi masa melibatkan penyelesaian persamaan kebezaan.

Persamaan kebezaan dikaji secara matematik dari pelbagai perspektif, lazimnya berkenaan dengan penyelesaiannya, iaitu set fungsi yang memuaskan persamaan itu. Hanya persamaan kebezaan yang teringkas membenarkan penyelesaian yang diberi oleh formula-formula yang tak tersirat; itupun, sesetengah ciri-ciri penyelesaian persamaan kebezaan tertentu boleh ditentukan tanpa mencari bentuk tepatnya. Seandainya tiada formula serba lengkap untuk penyelesaiannya, maka penyelesaiannya boleh dianggarkan angkanya dengan menggunakan komputer. Teori [[sistem dinamik]] menekankan analisa sistem secara kualitatif yang ditetapkan oleh persamaan kebezaan, manakala banyak [[kaedah berangka]] telah dimajukan untuk menentukan penyelesaian dengan setepat-setepatnya.

== Arah kajian ==
<!--
The study of differential equations is a wide field in [[pure mathematics|pure]] and [[applied mathematics]], [[physics]], [[meteorology]], and [[engineering]]. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have [[closed-form expression|closed form]] solutions. Instead, solutions can be approximated using [[Numerical ordinary differential equations|numerical methods]].

Mathematicians also study [[weak solution]]s (relying on [[weak derivative]]s), 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]].

== Tatanama ==
The theory of differential equations is quite developed and the methods 
used to study them vary significantly with the type of the equation.

* An [[ordinary differential equation]] (ODE) is a differential equation in which the unknown function (also known as the '''dependent variable''') is a function of a ''single'' independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be [[vector-valued function|vector-valued]] or [[matrix (mathematics)|matrix]]-valued: this corresponds to considering a system of ordinary differential equations for a single function. Ordinary differential equations are further classified according to the '''order''' of the highest derivative with respect to the dependent variable appearing in the equation. The most important cases for applications are first order and second order differential equations. In the classical literature also distinction is made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form.

* A [[partial differential equation]] (PDE)  is a differential equation in which the unknown function is a function of ''multiple''  independent variables and the equation involves its [[partial derivatives]]. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of '''mixed type'''.

Both ordinary and partial differential equations are broadly classified as '''linear''' and '''nonlinear'''. A differential equation is '''linear''' if the unknown function and its derivatives appear to the power 1 (products are not allowed) and '''nonlinear''' otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed  theory of linear differential equations. '''Homogeneous''' linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a '''constant coefficient linear differential equation'''. 

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 theory|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 [[linearization|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).

=== Contoh ===
In the first group of examples, let ''u'' be an unknown function of ''x'', and ''c'' and ''ω'' are known constants.  

* Inhomogeneous first order linear constant coefficient ordinary differential equation:

: <math> \frac{du}{dx} = cu+x^2. </math>

* Homogeneous second order linear ordinary differential equation: 

:<math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>

* Homogeneous second order constant coefficient linear ordinary differential equation describing the [[harmonic oscillator]]:

: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>

* First order nonlinear ordinary differential equation:

: <math> \frac{du}{dx} = u^2 + 1. </math>

* Second order nonlinear ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':

: <math> g\frac{d^2u}{dx^2} + L\sin u = 0. </math>

In the next group of examples, the unknown function ''u'' depends on two variables ''x'' and ''t'' or ''x'' and ''y''. 

* Homogeneous first order linear partial differential equation:

: <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math>

* Homogeneous second order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:

: <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math>

* Third order nonlinear partial differential equation, the [[Korteweg–de Vries equation]]:

: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>

== Konsep berkaitan ==
* A [[delay differential equation]] (DDE) is an equation for a function of a single variable, usually called '''time''', 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.

== Universality 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 [[mathematical modelling|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.

==Notable differential equations==
<div style="-moz-column-count:2; column-count:2;">
* [[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 equation]]
* [[Maxwell's equations]] in [[electromagnetism]]
* The [[heat equation]] in [[thermodynamics]]
* [[Laplace's equation]], which defines [[harmonic function]]s
* [[Poisson's equation]]
* [[Einstein's field equation]] in [[general relativity]]
* The [[Schrödinger equation]] in [[quantum mechanics]]
* The [[geodesic#(pseudo-)Riemannian geometry|geodesic 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]]
* The [[Poisson–Boltzmann equation]] in [[molecular dynamics]]
* The [[shallow water equations]]
* [[Universal differential equation]]
</div>

===Biologi===
*[[Verhulst equation]] – biological population growth
*[[Von_Bertalanffy#The_individual_growth_model|von Bertalanffy model]] – biological individual growth
*[[Lotka–Volterra equations]] – biological population dynamics
*[[Replicator dynamics]] – may be found in theoretical biology

===Ekonomi===
*[[Black–Scholes#The_Black–Scholes_PDE|The Black–Scholes PDE]]
*[[Exogenous growth model]]
*[[Logistic function|Verhulst's population model]]
*[[Malthusian growth model]]
*[[Sethi model|The Vidale-Wolfe advertising model]]

== Lihat juga ==
{{wikibooks|Differential Equations}}
*[[Complex differential equation]]
*[[Exact differential equation]]
*[[Integral equations]]
*[[Linear differential equation]]
*[[Picard-Lindelöf theorem]] on existence and uniqueness of solutions
-->{{more}}
== Rujukan ==
* 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]
* E.L. Ince, ''Ordinary Differential Equations'', Dover Publications, 1956
* E.A. Coddington and N. Levinson,  ''Theory of Ordinary Differential Equations'', McGraw-Hill, 1955
* P. Blanchard, R.L. Devaney, G.R. Hall, ''Differential Equations'', Thompson, 2006

== Pautan luar ==
* [http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.htm Lectures on Differential Equations] [[MIT]] Open CourseWare Video
* [http://tutorial.math.lamar.edu/classes/de/de.aspx Online Notes / Differential Equations] Paul Dawkins, [[Lamar University]]
* [http://www.sosmath.com/diffeq/diffeq.html Differential Equations], [[S.O.S. Mathematics]]
* [http://www.diptem.unige.it/patrone/differential_equations_intro.htm Introduction to modeling via differential equations] Introduction to modeling by means of differential equations, with critical remarks.
* [http://publicliterature.org/tools/differential_equation_solver/ Differential Equation Solver] Java applet tool used to solve differential equations.
* [http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode Mathematical Assistant on Web] Symbolic ODE tool, using [[Maxima (software)|Maxima]] 
* [http://eqworld.ipmnet.ru/en/solutions/ode.htm Exact Solutions of Ordinary Differential Equations]
* [http://www.maa.org/bll2/DIFFERENTIAL.htm A bibliography of books about differential equations], from the [[Mathematical Association of America]] 
* [http://www.hedengren.net/research/models.htm Collection of ODE and DAE models of physical systems] MATLAB models
* [http://www.jirka.org/diffyqs/ Notes on Diffy Qs: Differential Equations for Engineers] An introductory textbook on differential equations by Jiri Lebl of [[UIUC]]
{{Mathematics-footer}}

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