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= | {{Distinguish|Difference equation}} | ||
{{More footnotes|date=August 2009}} | |||
{{Differential equations}} | |||
[[File:Elmer-pump-heatequation.png|thumb|350px|Visualization of heat transfer in a pump casing, created by solving the [[heat equation]]. [[Heat]] is being generated internally in the casing and being cooled at the boundary, providing a [[steady state]] temperature distribution.]] | |||
A '''differential equation''' is a [[mathematics|mathematical]] [[equation]] that relates some [[function (mathematics)|function]] of one or more [[Variable (mathematics)|variables]] with its [[derivative]]s. Differential equations arise whenever a [[Deterministic system (mathematics)|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. Because such relations are extremely common, differential equations play a prominent role in many disciplines including [[engineering]], [[physics]], [[economics]], and [[biology]]. | |||
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. | |||
== | == Example == | ||
For example, in [[classical mechanics]], the motion of a body is described by its position and velocity as the time value varies. [[Newton's laws of motion|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 some cases, this differential equation (called an [[equations of motion|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 acceleration 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 a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation. | |||
== Directions of study == | |||
The study of differential equations is a wide field in [[pure mathematics|pure]] and [[applied mathematics]], [[physics]], 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. | |||
The study of the stability of solutions of differential equations is known as [[stability theory]]. | |||
== Nomenclature == | |||
The theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation. | |||
===Ordinary and partial=== | |||
* An [[ordinary differential equation]] (ODE) is a differential equation in which the unknown function (also known as the '''[[dependent variable#Use in mathematics|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. | |||
{{Anchor|first-order differential equation|second-order differential equation}} | |||
:Ordinary differential equations are further classified according to the '''order''' of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are '''first-order''' and '''second-order differential equations'''. For example, [[Bessel's differential equation]] | |||
:: <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0</math> | |||
:(in which ''y'' is the dependent variable) is a second-order differential equation. In the classical literature a distinction is also made between differential equations [[explicit function|explicitly solved]] with respect to the highest derivative and differential equations in an [[implicit derivative|implicit]] form. Also important is the '''degree''', or (highest) power, of the highest derivative(s) in the equation (''[[cf.]]'' : [[degree of a polynomial]]). A differential equation is called a '''[[nonlinear differential equation]]''' if its degree is not one (a [[sufficient condition|sufficient]] but [[necessary condition|unnecessary condition]]). | |||
* 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'''. | |||
===Linear and non-linear=== | |||
Both ordinary and partial differential equations are broadly classified as '''linear''' and '''nonlinear'''. | |||
*A [[Linear differential equation|differential equation is '''linear''']] if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and [[Nonlinear system#Nonlinear differential equations|'''nonlinear''']] otherwise. The characteristic property of linear equations is that their solutions form an [[affine space|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 solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular [[Symmetry|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]]). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.<ref>{{cite book | |||
| last = Boyce | |||
| first = William E. | |||
| last2 = DiPrima | |||
| first2 = Richard C. | |||
| authorlink = | |||
| title = Elementary Differential Equations and Boundary Value Problems | |||
| publisher =John Wiley & Sons | |||
| series = | |||
| volume = | |||
| edition = 4th | |||
| year = 1967 | |||
| location = | |||
| pages = 3 | |||
| language = | |||
| url = | |||
| doi = | |||
| id = | |||
| isbn = | |||
| mr = | |||
| zbl = | |||
| jfm = }}</ref> | |||
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). | |||
=== Examples === | |||
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 linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]: | |||
:: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math> | |||
* Inhomogeneous 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> L\frac{d^2u}{dx^2} + g\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> | |||
== Related concepts == | |||
* 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 turns out that many [[diffusion]] processes, while seemingly different, are described by the same equation; the [[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;"> | |||
===Physics and engineering=== | |||
* [[Newton's Second Law]] in [[dynamics (mechanics)]] | |||
* [[Euler–Lagrange equation]] in classical 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 [[Diffusion equation]] in [[stochastic processes]] | |||
* The [[Convection–diffusion equation]] in [[fluid dynamics]] | |||
* The [[Cauchy–Riemann equations]] in [[complex analysis]] | |||
* The [[Poisson–Boltzmann equation]] in [[molecular dynamics]] | |||
* The [[shallow water equations]] | |||
* [[Universal differential equation]] | |||
* The [[Lorenz attractor|Lorenz equations]] whose solutions exhibit chaotic flow. | |||
</div> | |||
===Biology=== | |||
*[[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]] – found in theoretical biology | |||
*[[Hodgkin–Huxley model]] – neural action potentials | |||
===Economics=== | |||
*[[Black–Scholes#The_Black–Scholes_PDE|The Black–Scholes PDE]] | |||
*[[Exogenous growth model]] | |||
*[[Malthusian growth model]] | |||
*[[Sethi model|The Vidale–Wolfe advertising model]] | |||
==See also== | |||
*[[Complex differential equation]] | |||
*[[Exact differential equation]] | |||
*[[Integral equations]] | |||
*[[Numerical_methods#Differential_equations|Numerical methods]] | |||
*[[Picard–Lindelöf theorem]] on existence and uniqueness of solutions | |||
*[[Recurrence relation]], also known as 'Difference Equation' | |||
== References == | |||
* P. Abbott and H. Neill, ''Teach Yourself Calculus'', 2003 pages 266-277 | |||
* P. Blanchard, R. L. Devaney, G. R. Hall, ''Differential Equations'', Thompson, 2006 | |||
* E. A. Coddington and N. Levinson, ''Theory of Ordinary Differential Equations'', McGraw-Hill, 1955 | |||
* E. L. Ince, ''Ordinary Differential Equations'', Dover Publications, 1956 | |||
* 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] | |||
* 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. | |||
* R. I. Porter, ''Further Elementary Analysis'', 1978, chapter XIX Differential Equations | |||
* {{Cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}} | |||
* D. Zwillinger, ''Handbook of Differential Equations (3rd edition)'', Academic Press, Boston, 1997. | |||
{{reflist}} | |||
==External links== | |||
{{wikibooks|Ordinary Differential Equations}} | |||
*[http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/ Lectures on Differential Equations] [[MIT]] Open CourseWare Videos | |||
*[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://publicliterature.org/tools/differential_equation_solver/ Differential Equation Solver] Java applet tool used to solve differential equations. | |||
*[http://www.fioravante.patrone.name/mat/u-u/en/differential_equations_intro.htm Introduction to modeling via differential equations] Introduction to modeling by means of differential equations, with critical remarks. | |||
* [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.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]] | |||
* [http://www.khanacademy.org/math/differential-equations Khan Academy Video playlist on differential equations ] Topics covered in a first year course in differential equations. | |||
*[http://math.rareinfos.com/category/courses/solutions-differential-equations/homogeneous-linear-systems/ MathDiscuss Video playlist on differential equations ] | |||
{{Mathematics-footer}} | |||
[[Category:Differential equations|*]] |
Revision as of 20:20, 21 January 2014
Template:Distinguish Template:More footnotes Template:Differential equations
A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives. Differential equations arise 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. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
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.
Example
For example, in classical mechanics, 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 some cases, 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 acceleration 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 a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation.
Directions of study
The study of differential equations is a wide field in pure and applied mathematics, physics, 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 solutions. Instead, solutions can be approximated using numerical methods.
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.
The study of the stability of solutions of differential equations is known as stability theory.
Nomenclature
The theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation.
Ordinary and partial
- 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 or 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 of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation
- (in which y is the dependent variable) is a second-order differential equation. In the classical literature a distinction is also made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form. Also important is the degree, or (highest) power, of the highest derivative(s) in the equation (cf. : degree of a polynomial). A differential equation is called a nonlinear differential equation if its degree is not one (a sufficient but unnecessary condition).
- 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.
Linear and non-linear
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 of the unknown function and its derivatives 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 solving nonlinear differential equations exactly; 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). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.[1]
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).
Examples
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:
- Homogeneous second-order linear ordinary differential equation:
- Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:
- Inhomogeneous first-order nonlinear ordinary differential equation:
- Second-order nonlinear ordinary differential equation describing the motion of a pendulum of length L:
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:
- Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:
- Third-order nonlinear partial differential equation, the Korteweg–de Vries equation:
Related concepts
- 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
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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 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 turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.
Notable differential equations
Physics and engineering
- Newton's Second Law in dynamics (mechanics)
- Euler–Lagrange equation in classical 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 functions
- Poisson's equation
- Einstein's field equation in general relativity
- The Schrödinger equation in quantum mechanics
- The geodesic equation
- The Navier–Stokes equations in fluid dynamics
- The Diffusion equation in stochastic processes
- The Convection–diffusion equation in fluid dynamics
- The Cauchy–Riemann equations in complex analysis
- The Poisson–Boltzmann equation in molecular dynamics
- The shallow water equations
- Universal differential equation
- The Lorenz equations whose solutions exhibit chaotic flow.
Biology
- Verhulst equation – biological population growth
- von Bertalanffy model – biological individual growth
- Lotka–Volterra equations – biological population dynamics
- Replicator dynamics – found in theoretical biology
- Hodgkin–Huxley model – neural action potentials
Economics
- The Black–Scholes PDE
- Exogenous growth model
- Malthusian growth model
- The Vidale–Wolfe advertising model
See also
- Complex differential equation
- Exact differential equation
- Integral equations
- Numerical methods
- Picard–Lindelöf theorem on existence and uniqueness of solutions
- Recurrence relation, also known as 'Difference Equation'
References
- P. Abbott and H. Neill, Teach Yourself Calculus, 2003 pages 266-277
- P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, Thompson, 2006
- E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955
- E. L. Ince, Ordinary Differential Equations, Dover Publications, 1956
- W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
- 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.
- R. I. Porter, Further Elementary Analysis, 1978, chapter XIX Differential Equations
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
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External links
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- Lectures on Differential Equations MIT Open CourseWare Videos
- Online Notes / Differential Equations Paul Dawkins, Lamar University
- Differential Equations, S.O.S. Mathematics
- Differential Equation Solver Java applet tool used to solve differential equations.
- Introduction to modeling via differential equations Introduction to modeling by means of differential equations, with critical remarks.
- Mathematical Assistant on Web Symbolic ODE tool, using Maxima
- Exact Solutions of Ordinary Differential Equations
- Collection of ODE and DAE models of physical systems MATLAB models
- Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC
- Khan Academy Video playlist on differential equations Topics covered in a first year course in differential equations.
- MathDiscuss Video playlist on differential equations
47 year-old Environmental Consultant Lester from Port Coquitlam, really loves skydiving, property developers in singapore and cake decorating. that included taking a trip to Ancient City of Ping Yao.
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- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534