Differential equation
Differential equations |
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In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives.[1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are soluble by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a
History
Differential equations came into existence with the
In all these cases, y is an unknown function of x (or of x1 and x2), and f is a given function.
He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.
Jacob Bernoulli proposed the Bernoulli differential equation in 1695.[3] This is an ordinary differential equation of the form
for which the following year Leibniz obtained solutions by simplifying it.[4]
Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.[5][6][7][8] In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.[9]
The
In 1822,
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 these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) 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 modeling 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 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 and verifying its validity.
Types
Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.
Ordinary differential equations
An
Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.
Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function).
As, in general, the solutions of a differential equation cannot be expressed by a
Partial differential equations
A
PDEs can be used to describe a wide variety of phenomena in nature such as
Non-linear differential equations
A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). 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 behaviour 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.[11]
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.
Equation order and degree
The order of the differential equation is the highest
When it is written as a
Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin-film equation, which is a fourth order partial differential equation.
Examples
In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.
- Heterogeneous 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:
- Heterogeneous first-order nonlinear ordinary differential equation:
- Second-order nonlinear (due to sine function) 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:
- Homogeneous third-order non-linear partial differential equation, the KdV equation:
Existence of solutions
For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point in the xy-plane, define some rectangular region , such that and is in the interior of . If we are given a differential equation and the condition that when , then there is locally a solution to this problem if and are both continuous on . This solution exists on some interval with its center at . The solution may not be unique. (See Ordinary differential equation for other results.)
However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:
such that
For any nonzero , if and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle g} are continuous on some interval containing , is unique and exists.[15]
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.
- Integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.[16]
- An integro-differential equation (IDE) is an equation that combines aspects of a differential equation and an integral equation.
- 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 stochastic partial differential equation (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in quantum field theory and statistical mechanics.
- An ultrametric differential operators.
- A differential algebraic equation(DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.
Connection to difference equations
The theory of differential equations is closely related to the theory of
Applications
The study of differential equations is a wide field in
Many fundamental laws of
The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See List of named differential equations.
Software
Some CAS software can solve differential equations. These are the commands used in the leading programs:
- Maple:[17]
dsolve
- Mathematica:[18]
DSolve[]
- Maxima:[19]
ode2(equation, y, x)
- SageMath:[20]
desolve()
- SymPy:[21]
sympy.solvers.ode.dsolve(equation)
- Xcas:[22]
desolve(y'=k*y,y)
See also
- Exact differential equation
- Functional differential equation
- Initial condition
- Integral equations
- Numerical methods for ordinary differential equations
- Numerical methods for partial differential equations
- Picard–Lindelöf theorem on existence and uniqueness of solutions
- Recurrence relation, also known as 'difference equation'
- Abstract differential equation
- System of differential equations
References
- ISBN 978-1-285-40110-2.
- ^ Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].
- ^ Bernoulli, Jacob (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis", Acta Eruditorum
- ISBN 978-3-540-56670-0
- ^ Frasier, Craig (July 1983). "Review of The evolution of dynamics, vibration theory from 1687 to 1742, by John T. Cannon and Sigalia Dostrovsky" (PDF). Bulletin of the American Mathematical Society. New Series. 9 (1).
- doi:10.1119/1.15311.
- ^ For a special collection of the 9 groundbreaking papers by the three authors, see First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings Archived 2020-02-09 at the Wayback Machine (retrieved 13 Nov 2012). Herman HJ Lynge and Son.
- ^ For de Lagrange's contributions to the acoustic wave equation, can consult Acoustics: An Introduction to Its Physical Principles and Applications Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
- ^ Speiser, David. Discovering the Principles of Mechanics 1600-1800, p. 191 (Basel: Birkhäuser, 2008).
- OCLC 2688081.
- ^ Boyce, William E.; DiPrima, Richard C. (1967). Elementary Differential Equations and Boundary Value Problems (4th ed.). John Wiley & Sons. p. 3.
- Weisstein, Eric W. "Ordinary Differential Equation Order." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html
- ^ Order and degree of a differential equation Archived 2016-04-01 at the Wayback Machine, accessed Dec 2015.
- ^ Elias Loomis (1887). Elements of the Differential and Integral Calculus (revised ed.). Harper & Bros. p. 247. Extract of page 247
- ISBN 0-534-37388-7.
- arXiv:1806.07366 [cs.LG].
- ^ "dsolve - Maple Programming Help". www.maplesoft.com. Retrieved 2020-05-09.
- ^ "DSolve - Wolfram Language Documentation". www.wolfram.com. Retrieved 2020-06-28.
- ^ Schelter, William F. Gaertner, Boris (ed.). "Differential Equations - Symbolic Solutions". The Computer Algebra Program Maxima - a Tutorial (in Maxima documentation on SourceForge). Archived from the original on 2022-10-04.
- ^ "Basic Algebra and Calculus — Sage Tutorial v9.0". doc.sagemath.org. Retrieved 2020-05-09.
- ^ "ODE". SymPy 1.11 documentation. 2022-08-22. Archived from the original on 2022-09-26.
- ^ "Symbolic algebra and Mathematics with Xcas" (PDF).
Further reading
- Abbott, P.; Neill, H. (2003). Teach Yourself Calculus. pp. 266–277.
- Blanchard, P.; Devaney, R. L.; Hall, G. R. (2006). Differential Equations. Thompson.
- Boyce, W.; DiPrima, R.; Meade, D. (2017). Elementary Differential Equations and Boundary Value Problems. Wiley.
- Coddington, E. A.; Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill.
- Ince, E. L. (1956). Ordinary Differential Equations. Dover.
- Johnson, W. (1913). A Treatise on Ordinary and Partial Differential Equations. John Wiley and Sons. In University of Michigan Historical Math Collection
- Polyanin, A. D.; Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.). Boca Raton: Chapman & Hall/CRC Press. ISBN 1-58488-297-2.
- Porter, R. I. (1978). "XIX Differential Equations". Further Elementary Analysis.
- ISBN 978-0-8218-8328-0.
- Daniel Zwillinger (12 May 2014). Handbook of Differential Equations. Elsevier Science. ISBN 978-1-4832-6396-0.
External links
- Media related to Differential equations at Wikimedia Commons
- Lectures on Differential Equations MITOpen CourseWare Videos
- Online Notes / Differential Equations Paul Dawkins, Lamar University
- Differential Equations, S.O.S. Mathematics
- 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 Archived 2008-12-19 at the Wayback Machine 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