Differential equation

Not to be confused with Difference equation. ter 2 of his 1671 work “Methodus fluxionum et Serierum
A differential equation is a mathematical equation Infinitarum”,[1] Isaac Newton listed three kinds of differ-
ential equations:

dy
= f (x)
dx
dy
= f (x, y)
dx
∂y ∂y
x1 + x2 =y
∂x1 ∂x2
He solves these examples and others using infinite series
and discusses the non-uniqueness of solutions.
Jacob Bernoulli proposed the Bernoulli differential equa-
tion in 1695.[2] This is an ordinary differential equation
of the form
Visualization of heat transfer in a pump casing, created by solv-
ing the heat equation. Heat is being generated internally in the
casing and being cooled at the boundary, providing a steady state y ′ + P (x)y = Q(x)y n
temperature distribution.
for which the following year Leibniz obtained solutions
[3]
that relates some function with its derivatives. In applica- by simplifying it.
tions, the functions usually represent physical quantities, Historically, the problem of a vibrating string such as
the derivatives represent their rates of change, and the that of a musical instrument was studied by Jean le
equation defines a relationship between the two. Because Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and
such relations are extremely common, differential equa- Joseph-Louis Lagrange.[4][5][6][7] In 1746, d’Alembert
tions play a prominent role in many disciplines including discovered the one-dimensional wave equation, and
engineering, physics, economics, and biology. within ten years Euler discovered the three-dimensional
[8]
In pure mathematics, differential equations are studied wave equation.
from several different perspectives, mostly concerned The Euler–Lagrange equation was developed in the 1750s
with their solutions—the set of functions that satisfy the by Euler and Lagrange in connection with their studies of
equation. Only the simplest differential equations are the tautochrone problem. This is the problem of deter-
solvable by explicit formulas; however, some properties mining a curve on which a weighted particle will fall to a
of solutions of a given differential equation may be de- fixed point in a fixed amount of time, independent of the
termined without finding their exact form. starting point.
If a self-contained formula for the solution is not avail- Lagrange solved this problem in 1755 and sent the solu-
able, the solution may be numerically approximated using tion to Euler. Both further developed Lagrange’s method
computers. The theory of dynamical systems puts em- and applied it to mechanics, which led to the formulation
phasis on qualitative analysis of systems described by dif- of Lagrangian mechanics.
ferential equations, while many numerical methods have
Fourier published his work on heat flow in Théorie ana-
been developed to determine solutions with a given de-
lytique de la chaleur (The Analytic Theory of Heat),[9] in
gree of accuracy.
which he based his reasoning on Newton’s law of cool-
ing, namely, that the flow of heat between two adjacent
molecules is proportional to the extremely small differ-
1 History ence of their temperatures. Contained in this book was
Fourier’s proposal of his heat equation for conductive dif-
Differential equations first came into existence with the fusion of heat. This partial differential equation is now
invention of calculus by Newton and Leibniz. In Chap- taught to every student of mathematical physics.

1
2 3 TYPES

2 Example functions in closed form: Instead, exact and analytic so-

lutions of ODEs are in series or integral form. Graphical
For example, in classical mechanics, the motion of a body and numerical methods, applied by hand or by computer,
is described by its position and velocity as the time value may approximate solutions of ODEs and perhaps yield
varies. Newton’s laws allow (given the position, velocity, useful information, often sufficing in the absence of ex-
acceleration and various forces acting on the body) one to act, analytic solutions.
express these variables dynamically as a differential equa-
tion for the unknown position of the body as a function
of time.
In some cases, this differential equation (called an 3.2 Partial differential equations
equation of motion) may be solved explicitly.
An example of modelling a real world problem using dif- Main article: Partial differential equation
ferential 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 A partial differential equation (PDE) is a differential
is the acceleration due to gravity minus the acceleration equation that contains unknown multivariable functions
due to air resistance. and their partial derivatives. (This is in contrast to
ordinary differential equations, which deal with functions
Gravity is considered constant, and air resistance may be of a single variable and their derivatives.) PDEs are used
modeled as proportional to the ball’s velocity. This means to formulate problems involving functions of several vari-
that the ball’s acceleration, which is a derivative of its ve- ables, and are either solved in closed form, or used to cre-
locity, depends on the velocity (and the velocity depends ate a relevant computer model.
on time). Finding the velocity as a function of time in-
volves solving a differential equation and verifying its va- PDEs can be used to describe a wide variety of phenom-
lidity. ena such as sound, heat, electrostatics, electrodynamics,
fluid flow, elasticity, or quantum mechanics. These
seemingly distinct physical phenomena can be for-
malised similarly in terms of PDEs. Just as ordi-
3 Types nary differential equations often model one-dimensional
dynamical systems, partial differential equations often
Differential equations can be divided into several types. model multidimensional systems. PDEs find their gen-
Apart from describing the properties of the equation eralisation in stochastic partial differential equations.
itself, these classes of differential equations can help
inform the choice of approach to a solution. Com-
monly used distinctions include whether the equation
is: Ordinary/Partial, Linear/Non-linear, and Homoge-
neous/Inhomogeneous. This list is far from exhaustive; 3.3 Linear differential equations
there are many other properties and subclasses of differ-
ential equations which can be very useful in specific con- Main article: Linear differential equation
texts.
A differential equation is linear if the unknown function
and its derivatives have degree 1 (products of the un-
3.1 Ordinary differential equations known function and its derivatives are not allowed) and
nonlinear otherwise. The characteristic property of lin-
Main article: Ordinary differential equation ear equations is that their solutions form an affine sub-
space of an appropriate function space, which results in
An ordinary differential equation (ODE) is an equation much more developed theory of linear differential equa-
containing a function of one independent variable and its tions.
derivatives. The term "ordinary" is used in contrast with Homogeneous linear differential equations are a subclass
the term partial differential equation which may be with of linear differential equations for which the space of
respect to more than one independent variable. solutions is a linear subspace i.e. the sum of any set
Linear differential equations, which have solutions that of solutions or multiples of solutions is also a solution.
can be added and multiplied by coefficients, are well- The coefficients of the unknown function and its deriva-
defined and understood, and exact closed-form solutions tives in a linear differential equation are allowed to be
are obtained. By contrast, ODEs that lack additive so- (known) functions of the independent variable or vari-
lutions are nonlinear, and solving them is far more in- ables; if these coefficients are constants then one speaks
tricate, as one can rarely represent them by elementary of a constant coefficient linear differential equation.
 3

3.4 Non-linear differential equations • Homogeneous second-order linear constant coeffi-

cient ordinary differential equation describing the
Non-linear differential equations are formed by the harmonic oscillator:
products of the unknown function and its derivatives are
allowed and its degree is > 1.There are very few methods
of solving nonlinear differential equations exactly; those
that are known typically depend on the equation having d2 u
+ ω 2 u = 0.
particular symmetries. Nonlinear differential equations dx2
can exhibit very complicated behavior over extended time
intervals, characteristic of chaos. Even the fundamen- • Inhomogeneous first-order nonlinear ordinary dif-
tal questions of existence, uniqueness, and extendabil- ferential equation:
ity of solutions for nonlinear differential equations, and
well-posedness of initial and boundary value problems for
nonlinear PDEs are hard problems and their resolution du
in special cases is considered to be a significant advance = u2 + 4.
dx
in the mathematical theory (cf. Navier–Stokes existence
and smoothness). However, if the differential equation • Second-order nonlinear (due to sine function) ordi-
is a correctly formulated representation of a meaningful nary differential equation describing the motion of
physical process, then one expects it to have a solution.[10] a pendulum of length L:
Linear differential equations frequently appear as
approximations to nonlinear equations. These approx-
imations are only valid under restricted conditions. d2 u
For example, the harmonic oscillator equation is an L 2 + g sin u = 0.
dx
approximation to the nonlinear pendulum equation that
is valid for small amplitude oscillations (see below). In the next group of examples, the unknown function u
depends on two variables x and t or x and y.
3.5 Equation order
• Homogeneous first-order linear partial differential
Differential equations are described by their order, de- equation:
termined by the term with the highest derivatives. An
equation containing only first derivatives is a first-order
differential equation, an equation containing the second ∂u ∂u
derivative is a second-order differential equation, and so +t = 0.
∂t ∂x
on.[11][12]
• Homogeneous second-order linear constant coeffi-
cient partial differential equation of elliptic type, the
3.6 Examples Laplace equation:

In the first group of examples, let u be an unknown func-

tion of x, and c and ω are known constants. Note both or-
dinary and partial differential equations are broadly clas- ∂2u ∂2u
+ 2 = 0.
sified as linear and nonlinear. ∂x2 ∂y

• Inhomogeneous first-order linear constant coeffi-

cient ordinary differential equation: ∂u ∂u ∂ 3 u
= 6u − .
∂t ∂x ∂x3

du 4 Existence of solutions
= cu + x2 .
dx
Solving differential equations is not like solving algebraic
• Homogeneous second-order linear ordinary differ- equations. Not only are their solutions oftentimes un-
ential equation: clear, but whether solutions are unique or exist at all are
also notable subjects of interest.
For first order initial value problems, the Peano existence
2
d u du theorem gives one set of circumstances in which a solu-
−x + u = 0. tion exists. Given any point (a, b) in the xy-plane, define
dx2 dx
4 7 APPLICATIONS

some rectangular region Z , such that Z = [l, m] × [n, p] 7 Applications

and (a, b) is in the interior of Z . If we are given a differ-
dy
ential equation dx = g(x, y) and the condition that y = b
The study of differential equations is a wide field in pure
when x = a , then there is locally a solution to this prob-
and applied mathematics, physics, and engineering. All
∂g
lem if g(x, y) and ∂x are both continuous on Z . This of these disciplines are concerned with the properties of
solution exists on some interval with its center at a . The
differential equations of various types. Pure mathemat-
solution may not be unique. (See Ordinary differential ics focuses on the existence and uniqueness of solutions,
equation for other results.) while applied mathematics emphasizes the rigorous justi-
However, this only helps us with first order initial value fication of the methods for approximating solutions. Dif-
problems. Suppose we had a linear initial value problem ferential equations play an important role in modelling
of the nth order: virtually every physical, technical, or biological process,
from celestial motion, to bridge design, to interactions be-
tween neurons. Differential equations such as those used
dn y dy to solve real-life problems may not necessarily be directly
fn (x) n + · · · + f1 (x) + f0 (x)y = g(x)
dx dx solvable, i.e. do not have closed form solutions. Instead,
such that solutions can be approximated using numerical methods.
Many fundamental laws of physics and chemistry can
be formulated as differential equations. In biology and
y(x0 ) = y0 , y ′ (x0 ) = y0′ , y ′′ (x0 ) = y0′′ , · · ·
economics, differential equations are used to model the
For any nonzero fn (x) , if {f0 , f1 , · · · } and g are con- behavior of complex systems. The mathematical theory
tinuous on some interval containing x0 , y is unique and of differential equations first developed together with the
exists.[13] sciences where the equations had originated and where
the results found application. However, diverse prob-
lems, sometimes originating in quite distinct scientific
5 Related concepts fields, may give rise to identical differential equations.
Whenever this happens, mathematical theory behind the
• A delay differential equation (DDE) is an equation equations can be viewed as a unifying principle behind di-
for a function of a single variable, usually called verse phenomena. As an example, consider propagation
time, in which the derivative of the function at a of light and sound in the atmosphere, and of waves on the
certain time is given in terms of the values of the surface of a pond. All of them may be described by the
function at earlier times. same second-order partial differential equation, the wave
equation, which allows us to think of light and sound as
• A stochastic differential equation (SDE) is an equa- forms of waves, much like familiar waves in the water.
tion in which the unknown quantity is a stochastic Conduction of heat, the theory of which was developed
process and the equation involves some known by Joseph Fourier, is governed by another second-order
stochastic processes, for example, the Wiener pro- partial differential equation, the heat equation. It turns
cess in the case of diffusion equations. out that many diffusion processes, while seemingly dif-
ferent, are described by the same equation; the Black–
• A differential algebraic equation (DAE) is a differ-
Scholes equation in finance is, for instance, related to the
ential equation comprising differential and algebraic
heat equation.
terms, given in implicit form.

7.1 Physics
6 Connection to difference equa-
tions • Euler–Lagrange equation in classical mechanics

• Hamilton’s equations in classical mechanics

See also: Time scale calculus
• Radioactive decay in nuclear physics
The theory of differential equations is closely related to • Newton’s law of cooling in thermodynamics
the theory of difference equations, in which the coordi-
nates assume only discrete values, and the relationship in- • The wave equation
volves values of the unknown function or functions and
• The heat equation in thermodynamics
values at nearby coordinates. Many methods to compute
numerical solutions of differential equations or study the • Laplace’s equation, which defines harmonic func-
properties of differential equations involve approxima- tions
tion of the solution of a differential equation by the solu-
tion of a corresponding difference equation. • Poisson’s equation
7.2 Biology 5

• The geodesic equation 7.1.4 Quantum mechanics

• The Navier–Stokes equations in fluid dynamics In quantum mechanics, the analogue of Newton’s law is
Schrödinger’s equation (a partial differential equation)
• The Diffusion equation in stochastic processes for a quantum system (usually atoms, molecules, and
subatomic particles whether free, bound, or localized).
• The Convection–diffusion equation in fluid dynam- It is not a simple algebraic equation, but in general a
ics linear partial differential equation, describing the time-
evolution of the system’s wave function (also called a
• The Cauchy–Riemann equations in complex analy- “state function”).[17]
sis

• The Poisson–Boltzmann equation in molecular dy- 7.2 Biology

namics
• Verhulst equation – biological population growth
• The shallow water equations
• von Bertalanffy model – biological individual growth
• Universal differential equation • Replicator dynamics – found in theoretical biology
• The Lorenz equations whose solutions exhibit • Hodgkin–Huxley model – neural action potentials
chaotic flow.

7.2.1 Predator-prey equations

7.1.1 Classical mechanics
The Lotka–Volterra equations, also known as the
So long as the force acting on a particle is known, predator–prey equations, are a pair of first-order, non-
Newton’s second law is sufficient to describe the motion linear, differential equations frequently used to describe
of a particle. Once independent relations for each force the dynamics of biological systems in which two species
acting on a particle are available, they can be substituted interact, one as a predator and the other as prey.
into Newton’s second law to obtain an ordinary differen-
tial equation, which is called the equation of motion.
7.3 Chemistry
The rate law or rate equation for a chemical reaction is
7.1.2 Electrodynamics
a differential equation that links the reaction rate with
concentrations or pressures of reactants and constant pa-
Maxwell’s equations are a set of partial differential equa-
rameters (normally rate coefficients and partial reaction
tions that, together with the Lorentz force law, form the
orders).[18] To determine the rate equation for a partic-
foundation of classical electrodynamics, classical optics,
ular system one combines the reaction rate with a mass
and electric circuits. These fields in turn underlie modern
balance for the system.[19]
electrical and communications technologies. Maxwell’s
equations describe how electric and magnetic fields are
generated and altered by each other and by charges and 7.4 Economics
currents. They are named after the Scottish physicist and
mathematician James Clerk Maxwell, who published an • The key equation of the Solow–Swan model is
early form of those equations between 1861 and 1862. ∂k(t)
∂t = s[k(t)] − δk(t)
α

• The Black–Scholes PDE

7.1.3 General relativity
• Malthusian growth model
The Einstein field equations (EFE; also known as “Ein- • The Vidale–Wolfe advertising model
stein’s equations”) are a set of ten partial differential
equations in Albert Einstein's general theory of rel-
ativity which describe the fundamental interaction of
gravitation as a result of spacetime being curved by matter
8 See also
and energy.[14] First published by Einstein in 1915[15]
as a tensor equation, the EFE equate local spacetime • Complex differential equation
curvature (expressed by the Einstein tensor) with the local • Exact differential equation
energy and momentum within that spacetime (expressed
by the stress–energy tensor).[16] • Initial condition
6 10 FURTHER READING

• Integral equations [11] Weisstein, Eric W. “Ordinary Differential Equa-

tion Order.” From MathWorld--A Wolfram
• Numerical methods Web Resource. http://mathworld.wolfram.com/
OrdinaryDifferentialEquationOrder.html
• Picard–Lindelöf theorem on existence and unique-
ness of solutions [12] Order and degree of a differential equation, accessed Dec
2015.
• Recurrence relation, also known as 'Difference
Equation' [13] Zill, Dennis G. A First Course in Differential Equations
(5th ed.). Brooks/Cole. ISBN 0-534-37388-7.

[14] Einstein, Albert (1916). “The Foundation of the

9 References General Theory of Relativity” (PDF). Annalen der
Physik. 354 (7): 769. Bibcode:1916AnP...354..769E.
doi:10.1002/andp.19163540702.
[1] Newton, Isaac. (c.1671). Methodus Fluxionum et Se-
rierum Infinitarum (The Method of Fluxions and Infinite [15] Einstein, Albert (November 25, 1915). “Die Feldgle-
Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66]. ichungen der Gravitation”. Sitzungsberichte der Preussis-
chen Akademie der Wissenschaften zu Berlin: 844–847.
[2] Bernoulli, Jacob (1695), “Explicationes, Annotationes & Retrieved 2006-09-12.
Additiones ad ea, quae in Actis sup. de Curva Elastica,
Isochrona Paracentrica, & Velaria, hinc inde memorata, [16] Misner, Charles W.; Thorne, Kip S.; Wheeler, John
& paratim controversa legundur; ubi de Linea mediarum Archibald (1973). Gravitation. San Francisco: W. H.
directionum, alliisque novis”, Acta Eruditorum Freeman. ISBN 978-0-7167-0344-0 Chapter 34, p. 916.
[3] Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard [17] Griffiths, David J. (2004), Introduction to Quantum Me-
(1993), Solving ordinary differential equations I: Nonstiff chanics (2nd ed.), Prentice Hall, pp. 1–2, ISBN 0-13-
problems, Berlin, New York: Springer-Verlag, ISBN 978- 111892-7
3-540-56670-0
[18] IUPAC Gold Book definition of rate law. See also: Ac-
[4] Cannon, John T.; Dostrovsky, Sigalia (1981). “The evo- cording to IUPAC Compendium of Chemical Terminol-
lution of dynamics, vibration theory from 1687 to 1742”. ogy.
Studies in the History of Mathematics and Physical Sci-
ences. 6. New York: Springer-Verlag: ix + 184 pp. ISBN [19] Kenneth A. Connors Chemical Kinetics, the study of reac-
0-3879-0626-6. GRAY, JW (July 1983). “BOOK RE- tion rates in solution, 1991, VCH Publishers.
VIEWS”. BULLETIN (New Series) OF THE AMERICAN
MATHEMATICAL SOCIETY. 9 (1). (retrieved 13 Nov
2012).
10 Further reading
[5] Wheeler, Gerard F.; Crummett, William P. (1987).
“The Vibrating String Controversy”. Am. J. Phys.
55 (1): 33–37. Bibcode:1987AmJPh..55...33W. • P. Abbott and H. Neill, Teach Yourself Calculus,
doi:10.1119/1.15311. 2003 pages 266-277

[6] For a special collection of the 9 groundbreaking papers • P. Blanchard, R. L. Devaney, G. R. Hall, Differential
by the three authors, see First Appearance of the wave Equations, Thompson, 2006
equation: D'Alembert, Leonhard Euler, Daniel Bernoulli.
- the controversy about vibrating strings (retrieved 13 Nov • E. A. Coddington and N. Levinson, Theory of Ordi-
2012). Herman HJ Lynge and Son. nary Differential Equations, McGraw-Hill, 1955

[7] For de Lagrange’s contributions to the acoustic wave equa- • E. L. Ince, Ordinary Differential Equations, Dover
tion, can consult Acoustics: An Introduction to Its Physi- Publications, 1956
cal Principles and Applications Allan D. Pierce, Acousti-
cal Soc of America, 1989; page 18.(retrieved 9 Dec 2012) • W. Johnson, A Treatise on Ordinary and Partial Dif-
ferential Equations, John Wiley and Sons, 1913, in
[8] Speiser, David. Discovering the Principles of Mechanics
University of Michigan Historical Math Collection
1600-1800, p. 191 (Basel: Birkhäuser, 2008).
• A. D. Polyanin and V. F. Zaitsev, Handbook of
[9] Fourier, Joseph (1822). Théorie analytique de la chaleur
(in French). Paris: Firmin Didot Père et Fils. OCLC Exact Solutions for Ordinary Differential Equations
2688081. (2nd edition), Chapman & Hall/CRC Press, Boca
Raton, 2003. ISBN 1-58488-297-2.
[10] Boyce, William E.; DiPrima, Richard C. (1967). Elemen-
tary Differential Equations and Boundary Value Problems • R. I. Porter, Further Elementary Analysis, 1978,
(4th ed.). John Wiley & Sons. p. 3. chapter XIX Differential Equations
 7

• Teschl, Gerald (2012). Ordinary Differential

Equations and Dynamical Systems. Providence:
American Mathematical Society. ISBN 978-0-
8218-8328-0.
• D. Zwillinger, Handbook of Differential Equations
(3rd edition), Academic Press, Boston, 1997.

11 External links
• Lectures on Differential Equations MIT Open
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 sys-
tems MATLAB models
• Notes on Diffy Qs: Differential Equations for Engi-
neers An introductory textbook on differential equa-
tions by Jiri Lebl of UIUC

• Khan Academy Video playlist on differential equa-

tions Topics covered in a first year course in differ-
ential equations.
• MathDiscuss Video playlist on differential equations
8 12 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

12 Text and image sources, contributors, and licenses

12.1 Text
• Differential equation Source: https://en.wikipedia.org/wiki/Differential_equation?oldid=745054994 Contributors: XJaM, JohnOwens,
Michael Hardy, Ahoerstemeier, Iulianu, Julesd, Andres, Charles Matthews, Dysprosia, Jitse Niesen, Phoebe, Lumos3, Gandalf61, Sverdrup,
Ruakh, KeithJonsn, Dbroadwell, Tea2min, Giftlite, Paul Richter, Jao, Lethe, Waltpohl, Siroxo, Andycjp, Geni, Bob.v.R, Yafujifide, Gauss,
Mysidia, Fintor, Wsears, Mh, EricBright, Mindspillage, Klaas van Aarsen, Mazi, Paul August, Bender235, Djordjes, Elwikipedista~enwiki,
MisterSheik, Liberatus, Chtito, Haham hanuka, Tsirel, Jérôme, Danski14, Eric Kvaalen, Davetcoleman, Grenavitar, Cmprince, TVBZ28,
Oleg Alexandrov, Tbsmith, Jak86, Linas, Skypher, Justinlebar, LOL, Mpatel, SDC, Waldir, Btyner, Mandarax, Dpv, MarSch, Salix alba,
Andrei Polyanin, Wihenao, Mathbot, Alfred Centauri, RexNL, M7bot, Chobot, Jersey Devil, Bgwhite, YurikBot, Wavelength, Texas-
Android, Baccala@freesoft.org, Brandon, DAJF, SFC9394, Bota47, Reyk, Willtron, Sardanaphalus, Attilios, SmackBot, K-UNIT, Pgk,
Gabrielleitao, Gilliam, Winterheart, DroEsperanto, Silly rabbit, Mohan1986, Rama’s Arrow, Kr5t, Berland, Vanished User 0001, Sun-
darBot, ConMan, Cybercobra, Bryanmcdonald, Bidabadi~enwiki, Lambiam, Donludwig, Romansanders, Wtwilson3, JorisvS, RichMorin,
Jim.belk, PseudoSudo, J arino, Mets501, Asyndeton, Norm mit, Madmath789, Spirits in the Material, Antonius Block~enwiki, Nkaye-
smith, Pahio, Slippyd, Paul Matthews, Rosasco, Lavateraguy, McVities, WeggeBot, Mtness~enwiki, AndrewHowse, Sam Staton, Martastic,
Christian75, After Midnight, Thijs!bot, Anupam, The Hybrid, Vthiru, CTZMSC3, Escarbot, AntiVandalBot, Seaphoto, Opelio, 17Drew,
Senoreuchrestud, LibLord, Qwerty Binary, Reallybored999, Haseeb Jamal, JamesBWatson, Soulbot, Baccyak4H, David Eppstein, User
A1, Martynas Patasius, DerHexer, Khalid Mahmood, Robin S, Matqkks, Jim.henderson, R'n'B, Tgeairn, DominiqueNC, J.delanoy, C. Tri-
fle, Maurice Carbonaro, Bygeorge2512, Jayden54, Gombang, NewEnglandYankee, Ilya Voyager, Ja 62, AlnoktaBOT, TXiKiBoT, Antoni
Barau, A4bot, Rei-bot, Andytalk, Anonymous Dissident, Krushia, Arcfrk, AlleborgoBot, Symane, SieBot, Gerakibot, Dr sarah madden,
Heikki m, Timelesseyes, Ddxc, Harry-, OKBot, Mattmnelson, Hamiltondaniel, Amahoney, Kayvan45622, ClueBot, Alarius, Fioravante
Patrone en, The Thing That Should Not Be, Drmies, Niceguyedc, Nik-renshaw, DragonBot, XLinkBot, Forbes72, Staticshakedown, Ap-
monitor, JinJian, Addbot, Cxz111, SmartPatrol, Haruth, Friend of the Facts, MrOllie, EconoPhysicist, Delaszk, SpBot, Jarble, Legobot,
Luckas-bot, Yobot, The Grumpy Hacker, Mossaiby, Estudiarme, Tannkrem, Nallimbot, KamikazeBot, AnomieBOT, ^musaz, Jim1138,
Materialscientist, Maxis ftw, ArthurBot, PavelSolin, Xqbot, Asdf39, Almabot, Qzd800, Kensaii, Tranum1234567890, Bejohns6, Fres-
coBot, Izodman2012, Iquseruniv, Pinethicket, Zepterfd, Jauhienij, Babayagagypsies, DixonDBot, Math.geek3.1415926, Callumds, Duo-
duoduo, Skakkle, Suffusion of Yellow, Snowjeep, Sampathsris, Tbhotch, Genedronek, Difu Wu, TjBot, Bento00, EmausBot, John of
Reading, BillyPreset, Dewritech, GoingBatty, Mduench, TuHan-Bot, Dcirovic, The Blade of the Northern Lights, AngryPhillip, Bigus-
bry, Érico, Wclxlus, Maschen, Parusaro, ChuispastonBot, TYelliot, Dmcourtn, Helpsome, Mankarse, ClueBot NG, Fioravante Patrone,
Frietjes, Rezabot, ‫ساجد امجد ساجد‬, Calabe1992, Bibcode Bot, BG19bot, Pine, Bolatbek, Jim Sukwutput, MusikAnimal, Cispyre, Sil-
vrous, FutureTrillionaire, Vishwanathnm, Soswow, Jeancey, F=q(E+v^B), Randomguess, Tuseroni, Ema--or, Toploftical, StarryGrandma,
Pratyya Ghosh, Brirush, Limit-theorem, Coginsys, Epicgenius, Eyesnore, Tentinator, SakeUPenn, Paul2520, Galobtter, Johndoeisnotmy-
name, Holmes1900, Flameturtle, Okopecz, Erin.Annette.Brown, Loraof, Kavya l, KasparBot, Ai295464684 and Anonymous: 278

12.2 Images
• File:Elmer-pump-heatequation.png Source: https://upload.wikimedia.org/wikipedia/commons/c/cd/Elmer-pump-heatequation.png
License: CC BY-SA 3.0 Contributors: ? Original artist: ?
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Brief description of the numerical method
The following code leverages some numerical methods to simulate the solution of the 2-dimensional Navier-Stokes equation.
We choose the simplified incompressible flow Navier-Stokes Equation as follows:

<img src='https://wikimedia.org/api/rest_v1/media/math/render/svg/1b352a66970b542690aff9810ff1514eca0952bd'
class='mwe-math-fallback-image-inline' aria-hidden='true' style='vertical-align: −2.505ex; width:27.565ex; height:6.176ex;'
alt='{\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t <i' />Original artist: $5+\mathbf {v} \cdot \nabla
\mathbf {v} \right)=\mu \nabla ^{2}\mathbf {v} .}">

The iterations here are based on the velocity change rate, which is given by

<img src='https://wikimedia.org/api/rest_v1/media/math/render/svg/84351a8157ffbc3af56ed19583c97d062bfd428d'
class='mwe-math-fallback-image-inline' aria-hidden='true' style='vertical-align: −2.338ex; width:23.36ex; height:5.843ex;'
alt='{\displaystyle {\frac {\partial \mathbf {v} }{\partial t <h2' />Content license $3={\frac {\mu }{\rho }}\nabla
^{2}\mathbf {v} -\mathbf {v} \cdot \nabla \mathbf {v} .}">

Or in X coordinates:

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d009b926bc255277cd55f75f9773d3721861f3ac"
class="mwe-math-fallback-image-inline” aria-hidden="true” style="vertical-align: −2.505ex; width:45.497ex;
height:6.343ex;" alt="{\displaystyle {\frac {\partial v_{x}}{\partial t}}={\frac {\mu }{\rho }}({\frac {\partial
^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y^{2}}})-v_{x}{\frac {\partial v_{x}}{\partial
x}}-v_{y}{\frac {\partial v_{x}}{\partial y}}.}">

The above equation gives the code. The case of Y is similar.|IkamusumeFan}}

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