A differential equation is an equation that relates one or more unknown functions to their derivatives. In applications the functions typically represent physical quantities and the derivatives their rates of change, so the branch covers ordinary and partial differential equations and the exact, qualitative, and numerical methods used to understand their solutions. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionGiven a rule that governs how a quantity changes rather than the quantity itself, can that quantity be recovered exactly, approximately, or merely shown to exist and be unique? 1 Key DebatePure mathematics and applied mathematics diverge in emphasis within the field: pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics is concerned with finding solutions, either directly or approximately. 1 Classification
Pure or AppliedBoth / Interdisciplinary 1 Differential Equations
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Associated With
Source Dynamical System (Wikipedia)
The Navier-Stokes equations are a system of partial differential equations governing fluid flow; the entity's own description opens by calling this "the most basic open question about the Navier-Stokes equations", already carrying In Branch edges to analysis and applied-and-computational-mathematics but none to differential-equations before this write.
Additional Source Clay Mathematics Institutehttps://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf
Includes
Source Paul Painlevé (Wikipedia)
Source Peano existence theorem, Wikipedia
Source Peter Lax (Wikipedia)
Source Pokhozhaev's identity - Wikipedia
Source Wikipedia: Rossler attractor
Source Wikipedia: Samuel Earnshaw
Source Sergei Bernstein, Biography (MacTutor History of Mathematics)
Source Sergei Sobolev (Wikipedia)
Source Shing-Tung Yau (Wikipedia)
Source Siméon Denis Poisson (Wikipedia)
Source Sofya Kovalevskaya (Wikipedia)
Source Sophus Lie (Wikipedia)
Source Spherical Harmonics (Wikipedia)
Source Wolfram MathWorld: Superposition Principle
Source Terence Tao (Wikipedia)
Source Vladimir Arnold, Biography (MacTutor History of Mathematics)
Showing 51 to 66 of 66 in Includes, page 2 of 2.
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1. Differential Equation (Wikipedia)
Wikimedia Foundationlead paragraph
Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics is concerned with finding solutions, either directly or approximately.
Lead section, pure or applied classification
Such relations are common in mathematical models and scientific laws; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
View the Source Dynamical System (Wikipedia)
Wikimedia FoundationAssociated With: Dynamical Systems, lead paragraphQuote, Associated With: Dynamical Systems, lead paragraph
a dynamical system is the description of how a system evolves in time
View the Source Clay Mathematics Institute
Clay Mathematics InstituteAssociated With: Navier-Stokes Existence and Smoothness, https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdfQuote, Associated With: Navier-Stokes Existence and Smoothness, https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf
Since we don't even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem.
View the Source Gronwall's inequality, Wikipedia
Includes: Gronwall's Inequality, LeadQuote, Includes: Gronwall's Inequality, Lead
allows one to bound a function that is known to satisfy a certain differential or integral inequality
View the Source Pokhozhaev's identity - Wikipedia
Includes: Pokhozhaev's Identity, LeadQuote, Includes: Pokhozhaev's Identity, Lead
integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein-Gordon equation
View the Source MacTutor History of Mathematics: Charles-François Sturm
Includes: Jacques Charles François Sturm, Biography textQuote, Includes: Jacques Charles François Sturm, Biography text
Papers of 1836 - 1837 by Sturm and Liouville on differential equations involved expansions of functions in series
View the Source Johann Friedrich Pfaff (MacTutor)
Includes: Johann Friedrich Pfaff, Biography, work in analysisQuote, Includes: Johann Friedrich Pfaff, Biography, work in analysis
Pfaff did important work in analysis working on partial differential equations, special functions and the theory of series.
View the Source Jean Leray, Wikipedia
Includes: Jean Leray, LeadQuote, Includes: Jean Leray, Lead
was a French mathematician, who worked on both partial differential equations and algebraic topology
View the Source Arnaud Denjoy (Wikipedia)
Includes: Arnaud Denjoy, Biography, contributions, differential equationsQuote, Includes: Arnaud Denjoy, Biography, contributions, differential equations
His contributions include work in harmonic analysis and differential equations
View the Source Balthasar van der Pol (MacTutor)
Includes: Balthasar van der Pol, Biography, contributionsQuote, Includes: Balthasar van der Pol, Biography, contributions
his papers covered number theory, special functions, operational calculus and nonlinear differential equations
View the Source Sergei Sobolev (Wikipedia)
Includes: Sergei Sobolev, LeadQuote, Includes: Sergei Sobolev, Lead
was a Soviet mathematician working in mathematical analysis and partial differential equations
View the Source Gabriel Lamé (Wikipedia)
Includes: Gabriel Lamé, LeadQuote, Includes: Gabriel Lamé, Lead
contributed to the theory of partial differential equations
View the Source Norman Levinson (Wikipedia)
Includes: Norman Levinson, Lead, fields of contribution (differential equations)Quote, Includes: Norman Levinson, Lead, fields of contribution (differential equations)
Some of his major contributions were in the study of Fourier transforms, complex analysis, non-linear differential equations, number theory, and signal processing.
View the Source Emil Hilb (Wikipedia)
Includes: Emil Hilb, LeadQuote, Includes: Emil Hilb, Lead
worked in the fields of special functions, differential equations, and difference equations
View the Source Maxime Bôcher (Wikipedia)
Includes: Maxime Bôcher, Lead sentenceQuote, Includes: Maxime Bôcher, Lead sentence
published about 100 papers on differential equations
View the Source Peter Lax (Wikipedia)
Includes: Peter Lax, CareerQuote, Includes: Peter Lax, Career
Lax made contributions to the theory of hyperbolic partial differential equations.
View the Source Cédric Villani (Wikipedia)
Includes: Cédric Villani, LeadQuote, Includes: Cédric Villani, Lead
a French mathematician and politician working primarily on partial differential equations, Riemannian geometry and mathematical physics
View the Source Aleksandr Nikolaevich Korkin (MacTutor)
Includes: Aleksandr Korkin, Biography, mathematical workQuote, Includes: Aleksandr Korkin, Biography, mathematical work
One of Korkin's major contributions was to the development of partial differential equations.
View the Source Anders Lindstedt (Wikipedia)
Includes: Anders Lindstedt, Life and work, celestial mechanicsQuote, Includes: Anders Lindstedt, Life and work, celestial mechanics
a technique for uniformly approximating periodic solutions to ordinary differential equations when regular perturbation approaches fail
View the Source Edward Burr Van Vleck (MacTutor)
Includes: Edward Burr Van Vleck, Biography, research papersQuote, Includes: Edward Burr Van Vleck, Biography, research papers
Almost all Van Vleck's research papers were in the fields of function theory and differential equations.
View the Source Joseph Ritt (Wikipedia)
Includes: Joseph Ritt, Lead, differential equationsQuote, Includes: Joseph Ritt, Lead, differential equations
his work on the theory of differential equations from the algebraic viewpoint, which he called differential algebra
View the Source Charles Riquier (Wikipedia)
Includes: Charles Riquier, Career, field of workQuote, Includes: Charles Riquier, Career, field of work
among the greatest pioneers of differential algebra and symbolic computation for systems of partial differential equations
View the Source Emanuel Beke (Wikipedia)
Includes: Emanuel Beke, Lead, field of workQuote, Includes: Emanuel Beke, Lead, field of work
specializing in differential equations, determinants, and mathematical physics
View the Source Eugenio Calabi (Wikipedia)
Includes: Eugenio Calabi, LeadQuote, Includes: Eugenio Calabi, Lead
specializing in differential geometry, partial differential equations and their applications
View the Source Mikhail Lavrentyev (Wikipedia)
Includes: Mikhail Lavrentyev, Mid careerQuote, Includes: Mikhail Lavrentyev, Mid career
His main contributions relate to conformal mappings and partial differential equations.
View the Source Sofya Kovalevskaya (Wikipedia)
Includes: Sofya Kovalevskaya, Lead, differential equationsQuote, Includes: Sofya Kovalevskaya, Lead, differential equations
partial differential equations and mechanics
View the Source Paul Painlevé (Wikipedia)
Includes: Paul Painlevé, Biography, early lifeQuote, Includes: Paul Painlevé, Biography, early life
Painlevé's mathematical work on differential equations
View the Source Mihailo Petrović Alas (Wikipedia)
Includes: Mihailo Petrović Alas, LeadQuote, Includes: Mihailo Petrović Alas, Lead
contributed significantly to the study of differential equations
View the Source Lazarus Fuchs (Wikipedia)
Includes: Lazarus Fuchs, LeadQuote, Includes: Lazarus Fuchs, Lead
important contributions to the field of linear differential equations
View the Source Joseph Petzval (Wikipedia)
Includes: Joseph Petzval, Academic work, acousticsQuote, Includes: Joseph Petzval, Academic work, acoustics
differential equations of the string oscillations
View the Source Oliver Heaviside (Wikipedia)
Includes: Oliver Heaviside, LeadQuote, Includes: Oliver Heaviside, Lead
who invented a new technique for solving differential equations
View the Source George William Hill (Wikipedia)
Includes: George William Hill, LeadQuote, Includes: George William Hill, Lead
to the theory of ordinary differential equations
View the Source Shing-Tung Yau (Wikipedia)
Includes: Shing-Tung Yau, LeadQuote, Includes: Shing-Tung Yau, Lead
in recognition of his contributions to partial differential equations
View the Source Olga Ladyzhenskaya (Wikipedia)
Includes: Olga Ladyzhenskaya, LeadQuote, Includes: Olga Ladyzhenskaya, Lead
who worked on partial differential equations, fluid dynamics
View the Source Lars Hörmander (Wikipedia)
Includes: Lars Hörmander, LeadQuote, Includes: Lars Hörmander, Lead
the foremost contributor to the modern theory of linear partial differential equations
View the Source Gabriele Manfredi (Wikipedia)
Includes: Gabriele Manfredi, BiografiaQuote, Includes: Gabriele Manfredi, Biografia
teoria delle equazioni differenziali
View the Source Calixto Pedro Calderón (MacTutor)
Includes: Calixto Calderón, Summary, differential equationsQuote, Includes: Calixto Calderón, Summary, differential equations
ordinary differential equations, partial differential equations, harmonic analysis, mathematical biology and the history of mathematics
View the Source Florimond de Beaune (Wikipedia)
Includes: Florimond de Beaune, Lead, field of workQuote, Includes: Florimond de Beaune, Lead, field of work
In a 1638 letter to Descartes, de Beaune posed a problem that can be reformulated in modern notation as solving the differential equation dydx=αy−x now seen as the first example of the inverse tangent method of deducing properties of a curve from its tangents.
View the Source Jacopo Riccati (Wikipedia)
Includes: Jacopo Riccati, Lead, field of workQuote, Includes: Jacopo Riccati, Lead, field of work
Jacopo Francesco Riccati (28 May 1676-15 April 1754) was a Venetian mathematician and jurist from Venice, known for his widely influential work on solving differential equations.
View the Source Nicolas Fatio de Duillier (Wikipedia)
Includes: Nicolas Fatio de Duillier, Lead, field of workQuote, Includes: Nicolas Fatio de Duillier, Lead, field of work
phenomenon of zodiacal light, for his "push" or "shadow" theory of gravitation, for inventing the integrating factor as a method for solving ordinary differential equations, for his close association with both Christiaan Huygens and Isaac Newton,
View the Source Nicolaus I Bernoulli (Wikipedia)
Includes: Nicolaus I Bernoulli, Lead, field of workQuote, Includes: Nicolaus I Bernoulli, Lead, field of work
In 1716 he obtained the Galileo-chair at the University of Padua, where he worked on differential equations and geometry.
View the Source Nicolaus II Bernoulli (Wikipedia)
Includes: Nicolaus II Bernoulli, Lead, field of workQuote, Includes: Nicolaus II Bernoulli, Lead, field of work
Nicolaus worked mostly on curves, differential equations, and probability.
View the Source Lyapunov Function (Wikipedia)
Includes: Lyapunov Function, Lead sentenceQuote, Includes: Lyapunov Function, Lead sentence
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions
View the Source Wikipedia: Rossler attractor
Wikipedia: Lotka-Volterra equations
Cauchy-Kovalevskaya Theorem (Wikipedia)
Wikimedia FoundationIncludes: Cauchy-Kovalevskaya Theorem, Lead sentenceQuote, Includes: Cauchy-Kovalevskaya Theorem, Lead sentence
tence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems.
View the Source Peano existence theorem, Wikipedia
Feynman-Kac formula, Wikipedia
Includes: Feynman-Kac Formula, Lead sentenceQuote, Includes: Feynman-Kac Formula, Lead sentence
Kac, establishes a link between parabolic partial differential equations and stochastic processes.
View the Source Sophus Lie (Wikipedia)
Includes: Sophus Lie, Lead paragraph [in-branch 2]Quote, Includes: Sophus Lie, Lead paragraph [in-branch 2]
differential equations
View the Source Ferdinand Georg Frobenius (Wikipedia)
Includes: Ferdinand Georg Frobenius, Lead paragraph [in-branch 3]Quote, Includes: Ferdinand Georg Frobenius, Lead paragraph [in-branch 3]
differential equations
View the Source Siméon Denis Poisson (Wikipedia)
Includes: Siméon Denis Poisson, Lead paragraph [in-branch 3]Quote, Includes: Siméon Denis Poisson, Lead paragraph [in-branch 3]
partial differential equations
View the Source Eugenio Elia Levi (Wikipedia)
Includes: Eugenio Elia Levi, Lead paragraph [in-branch 3]Quote, Includes: Eugenio Elia Levi, Lead paragraph [in-branch 3]
partial differential operators
View the Source Wikipedia: Bernard Malgrange
Includes: Bernard Malgrange, LeadQuote, Includes: Bernard Malgrange, Lead
worked on differential equations and singularity theory
View the Source Wikipedia: Oskar Perron
Includes: Oskar Perron, LeadQuote, Includes: Oskar Perron, Lead
He made numerous contributions to differential equations and partial differential equations
View the Source Wikipedia: Samuel Earnshaw
Includes: Samuel Earnshaw, LeadQuote, Includes: Samuel Earnshaw, Lead
Other topics included optics, calculus, trigonometry and partial differential equations in mathematics. == See also == Cotes's spiral == References == == External links == Samuel Earnshaw Archived 12 April 2016 at the Wayback Machine
View the Source Terence Tao (Wikipedia)
Includes: Terence Tao, Lead paragraphQuote, Includes: Terence Tao, Lead paragraph
He was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory.
View the Source Grothendieck-Katz p-curvature conjecture (Wikipedia)
Includes: Grothendieck-Katz p-curvature Conjecture, Lead paragraphQuote, Includes: Grothendieck-Katz p-curvature Conjecture, Lead paragraph
In mathematics, the Grothendieck-Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case.
View the Source Duffing equation (Wikipedia)
Includes: Duffing Equation, Lead paragraphQuote, Includes: Duffing Equation, Lead paragraph
The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861-1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators.
View the Source Wikipedia: Airy function
Includes: Airy Function, Lead paragraphQuote, Includes: Airy Function, Lead paragraph
In fact, the Airy equation is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential).
View the Source Spherical Harmonics (Wikipedia)
Includes: Spherical Harmonics, Lead paragraphQuote, Includes: Spherical Harmonics, Lead paragraph
They are often employed in solving partial differential equations in many scientific fields.
View the Source Lamé Function (Wikipedia)
Includes: Lamé Function, Lead paragraphQuote, Includes: Lamé Function, Lead paragraph
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation.
View the Source Black-Scholes model (Wikipedia)
Includes: Black-Scholes model, Lead paragraphQuote, Includes: Black-Scholes model, Lead paragraph
From the parabolic partial differential equation in the model, known as the Black-Scholes equation, one can deduce the Black-Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate).
View the Source Abel's identity - Wikipedia
Includes: Abel's Identity, Lead paragraphQuote, Includes: Abel's Identity, Lead paragraph
In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation.
View the Source Wolfram MathWorld: Characteristic
Wolfram MathWorldIncludes: Characteristic (algebra), Subject classificationsQuote, Includes: Characteristic (algebra), Subject classifications
Calculus and Analysis Differential Equations
View the Source Wolfram MathWorld: Superposition Principle
Wolfram MathWorldIncludes: Superposition Principle, Subject classificationsQuote, Includes: Superposition Principle, Subject classifications
Calculus and Analysis Differential Equations
View the Source MacTutor biography of Kurt Otto Friedrichs, St Andrews
Includes: Kurt Otto Friedrichs, SummaryQuote, Includes: Kurt Otto Friedrichs, Summary
Kurt Friedrichs was a German American mathematician who worked on many aspects of partial differential equations.
View the Source Marcel Riesz, Biography (MacTutor History of Mathematics)
Includes: Marcel Riesz, SummaryQuote, Includes: Marcel Riesz, Summary
Marcel Riesz was a Hungarian-born mathematician who worked on summation methods, potential theory and other parts of analysis, as well as number theory and partial differential equations.
View the Source Sergei Bernstein, Biography (MacTutor History of Mathematics)
Includes: Sergei Bernstein, SummaryQuote, Includes: Sergei Bernstein, Summary
Sergei Bernstein was a Ukrainian mathematician who made important contributions to partial differential equations, differential geometry, probability theory and approximation theory.
View the Source Vladimir Arnold, Biography (MacTutor History of Mathematics)
Includes: Vladimir Arnold, SummaryQuote, Includes: Vladimir Arnold, Summary
Vladimir Arnold is a Ukrainian-born mathematician who won a Wolf prize for his work on dynamical systems, differential equations and singularity theory.
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