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 Cauchy-Kovalevskaya Theorem (Wikipedia)
Source Eugenio Elia Levi (Wikipedia)
Source Ferdinand Georg Frobenius (Wikipedia)
Source Feynman-Kac formula, Wikipedia
Source Wikipedia: Lotka-Volterra equations
Source Lyapunov Function (Wikipedia)
Source Peano existence theorem, Wikipedia
Source Wikipedia: Rossler attractor
Source Siméon Denis Poisson (Wikipedia)
Source Sophus Lie (Wikipedia)
Sources
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
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.
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
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.
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.
Sophus Lie (Wikipedia)
Includes: Sophus Lie, Lead paragraph [in-branch 2]Quote, Includes: Sophus Lie, Lead paragraph [in-branch 2]
differential equations
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
Siméon Denis Poisson (Wikipedia)
Includes: Simeon Denis Poisson, Lead paragraph [in-branch 3]Quote, Includes: Simeon Denis Poisson, Lead paragraph [in-branch 3]
partial differential equations
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
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