Editing Partial differential equation (section)

{{short description|Type of differential equation}}
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[[File:Heat.gif|thumb|A visualisation of a solution to the two-dimensional [[heat equation]] with temperature represented by the vertical direction and color.]]
{{Differential equations}}

In [[mathematics]], a '''partial differential equation''' ('''PDE''') is an equation which involves a [[Function_of_several_real_variables|multivariable function]] and one or more of its [[partial derivative]]s.

The function is often thought of as an "unknown" that solves the equation, similar to how {{mvar|x}} is thought of as an unknown number solving, e.g., an [[algebraic equation]] like {{math|1=''x''<sup>2</sup> − 3''x'' + 2 = 0}}. However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to [[Numerical methods for partial differential equations|numerically approximate]] solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of [[pure mathematics|pure mathematical research]], in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability.<ref>{{Cite web |title=Regularity and singularities in elliptic PDE's: beyond monotonicity formulas {{!}} EllipticPDE Project {{!}} Fact Sheet {{!}} H2020 |url=https://cordis.europa.eu/project/id/801867 |access-date=2024-02-05 |website=CORDIS {{!}} European Commission |language=en}}</ref> Among the many open questions are the [[Navier–Stokes existence and smoothness|existence and smoothness]] of solutions to the [[Navier–Stokes equations]], named as one of the [[Millennium Prize Problems]] in 2000.

Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as [[physics]] and [[engineering]]. For instance, they are foundational in the modern scientific understanding of [[sound]], [[heat]], [[diffusion]], [[electrostatics]], [[electromagnetism|electrodynamics]], [[thermodynamics]], [[fluid dynamics]], [[elasticity (physics)|elasticity]], [[general relativity]], and [[quantum mechanics]] ([[Schrödinger equation]], [[Pauli equation]] etc.). They also arise from many purely mathematical considerations, such as [[differential geometry]] and the [[calculus of variations]]; among other notable applications, they are the fundamental tool in the proof of the [[Poincaré conjecture]] from [[geometric topology]].

Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, where the meaning of a solution depends on the context of the problem, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "universal theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.<ref>{{cite book |authorlink=Sergiu Klainerman |last=Klainerman |first=Sergiu |year=2010 |chapter=PDE as a Unified Subject |editor-last=Alon |editor-first=N. |editor2-last=Bourgain |editor2-first=J. |editor3-last=Connes |editor3-first=A. |editor4-last=Gromov |editor4-first=M. |editor5-last=Milman |editor5-first=V. |title=Visions in Mathematics |series=Modern Birkhäuser Classics |publisher=Birkhäuser |location=Basel |pages=279–315 |isbn=978-3-0346-0421-5 |doi=10.1007/978-3-0346-0422-2_10 }}</ref>

[[Ordinary differential equation]]s can be viewed as a subclass of partial differential equations, corresponding to [[Function of a real variable|functions of a single variable]]. [[Stochastic partial differential equation]]s and [[Fractional calculus|nonlocal equation]]s are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include [[elliptic partial differential equation|elliptic]] and [[parabolic partial differential equation|parabolic]] partial differential equations, [[fluid mechanics]], [[Boltzmann equation]]s, and [[dispersive partial differential equation]]s.<ref>{{Cite book |last=Erdoğan |first=M. Burak |url=https://www.cambridge.org/core/books/dispersive-partial-differential-equations/2DC65286BA080B54EB659E42A553CA88 |title=Dispersive Partial Differential Equations: Wellposedness and Applications |last2=Tzirakis |first2=Nikolaos |date=2016 |publisher=Cambridge University Press |isbn=978-1-107-14904-5 |series=London Mathematical Society Student Texts |location=Cambridge}}</ref>