Partial differential equation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a partial differential equation (PDE) is an equation which involves a <a href="/facts/Function_of_several_real_variables/cyujR2q2">multivariable function</a> and one or more of its <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a>.
The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an <a href="/facts/Algebraic_equation/1QauIGxs">algebraic equation</a> like x2 − 3x + 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 <a href="/facts/Numerical_methods_for_partial_differential_equations/69kwj7mL">numerically approximate</a> solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of <a href="/facts/Pure_mathematics/V29nQC1C">pure mathematical research</a>, 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. Among the many open questions are the <a href="/facts/Navier%E2%80%93Stokes_existence_and_smoothness/5aDcIPCw">existence and smoothness</a> of solutions to the <a href="/facts/Navier%E2%80%93Stokes_equations/48pQugEa">Navier–Stokes equations</a>, named as one of the <a href="/facts/Millennium_Prize_Problems/WWSB5ddp">Millennium Prize Problems</a> in 2000.
Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a>. For instance, they are foundational in the modern scientific understanding of <a href="/facts/Sound/DlKXlPuF">sound</a>, <a href="/facts/Heat/FjDGssCs">heat</a>, <a href="/facts/Diffusion/KbAxroBa">diffusion</a>, <a href="/facts/Electrostatics/bxkuW6s4">electrostatics</a>, <a href="/facts/Electromagnetism/oAkmALHT">electrodynamics</a>, <a href="/facts/Thermodynamics/q4hIx3yP">thermodynamics</a>, <a href="/facts/Fluid_dynamics/y7uzSCfg">fluid dynamics</a>, <a href="/facts/Elasticity_(physics)/wcJoIFg1">elasticity</a>, <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, and <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> (<a href="/facts/Schr%C3%B6dinger_equation/3csgSIWF">Schrödinger equation</a>, <a href="/facts/Pauli_equation/gdPIf0cw">Pauli equation</a> etc.). They also arise from many purely mathematical considerations, such as <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a> and the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>; among other notable applications, they are the fundamental tool in the proof of the <a href="/facts/Poincar%C3%A9_conjecture/mxk2Pju3">Poincaré conjecture</a> from <a href="/facts/Geometric_topology/NpsqSK2N">geometric topology</a>.
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.
<a href="/facts/Ordinary_differential_equation/ygKGQ8kD">Ordinary differential equations</a> can be viewed as a subclass of partial differential equations, corresponding to <a href="/facts/Function_of_a_real_variable/bbNG3DnN">functions of a single variable</a>. <a href="/facts/Stochastic_partial_differential_equation/3ARbPKSL">Stochastic partial differential equations</a> and <a href="/facts/Fractional_calculus/PxN1pMcj">nonlocal equations</a> are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include <a href="/facts/Elliptic_partial_differential_equation/pQx2wB7V">elliptic</a> and <a href="/facts/Parabolic_partial_differential_equation/gJQfRUlr">parabolic</a> partial differential equations, <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a>, <a href="/facts/Boltzmann_equation/6oRPrApT">Boltzmann equations</a>, and <a href="/facts/Dispersive_partial_differential_equation/SDjD3aVj">dispersive partial differential equations</a>.

Partial differential equation open-in-new

Partial differential equation