Definitions for partial differential equation
par·tial dif·fer·en·tial equa·tion

Princeton's WordNetRate this definition:0.0 / 0 votes

1. partial differential equationnoun

   a differential equation involving a functions of more than one variable

WiktionaryRate this definition:0.0 / 0 votes

1. partial differential equationnoun

   a differential equation that involves the partial derivatives of a function of several variables

WikipediaRate this definition:0.0 / 0 votes

1. Partial differential equation

   In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of 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. Among the many open questions are the 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, electrodynamics, thermodynamics, fluid dynamics, 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, 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 "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations 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 and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.

ChatGPTRate this definition:0.0 / 0 votes

1. partial differential equation

   A partial differential equation is a type of mathematical equation that involves unknown multivariable functions and their partial derivatives. These equations are used to formulate problems that involve functions of several variables and are typically applied in physics, engineering and computational modeling to represent the physics of various phenomena.

WikidataRate this definition:0.0 / 0 votes

1. Partial differential equation

   In mathematics, a partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, or elasticity. These seemingly distinct physical phenomena can be formalised identically in terms of PDEs, which shows that they are governed by the same underlying dynamic. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.

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1. • Differential Equation

How to pronounce partial differential equation?

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How to say partial differential equation in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of partial differential equation in Chaldean Numerology is: 7

2. Pythagorean Numerology

   The numerical value of partial differential equation in Pythagorean Numerology is: 9

Examples of partial differential equation in a Sentence

1. Clifford Truesdell:

   How did Biot arrive at the partial differential equation? [the heat conduction equation] . . . Perhaps Laplace gave Biot the equation and left him to sink or swim for a few years in trying to derive it. That would have been merely an instance of the way great mathematicians since the very beginnings of mathematical research have effortlessly maintained their superiority over ordinary mortals.

References

1. ^ Princeton's WordNet
   http://wordnetweb.princeton.edu/perl/webwn?s=partial differential equation
2. ^ Wiktionary
   https://en.wiktionary.org/wiki/Partial_Differential_Equation
3. ^ Wikipedia
   https://en.wikipedia.org/wiki/Partial_Differential_Equation
4. ^ ChatGPT
   https://chat.openai.com
5. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=partial differential equation