Partial differential equation

In mathematics, a partial differential equation PDE is an equation which imposes relations between a 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 + 2 = 0. However, it is ordinarily impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical & scientific research on methods to ] Among the numerous 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(a) 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 occur from numerous purely mathematical considerations, such(a) as differential geometry and the calculus of variations; among other notable applications, they are the necessary tool in the proof of the Poincaré conjecture from geometric topology.

Partly due to this species of sources, there is a wide spectrum of different manner of partial differential equations, and methods realize been developed for dealing with many of the individual equations which arise. As such, it is ordinarily acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided up up between several essentially distinct subfields.

Ordinary differential equations work a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are, as of 2020, especially 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.

Introduction

One says that a function of three variables is "Laplace equation" whether it satisfies the condition $${\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.}$$ Such functions were widely studied in the nineteenth century due to their relevance for classical mechanics, for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function. whether explicitly assumption a function, this is the usually a matter of straightforward computation to check whether or not it is for harmonic. For instance $${\displaystyle ux,y,z={\frac {1}{\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}}}$$ and $${\displaystyle ux,y,z=2x^{2}-y^{2}-z^{2}}$$ are both harmonic while $${\displaystyle ux,y,z=\sinxy+z}$$ is not. It may be surprising that the two given examples of harmonic functions are of such(a) a strikingly different form from one another. This is a reflection of the fact that they are not, in all immediate way, both special cases of a "general sum formula" of the Laplace equation. This is in striking contrast to the effect of roughly similar to the Laplace equation, with the intention of many introductory textbooks being to find algorithms leading to general total formulas. For the Laplace equation, as for a large number of partial differential equations, such solution formulas fail to exist.

The nature of this failure can be seen more concretely in the effect of the following PDE: for a function of two variables, consider the equation $${\displaystyle {\frac {\partial ^{2}v}{\partial x\partial y}}=0.}$$ It can be directly checked that all function v of the form , for any single-variable functions f and g whatsoever, will satisfy this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free selection of some numbers. In the inspect of PDE, one broadly has the free choice of functions.

The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of existence and uniqueness theorems for ODE can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any offered solution formula, while the uniqueness half is often only shown in the background in layout to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also necessary when implementation a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the data processor to calculate.

To discuss such existence and uniqueness theorems, it is necessary to be precise approximately the domain of the "unknown function." Otherwise, speaking only in terms such as "a function of two variables," it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.

The following gives two classic examples of such existence and uniqueness theorems. Even though the two PDE in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for thePDE, one has the free prescription of two functions.

Even more phenomena are possible. For instance, the coming after or as a result of. PDE, arising naturally in the field of differential geometry, illustrates an example where there is a simple and totally explicit solution formula, but with the free choice of only three numbers and non even one function.

In contrast to the earlier examples, this PDE is nonlinear, owing to the square roots and the squares. A linear PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and all fixed multiples of any solution is also a solution.