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ristics 4 3 integral transform 4 4 change of variables 4 5 fundamental solution 4 6 superposition principle 4 7 methods for non linear equations 4 8 lie group method 4 9 semi analytical methods 5 numerical solutions toggle numerical solutions subsection 5 1 finite element method 5 2 finite difference method 5 3 finite volume method 5 4 neural networks 6 weak solutions 7 theoretical studies toggle theoretical studies subsection 7 1 well posedness 7 2 regularity 8 see also 9 notes 10 references 11 further reading 12 external links toggle the table of contents partial differential equation 51 languages العربية الدارجة asturianu български বাংলা català čeština чӑвашла deutsch ελληνικά esperanto español eesti euskara فارسی suomi français galego עברית हिन्दी magyar հայերեն bahasa indonesia italiano 日本語 한국어 македонски bahasa melayu မြန်မာဘာသာ nederlands norsk bokmål polski português română русский scots srpskohrvatski српскохрватски simple english slovenčina slovenščina shqip српски srpski svenska ไทย tagalog türkçe українська oʻzbekcha ўзбекча tiếng việt 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikibooks wikiquote wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia type of differential equation this article includes a list of general references but lacks sufficient corresponding inline citations please help improve this article by introducing more precise citations march 2023 learn how and when to remove this message differential equations scope fields natural sciences engineering astronomy physics chemistry biology geology applied mathematics continuum mechanics chaos theory dynamical systems social sciences economics population dynamics classification types ordinary partial differential algebraic integro differential fractional linear non linear by variable type dependent and independent variables autonomous coupled decoupled exact homogeneous nonhomogeneous features order operator notation relation to processes difference discrete analogue stochastic stochastic partial delay solution existence and uniqueness well posed problem picard lindelöf theorem peano existence theorem carathéodory s existence theorem cauchy kovalevskaya theorem general topics initial values boundary values dirichlet neumann robin cauchy periodic wronskian abel s identity sturm liouville theory floquet theory phase portrait stability lyapunov asymptotic exponential series solutions rate of convergence asymptotic series special functions numerical integration dirac delta function solution methods inspection method of characteristics ansatz euler exponential response formula finite difference crank nicolson finite element infinite element finite volume galerkin petrov galerkin green s function integrating factor integral transforms perturbation theory reduction of order runge kutta separation of variables undetermined coefficients variation of parameters wkb approximation people list newton leibniz jacob bernoulli d alembert clairaut euler lagrange laplace wroński fourier cauchy green dirichlet sturm liouville neumann robin boole kovalevskaya runge kutta lipschitz lindelöf picard jeffreys nicolson crank hrennikoff courant named equations v t e a visualisation of a solution to the two dimensional heat equation with temperature represented by the vertical direction and color in mathematics a partial differential equation pde is an equation which involves a multivariable function and one or more of its partial derivatives the function is often thought of as an unknown that solves the equation however it is often impossible to write down explicit formulas for solutions of partial differential equations hence there is a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of partial differential equations using computers partial differential equations also occupy a large sector of pure mathematical research where the focus is on the qualitative features of solutions of various partial differential equations such as existence uniqueness regularity and stability 1 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 occur very widely 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 types of partial differential equations many different methods have been developed for dealing with the individual equations which arise as such there is no universal theory of partial differential equations with specialist knowledge being divided between several distinct subfields 2 ordinary differential equations can be viewed as a subclass of partial differential equations corresponding to functions of a single variable stochastic partial differential equations and nonlocal equations are 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 3 introduction and examples edit one of the most important partial differential equations with many applications is laplace s equation for a function u x y z of three variables laplace s equation is 2 u x 2 2 u y 2 2 u z 2 0 displaystyle frac partial 2 u partial x 2 frac partial 2 u partial y 2 frac partial 2 u partial z 2 0 a function that obeys this equation is called a harmonic function such functions were widely studied in the 19th century due to their relevance for classical mechanics for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function it is usually a matter of straightforward computation to check whether or not a given function is harmonic for instance u x y z 1 x 2 2 x y 2 z 2 1 displaystyle u x y z frac 1 sqrt x 2 2x y 2 z 2 1 u x y z e 5 x sin 3 y cos 4 z displaystyle u x y z e 5x sin 3y cos 4z and u x y z 2 x 2 y 2 z 2 displaystyle u x y z 2x 2 y 2 z 2 are all harmonic while u x y z sin x y z displaystyle u x y z sin xy z is not it may be surprising that these examples of harmonic functions are of such different forms this is a reflection of the fact that they are not special cases of a general solution formula of laplace s equation this is in striking contrast to the case of many ordinary differential equations odes where many introductory textbooks aim to find methods leading to general solutions for laplace s equation as for a large number of partial differential equations such solution formulas do not exist this can also be seen in the case of the following pde for a function v x y of two variables consider the equation 2 v x y 0 displaystyle frac partial 2 v partial x partial y 0 it can be directly checked that any function v of the form v x y f x g y for any single variable differentiable functions f and g whatsoever satisfies this condition this is far beyond the choices available in ode solution formulas which typically only allow the free choice of some constants in the study of pdes one generally 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 proposed solution formula while the uniqueness half is often only present in the background to ensure that a proposed solution formula is as general as possible by contrast for pdes 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 fundamental when carrying out 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 computer to calculate to discuss such existence and uniqueness theorems it is necessary to be precise about 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 provides two classic examples of such existence and uniqueness theorems even though the two pdes in question are so similar there is a major difference in behavior for the first pde one has the free prescription of a single function while for the second pde one has the free prescription of two functions let b denote the unit radius disk around the origin in the plane for any continuous function u on the unit circle there is exactly one function u on b such that 2 u x 2 2 u y 2 0 displaystyle frac partial 2 u partial x 2 frac partial 2 u partial y 2 0 and whose restriction to the unit circle is given by u for any functions f and g on the real line r there is exactly one function u on r 1 1 such that 2 u x 2 2 u y 2 0 displaystyle frac partial 2 u partial x 2 frac partial 2 u partial y 2 0 and with u x 0 f x and u y x 0 g x for all values of x even more phenomena are possible for instance the following pde arising naturally in the field of differential geometry illustrates an example where there is a simple and completely explicit solution formula but with the free choice of only three numbers and not even one function if u is a function on r 2 with x u x 1 u x 2 u y 2 y u y 1 u x 2 u y 2 0 displaystyle frac partial partial x frac frac partial u partial x sqrt 1 left frac partial u partial x right 2 left frac partial u partial y right 2 frac partial partial y frac frac partial u partial y sqrt 1 left frac partial u partial x right 2 left frac partial u partial y right 2 0 then there are numbers a b and c with u x y ax by c 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 any constant multiple of any solution is also a solution definition edit a partial differential equation is an equation that involves an unknown function of n 2 displaystyle n geq 2 variables and some of its partial derivatives 4 that is for the unknown function u u r displaystyle u u rightarrow mathbb r of variables x x 1 x n displaystyle x x_ 1 dots x_ n belonging to the open subset u displaystyle u of r n displaystyle mathbb r n the k t h displaystyle k th order partial differential equation is defined as f d k u d k 1 u d u u x 0 displaystyle f d k u d k 1 u dots du u x 0 where f r n k r n k 1 r n r u r displaystyle f mathbb r n k times mathbb r n k 1 dots times mathbb r n times mathbb r times u rightarrow mathbb r and d displaystyle d is the derivative operator notation edit main article notation for differentiation partial derivatives working in r n displaystyle mathbb r n the partial derivatives of a function u displaystyle u can be denoted by u x i displaystyle frac partial u partial x_ i or with a subscript u x i displaystyle u_ x_ i for multiple derivatives multi index notation can be used thus if α α 1 α n displaystyle alpha alpha _ 1 dots alpha _ n the length of α displaystyle alpha is denoted by α α 1 α n displaystyle alpha alpha _ 1 cdots alpha _ n and the iterated partial is denoted by d α u k u x 1 α 1 x n α n displaystyle d alpha u frac partial k u partial x_ 1 alpha _ 1 cdots partial x_ n alpha _ n in the above definition 5 of a partial differential equation the powers of d displaystyle d are the tensors whose components are the partial derivatives of u displaystyle u for example d k u displaystyle d k u is a tensor having n k displaystyle n k components that that are an arrangement of the set d α u α k displaystyle d alpha u mid alpha k after accounting for commutativity of partial derivatives the greek letter δ denotes the laplace operator if u is a function of n variables then δ u u 11 u 22 u n n displaystyle delta u u_ 11 u_ 22 cdots u_ nn in the physics literature the laplace operator is often denoted by 2 in the mathematics literature 2 u may also denote the hessian matrix of u which is here denoted by d 2 u displaystyle d 2 u classification edit linear and nonlinear equations edit a pde is called linear if it is linear in the unknown and its derivatives for example for a function u of x and y a second order linear pde is of the form a 1 x y u x x a 2 x y u x y a 3 x y u y x a 4 x y u y y a 5 x y u x a 6 x y u y a 7 x y u f x y displaystyle a_ 1 x y u_ xx a_ 2 x y u_ xy a_ 3 x y u_ yx a_ 4 x y u_ yy a_ 5 x y u_ x a_ 6 x y u_ y a_ 7 x y u f x y where a i and f are functions of the independent variables x and y only often the mixed partial derivatives u xy and u yx will be equated but this is not required for the discussion of linearity if the a i are constants independent of x and y then the pde is called linear with constant coefficients if f is zero everywhere then the linear pde is homogeneous otherwise it is inhomogeneous this is separate from asymptotic homogenization which studies the effects of high frequency oscillations in the coefficients upon solutions to pdes nearest to linear pdes are semi linear pdes where only the highest order derivatives appear as linear terms with coefficients that are functions of the independent variables the lower order derivatives and the unknown function may appear arbitrarily for example a general second order semi linear pde in two variables is a 1 x y u x x a 2 x y u x y a 3 x y u y x a 4 x y u y y f u x u y u x y 0 displaystyle a_ 1 x y u_ xx a_ 2 x y u_ xy a_ 3 x y u_ yx a_ 4 x y u_ yy f u_ x u_ y u x y 0 in a quasilinear pde the highest order derivatives likewise appear only as linear terms but with coefficients possibly functions of the unknown and lower order derivatives a 1 u x u y u x y u x x a 2 u x u y u x y u x y a 3 u x u y u x y u y x a 4 u x u y u x y u y y f u x u y u x y 0 displaystyle a_ 1 u_ x u_ y u x y u_ xx a_ 2 ...
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