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Inmathematics, apartial differential equation (PDE) is an equation which involves amultivariable function and one or more of itspartial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to howx is thought of as an unknown number solving, e.g., analgebraic equation likex2 − 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 tonumerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector ofpure 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.[1] Among the many open questions are theexistence and smoothness of solutions to theNavier–Stokes equations, named as one of theMillennium Prize Problems in 2000.
Partial differential equations are ubiquitous in mathematically oriented scientific fields, such asphysics andengineering. For instance, they are foundational in the modern scientific understanding ofsound,heat,diffusion,electrostatics,electrodynamics,thermodynamics,fluid dynamics,elasticity,general relativity, andquantum mechanics (Schrödinger equation,Pauli equation etc.). They also arise from many purely mathematical considerations, such asdifferential geometry and thecalculus of variations; among other notable applications, they are the fundamental tool in the proof of thePoincaré conjecture fromgeometric 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.[2]
Ordinary differential equations can be viewed as a subclass of partial differential equations, corresponding tofunctions of a single variable.Stochastic partial differential equations andnonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, includeelliptic andparabolic partial differential equations,fluid mechanics,Boltzmann equations, anddispersive partial differential equations.[3]
A functionu(x,y,z) of three variables is "harmonic" or "a solution of theLaplace equation" if it satisfies the conditionSuch functions were widely studied in the 19th century due to their relevance forclassical mechanics, for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function. If explicitly given a function, it is usually a matter of straightforward computation to check whether or not it is harmonic. For instance andare both harmonic whileis not. It may be surprising that the two examples of harmonic functions are of such strikingly different form. This is a reflection of the fact that they arenot, in any immediate way, special cases of a "general solution formula" of the Laplace equation. This is in striking contrast to the case ofordinary differential equations (ODEs)roughly similar to the Laplace equation, with the aim of many introductory textbooks being to find algorithms leading to general solution 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 case of the following PDE: for a functionv(x,y) of two variables, consider the equationIt can be directly checked that any functionv of the formv(x,y) =f(x) +g(y), for any single-variable functionsf andg whatsoever, will satisfy this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. 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 ofexistence 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 in order 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 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 thedomain 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 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 the second PDE, one has the free prescription of two functions.
Even more phenomena are possible. For instance, thefollowing PDE, arising naturally in the field ofdifferential 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.
In contrast to the earlier examples, this PDE isnonlinear, owing to the square roots and the squares. Alinear 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.
A partial differential equation is an equation that involves an unknown function of variables and (some of) its partial derivatives.[4] That is, for the unknown functionof variables belonging to the open subset of, the-order partial differential equation is defined aswhereand is the partial derivative operator.
When writing PDEs, it is common to denote partial derivatives using subscripts. For example:In the general situation thatu is a function ofn variables, thenui denotes the first partial derivative relative to thei-th input,uij denotes the second partial derivative relative to thei-th andj-th inputs, and so on.
The Greek letterΔ denotes theLaplace operator; ifu is a function ofn variables, thenIn the physics literature, the Laplace operator is often denoted by∇2; in the mathematics literature,∇2u may also denote theHessian matrix ofu.
A PDE is calledlinear if it is linear in the unknown and its derivatives. For example, for a functionu ofx andy, a second order linear PDE is of the formwhereai andf are functions of the independent variablesx andy only. (Often the mixed-partial derivativesuxy anduyx will be equated, but this is not required for the discussion of linearity.)If theai are constants (independent ofx andy) then the PDE is calledlinear with constant coefficients. Iff is zero everywhere then the linear PDE ishomogeneous, otherwise it isinhomogeneous. (This is separate fromasymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)
Nearest to linear PDEs aresemi-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
In aquasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:Many of the fundamental PDEs in physics are quasilinear, such as theEinstein equations ofgeneral relativity and theNavier–Stokes equations describing fluid motion.
A PDE without any linearity properties is calledfullynonlinear, and possesses nonlinearities on one or more of the highest-order derivatives. An example is theMonge–Ampère equation, which arises indifferential geometry.[5]
The elliptic/parabolic/hyperbolic classification provides a guide to appropriateinitial- andboundary conditions and to thesmoothness of the solutions. Assuminguxy =uyx, the general linear second-order PDE in two independent variables has the formwhere the coefficientsA,B,C... may depend uponx andy. IfA2 +B2 +C2 > 0 over a region of thexy-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
More precisely, replacing∂x byX, and likewise for other variables (formally this is done by aFourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (ahomogeneous polynomial, here aquadratic form) being most significant for the classification.
Just as one classifiesconic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on thediscriminantB2 − 4AC, the same can be done for a second-order PDE at a given point. However, thediscriminant in a PDE is given byB2 −AC due to the convention of thexy term being2B rather thanB; formally, the discriminant (of the associated quadratic form) is(2B)2 − 4AC = 4(B2 −AC), with the factor of 4 dropped for simplicity.
If there aren independent variablesx1,x2, …,xn, a general linear partial differential equation of second order has the form
The classification depends upon the signature of theeigenvalues of the coefficient matrixai,j.
The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of theLaplace equation, theheat equation, and thewave equation.
However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as theEuler–Tricomi equation; varying from elliptic to hyperbolic for differentregions of the domain, as well as higher-order PDEs, but such knowledge is more specialized.
The classification of partial differential equations can be extended to systems of first-order equations, where the unknownu is now avector withm components, and the coefficient matricesAν arem bym matrices forν = 1, 2, …,n. The partial differential equation takes the formwhere the coefficient matricesAν and the vectorB may depend uponx andu. If ahypersurfaceS is given in the implicit formwhereφ has a non-zero gradient, thenS is acharacteristic surface for theoperatorL at a given point if the characteristic form vanishes:
The geometric interpretation of this condition is as follows: if data foru are prescribed on the surfaceS, then it may be possible to determine the normal derivative ofu onS from the differential equation. If the data onS and the differential equation determine the normal derivative ofu onS, thenS is non-characteristic. If the data onS and the differential equationdo not determine the normal derivative ofu onS, then the surface ischaracteristic, and the differential equation restricts the data onS: the differential equation isinternal toS.
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it isthe solution (this also applies to ODEs). We assume as anansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.[8]
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.
This is possible for simple PDEs, which are calledseparable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond todiagonal matrices – thinking of "the value for fixedx" as a coordinate, each coordinate can be understood separately.
This generalizes to themethod of characteristics, and is also used inintegral transforms.
The characteristic surface inn =2-dimensional space is called acharacteristic curve.[9] In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called themethod of characteristics.
More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.
Anintegral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An important example of this isFourier analysis, which diagonalizes the heat equation using theeigenbasis of sinusoidal waves.
If the domain is finite or periodic, an infinite sum of solutions such as aFourier series is appropriate, but an integral of solutions such as aFourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.
Often a PDE can be reduced to a simpler form with a known solution by a suitablechange of variables. For example, theBlack–Scholes equationis reducible to theheat equationby the change of variables[10]
Inhomogeneous equations[clarification needed] can often be solved (for constant coefficient PDEs, always be solved) by finding thefundamental solution (the solution for a point source), then taking theconvolution with the boundary conditions to get the solution.
This is analogous insignal processing to understanding a filter by itsimpulse response.
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for examplesinx + sinx = 2 sinx. The same principle can be observed in PDEs where the solutions may be real or complex and additive. Ifu1 andu2 are solutions of linear PDE in some function spaceR, thenu =c1u1 +c2u2 with any constantsc1 andc2 are also a solution of that PDE in the same function space.
There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as theCauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part ofanalysis).
Nevertheless, some techniques can be used for several types of equations. Theh-principle is the most powerful method to solveunderdetermined equations. TheRiquier–Janet theory is an effective method for obtaining information about many analyticoverdetermined systems.
Themethod of characteristics can be used in some very special cases to solve nonlinear partial differential equations.[11]
In some cases, a PDE can be solved viaperturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives arenumerical analysis techniques from simplefinite difference schemes to the more maturemultigrid andfinite element methods. Many interesting problems in science and engineering are solved in this way usingcomputers, sometimes high performancesupercomputers.
From 1870Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now calledLie groups, be referred, to a common source; and that ordinary differential equations which admit the sameinfinitesimal transformations present comparable difficulties of integration. He also emphasized the subject oftransformations of contact.
A general approach to solving PDEs uses the symmetry property of differential equations, the continuousinfinitesimal transformations of solutions to solutions (Lie theory). Continuousgroup theory,Lie algebras anddifferential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find itsLax pairs, recursion operators,Bäcklund transform and finally finding exact analytic solutions to the PDE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
TheAdomian decomposition method,[12] theLyapunov artificial small parameter method, and hishomotopy perturbation method are all special cases of the more generalhomotopy analysis method.[13] These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well knownperturbation theory, thus giving these methods greater flexibility and solution generality.
The three most widely usednumerical methods to solve PDEs are thefinite element method (FEM),finite volume methods (FVM) andfinite difference methods (FDM), as well other kind of methods calledmeshfree methods, which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order versionhp-FEM. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM),extended finite element method (XFEM),spectral finite element method (SFEM),meshfree finite element method,discontinuous Galerkin finite element method (DGFEM),element-free Galerkin method (EFGM),interpolating element-free Galerkin method (IEFGM), etc.
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for approximating solutions of partial differential equations (PDE) as well as of integral equations using a finite set of functions.[14][15] The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.
Finite-difference methods are numerical methods for approximating the solutions to differential equations usingfinite difference equations to approximate derivatives.
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using thedivergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.
Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion ofdistributions.
An example[19] for the definition of a weak solution is as follows:
Consider the boundary-value problem given by: where denotes a second-order partial differential operator indivergence form.
We say a is a weak solution if for every, which can be derived by a formal integral by parts.
An example for a weak solution is as follows: is a weak solution satisfying in distributional sense, as formally,
As a branch of pure mathematics, the theoretical studies of PDEs focus on the criteria for a solution to exist, the properties of a solution, and finding its formula is often secondary.
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:
This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.
Regularity refers to the integrability and differentiability of weak solutions, which can often be represented bySobolev spaces.
This problem arise due to the difficulty in searching for classical solutions. Researchers often tend to find weak solutions at first and then find out whether it is smooth enough to be qualified as a classical solution.
Results fromfunctional analysis are often used in this field of study.
Some common PDEs
Types of boundary conditions
Various topics
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