Editing Partial differential equation (section)

== Classification ==

=== Linear and nonlinear equations ===

A PDE is called '''linear''' if it is linear in the unknown and its derivatives. For example, for a function {{mvar|u}} of {{mvar|x}} and {{mvar|y}}, a second order linear PDE is of the form
<math display="block"> 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)
</math>
where {{math|''a<sub>i</sub>''}} and {{mvar|''f''}} are functions of the independent variables {{mvar|x}} and {{mvar|y}} only. (Often the mixed-partial derivatives {{math|''u<sub>xy</sub>''}} and {{math|''u<sub>yx</sub>''}} will be equated, but this is not required for the discussion of linearity.)
If the {{math|''a<sub>i</sub>''}} are constants (independent of {{mvar|x}} and {{mvar|y}}) then the PDE is called '''linear with constant coefficients'''. If {{mvar|''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
<math display="block"> 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 </math>

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:
<math display="block"> a_1(u_x, u_y, u, x, y)u_{xx} + a_2(u_x, u_y, u, x, y)u_{xy} + a_3(u_x, u_y, u, x, y)u_{yx} + a_4(u_x, u_y, u, x, y)u_{yy} + f(u_x, u_y, u, x, y) = 0 </math>
Many of the fundamental PDEs in physics are quasilinear, such as the [[Einstein equations]] of [[general relativity]] and the [[Navier–Stokes equations]] describing fluid motion.

A PDE without any linearity properties is called '''fully [[Nonlinear partial differential equation|nonlinear]]''', and possesses nonlinearities on one or more of the highest-order derivatives. An example is the [[Monge–Ampère equation]], which arises in [[differential geometry]].<ref name="PrincetonCompanion">{{Citation|last = Klainerman|first = Sergiu|year = 2008|title =Partial Differential Equations|editor-last1 = Gowers|editor-first1 = Timothy|editor-last2 = Barrow-Green|editor-first2 = June|editor-last3 = Leader|editor-first3 = Imre|encyclopedia = The Princeton Companion to Mathematics|pages = 455–483|publisher = Princeton University Press}}</ref>

=== Second order equations ===
The elliptic/parabolic/hyperbolic classification provides a guide to appropriate [[Initial_condition|initial-]] and [[Boundary value problem|boundary conditions]] and to the [[smoothness]] of the solutions. Assuming {{math|1=''u<sub>xy</sub>'' = ''u<sub>yx</sub>''}}, the general linear second-order PDE in two independent variables has the form
<math display="block">Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0,</math>
where the coefficients {{mvar|A}}, {{mvar|B}}, {{mvar|C}}... may depend upon {{mvar|x}} and {{mvar|y}}. If {{math|''A''<sup>2</sup> + ''B''<sup>2</sup> + ''C''<sup>2</sup> > 0}} over a region of the {{mvar|xy}}-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
<math display="block">Ax^2 + 2Bxy + Cy^2 + \cdots = 0.</math>

More precisely, replacing {{math|∂<sub>''x''</sub>}} by {{mvar|X}}, and likewise for other variables (formally this is done by a [[Fourier transform]]), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a [[homogeneous polynomial]], here a [[quadratic form]]) being most significant for the classification.

Just as one classifies [[conic section]]s and quadratic forms into parabolic, hyperbolic, and elliptic based on the [[discriminant]] {{math|''B''<sup>2</sup> − 4''AC''}}, the same can be done for a second-order PDE at a given point. However, the [[discriminant]] in a PDE is given by {{math|''B''<sup>2</sup> − ''AC''}} due to the convention of the {{mvar|xy}} term being {{math|2''B''}} rather than {{mvar|B}}; formally, the discriminant (of the associated quadratic form) is {{math|1=(2''B'')<sup>2</sup> − 4''AC'' = 4(''B''<sup>2</sup> − ''AC'')}}, with the factor of 4 dropped for simplicity.

# {{math|''B''<sup>2</sup> − ''AC'' < 0}} (''[[elliptic partial differential equation]]''): Solutions of [[elliptic partial differential equation|elliptic PDEs]] are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of [[Laplace's equation]] are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where {{math|''x'' < 0}}. By change of variables, the equation can always be expressed in the form: <math display="block">u_{xx} + u_{yy} + \cdots = 0 ,  </math>where x and y correspond to changed variables. This justifies [[Laplace's equation|Laplace equation]] as an example of this type.<ref name=":0">{{Cite web |last=Levandosky |first=Julie |title=Classification of Second-Order Equations |url=https://web.stanford.edu/class/math220a/handouts/secondorder.pdf}}</ref>
# {{math|1=''B''<sup>2</sup> − ''AC'' = 0}} (''[[parabolic partial differential equation]]''): Equations that are [[parabolic partial differential equation|parabolic]] at every point can be transformed into a form analogous to the [[heat equation]] by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where {{math|1=''x'' = 0}}. By change of variables, the equation can always be expressed in the form: <math display="block">u_{xx} + \cdots = 0,</math>where x correspond to changed variables. This justifies [[heat equation]], which are of form <math display="inline">u_t - u_{xx} + \cdots = 0      </math>, as an example of this type.<ref name=":0" />
# {{math|''B''<sup>2</sup> − ''AC'' > 0}} (''[[hyperbolic partial differential equation]]''): [[hyperbolic partial differential equation|hyperbolic]] equations retain any discontinuities of functions or derivatives in the initial data. An example is the [[wave equation]]. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where {{math|''x'' > 0}}. By change of variables, the equation can always be expressed in the form: <math display="block">u_{xx} - u_{yy} + \cdots = 0,</math>where x and y correspond to changed variables. This justifies [[wave equation]] as an example of this type.<ref name=":0" />

If there are {{mvar|n}} independent variables {{math|''x''<sub>1</sub>, ''x''<sub>2 </sub>, …, ''x''<sub>''n''</sub>}}, a general linear partial differential equation of second order has the form
<math display="block">L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\partial^2 u}{\partial x_i \partial x_j} \quad+ \text{lower-order terms} = 0.</math>

The classification depends upon the signature of the [[eigenvalues]] of the coefficient matrix {{math|''a''<sub>''i'',''j''</sub>}}.

# Elliptic: the eigenvalues are all positive or all negative.
# Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
# Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
# Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.<ref>Courant and Hilbert (1962), p.182.</ref>

The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the [[Laplace equation]], the [[heat equation]], and the [[wave 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 the [[Euler–Tricomi equation]]; varying from elliptic to hyperbolic for different [[Region (mathematics)|regions]] of the domain, as well as higher-order PDEs, but such knowledge is more specialized.

=== Systems of first-order equations and characteristic surfaces ===
{{see also|First-order partial differential equation}}

The classification of partial differential equations can be extended to systems of first-order equations, where the unknown {{mvar|u}} is now a [[Euclidean vector|vector]] with {{mvar|m}} components, and the coefficient matrices {{mvar|A<sub>ν</sub>}} are {{mvar|m}} by {{mvar|m}} matrices for {{math|1=''ν'' = 1, 2, …, ''n''}}. The partial differential equation takes the form
<math display="block">Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0,</math>
where the coefficient matrices {{mvar|A<sub>ν</sub>}} and the vector {{mvar|B}} may depend upon {{mvar|x}} and {{mvar|u}}. If a [[hypersurface]] {{mvar|S}} is given in the implicit form
<math display="block">\varphi(x_1, x_2, \ldots, x_n)=0,</math>
where {{mvar|φ}} has a non-zero gradient, then {{mvar|S}} is a '''characteristic surface''' for the [[Differential_operator|operator]] {{mvar|L}} at a given point if the characteristic form vanishes:
<math display="block">Q\left(\frac{\partial\varphi}{\partial x_1}, \ldots, \frac{\partial\varphi}{\partial x_n}\right) = \det\left[\sum_{\nu=1}^n A_\nu \frac{\partial \varphi}{\partial x_\nu}\right] = 0.</math>

The geometric interpretation of this condition is as follows: if data for {{mvar|u}} are prescribed on the surface {{mvar|S}}, then it may be possible to determine the normal derivative of {{mvar|u}} on {{mvar|S}} from the differential equation. If the data on {{mvar|S}} and the differential equation determine the normal derivative of {{mvar|u}} on {{mvar|S}}, then {{mvar|S}} is non-characteristic. If the data on {{mvar|S}} and the differential equation ''do not'' determine the normal derivative of {{mvar|u}} on {{mvar|S}}, then the surface is '''characteristic''', and the differential equation restricts the data on {{mvar|S}}: the differential equation is ''internal'' to {{mvar|S}}.

# A first-order system {{math|1=''Lu'' = 0}} is ''elliptic'' if no surface is characteristic for {{mvar|L}}: the values of {{mvar|u}} on {{mvar|S}} and the differential equation always determine the normal derivative of {{mvar|u}} on {{mvar|S}}.
# A first-order system is ''hyperbolic'' at a point if there is a '''spacelike''' surface {{mvar|S}} with normal {{mvar|ξ}} at that point. This means that, given any non-trivial vector {{mvar|η}} orthogonal to {{mvar|ξ}}, and a scalar multiplier {{mvar|λ}}, the equation {{math|1=''Q''(''λξ'' + ''η'') = 0}} has {{mvar|m}} real roots {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>, …, ''λ''<sub>''m''</sub>}}. The system is '''strictly hyperbolic''' if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form {{math|1=''Q''(''ζ'') = 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has {{mvar|nm}} sheets, and the axis {{math|1=''ζ'' = ''λξ''}} runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
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