Editing Partial differential equation (section)

=== 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.