Editing Hyperbolic partial differential equation (section)

{{Short description|Type of partial differential equations}}
{{more footnotes needed|date=March 2012}}
In [[mathematics]], a '''hyperbolic partial differential equation''' of order <math>n</math> is a [[partial differential equation]] (PDE) that, roughly speaking, has a well-posed [[initial value problem]] for the first <math>n - 1</math> derivatives.{{citation needed|date=May 2024}} More precisely, the [[Cauchy problem]] can be locally solved for arbitrary initial data along any non-characteristic [[hypersurface]].  Many of the equations of [[mechanics]] are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the [[wave equation]].  In one spatial dimension, this is
<math display="block">\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} </math>
The equation has the property that, if {{mvar|''u''}} and its first time derivative are arbitrarily specified initial data on the line {{math|1=''t'' = 0}} (with sufficient smoothness properties), then there exists a solution for all time {{mvar|t}}.

The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once.  Relative to a fixed time coordinate, disturbances have a finite [[propagation speed]]. They travel along the [[method of characteristics|characteristics]] of the equation. This feature qualitatively distinguishes hyperbolic equations from [[elliptic partial differential equation]]s and [[parabolic partial differential equation]]s.  A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain.

Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration.  There is a well-developed theory for linear [[differential operators]], due to [[Lars Gårding]], in the context of [[microlocal analysis]].  Nonlinear differential equations are hyperbolic if their linearizations are hyperbolic in the sense of Gårding.  There is a somewhat different theory for first order systems of equations coming from systems of [[Conservation law (physics)|conservation law]]s.