View source for Hyperbolic partial differential equation

{{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.

== Definition ==
A partial differential equation is hyperbolic at a point <math>P</math> provided that the [[Cauchy problem]] is uniquely solvable in a neighborhood of <math>P</math> for any initial data given on a [[Method of characteristics|non-characteristic hypersurface]] passing through <math>P</math>.<ref name="Rozhdestvenskii">{{eom|id=H/h048300|first=B.L.|last= Rozhdestvenskii}}</ref>  Here the prescribed initial data consist of all (transverse) derivatives of the function on the surface up to one less than the order of the differential equation.

== Examples ==

By a linear change of variables, any equation of the form
<math display="block"> A\frac{\partial^2 u}{\partial x^2} + 2B\frac{\partial^2 u}{\partial x\partial y} + C\frac{\partial^2u}{\partial y^2} + \text{(lower order derivative terms)} = 0</math>
with
<math display="block"> B^2 - A C > 0</math>
can be transformed to the [[wave equation]], apart from lower order terms which are inessential for the qualitative understanding of the equation.<ref name="Evans 1998"/>{{rp|p=400}}  This definition is analogous to the definition of a planar [[Hyperbola#Quadratic equation|hyperbola]].

The one-dimensional [[wave equation]]:
<math display="block">\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0</math>
is an example of a hyperbolic equation. The two-dimensional and three-dimensional wave equations also fall into the category of hyperbolic PDE. This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.<ref name="Evans 1998">{{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | orig-year=1998 | url=https://www.worldcat.org/oclc/465190110 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=[[Graduate Studies in Mathematics]] | isbn=978-0-8218-4974-3 |mr=2597943 | year=2010 | volume=19 | doi=10.1090/gsm/019| oclc=465190110 }}</ref>{{rp|p=402}}

== Hyperbolic systems of first-order equations ==

The following is a system of first-order partial differential equations for <math>s</math> unknown [[function (mathematics)|function]]s {{nowrap|<math> \vec u = (u_1, \ldots, u_s) </math>,}} {{nowrap|<math> \vec u = \vec u (\vec x,t)</math>,}} where {{nowrap|<math>\vec x \in \mathbb{R}^d</math>:}}
{{NumBlk||<math display="block"> \frac{\partial \vec u}{\partial t}
 + \sum_{j=1}^d \frac{\partial}{\partial x_j}
 \vec {f}^j (\vec u) = 0,
</math>|{{EquationRef|∗}}}}

where <math>\vec {f}^j \in C^1(\mathbb{R}^s, \mathbb{R}^s)</math> are once [[Continuous function|continuously]] [[Differentiable function|differentiable]] functions, [[nonlinear]] in general.

Next, for each <math>\vec {f}^j</math> define the <math>s \times s</math> [[Jacobian matrix]]
<math display="block">A^j :=
\begin{pmatrix}
\frac{\partial f_1^j}{\partial u_1} & \cdots & \frac{\partial f_1^j}{\partial u_s} \\
\vdots & \ddots & \vdots \\
\frac{\partial f_s^j}{\partial u_1} & \cdots & \frac{\partial f_s^j}{\partial u_s}
\end{pmatrix}
,\text{ for }j = 1, \ldots, d.</math>

The system ({{EquationNote|∗}}) is '''hyperbolic''' if for all <math>\alpha_1, \ldots, \alpha_d \in \mathbb{R}</math> the matrix <math>A := \alpha_1 A^1 + \cdots + \alpha_d A^d</math>
has only [[Real number|real]] [[eigenvalue]]s and is [[Diagonalizable matrix|diagonalizable]].

If the matrix <math>A</math> has {{mvar|s}} ''distinct'' real eigenvalues, it follows that it is diagonalizable. In this case the system ({{EquationNote|∗}}) is called '''strictly hyperbolic'''.

If the matrix <math>A</math> is symmetric, it follows that it is diagonalizable and the eigenvalues are real. In this case the system ({{EquationNote|∗}}) is called '''symmetric hyperbolic'''.

=== Hyperbolic system and conservation laws ===

There is a connection between a hyperbolic system and a [[Conservation law (physics)|conservation law]]. Consider a hyperbolic system of one partial differential equation for one unknown function <math>u = u(\vec x, t)</math>. Then the system ({{EquationNote|∗}}) has the form
{{NumBlk||<math display="block"> \frac{\partial u}{\partial t}
 + \sum_{j=1}^d \frac{\partial}{\partial x_j}
 {f^j} (u) = 0.
</math>|{{EquationRef|∗∗}}}}

Here, <math>u</math> can be interpreted as a quantity that moves around according to the [[flux]] given by <math>\vec f = (f^1, \ldots, f^d)</math>. To see that the quantity <math>u</math> is conserved, [[Integral|integrate]] ({{EquationNote|∗∗}}) over a domain <math>\Omega</math>
<math display="block">\int_{\Omega} \frac{\partial u}{\partial t} \, d\Omega + \int_{\Omega} \nabla \cdot \vec f(u)\, d\Omega = 0.</math>

If <math>u</math> and <math>\vec f</math> are sufficiently smooth functions, we can use the [[divergence theorem]] and change the order of the integration and <math>\partial / \partial t</math> to get a conservation law for the quantity <math>u</math> in the general form
<math display="block">
\frac{ d}{ dt} \int_{\Omega} u \, d\Omega  
+ \int_{\partial\Omega} \vec f(u) \cdot \vec n \, d\Gamma = 0,
</math>
which means that the time rate of change of <math>u</math> in the domain <math>\Omega</math> is equal to the net flux of <math>u</math> through its boundary <math>\partial\Omega</math>. Since this is an equality, it can be concluded that <math>u</math> is conserved within <math>\Omega</math>.

== See also ==
* [[Elliptic partial differential equation]]
* [[Hypoelliptic operator]]
* [[Parabolic partial differential equation]]

== References ==
{{Reflist}}

== Further reading ==
* A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. {{ISBN|1-58488-299-9}}
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== External links ==
* {{springer|title=Hyperbolic partial differential equation, numerical methods|id=p/h048310}}
* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc2.pdf Linear Hyperbolic Equations] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc2.pdf Nonlinear Hyperbolic Equations] at EqWorld: The World of Mathematical Equations.

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[[Category:Hyperbolic partial differential equations| ]]