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In mathematics, a **hyperbolic partial differential equation** of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives.^{[citation needed]} 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

The equation has the property that, if

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 characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and parabolic partial differential equations. 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 laws.

A partial differential equation is hyperbolic at a point provided that the Cauchy problem is uniquely solvable in a neighborhood of for any initial data given on a non-characteristic hypersurface passing through .^{[1]} 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.

By a linear change of variables, any equation of the form

The one-dimensional wave equation:

The following is a system of first order partial differential equations for unknown functions , , where :

(∗) |

where are once continuously differentiable functions, nonlinear in general.

Next, for each define the Jacobian matrix

The system (**∗**) is **hyperbolic** if for all the matrix
has only real eigenvalues and is diagonalizable.

If the matrix has s *distinct* real eigenvalues, it follows that it is diagonalizable. In this case the system (**∗**) is called **strictly hyperbolic**.

If the matrix is symmetric, it follows that it is diagonalizable and the eigenvalues are real. In this case the system (**∗**) is called **symmetric hyperbolic**.

There is a connection between a hyperbolic system and a conservation law. Consider a hyperbolic system of one partial differential equation for one unknown function . Then the system (**∗**) has the form

(∗∗) |

Here, can be interpreted as a quantity that moves around according to the flux given by . To see that the quantity is conserved, integrate (**∗∗**) over a domain

If and are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and to get a conservation law for the quantity in the general form

**^**Rozhdestvenskii, B.L. (2001) [1994], "Hyperbolic partial differential equation",*Encyclopedia of Mathematics*, EMS Press- ^
^{a}^{b}Evans, Lawrence C. (2010) [1998],*Partial differential equations*, Graduate Studies in Mathematics, vol. 19 (2nd ed.), Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/019, ISBN 978-0-8218-4974-3, MR 2597943, OCLC 465190110

- 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

- "Hyperbolic partial differential equation, numerical methods",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Linear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.
- Nonlinear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.