A hyperbolic partial differential equation is usually a second-order partial differential equation of the form

Au_xx + 2Bu_xy + Cu_yy + Du_x + Eu_y + F = 0

with . The wave equation: The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. ...

is such a hyperbolic equation.

This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.

Hyperbolic system of partial differential equations

Consider the following system of s first order partial differential equations for s unknown functions , , where In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

are once continuously differentiable functions, nonlinear in general. In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...

Now define for each a matrix Wiktionary has a definition of: Matrix The word matrix (plural matrices, or less often, matrixes) has several meanings. ...

, for each .

We say that the system ( * ) is hyperbolic if for all the matrix has only real eigenvalues and is diagonalizable. The text or formatting below is generated by a template which has been proposed for deletion. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

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

Hyperbolic system and conservation laws

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 In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

Now u can be some quantity with a flux .To show that this quantity is conserved, integrate ( * * ) over a domain Ω The flux visualized. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

If u and are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and and we get a conservation law for the quantity u in a common form In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

External links

• Linear Hyperbolic Equations (http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc2.pdf) at EqWorld: The World of Mathematical Equations.
• Nonlinear Hyperbolic Equations (http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc2.pdf) at EqWorld: The World of Mathematical Equations.

Bibliography

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.