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# Hyperbolic partial differential equation

In mathematics, a hyperbolic partial differential equation is usually a second-order partial differential equation (PDE) of the form

$A u_\left\{xx\right\} + 2 B u_\left\{xy\right\} + C u_\left\{yy\right\} + D u_x + E u_y + F = 0$

with

$det begin\left\{pmatrix\right\} A & B B & C end\left\{pmatrix\right\} = A C - B^2 < 0.$

The one-dimensional wave equation:

$frac\left\{partial^2 u\right\}\left\{partial t^2\right\} - c^2frac\left\{partial^2 u\right\}\left\{partial x^2\right\} = 0$

is an example of 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.

## Hyperbolic system of partial differential equations

Consider the following system of $s$ first order partial differential equations for $s$ unknown functions $vec u = \left(u_1, ldots, u_s\right)$, $vec u =vec u \left(vec x,t\right)$, where $vec x in mathbb\left\{R\right\}^d$

$\left(*\right) quad frac\left\{partial vec u\right\}\left\{partial t\right\}$
+ sum_{j=1}^d frac{partial}{partial x_j} vec {f^j} (vec u) = 0,

$vec \left\{f^j\right\} in C^1\left(mathbb\left\{R\right\}^s, mathbb\left\{R\right\}^s\right), j = 1, ldots, d$ are once continuously differentiable functions, nonlinear in general.

Now define for each $vec \left\{f^j\right\}$ a matrix $s times s$

$A^j:=$
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.

We say that the system $\left(*\right)$ is hyperbolic if for all $alpha_1, ldots, alpha_d in mathbb\left\{R\right\}$ the matrix $A := alpha_1 A^1 + cdots + alpha_d A^d$ has only real eigenvalues and is diagonalizable.

If the matrix $A$ has distinct real eigenvalues, it follows that it's diagonalizable. In this case the system $\left(*\right)$ 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 $u = u\left(vec x, t\right)$. Then the system $\left(*\right)$ has the form

$\left(**\right) quad frac\left\{partial u\right\}\left\{partial t\right\}$
+ sum_{j=1}^d frac{partial}{partial x_j} {f^j} (u) = 0,

Now $u$ can be some quantity with a flux $vec f = \left(f^1, ldots, f^d\right)$. To show that this quantity is conserved, integrate $\left(**\right)$ over a domain $Omega$

$int_\left\{Omega\right\} frac\left\{partial u\right\}\left\{partial t\right\} dOmega + int_\left\{Omega\right\} nabla cdot vec f\left(u\right) dOmega = 0.$

If $u$ and $vec f$ are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and $partial / partial t$ to get a conservation law for the quantity $u$ in the general form

$frac\left\{d\right\}\left\{dt\right\} int_\left\{Omega\right\} u dOmega + int_\left\{Gamma\right\} vec f\left(u\right) cdot vec n dGamma = 0,$
which means that the time rate of change of $u$ in the domain $Omega$ is equal to the net flux of $u$ through its boundary $Gamma$. Since this is an equality, it can be concluded that $u$ is conserved within $Omega$.