Hyperbolic partial differential equation

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

:$A\; u\_\{xx\}\; +\; 2\; B\; u\_\{xy\}\; +\; C\; u\_\{yy\}\; +\; D\; u\_x\; +\; E\; u\_y\; +\; F\; =\; 0$

with

:

The one-dimensional wave equation:

:$frac\{partial^2\; u\}\{partial\; t^2\}\; -\; c^2frac\{partial^2\; u\}\{partial\; x^2\}\; =\; 0$

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 $vec\; u\; =\; (u\_1,\; ldots,\; u\_s)$, $vec\; u\; =vec\; u\; (vec\; x,t)$, where $vec\; x\; in\; mathbb\{R\}^d$

:$(*)\; quad\; frac\{partial\; vec\; u\}\{partial\; t\}\; +\; sum\_\{j=1\}^d\; frac\{partial\}\{partial\; x\_j\}\; vec\; \{f^j\}\; (vec\; u)\; =\; 0,$

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

Now define for each $vec\; \{f^j\}$ a matrix $s\; imes\; s$

:

We say that the system $(*)$ is hyperbolic if for all $alpha\_1,\; ldots,\; alpha\_d\; in\; mathbb\{R\}$ 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 $(*)$ 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 $u\; =\; u(vec\; x,\; t)$. Then the system $(*)$ has the form

:$(**)\; quad\; frac\{partial\; u\}\{partial\; t\}\; +\; sum\_\{j=1\}^d\; frac\{partial\}\{partial\; x\_j\}\; \{f^j\}\; (u)\; =\; 0,$

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

:$int\_\{Omega\}\; frac\{partial\; u\}\{partial\; t\}\; dOmega\; +\; int\_\{Omega\}\; abla\; cdot\; vec\; f(u)\; dOmega\; =\; 0.$

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

:$frac\{d\}\{dt\}\; int\_\{Omega\}\; u\; dOmega\; +\; int\_\{Gamma\}\; vec\; f(u)\; cdot\; vec\; n\; dGamma\; =\; 0,$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 $Omega$.

See also

* Elliptic partial differential equation
* Parabolic partial differential equation
* Hypoelliptic operator

Bibliography

* 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

External links

* [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.