Hyperbolic partial differential equation

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

$Au_{xx} + Bu_{xy} + Cu_{yy} + \cdots = 0$

with

B - 4 A C > 0

This definition is analogous to the definition of a planar hyperbola.

The one-dimensional wave equation:

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

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.

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 $s \times s$

$A^j:= \begin{pmatrix} \frac{\partial f_1^j}{\partial u_1} & \cdots & \frac{\partial f_1^j}{\partial u_s} \\\vdots & \ddots & \vdots \\\frac{\partial f_s^j}{\partial u_1} & \cdots & \frac{\partial f_s^j}{\partial u_s} \end{pmatrix} ,\text{ for }j = 1, \ldots, d.$

We say that the system $( * )$ 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 $Ω$

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

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

$\frac{d}{dt} \int_{\Omega} u d\Omega+ \int_{\Gamma} \vec f(u) \cdot \vec n d\Gamma = 0,$

which means that the time rate of change of $u$ in the domain $Ω$ is equal to the net flux of $u$ through its boundary $Γ$ . Since this is an equality, it can be concluded that $u$ is conserved within $Ω$ .

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

Linear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.
Nonlinear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.

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Published in July 2009.

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Hyperbolic system of partial differential equations

Hyperbolic system and conservation laws

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