Hyperbolic partial differential equation: Information from Answers.com

In mathematics, a hyperbolic partial differential equation is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem. 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

$u_{tt} - u_{xx} = 0.\,$

The equation has the property that, if u and its first time derivative are arbitrarily specified initial data on the initial line t = 0 (with sufficient smoothness properties), then there exists a solution for all time.

The precise definition of hyperbolicity depends 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. There is also a somewhat different theory for first order systems of equations coming from systems of conservation laws.

1 Examples
2 Hyperbolic system of partial differential equations
3 Hyperbolic system and conservation laws
4 See also
5 Bibliography
6 External links

Examples

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

A u x x + B u x y + C u y y + (lower order terms) = 0

with

B 2 - 4 A C > 0

can be transformed to the wave equation, apart from lower order terms which are inessential for the qualitative understanding of the equation. 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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Hyperbolic partial differential equation

Contents

Examples

Hyperbolic system of partial differential equations

Hyperbolic system and conservation laws

See also

Bibliography

External links