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

Auxx + 2Buxy + Cuyy + Dux + Euy + F = 0

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

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 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.

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[edit] 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} , for each 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 it's diagonalizable. In this case the system ( * ) is called strictly hyperbolic.

[edit] 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.

[edit] See also

[edit] External links

[edit] 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