Hyperbolic partial differential equation
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A hyperbolic partial differential equation is usually a second-order partial differential equation of the form
- <math>A u_{xx} + 2 B u_{xy} + C u_{yy} + D u_x + E u_y + F = 0</math>
with <math>\det \begin{pmatrix} A & B \\ B & C \end{pmatrix} = A C - B^2 < 0</math>. The wave equation:
- <math>\frac{\partial^2 u}{\partial t^2} - \Delta u = 0</math>
is such a hyperbolic equation.
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 <math>s</math> first order partial differential equations for <math>s</math> unknown functions <math> \vec u = (u_1, \ldots, u_s) </math>, <math> \vec u =\vec u (\vec x,t)</math>, where <math>\vec x \in \mathbb{R}^d</math>
- <math>(*) \quad \frac{\partial \vec u}{\partial t}
+ \sum_{j=1}^d \frac{\partial}{\partial x_j}
\vec {f^j} (\vec u) = 0,
</math>
<math>\vec {f^j} \in C^1(\mathbb{R}^s, \mathbb{R}^s), j = 1, \ldots, d</math> are once continuously differentiable functions, nonlinear in general.
Now define for each <math>\vec {f^j}</math> a matrix <math>s \times s</math>
- <math>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} </math>, for each <math>j = 1, \ldots, d</math>.
We say that the system <math>(*)</math> is hyperbolic if for all <math>\alpha_1, \ldots, \alpha_d \in \mathbb{R}</math> the matrix <math>A := \alpha_1 A^1 + \cdots + \alpha_d A^d</math> has only real eigenvalues and is diagonalizable.
If the matrix <math>A</math> has distinct real eigenvalues, it follows it's diagonalizable. In this case the system <math>(*)</math> 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 <math>u = u(\vec x, t)</math>. Then the system <math>(*)</math> has the form
- <math>(**) \quad \frac{\partial u}{\partial t}
+ \sum_{j=1}^d \frac{\partial}{\partial x_j}
{f^j} (u) = 0,
</math>
Now <math>u</math> can be some quantity with a flux <math>\vec f = (f^1, \ldots, f^d)</math>.To show that this quantity is conserved, integrate <math>(**)</math> over a domain <math>\Omega</math>
- <math>\int_{\Omega} \frac{\partial u}{\partial t} + \int_{\Omega} \nabla \cdot \vec f(u) = 0</math>
If <math>u</math> and <math>\vec f</math> are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and <math>\partial / \partial t</math> and we get a conservation law for the quantity <math>u</math> in a common form
- <math>\frac{\partial}{\partial t} \int_{\Omega} u + \int_{\partial \Omega} \vec f(u) \cdot \vec n = 0</math>
[edit] See also
[edit] External links
- Linear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.
- Nonlinear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.
[edit] Bibliography
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9cs:Hyperbolická diferenciální rovnice
