Hyperbolic partial differential equation

A hyperbolic partial differential equation is usually a second-order partial differential equation of the form

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} – \nabla^2 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.

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.

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 . \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) . \vec n = 0<math>

External links

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

Categories: Partial differential equations