Editing Hyperbolic partial differential equation (section)

== Hyperbolic systems of first-order equations ==

The following is a system of first-order partial differential equations for <math>s</math> unknown [[function (mathematics)|function]]s {{nowrap|<math> \vec u = (u_1, \ldots, u_s) </math>,}} {{nowrap|<math> \vec u = \vec u (\vec x,t)</math>,}} where {{nowrap|<math>\vec x \in \mathbb{R}^d</math>:}}
{{NumBlk||<math display="block"> \frac{\partial \vec u}{\partial t}
 + \sum_{j=1}^d \frac{\partial}{\partial x_j}
 \vec {f}^j (\vec u) = 0,
</math>|{{EquationRef|∗}}}}

where <math>\vec {f}^j \in C^1(\mathbb{R}^s, \mathbb{R}^s)</math> are once [[Continuous function|continuously]] [[Differentiable function|differentiable]] functions, [[nonlinear]] in general.

Next, for each <math>\vec {f}^j</math> define the <math>s \times s</math> [[Jacobian matrix]]
<math display="block">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.</math>

The system ({{EquationNote|∗}}) 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 number|real]] [[eigenvalue]]s and is [[Diagonalizable matrix|diagonalizable]].

If the matrix <math>A</math> has {{mvar|s}} ''distinct'' real eigenvalues, it follows that it is diagonalizable. In this case the system ({{EquationNote|∗}}) is called '''strictly hyperbolic'''.

If the matrix <math>A</math> is symmetric, it follows that it is diagonalizable and the eigenvalues are real. In this case the system ({{EquationNote|∗}}) is called '''symmetric hyperbolic'''.

=== Hyperbolic system and conservation laws ===

There is a connection between a hyperbolic system and a [[Conservation law (physics)|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 ({{EquationNote|∗}}) has the form
{{NumBlk||<math display="block"> \frac{\partial u}{\partial t}
 + \sum_{j=1}^d \frac{\partial}{\partial x_j}
 {f^j} (u) = 0.
</math>|{{EquationRef|∗∗}}}}

Here, <math>u</math> can be interpreted as a quantity that moves around according to the [[flux]] given by <math>\vec f = (f^1, \ldots, f^d)</math>. To see that the quantity <math>u</math> is conserved, [[Integral|integrate]] ({{EquationNote|∗∗}}) over a domain <math>\Omega</math>
<math display="block">\int_{\Omega} \frac{\partial u}{\partial t} \, d\Omega + \int_{\Omega} \nabla \cdot \vec f(u)\, d\Omega = 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> to get a conservation law for the quantity <math>u</math> in the general form
<math display="block">
\frac{ d}{ dt} \int_{\Omega} u \, d\Omega  
+ \int_{\partial\Omega} \vec f(u) \cdot \vec n \, d\Gamma = 0,
</math>
which means that the time rate of change of <math>u</math> in the domain <math>\Omega</math> is equal to the net flux of <math>u</math> through its boundary <math>\partial\Omega</math>. Since this is an equality, it can be concluded that <math>u</math> is conserved within <math>\Omega</math>.