1.5.2 Quasi-linear system

In Eq. (41) we can compute the spatial derivative and get

$\begin{displaymath} \;\! \frac{\partial }{\partial t} \ensuremath{\mathchoice{\... ...th$\scriptstyle 0$}} {\mbox{\boldmath$\scriptscriptstyle 0$}}} \end{displaymath}$

(43)

with the Jacobian

$\begin{displaymath} \bar{\bar{\ensuremath{\mathsf{A}}}} = \frac{\partial \ensur... ...e q$}} {\mbox{\boldmath$\scriptscriptstyle q$}}}} \enspace . \end{displaymath}$

(44)

A system of partial differential equations in the form of Eq. (43) is called quasi-linear if

$\begin{displaymath} \bar{\bar{\ensuremath{\mathsf{A}}}} = \bar{\bar{\ensuremath... ...x{\boldmath$\scriptscriptstyle q$}}}, x, t \right) \enspace . \end{displaymath}$

(45)

It is linear if $\bgroup\color{DEFcolor}$\bar{\bar{\ensuremath{\mathsf{A}}}}$\egroup$ is constant.