diff --git a/chapters/classes.tex b/chapters/classes.tex index d26725058..7b3484e6a 100644 --- a/chapters/classes.tex +++ b/chapters/classes.tex @@ -1142,11 +1142,11 @@ \section{Balanced Models}\doublelabel{balanced-models} %TODO-FORMAT Should this be verbatim code instead? Local equations: -\begin{eqnarray*} -0 &=& p.i + n.i;\\ -u &=& p.v - n.v;\\ -C \cdot \text{der}(u) &=& p.i; -\end{eqnarray*} +\begin{align*} +0 &= p.i + n.i;\\ +u &= p.v - n.v;\\ +C \cdot \text{der}(u) &= p.i; +\end{align*} and 2 equations corresponding to the 2 flow-variables \lstinline!p.i! and \lstinline!n.i!. These are 5 equations in 5 unknowns (\textbf{locally} @@ -1213,13 +1213,13 @@ \section{Balanced Models}\doublelabel{balanced-models} flow variables for \lstinline!t! (\lstinline!t.p.i!, \lstinline!t.n.i!), and 2 flow variables for \lstinline!c! (\lstinline!c.p.i!, \lstinline!c.n.i!). Local equations: -\begin{eqnarray*} \text{p.v} &=& \text{t.p.v};\\ -0 &=& \text{p.i}-\text{t.p.i};\\ -\text{c.p.v} &=& \text{t.n.v};\\ -0 &=& \text{c.p.i}+\text{t.n.i};\\ -\text{n.v} &=& \text{c.n.v};\\ -0 &=& \text{n.i}-\text{c.n.i}; -\end{eqnarray*} +\begin{align*} \text{p.v} &= \text{t.p.v};\\ +0 &= \text{p.i}-\text{t.p.i};\\ +\text{c.p.v} &= \text{t.n.v};\\ +0 &= \text{c.p.i}+\text{t.n.i};\\ +\text{n.v} &= \text{c.n.v};\\ +0 &= \text{n.i}-\text{c.n.i}; +\end{align*} and 2 equation corresponding to the flow variables \lstinline!p.i!, \lstinline!n.i!. In total we have 8 scalar unknowns and 8 scalar equations, i.e., a diff --git a/chapters/operatorsandexpressions.tex b/chapters/operatorsandexpressions.tex index 5c0d03677..a14723882 100644 --- a/chapters/operatorsandexpressions.tex +++ b/chapters/operatorsandexpressions.tex @@ -714,11 +714,11 @@ \subsubsection{spatialDistribution}\doublelabel{spatialdistribution} The \lstinline!spatialDistribution!() operator allows to approximate efficiently the solution of the infinite-dimensional problem -\begin{eqnarray*} -\frac{\partial z(y,t)}{\partial t}+v(t)\frac{\partial z(y,t)}{\partial y}&=&0.0\\ -z(0.0, t)=\mathrm{in}_0(t) \text{ if $v\ge 0$}\\ -z(1.0, t)=\mathrm{in}_1(t) \text{ if $v<0$} -\end{eqnarray*} +\begin{align*} +\frac{\partial z(y,t)}{\partial t}+v(t)\frac{\partial z(y,t)}{\partial y} &= 0.0\\ +z(0.0, t) &= \mathrm{in}_0(t) \text{ if $v\ge 0$}\\ +z(1.0, t) &= \mathrm{in}_1(t) \text{ if $v<0$} +\end{align*} where $z(y, t)$ is the transported quantity, $y$ is the normalized spatial coordinate ($0.0 \le y \le 1.0$), $t$ is the time, $v(t)=\mathrm{der}(x)$ is the normalized