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\begin_body

\begin_layout Title
Boundary layer flow
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\frac{\partial u}{\partial t}-\frac{1}{s^{2}}\frac{\partial^{2}u}{\partial y^{2}}=K(t)
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $y=0:u=0$
\end_inset

, 
\begin_inset Formula $y=1,\frac{\partial u}{\partial y}=0$
\end_inset


\end_layout

\begin_layout Standard
Discretization, FTCD:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}-\frac{1}{s^{2}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{\Delta y^{2}}=K^{n}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Rewriting:
\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset Formula $u_{i}^{n+1}-u_{i}^{n}-\frac{\Delta t}{\Delta y^{2}s^{2}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{}=K^{n}\Delta t$
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Formula $u_{i}^{n+1}=\Delta tK^{n}+u_{i}^{n}+\frac{\Delta t}{\Delta y^{2}s^{2}}\left(u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}\right)$
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Formula $u_{0}=0$
\end_inset


\end_layout

\begin_layout Plain Layout
and
\end_layout

\begin_layout Plain Layout
\begin_inset Formula $u_{N}-u_{n-1}=0$
\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\]

\end_inset


\end_layout

\end_body
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