33 lines
1.0 KiB
TeX
33 lines
1.0 KiB
TeX
%--------------------------------------------------------------
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\begin{frame}
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\begin{center}
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\Huge Non-reflecting Boundary Conditions
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\end{center}
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\end{frame}
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\begin{frame}
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\scriptsize
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\frametitle{Lemma [needed to compute LT of $\Psi_\ell(t,r)$]}
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\begin{figure}
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\includegraphics[width=10cm]{PDFfigs/rline.pdf}
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\caption{
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$\color{ec} D$ and open region $\color{ec}r>r_B-\delta$}
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\end{figure}
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Assume $\color{ec} f(u)$ supported on $ \color{ec} [-r_B+\delta, -r_0-\delta]=D$,
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$\color{ec} f\in C_0^\infty (D)$.
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\begin{block}{}
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For $\color{eqncolor}r > r_B - \delta$, Fourier-Laplace transform of $\color{eqncolor}f^{(\ell - k)}(t-r)$ is
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$$\color{eqncolor}
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s^{\ell-k}e^{-sr}a(s), \hspace{2em} a(s) = \int_{-r_B+ \delta}^{-r_0-\delta} e^{-su} f(u) du.
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$$
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\end{block}
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\begin{block}{proof}
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$$\color{eqncolor}
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\int_0^\infty e^{-st} f^{(\ell-k)}(t-r)dt = e^{-sr} \int_{-r}^\infty e^{-su}f^{(\ell-k)}(u) du
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$$
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$$\color{eqncolor}
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= e^{-sr} \int_{-r_B+\delta}^{-r_0-\delta} e^{-su}f^{(\ell-k)}(u) du, \hspace{2em} -r<-r_B+\delta
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$$
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\end{block}
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\end{frame}
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