laplaceTransformVec.tex

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% Copyright © 2014 Peeter Joot.  All Rights Reserved.
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%\beginArtNoToc
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%\generatetitle{Laplace transform refresher}
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Laplace transforms \index{Laplace transform} were used to solve the MNA equations for time dependent systems, and to find the moments used to in MOR.
\index{model order reduction}
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For the record, the Laplace transform is defined as:
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%\begin{equation}\label{eqn:laplaceTransformVec:20}
\boxedEquation{eqn:laplaceTransformVec:20}{
\LL( f(t) ) =
\int_0^\infty e^{-s t} f(t) dt.
}
%\end{equation}
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The only Laplace transform pair used in the lectures is that of the first derivative
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\begin{equation}\label{eqn:laplaceTransformVec:40}
\begin{aligned}
\LL(f'(t)) &=
\int_0^\infty e^{-s t} \ddt{f(t)} dt
\\ &=
\evalrange{e^{-s t} f(t)}{0}{\infty} - (-s) \int_0^\infty e^{-s t} f(t) dt
\\ &=
-f(0) + s \LL(f(t)).
\end{aligned}
\end{equation}
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Here it is loosely assumed that the real part of \( s \) is positive, and that \( f(t) \) is ``well defined'' enough that \( e^{-s \infty } f(\infty) \rightarrow 0 \).
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Where used in the lectures, the Laplace transforms were of vectors such as the matrix vector product \( \LL(\BG \Bx(t)) \).  Because such a product is linear, observe that it can be expressed as the original matrix times a vector of Laplace transforms
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\begin{equation}\label{eqn:laplaceTransformVec:60}
\begin{aligned}
\LL( \BG \Bx(t) )
&=
\LL {\begin{bmatrix}
G_{i k} x_k(t)
\end{bmatrix}}_i
\\ &=
{\begin{bmatrix}
G_{i k} \LL x_k(t)
\end{bmatrix}}_i
\\ &=
\BG
{\begin{bmatrix}
\LL x_i(t)
\end{bmatrix}}_i.
\end{aligned}
\end{equation}
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The following notation was used in the lectures for such a vector of Laplace transforms
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\begin{equation}\label{eqn:laplaceTransformVec:80}
\BX(s) = \LL \Bx(t) =
{\begin{bmatrix}
\LL x_i(t)
\end{bmatrix}}_i.
\end{equation}
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