
% extra credit ch7X 2280, S2018
% Challenging Laplace problems
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\input{linalg.sty}
\def\rf#1{(\ref{#1})}
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{\large\sf Problem #1}. {\bf #2}\par}
\parskip=3pt
\def\MAPLE{{\tt maple}}
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\begin{document}                
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2280 Extra Credit Problems Chapter 7X \\
\bf Challenging Laplace Applications S2018
\end{center}


{\bf Submitted work}. Please submit one stapled package with this sheet
on top. Kindly check-mark the problems submitted and
label the paper \fbox{\bf Extra Credit}. Label each solved problem with
its corresponding problem number, e.g., \fbox{Xc7.3-20}.


\Prob{Xc7X.1}{(Inverse transform)}
Solve for $f(t)$, given
$\L(f(t)) = e^{-2s}\dd\frac{s}{(s+1)(s^2+1)}$.

\Prob{Xc7X.2}{(Inverse transform)}
Solve for $f(t)$, given
$\L(f(t)) = \dd\frac{d}{ds}\left(e^{-2s}\frac{s+1}{s^2(s^2+1)}\right)$.


\Prob{Xc7X.3}{(Inverse transform)}
Solve for $f(t)$, given
$\L(f(t)) = \dd\frac{s+4}{s^2(s^2+2s+2)} +
\frac{e^{-2s}}{s(1-e^{-2s})}$.
Hint: Look on the inside cover of Edwards-Penney.

\Prob{Xc7X.4}{(Resolvent Equation and $e^{At}$)}
Leverrier and Faddeeva derived the following recursions for the
coefficients in the expansion
$$ (sI - A)^{-1} =\sum_{k=0}^{n-1} \frac{s^{n-k-1}}{\det(sI-A)}A_k.$$

$$ \det(sI-A)=s^n-\sum_{k=0}^{n-1} c_k s^k,\quad
A_0=I,\quad A_k=A_{k-1}A-c_{n-k}I.$$
Then
$$ e^{At} = \L^{-1}\left((sI-A)^{-1}\right)=\sum_{k=0}^{n-1}
\L^{-1}\left(\frac{s^{n-k-1}}{\det(sI-A)}\right) A_k.$$
(a) Write out $A_0$, $A_1$ for a $2\times 2$ matrix $A$, given
$\det(sI-A)=s^2-c_1s-c_0$.

(b) Write out $A_0$, $A_1$, $A_2$ for a $3\times 3$
matrix $A$, given $\det(sI-A)=s^3-c_2s^2-c_2s-c_0$.

(c) Apply the Leverrier-Faddeeva formulas to find the matrix
exponential $e^{At}$ for the special case
$$A = \threebythree{2&0&0}{0&1&1}{0&0&1}.$$

(d) Check the answer for $e^{At}$ in a computer algebra system.
% See Faddeeva, Computational Linear Algebra.

\Prob{Xc7X.5}{(Laplace transform)}
    Find $\L(f(t))$, given $f(t)=\dd\frac{\sinh(t)}{t}$.

\Prob{Xc7X.6}{(Laplace transform)}
    Find $\L(f(t))$, given $f(t)=\dd\frac{d}{dt}(\sinh(t)\cosh(t))$.

\newpage

\Prob{Xc7X.7}{(Laplace's method)}
Solve by Laplace's method the initial value problem $tx''(t)-2x'(t) +
tx(t)=0$, $x(0)=1$, $x'(0)=0$. Check the answer in maple.

\Prob{Xc7X.8}{(Laplace's method)}
Solve by Laplace's method the initial value problem $tx''(t)+2x'(t) +
x(t)=\delta(t)+\delta(t-1)+\delta(t-2)$, $x(0)=0$, $x'(0)=1$. Check the answer in maple
using \fbox{\tt dsolve(\{de,ic\},x(t),method=laplace)}.

\Prob{Xc7X.9}{(Backward table)} Solve for $f(t)$, given $F(s)=\L(f(t))$.
$$F(s) = \frac{5s^4+16s^3+2560s^2+1600s+327680}{(s^2+100)(s^2+256)^2}.$$


\Prob{Xc7X.10}{(Backward table)} Solve for $f(t)$, given $F(s)=\L(f(t))$.
$$F(s) = \frac{5s^4 - 24s^3 + 80s^2 -32s + 256}{(s-2)^3(s^2+16)^2}.$$


 \vfill {\bf End of extra credit problems chapter 7X.}

\end{document}
