5.5: Exercises- Matrix Methods for Dynamical Systems
Compute, without the aid of a machine, the Laplace transforms of \(e^t\) and \(te^{-t}\). Show ALL of your work.
Extract from
fib3.m
analytical expressions for \(x_2\) and \(x_{3}\)
Use
eig
to compute the eigenvalues of \(B = \begin{pmatrix} {2}&{-1}\\ {-1}&{2} \end{pmatrix}\). Use
det
to compute the characteristic polynomial of \(B\)
roots
to compute the roots of this characteristic polynomial. Compare these to the results of
eig
. How does Matlab compute the roots of a polynomial? (type
help roots
for the answer).
Adapt the Backward Euler portion of
fib3.m
so that one may specify an arbitrary number of compartments, as in
fib1.m
. Submit your well documented M-file along with a plot of \(x_{1}\) and \(x_{10}\)
versus
time (on the same well labeled graph) for a nine compartment fiber of length \(l = 1cm\).
Derive \(\frac{\tilde{x}(t)-\tilde{x}(t-dt)}{dt} = B \tilde{x}(t)+g(t)\) from \(\textbf{x}' = B \textbf{x}+\textbf{g}\), by working backwards toward \(x(0)\). Along the way you should explain why
\(\frac{(\frac{I}{d(t)}-B)^{-1}}{d(t)} = (I-d(t)B)^{-1}\)