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{\bf Math 269A.} {\bf HW \#6.} {\bf Due: Wednesday, Nov. 28, or Friday, Nov. 30}
\bigbreak
{\bf[1]}
(a) Give the definition of an A-stable method.
(b) Determine all values of $\theta$ such that the theta method given below is
A-stable.
$$y_{i+1}=y_i+h\Big[\theta f(x_i,y_i)+(1-\theta)f(x_{i+1},y_{i+1})\Big],\ \
i=0,1,...$$
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{\bf[2]} The two-step method $y_{i+1}=y_{i-1}+2hf(x_{i},y_{i})$ is called
the {\it explicit midpoint rule}.
(a) Implement this two-step method for the very simple differential equation
$y'=-y$, $y(0)=1$ (the exact solution is $e^{-x}$). Use $y_1=y(h)=e^{-h}$
and the values $h=1/2$, $h=1/4$, $h=1/8$, $h=1/16$. Plot the exact solution and
the numerical approximations on the interval [0,8]. You should turn in the
code and the plot of values.
(b) Show that the region of absolute stability for the explicit midpoint rule
is the empty set.
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{\bf[3]} Consider the following (IVP):
\begin{eqnarray*}
y_1' & = & 198y_1+199y_2, \ \ \ \ y_1(0)=1\\
y_2' & = & -398y_1-399y_2, \ \ y_2(0)=-1,
\end{eqnarray*}
that we write in matrix-vector form $\vec{y}'=A\vec{y}$,
$\vec{y}(0)=\vec{y}_0$.
(a) Find the exact solution of this autonomous linear system. What is its
asymptotic behavior, as $x\rightarrow\infty$ ?
(b) Compute the eigenvalues $\lambda_1$ and $\lambda_2$ of the matrix $A$ and
the corresponding matrix $P$ of eigenvectors. What relation exists between
$A$, $P$ and $\Lambda=diag(\lambda_1,\lambda_2)$ ?
(c) Is this a stiff system of ODE's ? If yes, what is the stiffness ratio ?
Explain.
(d) Apply the trapezoidal rule to this system following the steps:
(i) Express $\vec{y}_{j+1}$ function of $\vec{y}_j$, by a recurrence formula
given in matrix-vector form.
(ii) If $\vec{z}_j$ is defined such that $\vec{y}_j=P\vec{z}_j$, express
$\vec{z}_{j+1}$ function of
$\vec{z}_j$ in matrix-vector form and the associated
scalar recursions for each component of $\vec{z}_j$.
(e) If the system is solved using the trapezoidal method, what restriction,
if any, has to be imposed on the stepsize $h$ to obtain a correct qualitative
behavior ?
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{\bf [4]} (a) Give the solution (e.g. explicit formulas for $y_1(x)$ and $y_2(x)$)
to
$$
\frac{\partial}{\partial x}
\left(
\begin{array}{c}
y_1\\
y_2
\end{array}
\right)
=
\left(
\begin{array}{cc}
-2 & -2\\
2 & -2
\end{array}
\right)
\left(
\begin{array}{c}
y_1\\
y_2
\end{array}
\right),
\ \ \ \ \
\left(
\begin{array}{c}
y_1(0)\\
y_2(0)
\end{array}
\right)
=
\left(
\begin{array}{c}
-\sqrt{2}\\
\sqrt{2}
\end{array}
\right).
$$
Give an estimate of the stepsize required to obtain a qualitatively correct
solution if one is using Euler's method.
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{\bf [5]} Consider the second order differential equation
$$y''+19y'-20y=0.$$
(a) Give an equivalent first order system for this equation.
(b) Give the stability stepsize restriction if backward Euler is used to
compute solutions to the first order system.
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