$\DeclareMathOperator{\prob}{Pr}$
$\newcommand{\paren}[1]{\left(#1\right)}$
$\newcommand{\brac}[1]{\left[#1\right]}$
$\newcommand{\dt}{\Delta t}$
Homework 2 (Due 10/12 in class)¶
For your reference, the ODE material we've been covering, and will cover next week in class, comes from:
Leveque: Ch 55.8, Ch 6.3.3, 6.3.4, 7.17.2, 7.47.5, 88.1, 8.3.1.
Part I Problems:¶

What is the order of $f(x)=\exp(x^2)  1 x^2$ in $x$ as $x \to 0$? What is the order of $g(\Delta t)=\frac{1}{1\Delta t}1$ in $\Delta t$ as $\Delta t \to 0$?

The theta method for solving the ODE $y' = f(y)$ is given by
$$
y^{n+1} = y^{n} + \Delta t \left[ \theta \, f(y^{n})+ (1\theta) \, f(y^{n+1}) \right].
$$
Rewriting this equation to look like an approximation of the ODE we define the truncation error,
\begin{align*}
\tau^{n+1} = \frac{y(t_{n+1})y(t_{n})}{\Delta t}  \Big[\theta \, f(y(t_{n}))+ (1\theta) \, f(y(t_{n+1}))\Big].
\end{align*}
By Taylor series expanding about $t_{n}$, show that the method is first order for $0 \leq \theta \leq 1$, except for $\theta = \frac{1}{2}$ when it is second order. What methods do the theta method reduce to when $\theta =0$, $\theta = \frac{1}{2}$ and $\theta=1$?
Part II Problems:¶

In class we derived an ODE for $O(t)$ by writing an equation for $O(t + \dt)$:
\begin{align*}
O(t + \dt) &= \prob \brac{D(t+\dt)=1}\\
&= \prob\brac{D(t+\dt)=1 \text{ and } D(t)=1} + \prob\brac{D(t+\dt)=1 \text{ and } D(t)=0}\\
&= I + II,
\end{align*}
simplifying $I$ and $II$ using Baye's Law, and then rearranging and taking the limit that $\dt \to 0$.
We found that
\begin{equation*}
\frac{dO}{dt} = k_c O + k_o C.
\end{equation*}
For this problem, derive an equation for $C(t+\dt)$ in terms of $O(t)$ and $C(t)$. Then rearrange and
take the limit as $\dt \to 0$ to derive the ODE that $C(t)$ satisfies,
\begin{equation*}
\frac{dC}{dt} = k_c O  k_o C.
\end{equation*}

In the ODEs for $O(t)$ and $C(t)$, $O(t)$ represents the probability the DNA is in the open state at time $t$ and
$C(t)$ the probability the DNA is in the closed state at time $t$.

Using that $O(t)+C(t)=1$, eliminate the $C$ equation and find a single equation that $O(t)$ satisfies.

Solve for the steady state of this equation, $O_{SS}$ and find $C_{SS}$.

Solve for $O(t)$ in terms of $O_{SS}$ and $O(0)$.

Write down Euler's method applied to the ODE for $O(t)$, i.e. $O^{n+1} = \,$?

By iterating Euler's method backwards until you reach $O^{0}=O(0)$, solve for $O^n$ in terms of just the ODE parameters, $\dt$, and $n$.

Assume $O(0) = 0$. Take the limit that $\dt \to 0$ and $n \to \infty$ in the formula you just derived with $t=n\dt$ fixed. Show that your formula for $O^n$ converges to the exact solution $O(t)$ you found above in this limit.
(Hint for the last parts, look at how we showed Euler's method converged for $y'=\lambda y$ and the general proof of convergence for Euler's method.)