Problem Sheet 4 - Multistep Methods#

Question 1#

  1. a. Apply the 3-step Adams-Bashforth to approximate the solution of the given initial value problems using the indicated number of time steps. Compare the approximate solution with the given exact solution:

    (308)#\[\begin{equation} y'=t-y, \ \ (0\leq t \leq 4),\end{equation}\]

    with the initial condition \(y(0)=1,\) Let \(N=4\), with the exact solution

    (309)#\[\begin{equation}y(t)=2e^{-t}+t-1.\end{equation}\]

    b. Apply the 3-step Adams-Bashforth to approximate the solution of the given initial value problems using the indicated number of time steps. Compare the approximate solution with the given exact solution:

    (310)#\[\begin{equation}y'=y-t, \ \ (0\leq t \leq 2),\end{equation}\]

    with the initial condition \(y(0)=2,\) Let \(N=4\), with the exact solution

    (311)#\[\begin{equation}y(t)=e^{t}+t+1.\end{equation}\]

Question 2#

  1. a. Apply the 2-step Adams-Moulton Method to approximate the solution of the given initial value problems using the indicated number of time steps. Compare the approximate solution with the given exact solution:

    (312)#\[\begin{equation} y'=t-y, \ \ (0\leq t \leq 4),\end{equation}\]

    with the initial condition \(y(0)=1,\) Let \(N=4\), with the exact solution

    (313)#\[\begin{equation}y(t)=2e^{-t}+t-1.\end{equation}\]

    b. Apply the 3-step Adams-Moulton Method to approximate the solution of the given initial value problems using the indicated number of time steps. Compare the approximate solution with the given exact solution:

    (314)#\[\begin{equation}y'=y-t, \ \ (0\leq t \leq 2),\end{equation}\]

    with the initial condition \(y(0)=2,\) Let \(N=4\), with the exact solution

    (315)#\[\begin{equation}y(t)=e^{t}+t+1.\end{equation}\]

Question 3#

  1. Derive the difference equation for the 1-step Adams-Bashforth method:

(316)#\[\begin{equation} w_{n+1}=w_n+hf(t_{n},w_{n}),\end{equation}\]

with the local truncation error

(317)#\[\begin{equation} \tau_{n+1}(h)=\frac{h}{2}y^{2}(\mu_n),\end{equation}\]

where \(\mu_n \in (t_{n},t_{n+1})\).

Question 4#

  1. Derive the difference equation for the 2-step Adams-Bashforth method:

(318)#\[\begin{equation} w_{n+1}=w_n+(\frac{3}{2}hf(t_{n},w_{n})-\frac{1}{2}hf(t_{n-1},w_{n-1})),\end{equation}\]

with the local truncation error

(319)#\[\begin{equation} \tau_{n+1}(h)=\frac{5h^2}{12}y^{3}(\mu_n),\end{equation}\]

where \(\mu_n \in (t_{n-1},t_{n+1})\).

Question 5#

  1. Derive the difference equation for the 2-step Adams-Moulton method:

(320)#\[\begin{equation} w_{n+1}=w_n+hf(t_{n+1},w_{n+1}),\end{equation}\]

with the local truncation error

(321)#\[\begin{equation}\tau_{n+1}(h)=-\frac{h}{2}y^{2}(\mu_n),\end{equation}\]

where \(\mu_n \in (t_{n-2},t_{n+1})\).

Question 6#

  1. Adapt the Python code for the 2-step Adams-Bashforth method provided to approximate solution of the integrate and fire differential equation:

    (322)#\[\begin{equation} \tau_m\frac{dV}{dt} = -(V-E_L) + R_mI(t), \ \ -50\leq t \leq 400, \end{equation}\]

    where \(E_L = -75\), \(\tau_m = 10\), \(R_m = 10\) and \(I(t)=0.01t\) and the initial condition \(V(-50) = -75\) using a stepsize of \(h=0.5\).

Question 7#

a. Define in your own words the terms strongly stable, weakly stable and unstable with respect to the characteristic equation.

b. Show that the two step Adams-Bashforth method is strongly stable.

Question 8#

  1. Apply the Predictor-Corrector method to numerically approximate the solution at \(x=1.0\) of the Initial Value Problem

(323)#\[\begin{equation}\frac{dy}{dx}=-y+x^2, \ \ 0\leq x \leq 1, \ \ \ y(0) = 1, \ \ \ y(0.25)=0.65, \end{equation}\]

using the two step Adams-Bashforth method and the one step Adams-Moulton method

(324)#\[\begin{equation} w_{i+1}=w_i + \frac{h}{2}[f(x_{i+1},w_{i+1})+f(x_{i},w_{i})], \end{equation}\]

with a step size of \(h=0.25\).

Question 9#

  1. Describe in your own words how the predictor-corrector technique can be used for error-control.