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:

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

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

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

  1. 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:

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

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

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

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:

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

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

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

  1. b) 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:

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

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

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

Question 3#

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

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

with the local truncation error

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

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

Question 4#

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

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

with the local truncation error

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

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

Question 5#

  1. Derive the difference equation for the 3-step Adams-Bashforth method:

(499)#\[\begin{equation} w_{n+1}=w_n+(\frac{23}{12}hf(t_{n},w_{n})-\frac{4}{3}hf(t_{n-1},w_{n-1})+\frac{5}{12}hf(t_{n-2},w_{n-2})),\end{equation}\]

with the local truncation error

(500)#\[\begin{equation} \tau_{n+1}(h)=\frac{9h^3}{24}y^{4}(\mu_n),\end{equation}\]

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

Question 6#

  1. Derive the difference equation for the 0-step Adams-Moulton method:

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

with the local truncation error

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

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

Question 7#

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

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

with the local truncation error

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

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

Question 8#

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

(505)#\[\begin{equation} w_{n+1}=w_n+\frac{5}{12}hf(t_{n+1},w_{n+1})+\frac{8}{12}hf(t_{n},w_{n})-\frac{1}{12}hf(t_{n-1},w_{n-1}),\end{equation}\]

with the local truncation error

(506)#\[\begin{equation} \tau_{n+1}(h)=-\frac{h^3}{24}y^{4}(\mu_n),\end{equation}\]

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

Question 9#

  1. Derive the difference equation for the 3-step Adams-Moulton method:\

(507)#\[\begin{equation} w_{n+1}=w_n+\frac{9}{24}hf(t_{n+1},w_{n+1})+\frac{19}{24}hf(t_{n},w_{n})-\frac{5}{24}hf(t_{n-1},w_{n-1})+\frac{1}{24}hf(t_{n-2},w_{n-2}),\end{equation}\]

with the local truncation error

(508)#\[\begin{equation} \tau_{n+1}(h)=-\frac{h^4}{720}y^{5}(\mu_n),\end{equation}\]

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