Problem Sheet 4 - Multistep Methods

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: $\( y'=t-y, \ \ (0\leq t \leq 4),\)\( with the initial condition \)y(0)=1,\( Let \)N=4\(, with the exact solution \)\(y(t)=2e^{-t}+t-1.\)$

2. 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: $\(y'=y-t, \ \ (0\leq t \leq 2),\)\( with the initial condition \)y(0)=2,\( Let \)N=4\(, with the exact solution \)\(y(t)=e^{t}+t+1.\)$

3. 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: $\( y'=t-y, \ \ (0\leq t \leq 4),\)\( with the initial condition \)y(0)=1,\( Let \)N=4\(, with the exact solution \)\(y(t)=2e^{-t}+t-1.\)$

4. 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: $\(y'=y-t, \ \ (0\leq t \leq 2),\)\( with the initial condition \)y(0)=2,\( Let \)N=4\(, with the exact solution \)\(y(t)=e^{t}+t+1.\)$

5. Derive the difference equation for the 1-step Adams-Bashforth method: $\( w_{n+1}=w_n+hf(t_{n},w_{n}),\)\( with the local truncation error \)\( \tau_{n+1}(h)=\frac{h}{2}y^{2}(\mu_n),\)\( where \)\mu_n \in (t_{n},t_{n+1})$.

6. Derive the difference equation for the 2-step Adams-Bashforth method: $\( w_{n+1}=w_n+(\frac{3}{2}hf(t_{n},w_{n})-\frac{1}{2}hf(t_{n-1},w_{n-1})),\)\( with the local truncation error \)\( \tau_{n+1}(h)=\frac{5h^2}{12}y^{3}(\mu_n),\)\( where \)\mu_n \in (t_{n-1},t_{n+1})$.

7. Derive the difference equation for the 3-step Adams-Bashforth method: $\( w_{n+1}=w_n+(\frac{23}{12}hf(t_{n},w_{n})-\frac{4}{3}hf(t_{n-1},w_{n-1})+\frac{5}{12}hf(t_{n-2},w_{n-2})),\)\( with the local truncation error \)\( \tau_{n+1}(h)=\frac{9h^3}{24}y^{4}(\mu_n),\)\( where \)\mu_n \in (t_{n-2},t_{n+1})$.

8. Derive the difference equation for the 0-step Adams-Moulton method: $\( w_{n+1}=w_n+hf(t_{n+1},w_{n+1}),\)\( with the local truncation error \)\(\tau_{n+1}(h)=-\frac{h}{2}y^{2}(\mu_n),\)\( where \)\mu_n \in (t_{n-2},t_{n+1})$.

9. Derive the difference equation for the 1-step Adams-Moulton method: $\( w_{n+1}=w_n+\frac{1}{2}hf(t_{n+1},w_{n+1})+\frac{1}{2}hf(t_{n},w_{n}),\)\( with the local truncation error \)\( \tau_{n+1}(h)=-\frac{h^2}{12}y^{3}(\mu_n),\)\( where \)\mu_n \in (t_{n},t_{n+1})$.

10. Derive the difference equation for the 2-step Adams-Moulton method: $\( w_{n+1}=w_n+\frac{5}{12}hf(t_{n+1},w_{n+1})+\frac{8}{12}hf(t_{n},w_{n})-\frac{1}{12}hf(t_{n-1},w_{n-1}),\)\( with the local truncation error \)\( \tau_{n+1}(h)=-\frac{h^3}{24}y^{4}(\mu_n),\)\( where \)\mu_n \in (t_{n-1},t_{n+1})$.

11. Derive the difference equation for the 3-step Adams-Moulton method:\ $\( w_{n+1}=w_n+\frac{9}{24}hf(t_{n+1},w_{n+1})+\frac{19}{24}hf(t_{n},w_{n})-\frac{5}{24}hf(t_{n-1},w_{n-1})+\frac{1}{24}hf(t_{n-2},w_{n-2}),\)\( with the local truncation error \)\( \tau_{n+1}(h)=-\frac{h^4}{720}y^{5}(\mu_n),\)\( where \)\mu_n \in (t_{n-2},t_{n+1})$.