Math428: Problem Set #4

1.

Derive the optimal coefficients for 3rd order Adams-Bashforth.

2.

[KC 8.4:4] Use the method of undetermined coefficients to derive the fourth-order Adams-Bashforth formula

$$x_{n+1} = x_n + \frac{h}{24}\left[55 f_n - 59 f_{n-1} + 37 f_{n-2} - 9 f_{n-3}\right]$$

3.

[KC 8.4:5] Derive the fourth-order Adams-Moulton formula

$$x_{n+1} = x_n + \frac{h}{24}\left[9 f_{n+1} + 19 f_n - 5 f_{n-1} + f_{n-2}\right]$$

4.

Consider the ``poor man's orbital problem''

$$x' = -y, \ \ \ \ y' = x \ \ \ \ x(0) = 1, y(0) = 0.$$

Solve the problem for $0 \leq t \leq 8 \pi$ (four revolutions) with $h=\pi/20$ and $h=\pi/40$ using forward Euler, backward Euler, 3rd order Adams-Bashforth, 3rd order Adams-Bashforth-Moulton. Use an appropriate Taylor series start-up procedure.