Multi-step Methods

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Differential Equations › Multi-step Methods

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The two-step Adams-Bashforth method of approximation uses the approximation scheme $y_{i+2} = y_{i+1} - \frac{1}{2}hf(t_i, y_i) + \frac{3}{2}hf(t_{i+1}, y_{i+1})$ .

Given that and , use the Adams-Bashforth method to approximate for with a step size of

$-\frac{159}{8}$

$\frac{97}{8}$

$\frac{55}{8}$

$\frac{33}{4}$

$\frac{133}{8}$

Explanation

In this problem, we're given two points, so we can start plugging in immediately. If we were not, we could approximate $y_{i+1}$ by using the explicit Euler method on $y_{i}$ .

Plugging into $y_{i+2} = y_{i+1} - \frac{1}{2}hf(t_i, y_i) + \frac{3}{2}hf(t_{i+1}, y_{i+1})$ , we have

$y(2) \approx \mu_2 = 2 -\frac{1}{2}\cdot 1 \cdot -1^2 + \frac{3}{2}\cdot 1 \cdot -2^2 = 2 +\frac{1}{2} - 6 = -\frac{7}{2}$

$y(3)\approx \mu_3 = -\frac{7}{2} + \frac{1}{2}\cdot4 - \frac{3}{2}\cdot \frac{49}{4} =-\frac{159}{8}$ .

Note, our approximation likely won't be very good with such large a time step, but the process doesn't change regardless of the accuracy.

$y(3)\approx -\frac{159}{8}$