I am working on the following problem:
Derive the Adams-Bashforth Three-Step method by the following method.
Set y $(t_{i+1}) = y (t_i) + ah f (t_i, y (t_i)) + bhf (t_{i−1}, y (t_{i−1})) + ch f (t_{i−2}, y (t_{i−2})$
I am told to expand y $(t_{i+1}) , f (t_{i−2}, y (t_{i−2})) , and f (t_{i−1}, y (t_{i−1}))$ in Taylor series about $(t_i, y (t_i))$ , and equate the coefficients of h, h2 and h3 to obtain a, b, and c.
Now Im familiar with the Taylor expansion, which has the form :
$$\sum_{i=0}^{\infty} \frac{(f^n)(a)}{n!}(x-a)^n$$
But I dont know how to do the expansions required to solve this problem. Help would definitely be appreciated.