9.3 Higher-order Taylor methods
Keeping more terms of the Taylor series gives higher order. Since \(y' = f(t,y)\), the higher derivatives of \(y\) are obtained by differentiating \(f\) along the solution: \(y'' = f'(t,y) = \frac {\partial f}{\partial t}+\frac {\partial f}{\partial y}f\), and so on. The Taylor method of order \(n\) is \[w_{i+1} = w_i+hT^{(n)}(t_i,w_i),\qquad T^{(n)} = f+\frac h2f'+\dots +\frac {h^{n-1}}{n!}f^{(n-1)},\] with local truncation error \(O(h^n)\).
Solution. Here \(f' = \frac {d}{dt}(y-t^2+1) = y'-2t = y-t^2+1-2t\), so \[w_{i+1} = w_i+h\left [(w_i-t_i^2+1)+\frac h2(w_i-t_i^2+1-2t_i)\right ] .\] The first step is \(w_1 = 0.5+0.2[1.5+0.1(1.5)] = 0.83\), against \(y(0.2) = 0.8292986\). At \(t = 1\) and \(t = 2\) the values are \(2.6486459\) and \(5.3476843\), with errors \(7.8\times 10^{-3}\) and \(4.2\times 10^{-2}\): ten times smaller than Euler’s.
Taylor methods need the derivatives of \(f\), which are often complicated. The methods of the next subsection reach the same order using only values of \(f\).
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