Q10Advanced Engineering Mathematics

Milne's method is a sophisticated multi-step numerical algorithm for solving ordinary differential equations. It operates in two phases. First, it utilizes an open-type extrapolation Predictor formula: $y_{n+1}^{(P)}=y_{n-3}+\frac{4h}{3}(2f_{n-2}-f_{n-1}+2f_n)$. Second, it immediately refines this prediction using a closed-type Simpson's based Corrector formula: $y_{n+1}^{(C)}=y_{n-1}+\frac{h}{3}(f_{n-1}+4f_n+f_{n+1}^{(P)})$.