A class of two-step implicit methods involving higher-order derivatives of y for initial value problems of the form y? = f(t, y, y?)is developed. The methods involve arbitrary parameters p and q, which are determined so that the methods become absolutely stable when applied to the test equation y? + ?y? + ?y = 0. Numerical results for Bessel's and general second-order differential equations are presented to illustrate that the methods are absolutely stable and are of order O(h4), O(h6) and O(h8).

Sesappa Rai, A.Ananthakrishnaiah, U.2026-02-05ObrechkoffJournal of Computational and Applied Mathematics, 1997, 79, 2, pp. 167-1823770427https://doi.org/10.1016/S0377-0427(96)00132-Xhttps://idr.nitk.ac.in/handle/123456789/280481997Computational methodsConvergence of numerical methodsNumerical analysisProblem solvingInitial value problemsObrechkoff methodsDifferential equationsA class of two-step implicit methods involving higher-order derivatives of y for initial value problems of the form y? = f(t, y, y?)is developed. The methods involve arbitrary parameters p and q, which are determined so that the methods become absolutely stable when applied to the test equation y? + ?y? + ?y = 0. Numerical results for Bessel's and general second-order differential equations are presented to illustrate that the methods are absolutely stable and are of order O(h4), O(h6) and O(h8).