A modification of the Stiefel-Bettis method for nonlinearly damped oscillators. (English) Zbl 0646.65063

This paper presents a modification of the method of E. Stiefel and D. G. Bettis [Numer. Math. 13, 154-175 (1969; Zbl 0219.65062)] for the numerical solution of initial value problems for second order differential equations of the type: \(y''+w^ 2y=f(t,y,y')\), where w is constant and the perturbed term f(t,y,y’) is much smaller than \(w^ 2y\). In order to derive such a formula, the differential equation is replaced by an equivalent integro-difference equation and then the integral is approximated by a three point integration formula. Thus the author obtains an implicit method which integrates exactly the functions 1, t, \(t^ 2\), \(t^ 3\), sin wt, cos wt and has a truncation error of \(O(h^ 6)\). Furthermore if f is independent of y’, the author proposes an explicit predictor which together with the modified Stiefel-Bettis method has trigonometric order one and truncation error of \(O(h^ 6)\). Finally some numerical experiments are presented.
Reviewer: M.Calvo


65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations


Zbl 0219.65062
Full Text: DOI


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