Comparison of two iteration procedures for a class of nonlinear jerk equations

Abstract

A modified Michens iteration procedure and a direct Michens iteration procedure are applied to determine approximate periods and analytical approximate periodic solutions of a class of nonlinear jerk equations. When we use the modified Michens iteration procedure to deal with the nonlinear jerk equations, we need to solve the nonlinear algebra equation for determining the approximate angular frequency. The higher the number of iterations, the more difficult it is to deal with the nonlinear algebra equation. This is because the kth-order approximate solution obtained by the modified Michens iteration procedure is a function of the angular frequency. However, since the kth-order approximate solution obtained by the direct Michens iteration procedure is independent on the kth-order approximate angular frequency, the form of nonlinear algebra equation that determines the kth-order approximate angular frequency of nonlinear jerk equation is similar. The second-order approximate period obtained by the direct Michens iteration procedure provides very accurate results for the large initial velocity amplitudes. But the second-order approximate period obtained by the modified Michens iteration procedure is invalid for all amplitudes B of the initial velocity. A comparison of the first- and second-order approximate periodic solutions obtained by the direct Michens iteration procedure with the numerically exact solutions shows that the second-order approximate periodic solution is much more accurate than the first one. Thus, the direct Michens iteration procedure is very effective for the class of nonlinear jerk equations.