Hybrid, phase–fitted, four–step methods of seventh order for solving x″(t) = f(t,x)

Abstract

A four‐step method of seventh algebraic order is presented. It is tuned for addressing the special second order initial value problem. The new method is hybrid, explicit, and uses three stages per step. In addition is phase fitted. In consequence it uses variable coefficients that depend on the magnitude of the step‐size. We also present numerical tests on a set of standard problems that illustrate the efficiency of the derived method over older ones given in the relevant literature.

Phase‐fitted, six‐step methods for solving x′′ = f(t,x)

“…As companion of these four-step methods, we presented their trigonometric fitted and phase-fitted counterparts in the previous studies. 27,28 Recently, we dealt with six-steps methods of the form (1-2). As first result, we presented a two-stage, sixth-order method in Fang et al 29 Here, we indent to derive its phase-fitted version that is tuned for integrating problems with oscillatory solutions.…”

Explicit hybrid six–step, sixth order, fully symmetric methods for solving y ″ = f (x,y)

“…In Medvedev et al and Simos and Tsitouras,24,25 we proposed two-stage and three-stage methods of orders six and seven, respectively, based on an interpolatory approach. In addition, phase-fitted and trigonometric fitted four-step methods were examined in Medvedev et al 26,27 Lambert and Watson presented the following six-step, sixth order method: 28…”