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In the present paper, we will investigate a family of explicit four-step methods first introduced by Anastassi and Simos [1] for the case of vanishing of phase-lag and its first derivative. These methods are efficient for the numerical solution of the Schr¨odinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions. As we mentioned before the main scope of this paper is the study of the elimination of the phase-lag and its first derivative of the family of the methods to be investigated. A comparative error and stability analysis will be presented for the studied family of four-step explicit methods. The new obtained methods will finally be tested on the resonance problem of the Schr¨odinger equation in order to examine their efficiency.

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