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A hybrid type four-step method with vanished phase-lag and its first, second and third derivatives for each level for the numerical integration of the Schrödinger equation

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Abstract

In this paper, we study the effects of the vanishing of the phase-lag and its first, second and third derivatives on the effectiveness of a four-step hybrid type method of sixth algebraic order. As a result of the above described study, a Hybrid type of three level four-step method of sixth algebraic order is obtained. We investigate the new produced method theoretically and computationally. The theoretical investigation of the new hybrid method consists of:

  • The development of the new method.

  • The computation of the Local Truncation Error.

  • The Comparison of the Local Truncation Error analysis with other known methods of the same form.

  • The Stability Analysis.

The computational investigation consists of the application of the new obtained hybrid method to the numerical solution of the resonance problem of the radial time independent Schrödinger equation.

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Notes

  1. Where \(S\) is a set of distinct points.

  2. With the term classical we mean the method of Sect. 4 with constant coefficients.

  3. The reference values are computed using the well known two-step method of Chawla and Rao [23] with small step size for the integration

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    Ibraheem Alolyan & T. E. Simos

  2. Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00 , Tripolis, Greece

    T. E. Simos

  3. 10 Konitsis Street, Amfithea-Paleon Faliron, 175 64 , Athens, Greece

    T. E. Simos

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  1. Ibraheem Alolyan
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Highly Cited Researcher (http://isihighlycited.com/), Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding Member of European Academy of Arts, Sciences and Humanities.

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Alolyan, I., Simos, T.E. A hybrid type four-step method with vanished phase-lag and its first, second and third derivatives for each level for the numerical integration of the Schrödinger equation. J Math Chem 52, 2334–2379 (2014). https://doi.org/10.1007/s10910-014-0375-4

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