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A predictor–corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation

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Abstract

A predictor–corrector explicit four-step method of sixth algebraic order is investigated in this paper. More specifically, we investigate the results of the elimination of the phase-lag and its first, second and third derivatives on the efficiency of the proposed method. The resultant method is studied theoretically and computationally. The theoretical investigation of the new hybrid method consists of: (1) the construction of the new method, (2) the definition (calculation) of the local truncation error, (3) the comparative local truncation error analysis (with other known methods of the same form), (4) the stability analysis using scalar test equation with frequency different than the frequency of the phase-lag analysis. Finally, we will study computationally the new obtained method. This study is based on the application of the new produced predictor–corrector explicit four-step method to the approximate 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 [24] 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, Tripoli, Greece

    T. E. Simos

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

    T. E. Simos

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Alolyan, I., Simos, T.E. A predictor–corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation. J Math Chem 53, 685–717 (2015). https://doi.org/10.1007/s10910-014-0449-3

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