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A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation

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Abstract

A low computational cost eighth algebraic order hybrid two-step method with vanished phase-lag and its first, second, third and fourth derivatives is developed in this paper. We also investigate the local truncation error, the stability and the result of the elimination of the phase-lag and its derivatives on the effectiveness of the produced method.

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Notes

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

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Conflict of interest

The authors declare that they have no conflict of interest.

Author information

Authors and Affiliations

  1. School of information engineering, Chang’an University, Xi’an, 710064, China

    Hang Ning

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

    T. E. Simos

  3. 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

  4. 10 Konitsis Street, Amfithea - Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

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Appendix: Formulae of the derivatives of \(q_{n}\)

Appendix: Formulae of the derivatives of \(q_{n}\)

Formulae of the derivatives which presented in the formulae of the LTEs:

$$\begin{aligned} y_{n}^{(2)} \,= & {} \, \left( V(x)-V_{c} + G\right) \, y(x) \\ y_{n}^{(3)} \,= & {} \, \left( {\frac{d}{dx}}g \left( x \right) \right) y \left( x \right) + \left( g \left( x \right) +G \right) {\frac{d}{dx}}y \left( x \right) \\ y_{n}^{(4)} \,= & {} \, \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) y \left( x \right) +2\, \left( {\frac{d}{dx}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) \\&+ \left( g \left( x \right) +G \right) ^{2}y \left( x \right) \\ y_{n}^{(5)} \,= & {} \, \left( {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \right) y \left( x \right) +3\, \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) \\&+\,4\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d}{dx}}g \left( x \right) + \left( g \left( x \right) +G \right) ^{2}{\frac{d}{dx}}y \left( x \right) \\ y_{n}^{(6)} \,= & {} \, \left( {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \right) y \left( x \right) +4\, \left( {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) \\&+\,7\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) +4\, \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}y \left( x \right) \\&+\,6\, \left( g \left( x \right) +G \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d}{dx}}g \left( x \right) + \left( g \left( x \right) +G \right) ^{3}y \left( x \right) \\ y_{n}^{(7)} \,= & {} \, \left( {\frac{d^{5}}{d{x}^{5}}}g \left( x \right) \right) y \left( x \right) +\,5\, \left( {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) \\&+\,11\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) +\,15\, \left( {\frac{d}{dx}}g \left( x \right) \right) y \left( x \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+\,13\, \left( g \left( x \right) +G \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+\,10\, \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}{\frac{d}{dx}}y \left( x \right) +9\, \left( g \left( x \right) +G \right) ^{2}y \left( x \right) {\frac{d}{dx}}g \left( x \right) \\&+\, \left( g \left( x \right) +G \right) ^{3}{\frac{d}{dx}}y \left( x \right) \end{aligned}$$
$$\begin{aligned} y_{n}^{(8)} \,= & {} \, \left( {\frac{d^{6}}{d{x}^{6}}}g \left( x \right) \right) y \left( x \right) +6\, \left( {\frac{d^{5}}{d{x}^{5}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) \\&+\,16\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) +\,26\, \left( {\frac{d}{dx}}g \left( x \right) \right) y \left( x \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \\&+\,24\, \left( g \left( x \right) +G \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \\&+\,15\, \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) ^{2}y \left( x \right) +48\, \left( {\frac{d}{dx}}g \left( x \right) \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+\,22\, \left( g \left( x \right) +G \right) ^{2}y \left( x \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) +28\, \left( g \left( x \right) +G \right) y \left( x \right) \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}\\&+\,12\, \left( g \left( x \right) +G \right) ^{2} \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d}{dx}}g \left( x \right) + \left( g \left( x \right) +G \right) ^{4}y \left( x \right) \ldots \end{aligned}$$

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Ning, H., Simos, T. A low computational cost eight algebraic order hybrid method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation. J Math Chem 53, 1295–1312 (2015). https://doi.org/10.1007/s10910-015-0489-3

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