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Journal of Mathematical Chemistry

, Volume 56, Issue 8, pp 2267–2301 | Cite as

New three-stages symmetric six-step finite difference method with vanished phase-lag and its derivatives up to sixth derivative for second order initial and/or boundary value problems with periodical and/or oscillating solutions

  • Ibraheem Alolyan
  • T. E. Simos
Original Paper
  • 55 Downloads

Abstract

In this paper, for the first time in the literature, we develop a symmetric three-stages six-step method with the following characteristics; the method
  1. 1.

    is a symmetric hybrid (multistages) six-step method,

     
  2. 2.

    is of three-stages,

     
  3. 3.

    is of twelfth algebraic order,

     
  4. 4.

    has vanished the phase-lag and

     
  5. 5.

    has vanished the derivatives of the phase-lag up to order six.

     
A detailed theoretical, numerical and computational analysis is also presented. The above analyses consist of:
  • the construction of the new six-step pair,

  • the presentation of the computed local truncation error of the new six-step pair,

  • the comparative error analysis of the new six-step pair with other six-step pairs of the same family which are:
    • the classical six-step pair of the family (i.e. the six-step pair with constant coefficients),

    • the recently proposed six-step pair of the same family with vanished phase-lag and its first derivative,

    • the recently proposed six-step pair of the same family with vanished phase-lag and its first and second derivatives,

    • the recently proposed six-step pair of the same family with vanished phase-lag and its first, second and third derivatives,

    • the recently proposed six-step pair of the same family with vanished phase-lag and its first, second, third and fourth derivatives and finally,

    • the recently proposed six-step pair of the same family with vanished phase-lag and its first, second, third, fourth and fifth derivatives

  • the stability and the interval of periodicity analysis for the new obtained six-step pair and finally

  • the investigation of the accuracy and computational efficiency of the new developed six-step pair for the solution of the Schrödinger equation.

The theoretical, numerical and computational achievements lead to the conclusion that the new produced three-stages symmetric six-step pair is more efficient than other known or recently developed finite difference pairs of the literature.

Keywords

Schrödinger equation Multistep methods Multistage methods Interval of periodicity Phase-lag Phase-fitted Derivatives of the phase-lag 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

10910_2018_888_MOESM1_ESM.pdf (212 kb)
Supplementary material 1 (pdf 212 KB)

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
  2. 2.Department of Automation EngineeringTEI of Sterea HellasPsachnaGreece
  3. 3.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece
  4. 4.AthensGreece

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