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A new two stages tenth algebraic order symmetric six-step method with vanished phase-lag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems

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Abstract

The presentation, development and analysis of a new two-stages tenth algebraic order symmetric six-step method is introduced, for the first time in the literature, in this paper. More specifically, we present the development of the new method (requesting the highest algebraic order and the elimination of the phase-lag and its first and second derivatives), the analysis (error analysis and stability and interval of periodicity analysis) and the evaluation of the new developed method comparing its efficiency with the efficiency of well known methods and very recently produced methods in the literature on the approximate solution of the resonance problem of the one dimensional (or radial) Schrödinger equation. From the developments achieved and the results presented, we prove that the new obtained method is most more effective than other well known or recently developed methods of the literature.

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Notes

  1. where S is a set of distinct points

References

  1. L.G. Ixaru, M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)

    Google Scholar 

  2. L.D. Landau, F.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965)

    Google Scholar 

  3. I. Prigogine, S. Rice (eds.), Advances in Chemical Physics Vol. 93. New Methods in Computational Quantum Mechanics, vol. 93 (Wiley, New York, 1997)

    Google Scholar 

  4. G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, Toronto, 1950)

    Google Scholar 

  5. T.E. Simos, J. Vigo-Aguiar, A modified phase-fitted Runge–Kutta method for the numerical solution of the Schrödinger equation. J. Math. Chem. 30(1), 121–131 (2001)

    Article  CAS  Google Scholar 

  6. K. Tselios, T.E. Simos, Runge–Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. J. Comput. Appl. Math. 175(1), 173–181 (2005)

    Article  Google Scholar 

  7. Z.A. Anastassi, T.E. Simos, An optimized Runge–Kutta method for the solution of orbital problems. J. Comput. Appl. Math. 175(1), 1–9 (2005)

    Article  Google Scholar 

  8. D.F. Papadopoulos, T.E. Simos, A new methodology for the construction of optimized Runge–Kutta–Nyström methods. Int. J. Modern Phys. C 22(6), 623–634 (2011)

    Article  Google Scholar 

  9. D.F. Papadopoulos, T.E. Simos, A modified Runge–Kutta–Nyström method by using phase lag properties for the numerical solution of orbital problems. Appl. Math. Inform. Sci. 7(2), 433–437 (2013)

    Article  Google Scholar 

  10. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inform. Sci. 7(1), 81–85 (2013)

    Article  Google Scholar 

  11. A.A. Kosti, Z.A. Anastassi, T.E. Simos, Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems. Comput. Math. Appl. 61(11), 3381–3390 (2011)

    Article  Google Scholar 

  12. Z. Kalogiratou, T. Monovasilis, G. Psihoyios, T.E. Simos, Runge-Kutta type methods with special properties for the numerical integration of ordinary differential equations. Phys. Rep. Rev. Sect. Phys. Lett. 536(3), 75–146 (2014)

    Google Scholar 

  13. Z. Kalogiratou, T. Monovasilis, T.E. Simos, A fourth order modified trigonometrically fitted symplectic Runge–Kutta–Nyström method. Comput. Phys. Commun. 185(12), 3151–3155 (2014)

    Article  CAS  Google Scholar 

  14. A.A. Kosti, Z.A. Anastassi, T.E. Simos, An optimized explicit Runge–Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 47(1), 315–330 (2010)

    Article  CAS  Google Scholar 

  15. Z. Kalogiratou, T.E. Simos, Construction of trigonometrically and exponentially fitted Runge–Kutta–Nyström methods for the numerical solution of the Schrödinger equation and related problems a method of 8th algebraic order. J. Math. Chem 31(2), 211–232 (2002)

    Article  CAS  Google Scholar 

  16. T.E. Simos, A fourth algebraic order exponentially-fitted Runge–Kutta method for the numerical solution of the Schrödinger equation. IMA J. Numer. Anal. 21(4), 919–931 (2001)

    Article  Google Scholar 

  17. T.E. Simos, Exponentially-fitted Runge–Kutta–Nyström method for the numerical solution of initial-value problems with oscillating solutions. Appl. Math. Lett. 15(2), 217–225 (2002)

    Article  Google Scholar 

  18. C. Tsitouras, T.E. Simos, Optimized Runge–Kutta pairs for problems with oscillating solutions. J. Comput. Appl. Math. 147(2), 397–409 (2002)

    Article  Google Scholar 

  19. Z.A. Anastassi, T.E. Simos, Trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 281–293 (2005)

    Article  CAS  Google Scholar 

  20. Z.A. Anastassi, T.E. Simos, A family of exponentially-fitted Runge–Kutta methods with exponential order up to three for the numerical solution of the Schrödinger equation. J. Math. Chem 41(1), 79–100 (2007)

    Article  CAS  Google Scholar 

  21. J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)

    Article  Google Scholar 

  22. G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)

    Article  Google Scholar 

  23. H. Ramos, Z. Kalogiratou, T. Monovasilis, T.E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms 72(4), 1089–1102 (2016)

  24. Z. Kalogiratou, T. Monovasilis, H. Ramos, T.E. Simos, A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comp. Appl. Math. 303 (2016)

  25. C. Tsitouras, I.T. Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62(4), 2101–2111 (2011)

    Article  Google Scholar 

  26. http://burtleburtle.net/bob/math/multistep.html

  27. G. Avdelas, A. Konguetsof, T.E. Simos, A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 1. Development of the basic method. J. Math. Chem 29(4), 281–291 (2001)

    Article  CAS  Google Scholar 

  28. M.M. Chawla, P.S. Rao, An explicit sixth-order method with phase-lag of order eight for \(y^{\prime \prime }=f(t, y)\). J. Comput. Appl. Math. 17, 363–368 (1987)

    Google Scholar 

  29. M.M. Chawla, P.S. Rao, An Noumerov-typ method with minimal phase-lag for the integration of second order periodic initial-value problems II. Explicit method. J. Comput. Appl. Math. 15, 329–337 (1986)

    Article  Google Scholar 

  30. T.E. Simos, P.S. Williams, A finite difference method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189–205 (1997)

    Article  Google Scholar 

  31. G. Avdelas, A. Konguetsof, T.E. Simos, A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrödinger equation. Part 2. Development of the generator; optimization of the generator and numerical results. J. Math. Chem 29(4), 293–305 (2001)

    Article  CAS  Google Scholar 

  32. T.E. Simos, J. Vigo-Aguiar, Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation. J. Math. Chem 31(2), 135–144 (2002)

    Article  CAS  Google Scholar 

  33. A. Konguetsof, T.E. Simos, A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 93–106 (2003)

    Article  Google Scholar 

  34. T.E. Simos, I.T. Famelis, C. Tsitouras, Zero dissipative. Explicit numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms 34(1), 27–40 (2003)

    Article  Google Scholar 

  35. D.P. Sakas, T.E. Simos, Multiderivative methods of eighth algrebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation. J. Comput. Appl. Math. 175(1), 161–172 (2005)

    Article  Google Scholar 

  36. E.T. Simos, Optimizing a class of linear multi-step methods for the approximate solution of the radial Schrödinger equation and related problems with respect to phase-lag. Cent. Eur. J. Phys. 9(6), 1518–1535 (2011)

    Google Scholar 

  37. D.P. Sakas, T.E. Simos, A family of multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 317–331 (2005)

    Article  CAS  Google Scholar 

  38. H. Van de Vyver, Phase-fitted and amplification-fitted two-step hybrid methods for \(y^{\prime \prime }=f(x, y)\). J. Comput. Appl. Math. 209(1), 33–53 (2007)

    Article  Google Scholar 

  39. H. Van de Vyver, An explicit Numerov-type method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53, 1339–1348 (2007)

    Article  Google Scholar 

  40. T.E. Simos, A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem. 46(3), 981–1007 (2009)

    Article  CAS  Google Scholar 

  41. G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A new symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems. Int. J. Mod. Phys. C 22(2), 133–153 (2011)

    Article  Google Scholar 

  42. G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions. Comput. Phys. Commun. 182(8), 1626–1637 (2011)

    Article  CAS  Google Scholar 

  43. T. E. Simos, Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag. J. Appl. Math. Article ID 420387, Volume 2012 (2012)

  44. T.E. Simos, A two-step method with vanished phase-lag and its first two derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 49(10), 2486–2518 (2011)

    Article  CAS  Google Scholar 

  45. I. Alolyan, T.E. Simos, A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 62(10), 3756–3774 (2011)

    Article  Google Scholar 

  46. I. Alolyan, T.E. Simos, A new four-step hybrid type method with vanished phase-lag and its first derivatives for each level for the approximate integration of the Schrödinger equation. J. Math. Chem. 51, 2542–2571 (2013)

    Article  CAS  Google Scholar 

  47. I. Alolyan, T.E. Simos, A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem. 52, 917–947 (2014)

    Article  CAS  Google Scholar 

  48. I. Alolyan, T.E. Simos, Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53, 1808–1834 (2015)

    Article  CAS  Google Scholar 

  49. I. Alolyan, T.E. Simos, Family of symmetric linear six-step methods with vanished phase-lag and its derivatives and their application to the radial Schrödinger equation and related problems. J. Math. Chem. 54, 466–502 (2016)

    Article  CAS  Google Scholar 

  50. I. Alolyan, T.E. Simos. A family of two stages tenth algebraic order symmetric six-step methods with vanished phase–lag and its first derivatives for the numerical solution of the radial Schrödinger equation and related problems. J. Math. Chem. (2016). doi:10.1007/s10910-016-0654-3

  51. I. Alolyan, Z.A. Anastassi, T.E. Simos, A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput. 218(9), 5370–5382 (2012)

    Google Scholar 

  52. Z.A. Anastassi, T.E. Simos, A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems. J. Comput. Appl. Math. 236(16), 3880–3889 (2012)

    Article  Google Scholar 

  53. G.A. Panopoulos, T.E. Simos, An optimized symmetric 8-step semi-embedded predictor-corrector method for IVPs with oscillating solutions. Appl. Math. Inform. Sci. 7(1), 73–80 (2013)

    Article  Google Scholar 

  54. G. A. Panopoulos, Z. A. Anastassi and T.E. Simos, A new Eight-step symmetric embedded predictor-corrector method (EPCM) for orbital problems and related IVPs with oscillatory solutions. Astron. J. 145(3) Article Number: 75 (2013). doi:10.1088/0004-6256/145/3/75

  55. T.E. Simos, New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. Construction and theoretical analysis. J. Math. Chem. 51(1), 194–226 (2013)

    Article  CAS  Google Scholar 

  56. T.E. Simos, On the explicit four-step methods with vanished phase-lag and its first derivative. Appl. Math. Inform. Sci. 8(2), 447–458 (2014)

    Article  Google Scholar 

  57. G.A. Panopoulos, T.E. Simos, A new optimized symmetric embedded predictor-corrector method (EPCM) for initial-value problems with oscillatory solutions. Appl. Math. Inform. Sci. 8(2), 703–713 (2014)

    Article  Google Scholar 

  58. T.E. Simos, An explicit four-step method with vanished phase-lag and its first and second derivatives. J. Math. Chem. 52(3), 833–855 (2014)

    Article  CAS  Google Scholar 

  59. T.E. Simos, An explicit linear six-step method with vanished phase-lag and its first derivative. J. Math. Chem. 52(7), 1895–1920 (2014)

    Article  CAS  Google Scholar 

  60. T.E. Simos, A new explicit hybrid four-step method with vanished phase-lag and its derivatives. J. Math. Chem. 52(7), 1690–1716 (2014)

    Article  CAS  Google Scholar 

  61. I. Alolyan, T.E. Simos, A family of explicit linear six-step methods with vanished phase-lag and its first derivative. J. Math. Chem. 52(8), 2087–2118 (2014)

    Article  CAS  Google Scholar 

  62. I. Alolyan, T.E. Simos, 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(9), 2334–2379 (2014)

    Article  CAS  Google Scholar 

  63. E. Thedore, Multistage symmetric two-step p-stable method with vanished phase-lag and its first, second and third derivatives. Appl. Comput. Math. 14(3), 296–315 (2015)

    Google Scholar 

  64. F. Hui, T.E. Simos, Four stages symmetric two-step p-stable method with vanished phase-lag and its first, second, third and fourth derivatives. Appl. Comput. Math. 15(2), 220–238 (2016)

    Google Scholar 

  65. I. Alolyan, T.E. Simos, A family of embedded explicit six-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation: development and theoretical analysis. J. Math. Chem. 54(5), 1159–1186 (2016)

    Article  CAS  Google Scholar 

  66. M. Liang, T.E. Simos, A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation. J. Math. Chem. 54(5), 1187–1211 (2016)

    Article  CAS  Google Scholar 

  67. I. Alolyan, T.E. Simos, An implicit symmetric linear six-step methods with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation and related problems. J. Math. Chem. 54(4), 1010–1040 (2016)

    Article  CAS  Google Scholar 

  68. Z. Zhou, T.E. Simos, A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 54(2), 442–465 (2016)

    Article  CAS  Google Scholar 

  69. F. Hui, T.E. Simos, A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(10), 2191–2213 (2015)

    Article  CAS  Google Scholar 

  70. I. Alolyan, T.E. Simos, A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem. 53(8), 1915–1942 (2015)

    Article  CAS  Google Scholar 

  71. I. Alolyan, T.E. Simos, A high algebraic order predictor-corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem. 53(7), 1495–1522 (2015)

    Article  CAS  Google Scholar 

  72. K. Mu, T.E. Simos, A Runge-Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation. J. Math. Chem. 53(5), 1239–1256 (2015)

    Article  CAS  Google Scholar 

  73. I. Alolyan, T.E. Simos, 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(2), 685–717 (2015)

    Article  CAS  Google Scholar 

  74. T.E. Simos, A new explicit four-step method with vanished phase-lag and its first and second derivatives. J. Math. Chem. 53(1), 402–429 (2015)

    Article  CAS  Google Scholar 

  75. G.A. Panopoulos, T.E. Simos, An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math. 290, 1–15 (2015)

    Article  Google Scholar 

  76. A. Konguetsof, A new two-step hybrid method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871–890 (2010)

    Article  CAS  Google Scholar 

  77. K. Tselios, T.E. Simos, Symplectic methods for the numerical solution of the radial Shrödinger equation. J. Math. Chem 34(1–2), 83–94 (2003)

    Article  CAS  Google Scholar 

  78. K. Tselios, T.E. Simos, Symplectic methods of fifth order for the numerical solution of the radial Shrodinger equation. J. Math. Chem 35(1), 55–63 (2004)

    Article  CAS  Google Scholar 

  79. T. Monovasilis, T.E. Simos, New second-order exponentially and trigonometrically fitted symplectic integrators for the numerical solution of the time-independent Schrödinger equation. J. Math. Chem. 42(3), 535–545 (2007)

    Article  CAS  Google Scholar 

  80. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 37(3), 263–270 (2005)

    Article  CAS  Google Scholar 

  81. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Trigonometrically fitted and exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 40(3), 257–267 (2006)

    Article  CAS  Google Scholar 

  82. Z. Kalogiratou, T. Monovasilis, T.E. Simos, Symplectic integrators for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 158(1), 83–92 (2003)

    Article  Google Scholar 

  83. T.E. Simos, Closed Newton-cotes trigonometrically-fitted formulae of high-order for long-time integration of orbital problems. Appl. Math. Lett. 22(10), 1616–1621 (2009)

    Article  Google Scholar 

  84. Z. Kalogiratou, T.E. Simos, Newton-Cotes formulae for long-time integration. J. Comput. Appl. Math. 158(1), 75–82 (2003)

    Article  Google Scholar 

  85. T.E. Simos, High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 209(1), 137–151 (2009)

    Google Scholar 

  86. T.E. Simos, Closed Newton-Cotes trigonometrically-fitted formulae for the solution of the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 787–801 (2008)

    CAS  Google Scholar 

  87. T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae of high order for the numerical integration of the Schrödinger equation. J. Math. Chem. 44(2), 483–499 (2008)

    Article  CAS  Google Scholar 

  88. T.E. Simos, High-order closed Newton–Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Comput. Phys. Commun. 178(3), 199–207 (2008)

    Article  CAS  Google Scholar 

  89. T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae for numerical integration of the Schrödinger equation. Comput. Lett. 3(1), 45–57 (2007)

    Article  Google Scholar 

  90. T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae for long-time integration of orbital problems. RevMexAA 42(2), 167–177 (2006)

    Google Scholar 

  91. T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae for long-time integration. Int. J. Mod. Phys. C 14(8), 1061–1074 (2003)

    Article  Google Scholar 

  92. T.E. Simos. New closed newton-cotes type formulae as multilayer symplectic integrators. J. Chem. Phys. 133(10) Article Number: 104108 (2010)

  93. T.E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration. Abstr. Appl. Anal. Article Number: 182536, (2012). doi:10.1155/2012/182536

  94. T.E. Simos, High order closed Newton–Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation. J. Math. Chem. 50(5), 1224–1261 (2012)

    Article  CAS  Google Scholar 

  95. T.E. Simos, Accurately closed newton-cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Int. J. Modern Phys. C 24(3), (2013). doi:10.1142/S0129183113500149

  96. T.E. Simos, New open modified Newton–Cotes type formulae as multilayer symplectic integrators. Appl. Math. Model. 37(4), 1983–1991 (2013)

    Article  Google Scholar 

  97. G. Vanden Berghe, M. Van Daele, Exponentially fitted open Newton–Cotes differential methods as multilayer symplectic integrators. J. Chem. Phys. 132, 204107 (2010)

    Article  CAS  Google Scholar 

  98. Z. Kalogiratou, T. Monovasilis, T.E. Simos, A fifth-order symplectic trigonometrically fitted partitioned Runge–Kutta method. International Conference on Numerical Analysis and Applied Mathematics, Sep 16–20, 2007 Corfu, GREECE, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings vol. 936, pp 313–317 (2007)

  99. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems. Comput. Phys. Commun. 177(10), 757–763 (2007)

    Article  CAS  Google Scholar 

  100. T. Monovasilis, T.E. Simos, Symplectic methods for the numerical integration of the Schrödinger equation. Comput. Mater. Sci. 38(3), 526–532 (2007)

    Article  CAS  Google Scholar 

  101. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge–Kutta methods. Phys. Lett. A 372(5), 569–573 (2008)

    Article  CAS  Google Scholar 

  102. Z. Kalogiratou, T. Monovasilis, T.E. Simos, New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation. Comput. Math. Appl. 60(6), 1639–1647 (2010)

    Article  Google Scholar 

  103. T. Monovasilis, Z. Kalogiratou, T.E. Simos, A family of trigonometrically fitted partitioned Runge–Kutta symplectic methods. Appl. Math. Comput. 209(1), 91–96 (2009)

    Google Scholar 

  104. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Construction of exponentially fitted symplectic Runge–Kutta–Nyström methods from partitioned Runge–Kutta methods. Mediterr. J. Math. (2015). doi:10.1007/s00009-015-0587-2

  105. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Two new phase-fitted symplectic partitioned Runge–Kutta methods. Int. J. Mod. Phys. C 22(12), 1343–1355 (2011)

    Article  Google Scholar 

  106. K. Tselios, T.E. Simos, Optimized fifth order symplectic integrators for orbital problems. Rev. Mex. Astron. Astrofis. 49(1), 11–24 (2013)

    Google Scholar 

  107. T. Monovasilis, Z. Kalogiratou, T.E. Simos, Symplectic partitioned Runge–Kutta methods with minimal phase-lag. Comput. Phys. Commun. 181(7), 1251–1254 (2010)

    Article  CAS  Google Scholar 

  108. L.G. Ixaru, M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19, 23–27 (1980)

    Article  Google Scholar 

  109. A.D. Raptis, A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)

    Article  Google Scholar 

  110. J. Vigo-Aguiar, T.E. Simos, Family of twelve steps exponential fitting symmetric multistep methods for the numerical solution of the Schrödinger equation. J. Math. Chem 32(3), 257–270 (2002)

    Article  CAS  Google Scholar 

  111. G. Psihoyios, T.E. Simos, Trigonometrically fitted predictor-corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math. 158(1), 135–144 (2003)

    Article  Google Scholar 

  112. G. Psihoyios, T.E. Simos, A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions. J. Comput. Appl. Math. 175(1), 137–147 (2005)

    Article  Google Scholar 

  113. T.E. Simos, Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004)

    Article  Google Scholar 

  114. T.E. Simos, Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Appl. Math. 110(3), 1331–1352 (2010)

    Article  Google Scholar 

  115. G. Avdelas, E. Kefalidis, T.E. Simos, New P-stable eighth algebraic order exponentially-fitted methods for the numerical integration of the Schrödinger equation. J. Math. Chem. 31(4), 371–404 (2002)

    Article  CAS  Google Scholar 

  116. T.E. Simos, A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrödinger equation and related problems. J. Math. Chem. 34(1–2), 39–58 (2003)

    Article  CAS  Google Scholar 

  117. T.E. Simos, Exponentially—fitted multiderivative methods for the numerical solution of the Schrödinger equation. J. Math. Chem. 36(1), 13–27 (2004)

    Article  CAS  Google Scholar 

  118. T.E. Simos, A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem. 40(3), 305–318 (2006)

    Article  CAS  Google Scholar 

  119. H. Van de Vyver, A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems. Appl. Math. Comput. 189(1), 178–185 (2007)

    Google Scholar 

  120. T.E. Simos, A family of four-step trigonometrically-fitted methods and its application to the Schrodinger equation. J. Math. Chem. 44(2), 447–466 (2009)

    Article  CAS  Google Scholar 

  121. Z.A. Anastassi, T.E. Simos, A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution. J. Math. Chem. 45(4), 1102–1129 (2009)

    Article  CAS  Google Scholar 

  122. G. Psihoyios, T.E. Simos, Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 37(3), 295–316 (2005)

    Article  CAS  Google Scholar 

  123. G. Psihoyios, T.E. Simos, The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order predictor-corrector methods. J. Math. Chem. 40(3), 269–293 (2006)

    Article  CAS  Google Scholar 

  124. Z. Wang, P-stable linear symmetric multistep methods for periodic initial-value problems. Comput. Phys. Commun. 171(3), 162–174 (2005)

    Article  CAS  Google Scholar 

  125. T.E. Simos, A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation. J. Math. Chem. 27(4), 343–356 (2000)

    Article  CAS  Google Scholar 

  126. Z.A. Anastassi, T.E. Simos, A family of two-stage two-step methods for the numerical integration of the Schrödinger equation and related IVPs with oscillating solution. J. Math. Chem. 45(4), 1102–1129 (2009)

    Article  CAS  Google Scholar 

  127. C. Tang, W. Wang, H. Yan, Z. Chen, High-order predictor-corrector of exponential fitting for the N-body problems. J. Comput. Phys. 214(2), 505–520 (2006)

    Article  CAS  Google Scholar 

  128. G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions. J. Math. Chem. 46(2), 604–620 (2009)

    Article  CAS  Google Scholar 

  129. S. Stavroyiannis, T.E. Simos, Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Numer. Math. 59(10), 2467–2474 (2009)

    Article  Google Scholar 

  130. S. Stavroyiannis, T.E. Simos, A nonlinear explicit two-step fourth algebraic order method of order infinity for linear periodic initial value problems. Comput. Phys. Commun. 181(8), 1362–1368 (2010)

    Article  CAS  Google Scholar 

  131. Z.A. Anastassi, T.E. Simos, Numerical multistep methods for the efficient solution of quantum mechanics and related problems. Phys. Rep. 482, 1–240 (2009)

    Article  CAS  Google Scholar 

  132. R. Vujasin, M. Sencanski, J. Radic-Peric, M. Peric, A comparison of various variational approaches for solving the one-dimensional vibrational Schrödinger equation. MATCH Commun. Math. Comput. Chem. 63(2), 363–378 (2010)

    CAS  Google Scholar 

  133. T.E. Simos, P.S. Williams, On finite difference methods for the solution of the Schrödinger equation. Comput. Chem. 23, 513–554 (1999)

    Article  CAS  Google Scholar 

  134. L.G. Ixaru, M. Rizea, Comparison of some four-step methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985)

    Article  CAS  Google Scholar 

  135. J. Vigo-Aguiar, T.E. Simos, Review of multistep methods for the numerical solution of the radial Schrödinger equation. Int. J. Quant. Chem. 103(3), 278–290 (2005)

    Article  CAS  Google Scholar 

  136. T.E. Simos and G. Psihoyios, Special issue: The International Conference on Computational Methods in Sciences and Engineering 2004 - Preface, J. Comput. Appl. Math. 191(2) 165–165 (2006)

  137. T.E. Simos, G. Psihoyios, Special issue—selected papers of the international conference on computational methods in sciences and engineering (ICCMSE 2003) Kastoria, Greece, 12–16 September 2003—Preface. J. Comput. Appl. Math 175(1), IX-IX (2005)

  138. T.E. Simos, J. Vigo-Aguiar, Special Issue—selected papers from the conference on computational and mathematical methods for science and engineering (CMMSE-2002) - Alicante University, Spain, 20–25 September 2002—Preface. J. Comput. Appl. Math. 158(1) IX-IX (2003)

  139. T.E. Simos, C. Tsitouras, I. Gutman: Preface for the special issue numerical methods in chemistry. MATCH Commun. Math. Comput. Chem. 60(3) (2008)

  140. T.E. Simos, I. Gutman. Papers presented on the international conference on computational methods in sciences and engineering (Castoria, Greece, September 12–16, 2003), MATCH Commun. Math. Comput. Chem. 53(2) A3–A4 (2005)

  141. J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of Runge–Kutta–Nyström formulae. IMA J. Numer. Anal. 7, 235–250 (1987)

    Article  Google Scholar 

  142. J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  Google Scholar 

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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, Amphithea - Paleon Phaleron, 175 64, Athens, Greece

    T. E. Simos

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  1. Ibraheem Alolyan
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T.E. Simos: 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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Appendix

Appendix

See Table 3.

Table 3 Constant coefficients for the symmetric two stages Six-Step Methods of the same form
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Alolyan, I., Simos, T.E. A new two stages tenth algebraic order symmetric six-step method with vanished phase-lag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems. J Math Chem 55, 105–131 (2017). https://doi.org/10.1007/s10910-016-0674-z

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