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A phase fitted FiniteDiffr process for DiffrntEqutns in chemistry

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Abstract

A new FiniteDiffr (= Finite Difference) process is introduced or the effectual solution of the DiffrntEqutns (= Differential Equations) in Chemistry.

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References

  1. A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)

    Google Scholar 

  2. C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)

    Google Scholar 

  3. F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)

    Google Scholar 

  4. A.R. Leach, Molecular Modelling: Principles and Applications (Pearson, London, 2001)

    Google Scholar 

  5. P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford Univ. Press, Oxford, 2011)

    Google Scholar 

  6. 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, 1239–1256 (2015)

    CAS  Google Scholar 

  7. T.E. Simos, Exponentially fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation and related problems. Comput. Mater. Sci. 18, 315–332 (2000)

    Google Scholar 

  8. V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Y.A. Khakhalev, A.N. Zolotov, Numerical research of turbulent boundary layer based on the fractal dimension of pressure fluctuations. AIP Conf. Proc. 738, 480004 (2016)

    Google Scholar 

  9. V.N. Kovalnogov, R.V. Fedorov, T.V. Karpukhina, E.V. Tsvetova, Numerical analysis of the temperature stratification of the disperse flow. AIP Conf. Proc. 1648, 850033 (2015)

    Google Scholar 

  10. N. Kovalnogov, E. Nadyseva, O. Shakhov, V. Kovalnogov, Control of turbulent transfer in the boundary layer through applied periodic effects. Izvestiya Vysshikh Uchebnykh Zavedenii Aviatsionaya Tekhnika 1, 49–53 (1998)

    Google Scholar 

  11. V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Modeling and development of cooling technology of turbine engine blades. Int. Rev. Mech. Eng. 9(4), 331–335 (2015)

    Google Scholar 

  12. B.F. Hu, H. Wang, X.M. Yu, W.H. Yuan, T.W. He, Sparse network embedding for community detection and sign prediction in signed social networks. J. Ambient Intell. Humaniz. Comput. 10(1), 175–186 (2019). https://doi.org/10.1007/s12652-017-0630-1

    Article  Google Scholar 

  13. Z.H. Wang, J.J. Xu, X.M. Song, H.X. Zhang, Consensus problem in multi-agent systems under delayed information. Neurocomputing 316, 277–283 (2018). https://doi.org/10.1016/j.neucom.2018.08.002

    Article  Google Scholar 

  14. H. Liu, B. Xu, D.J. Lu, G.J. Zhang, A path planning approach for crowd evacuation in buildings based on improved artificial bee colony algorithm. Appl. Soft Comput. 68, 360–376 (2018). https://doi.org/10.1016/j.asoc.2018.04.015

    Article  Google Scholar 

  15. H. Liu, B. Liu, H. Zhang, L. Li, X. Qin, G. Zhan, Crowd evacuation simulation approach based on navigation knowledge and two-layer control mechanism. Inform. Sci. 436–437, 247–267 (2018)

    Google Scholar 

  16. J.X. Zong, L.L. Meng, H.X. Zhang, W.B. Wan, JND-based multiple description image coding. KSII Trans. Internet Inf. Syst. 11(8), 3935–3949 (2017). https://doi.org/10.3837/tiis.2017.08.010

    Article  Google Scholar 

  17. L. Zhai, H. Wang, Crowdsensing task assignment based on particle swarm optimization in cognitive radio networks. Wirel. Commun. Mob. Comput. 2017, 9 (2017). https://doi.org/10.1155/2017/4687974. Article ID 4687974

    Article  Google Scholar 

  18. S. Kottwitz, LaTeX Cookbook (Packt Publishing Ltd., Birmingham, 2015), pp. 231–236

    Google Scholar 

  19. M.A. Medvedeva, T.E. Simos, An accomplished phase FD process for DEs in chemistry. J. Math. Chem. 57, 2208–2228 (2019). https://doi.org/10.1007/s10910-019-01067-4

    Article  CAS  Google Scholar 

  20. 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)

    Google Scholar 

  21. Y.-Y. Ma, C.-L. Lin, T.E. Simos, An integrated in phase FD procedure for DiffEqns in chemical problems. J. Math. Chem. 58, 6–28 (2020). https://doi.org/10.1007/s10910-019-01070-9

    Article  CAS  Google Scholar 

  22. S. Hao, T.E. Simos, A phase fitted FinDiff process for DifEquns in quantum chemistry. J. Math. Chem. 58, 353–381 (2020). https://doi.org/10.1007/s10910-019-01081-6

    Article  CAS  Google Scholar 

  23. X. Tong, T.E. Simos, A complete in phase FinitDiff procedure for DiffEquns in chemistry. J. Math. Chem. 58, 407–438 (2020). https://doi.org/10.1007/s10910-019-01095-0

    CAS  Google Scholar 

  24. K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem. 56, 170–192 (2018). https://doi.org/10.1007/s10910-017-0787-z

    Article  CAS  Google Scholar 

  25. M.M. Chawla, S.R. Sharma, Families of 5th order Nyström methods for \(\text{Y}^{\prime \prime }=\text{F}(\text{X},\text{Y})\) and intervals of periodicity. Computing 26(3), 247–256 (1981)

    Google Scholar 

  26. J.M. Franco, M. Palacios, High-order P-stable multistep methods. J. Comput. Appl. Math. 30, 1–10 (1990)

    Google Scholar 

  27. J.D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem (Wiley, Hoboken, 1991), pp. 104–107

    Google Scholar 

  28. E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math. 13, 154–175 (1969)

    Google Scholar 

  29. M.M. Chawla, S.R. Sharma, Intervals of periodicity and absolute stability of explicit Nyström methods for \(\text{Y}^{\prime \prime }=\text{F}(\text{X},\text{Y})\). Bit 21(4), 455–464 (1981)

    Google Scholar 

  30. M.M. Chawla, Unconditionally stable Noumerov-type methods for 2nd order differential-equations. Bit 23(4), 541–542 (1983)

    Google Scholar 

  31. http://www.burtleburtle.net/bob/math/multistep.html

  32. M.M. Chawla, P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of 2nd order periodic initial-value problems. J. Comput. Appl. Math. 11(3), 277–281 (1984)

    Google Scholar 

  33. M.M. Chawla, Numerov made explicit has better stability. Bit 24(1), 117–118 (1984)

    Google Scholar 

  34. M.M. Chawla, P.S. Rao, High-accuracy P-stable methods for \(\text{Y}^{\prime \prime } =\text{F}(\text{T},\text{Y})\). IMA J. Numer. Anal. 5(2), 215–220 (1985); M.M Chawla, Correction. IMA J. Numer. Anal. 6(2), 252–252 (1986)

  35. T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Numer. Math. 19, 65–75 (1972)

    Google Scholar 

  36. R.M. Thomas, Phase properties of high order almost P-stable formulae. Bit 24, 225–238 (1984)

    Google Scholar 

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

    Google Scholar 

  38. M.M. Chawla, A new class of explicit 2-step 4th order methods for \(\text{Y}^{\prime \prime }= \text{F}(\text{T}, \text{Y})\) with extended intervals of periodicity. J. Comput. Appl. Math. 14(3), 467–470 (1986)

    Google Scholar 

  39. M.M. Chawla, B. Neta, Families of 2-step 4th-order p-stable methods for 2nd-order differential-equations. J. Comput. Appl. Math. 15(2), 213–223 (1986)

    Google Scholar 

  40. M.M. Chawla, P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of 2nd-order periodic initial-value problems. 2. Explicit method. J. Comput. Appl. Math. 15(3), 329–337 (1986)

    Google Scholar 

  41. M.M. Chawla, P.S. Rao, B. Neta, 2-Step 4th-order P-stable methods with phase-lag of order 6 for \(\text{Y}^{\prime \prime }=\text{F}(\text{T},\text{Y})\). J. Comput. Appl. Math. 16(2), 233–236 (1986)

    Google Scholar 

  42. M.M. Chawla, P.S. Rao, An explicit 6th-order method with phase-lag of order 8 for \(\text{Y}^{\prime \prime }=\text{F}(\text{T}, \text{Y})\). J. Comput. Appl. Math. 17(3), 365–368 (1987)

    Google Scholar 

  43. M.M. Chawla, M.A. Al-Zanaidi, Non-dissipative extended one-step methods for oscillatory problems. Int. J. Comput. Math. 69(1–2), 85–100 (1998)

    Google Scholar 

  44. M.M. Chawla, M.A. Al-Zanaidi, A two-stage fourth-order almost P-stable method for oscillatory problems. J. Comput. Appl. Math. 89(1), 115–118 (1998)

    Google Scholar 

  45. M.M. Chawla, M.A. Al-Zanaidi, S.S. Al-Ghonaim, Singly-implicit stabilized extended one-step methods for second-order initial-value problems with oscillating solutions. Math. Comput. Model. 29(2), 63–72 (1999)

    Google Scholar 

  46. J.P. Coleman, Numerical-methods for \(\text{Y}^{\prime \prime }=\text{F}(\text{X},\text{Y})\) via rational-approximations for the cosine. IMA J. Numer. Anal. 9(2), 145–165 (1989)

    Google Scholar 

  47. J.P. Coleman, A.S. Booth, Analysis of a family of Chebyshev methods for \(\text{Y}^{\prime \prime } = \text{F}(\text{X}, \text{Y})\). J. Comput. Appl. Math. 44(1), 95–114 (1992)

    Google Scholar 

  48. J.P. Coleman, L.Gr. Ixaru, P-stability and exponential-fitting methods for \(\text{Y}^{\prime \prime }=\text{F}(\text{X}, \text{Y})\). IMA J. Numer. Anal. 16(2), 179–199 (1996)

    Google Scholar 

  49. J.P. Coleman, S.C. Duxbury, Mixed collocation methods for \(\text{Y}^{\prime \prime } = \text{F}(\text{X}, \text{Y})\). J. Comput. Appl. Math. 126(1–2), 47–75 (2000)

    Google Scholar 

  50. L.Gr. Ixaru, S. Berceanu, Coleman method maximally adapted to the Schrödinger-equation. Comput. Phys. Commun. 44(1–2), 11–20 (1987)

    Google Scholar 

  51. L.Gr. Ixaru, The Numerov method and singular potentials. J. Comput. Phys. 72(1), 270–274 (1987)

    CAS  Google Scholar 

  52. L.Gr. Ixaru, M. Rizea, Numerov method maximally adapted to the Schrödinger-equation. J. Comput. Phys. 73(2), 306–324 (1987)

    Google Scholar 

  53. L.Gr. Ixaru, H. De Meyer, G. Vanden Berghe, M. Van Daele, Expfit4: a fortran program for the numerical solution of systems of nonlinear second-order initial-value problems. Comput. Phys. Commun. 100(1–2), 71–80 (1997)

    CAS  Google Scholar 

  54. L.Gr. Ixaru, G. Vanden Berghe, H. De Meyer, M. Van Daele, Four-step exponential-fitted methods for nonlinear physical problems. Comput. Phys. Commun. 100(1–2), 56–70 (1997)

    CAS  Google Scholar 

  55. L.Gr. Ixaru, M. Rizea, Four step methods for \(\text{Y}^{\prime \prime }=\text{F}(\text{X},\text{Y})\). J. Comput. Appl. Math. 79(1), 87–99 (1997)

    Google Scholar 

  56. M. Van Daele, G. Vanden Berghe, H. De Meyer, L.Gr. Ixaru, Exponential-fitted four-step methods for \(\text{Y} ^{\prime \prime }=\text{F}(\text{X}, \text{Y})\). Int. J. Comput. Math. 66(3–4), 299–309 (1998)

    Google Scholar 

  57. L.Gr. Ixaru, B. Paternoster, A conditionally P-stable fourth-order exponential-fitting method for \(\text{Y}^{\prime \prime } = \text{F}(\text{X}, \text{Y})\). J. Comput. Appl. Math. 106(1), 87–98 (1999)

    Google Scholar 

  58. L.Gr. Ixaru, Numerical operations on oscillatory functions. Comput. Chem. 25(1), 39–53 (2001)

    CAS  PubMed  Google Scholar 

  59. L.Gr. Ixaru, G. Vanden Berghe, H. De Meyer, Exponentially fitted variable two-step BDF algorithm for first order odes. Comput. Phys. Commun. 150(2), 116–128 (2003)

    CAS  Google Scholar 

  60. M.A. Medvedev, T.E. Simos, A three-stages multistep teeming in phase algorithm for computational problems in chemistry. J. Math. Chem. 57(6), 1598–1617 (2019)

    CAS  Google Scholar 

  61. M. Xu, T.E. Simos, A multistage two-step fraught in phase scheme for problems in mathematical chemistry. J. Math. Chem. 57(7), 1710–1731 (2019)

    CAS  Google Scholar 

  62. J. Lv, T.E. Simos, A Runge–Kutta type crowded in phase algorithm for quantum chemistry problems. J. Math. Chem 57(8), 1983–2006 (2019)

    CAS  Google Scholar 

  63. X. Zhang, T.E. Simos, A multiple stage absolute in phase scheme for chemistry problems. J. Math. Chem. 57(9), 2049–2074 (2019)

    CAS  Google Scholar 

  64. J. Qiu, J. Huang, T.E. Simos, A perfect in phase FD algorithm for problems in quantum chemistry. J. Math. Chem. 57(9), 2019–2048 (2019)

    CAS  Google Scholar 

  65. 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)

    CAS  Google Scholar 

  66. L.Gr. 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)

    CAS  Google Scholar 

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

    Google Scholar 

  68. L.Gr. 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)

    Google Scholar 

  69. 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)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  73. M.M. Chawla, P.S. Rao, An Noumerov-type 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)

    Google Scholar 

  74. 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 

  75. M. Rizea, Exponential fitting method for the time-dependent Schrödinger equation. J. Math. Chem. 48(1), 55–65 (2010)

    CAS  Google Scholar 

  76. A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224–252 (2010)

    CAS  Google Scholar 

  77. A.D. Raptis, J.R. Cash, A variable step method for the numerical integration of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 36, 113–119 (1985)

    CAS  Google Scholar 

  78. R.B. Bernstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules. Proc. R. Soc. Ser. A 274, 427–442 (1963)

    CAS  Google Scholar 

  79. R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)

    CAS  Google Scholar 

  80. M. Rizea, V. Ledoux, M. Van Daele, G. Vanden Berghe, N. Carjan, Finite difference approach for the two-dimensional Schrödinger equation with application to scission-neutron emission. Comput. Phys. Commun. 179(7), 466–478 (2008)

    CAS  Google Scholar 

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

    Google Scholar 

  82. L.Gr. Ixaru, M. Rizea, G. Vanden Berghe, H. De Meyer, Weights of the exponential fitting multistep algorithms for first-order odes. J. Comput. Appl. Math. 132(1), 83–93 (2001)

    Google Scholar 

  83. A.D. Raptis, J.R. Cash, Exponential and Bessel fitting methods for the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 44(1–2), 95–103 (1987)

    Google Scholar 

  84. C.D. Papageorgiou, A.D. Raptis, A method for the solution of the Schrödinger-equation. Comput. Phys. Commun. 43(3), 325–328 (1987)

    CAS  Google Scholar 

  85. 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, 442–465 (2016)

    CAS  Google Scholar 

  86. A.D. Raptis, Exponential multistep methods for ordinary differential equations. Bull. Greek Math. Soc. 25, 113–126 (1984)

    Google Scholar 

  87. H. Ning, T.E. Simos, 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(6), 1295–1312 (2015)

    CAS  Google Scholar 

  88. Z. Wang, T.E. Simos, An economical eighth-order method for the approximation of the solution of the Schrödinger equation. J. Math. Chem. 55, 717–733 (2017)

    CAS  Google Scholar 

  89. J.R. Cash, A.D. Raptis, A high-order method for the numerical-integration of the one-dimensional Schrödinger-equation. Comput. Phys. Commun. 33(4), 299–304 (1984)

    CAS  Google Scholar 

  90. A.D. Raptis, Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control. Comput. Phys. Commun. 28(4), 427–431 (1983)

    Google Scholar 

  91. A.D. Raptis, 2-Step methods for the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 28(4), 373–378 (1982)

    Google Scholar 

  92. A.D. Raptis, On the numerical-solution of the Schrödinger-equation. Comput. Phys. Commun. 24(1), 1–4 (1981)

    Google Scholar 

  93. A.D. Raptis, Exponential-fitting methods for the numerical-integration of the 4th-order differential-equation \(\text{Y}^{iv}+\text{F}{.}\text{Y}=\text{ G }\). Computing 24(2–3), 241–250 (1980)

    Google Scholar 

  94. H. Van De Vyver, A symplectic exponentially fitted modified Runge–Kutta–Nyström method for the numerical integration of orbital problems. New Astron. 10(4), 261–269 (2005)

    Google Scholar 

  95. H. Van De Vyver, On the generation of P-stable exponentially fitted Runge–Kutta–Nyström methods by exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 188(2), 309–318 (2006)

    Google Scholar 

  96. M. Van Daele, G. Vanden Berghe, P-stable Obrechkoff methods of arbitrary order for second-order differential equations. Numer. Algorithms 44(2), 115–131 (2007)

    Google Scholar 

  97. M. Van DAELE, G. Vanden BERGHE, P-stable exponentially-fitted Obrechkoff methods of arbitrary order for second-order differential equations. Numer. Algorithms 46(4), 333–350 (2007)

    Google Scholar 

  98. Y. Fang, X. Wu, A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions. Appl. Numer. Math. 58(3), 341–351 (2008)

    Google Scholar 

  99. G. Vanden Berghe, M. Van Daele, Exponentially-fitted Obrechkoff methods for second-order differential equations. Appl. Numer. Math. 59(3–4), 815–829 (2009)

    Google Scholar 

  100. D. Hollevoet, M. Van Daele, G. Vanden Berghe, The optimal exponentially-fitted Numerov method for solving two-point boundary value problems. J. Comput. Appl. Math. 230(1), 260–269 (2009)

    Google Scholar 

  101. J.M. Franco, L. Rández, Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs. Appl. Math. Comput. 273, 493–505 (2016)

    Google Scholar 

  102. J.M. Franco, Y. Khiar, L. Rández, Two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems. Appl. Math. Comput. 252, 45–57 (2015)

    Google Scholar 

  103. J.M. Franco, I. Gomez, L. Rández, Optimization of explicit two-step hybrid methods for solving orbital and oscillatory problems. Comput. Phys. Commun. 185(10), 2527–2537 (2014)

    CAS  Google Scholar 

  104. J.M. Franco, I. Gomez, Trigonometrically fitted nonlinear two-step methods for solving second order oscillatory IVPs. Appl. Math. Comput. 232, 643–657 (2014)

    Google Scholar 

  105. J.M. Franco, I. Gomez, Symplectic explicit methods of Runge–Kutta–Nyström type for solving perturbed oscillators. J. Comput. Appl. Math. 260, 482–493 (2014)

    Google Scholar 

  106. J.M. Franco, I. Gomez, Some procedures for the construction of high-order exponentially fitted Runge–Kutta–Nyström methods of explicit type. Comput. Phys. Commun. 184(4), 1310–1321 (2013)

    CAS  Google Scholar 

  107. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, On some new low storage implementations of time advancing Runge–Kutta methods. J. Comput. Appl. Math. 236(15), 3665–3675 (2012)

    Google Scholar 

  108. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Symmetric and symplectic exponentially fitted Runge–Kutta methods of high order. Comput. Phys. Commun. 181(12), 2044–2056 (2010)

    CAS  Google Scholar 

  109. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, On high order symmetric and symplectic trigonometrically fitted Runge–Kutta methods with an even number of stages. BIT Numer. Math. 50(1), 3–21 (2010)

    Google Scholar 

  110. J.M. Franco, I. Gomez, Accuracy and linear stability of RKN methods for solving second-order stiff problems. Appl. Numer. Math. 59(5), 959–975 (2009)

    Google Scholar 

  111. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Sixth-order symmetric and symplectic exponentially fitted Runge–Kutta methods of the Gauss type. J. Comput. Appl. Math. 223(1), 387–398 (2009)

    Google Scholar 

  112. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Structure preservation of exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 218(2), 421–434 (2008)

    Google Scholar 

  113. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Sixth-order symmetric and symplectic exponentially fitted modified Runge–Kutta methods of Gauss type. Comput. Phys. Commun. 178(10), 732–744 (2008)

    CAS  Google Scholar 

  114. J.M. Franco, Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Commun. 177(6), 479–492 (2007)

    CAS  Google Scholar 

  115. J.M. Franco, New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56(8), 1040–1053 (2006)

    Google Scholar 

  116. J.M. Franco, Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)

    CAS  Google Scholar 

  117. J.M. Franco, Stability of explicit ARKN methods for perturbed oscillators. J. Comput. Appl. Math. 173, 389–396 (2005)

    Google Scholar 

  118. X.Y. Wu, X. You, J.Y. Li, Note on derivation of order conditions for ARKN methods for perturbed oscillators. Comput. Phys. Commun. 180, 1545–1549 (2009)

    CAS  Google Scholar 

  119. A. Tocino, J. Vigo-Aguiar, Symplectic conditions for exponential fitting Runge–Kutta–Nyström methods. Math. Comput. Model. 42, 873–876 (2005)

    Google Scholar 

  120. L. Brugnano, F. Iavernaro, D. Trigiante, Hamiltonian boundary value methods (energy preserving discrete line integral methods). J. Numer. Anal. Ind. Appl. Math. 5, 17–37 (2010)

    Google Scholar 

  121. F. Iavernaro, D. Trigiante, High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. J. Numer. Anal. Ind. Appl. Math. 4, 87–101 (2009)

    Google Scholar 

  122. A. Konguetsof, A generator of families of two-step numerical methods with free parameters and minimal phase-lag. J. Math. Chem. 55(9), 1808–1832 (2017)

    CAS  Google Scholar 

  123. A. Konguetsof, A hybrid method with phase-lag and derivatives equal to zero for the numerical integration of the Schrödinger equation. J. Math. Chem. 49(7), 1330–1356 (2011)

    CAS  Google Scholar 

  124. H. Van de Vyver, A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems. Int. J. Mod. Phys. C 17(5), 663–675 (2006)

    Google Scholar 

  125. H. Van de Vyver, An explicit Numerov-type method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53(9), 1339–1348 (2007)

    Google Scholar 

  126. Y. Fang, X. Wu, A trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems. Appl. Math. Comput. 189(1), 178–185 (2007)

    Google Scholar 

  127. B. Neta, P-stable high-order super-implicit and Obrechkoff methods for periodic initial value problems. Comput. Math. Appl. 54(1), 117–126 (2007)

    Google Scholar 

  128. H. Van de Vyver, Phase-fitted and amplification-fitted two-step hybrid methods for \(\text{Y}^{\prime \prime } = \text{F} (\text{X}, \text{Y})\). J. Comput. Appl. Math. 209(1), 33–53 (2007)

    Google Scholar 

  129. H. Van de Vyver, Efficient one-step methods for the Schrödinger equation. MATCH Commun. Math. Comput. Chem. 60(3), 711–732 (2008)

    Google Scholar 

  130. J. Martín-Vaquero, J. Vigo-Aguiar, Exponential fitted Gauss, Radau and Lobatto methods of low order. Numer. Algorithms 48(4), 327–346 (2008)

    Google Scholar 

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

    CAS  Google Scholar 

  132. F.A. Hendi, P-stable higher derivative methods with minimal phase-lag for solving second order differential equations. J. Appl. Math. 2011, 407151 (2011)

    Google Scholar 

  133. H. Van de Vyver, Comparison of some special optimized fourth-order Runge–Kutta methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 166(2), 109–122 (2005)

    Google Scholar 

  134. Z. Wang, D. Zhao, Y. Dai, D. Wu, An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial-value problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 461(2058), 1639–1658 (2005)

    Google Scholar 

  135. M. Van Daele, G. Vanden Berghe, H. De Meyer, Properties and implementation of \(\rho \)-Adams methods based on mixed-type interpolation. Comput. Math. Appl. 30(10), 37–54 (1995)

    Google Scholar 

  136. J. Vigo-Aguiar, L.M. Quintales, A parallel ODE solver adapted to oscillatory problems. J. Supercomput. 19(2), 163–171 (2001)

    Google Scholar 

  137. Z. Wang, Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Duffing equation. Comput. Phys. Commun. 174(2), 109–118 (2006)

    CAS  Google Scholar 

  138. Z. Wang, Trigonometrically-fitted method for a periodic initial value problem with two frequencies. Comput. Phys. Commun. 175(4), 241–249 (2006)

    CAS  Google Scholar 

  139. J. Vigo-Aguiar, J.M. Ferrandiz, A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems. SIAM J. Numer. Anal. 35(4), 1684–1708 (1998)

    Google Scholar 

  140. J. Vigo-Aguiar, H. Ramos, Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations. J. Comput. Appl. Math. 158(1), 187–211 (2003)

    Google Scholar 

  141. J. Vigo-Aguiar, S. Natesan, A parallel boundary value technique for singularly perturbed two-point boundary value problems. J. Supercomput. 27(2), 195–206 (2004)

    Google Scholar 

  142. C. Tang, H. Yan, H. Zhang, W.R. Li, The various order explicit multistep exponential fitting for systems of ordinary differential equations. J. Comput. Appl. Math. 169(1), 171–182 (2004)

    Google Scholar 

  143. C. Tang, H. Yan, H. Zhang, Z. Chen, M. Liu, G. Zhang, The arbitrary order implicit multistep schemes of exponential fitting and their applications. J. Comput. Appl. Math. 173(1), 155–168 (2005)

    Google Scholar 

  144. H. Van de Vyver, Frequency evaluation for exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 184(2), 442–463 (2005)

    Google Scholar 

  145. J.P. Coleman, L.Gr. Ixaru, Truncation errors in exponential fitting for oscillatory problems. SIAM J. Numer. Anal. 44(4), 1441–1465 (2006)

    Google Scholar 

  146. J. Martín-Vaquero, J. Vigo-Aguiar, Adapted BDF algorithms: higher-order methods and their stability. J. Sci. Comput. 32(2), 287–313 (2007)

    Google Scholar 

  147. J. Vigo-Aguiar, J. Martín-Vaquero, H. Ramos, Exponential fitting BDF–Runge–Kutta algorithms. Comput. Phys. Commun. 178(1), 15–34 (2008)

    CAS  Google Scholar 

  148. B. Paternoster, Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday. Comput. Phys. Commun. 183(12), 2499–2512 (2012)

    CAS  Google Scholar 

  149. Z. Wang, Obrechkoff one-step method fitted with Fourier spectrum for undamped Duffing equation. Comput. Phys. Commun. 175(11–12), 692–699 (2006)

    CAS  Google Scholar 

  150. C. Wang, Z. Wang, A P-stable eighteenth-order six-step method for periodic initial value problems. Int. J. Mod. Phys. C 18(3), 419–431 (2007)

    Google Scholar 

  151. J. Chen, Z. Wang, H. Shao, H. Hao, Highly-accurate ground state energies of the He atom and the He-like ions by Hartree SCF calculation with Obrechkoff method. Comput. Phys. Commun. 179(7), 486–491 (2008)

    CAS  Google Scholar 

  152. H. Shao, Z. Wang, Arbitrarily precise numerical solutions of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 180(1), 1–7 (2009)

    CAS  Google Scholar 

  153. H. Shao, Z. Wang, Numerical solutions of the time-dependent Schrödinger equation: reduction of the error due to space discretization. Phys. Rev. E 79(5), 056705 (2009)

    Google Scholar 

  154. Z. Wang, H. Shao, A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. Comput. Phys. Commun. 180(6), 842–849 (2009)

    CAS  Google Scholar 

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Chen, X., Simos, T.E. A phase fitted FiniteDiffr process for DiffrntEqutns in chemistry. J Math Chem 58, 1059–1090 (2020). https://doi.org/10.1007/s10910-020-01104-7

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