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An integrated in phase FD procedure for DiffEqns in chemical problems

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Abstract

A newly FD procedure is perused for the effective application on the DiffEqns in Chemical problems.

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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, Essex, 2001)

    Google Scholar 

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

    Google Scholar 

  6. T.E. Kenan Mu, A. Simos, 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)

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

    Google Scholar 

  13. M.A. Medvedeva, T.E. Simos, An accomplished phase FD process for DEs in chemistry. J. Math. Chem. 57(10), 2208–2228 (2019)

    CAS  Google Scholar 

  14. 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 

  15. K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem. 56(1), 170–192 (2018)

    CAS  Google Scholar 

  16. M.M. Chawla, S.R. Sharma, Families of 5Th order Nyström methods for Y”=F(X, Y) and intervals of periodicity. Computing 26(3), 247–256 (1981)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  20. M.M. Chawla, S.R. Sharma, Intervals of periodicity and absolute stability of explicit Nyström methods for Y”=F(X, Y). BIT Numer. Math. 21(4), 455–464 (1981)

    Google Scholar 

  21. M.M. Chawla, Unconditionally stable noumerov-type methods for 2nd order differential-equations. BIT Numer. Math. 23(4), 541–542 (1983)

    Google Scholar 

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

  23. 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 

  24. M.M. Chawla, Numerov made explicit has better stability. BIT Numer. Math. 24(1), 117–118 (1984)

    Google Scholar 

  25. M.M. Chawla, P.S. Rao, High-accuracy P-stable methods for Y” = F(T, Y). IMA J. Numer. Anal. 5(2), 215–220 (1985)

    Google Scholar 

  26. M.M. Chawla, Correction. IMA J. Numer. Anal. 6(2), 252–252 (1986)

    Google Scholar 

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

    Google Scholar 

  28. R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT Numer. Math. 24, 225–238 (1984)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  31. 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 

  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 2. Explicit method. J. Comput. Appl. Math. 15(3), 329–337 (1986)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  35. 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 

  36. 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 

  37. 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. Modell. 29(2), 63–72 (1999)

    Google Scholar 

  38. J.P. Coleman, Numerical-methods for Y”=F(X, Y) via rational-approximations for the cosine. IMA J. Numer. Anal. 9(2), 145–165 (1989)

    Google Scholar 

  39. J.P. Coleman, A.S. Booth, Analysis of a family of Chebyshev methods for Y” = F(X, Y). J. Comput. Appl. Math. 44(1), 95–114 (1992)

    Google Scholar 

  40. J.P. Coleman, L.G. Ixaru, P-stability and exponential-fitting methods for Y”=F(X, Y). IMA J. Numer. Anal. 16(2), 179–199 (1996)

    Google Scholar 

  41. J.P. Coleman, S.C. Duxbury, Mixed collocation methods for Y ” = F(X, Y). J. Comput. Appl. Math. 126(1–2), 47–75 (2000)

    Google Scholar 

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

    Google Scholar 

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

    CAS  Google Scholar 

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

    Google Scholar 

  45. L.G. 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 

  46. L.G. 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 

  47. L.G. Ixaru, M. Rizea, Four step methods for Y”=F(X,Y). J. Comput. Appl. Math. 79(1), 87–99 (1997)

    Google Scholar 

  48. M. Van Daele, G. Vanden Berghe, H. De Meyer, L.G. Ixaru, Exponential-fitted four-step methods for Y ”=F(X,Y). Int. J. Comput. Math. 66(3–4), 299–309 (1998)

    Google Scholar 

  49. L.G. Ixaru, B. Paternoster, A conditionally p-stable fourth-order exponential-fitting method for Y ” = F(X, Y). J. Comput. Appl. Math. 106(1), 87–98 (1999)

    Google Scholar 

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

    CAS  PubMed  Google Scholar 

  51. L.G. 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 

  52. 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 

  53. 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 

  54. 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 

  55. 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)

    CAS  Google Scholar 

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

    Google Scholar 

  57. 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)

    Google Scholar 

  58. 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 

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

    Google Scholar 

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

    Google Scholar 

  61. 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 

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

    CAS  Google Scholar 

  63. 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 

  64. 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 

  65. 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 

  66. 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 

  67. 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 

  68. L.G. 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 

  69. 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 

  70. 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 

  71. 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 

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

    Google Scholar 

  73. 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 

  74. 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 

  75. 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 

  76. 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 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  80. 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 

  81. 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 

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

    Google Scholar 

  83. 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 

  84. Y. Fang, W. Xinyuan, 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 

  85. 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 

  86. 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 

  87. 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 

  88. 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 

  89. 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 

  90. 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 

  91. 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 

  92. 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 

  93. 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 

  94. 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 

  95. 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 

  96. 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 

  97. 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 

  98. 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 

  99. 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 

  100. 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 

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

    Google Scholar 

  102. 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 

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

    Google Scholar 

  104. 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 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  108. 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 

  109. 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 

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

    Google Scholar 

  111. 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 

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

    Google Scholar 

  113. 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 

  114. 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 

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

    Google Scholar 

  116. 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 

  117. 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 

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

  119. 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 

  120. Z. Wang, D. Zhao, Y. Dai, W. Dongmei, 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 

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

    Google Scholar 

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

    Google Scholar 

  123. 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 

  124. 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 

  125. 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 

  126. 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 

  127. 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 

  128. 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 

  129. 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 

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

    Google Scholar 

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

    Google Scholar 

  132. 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 

  133. 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 

  134. 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 

  135. 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 

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

    Google Scholar 

  137. 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 

  138. 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 

  139. 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) Article Number: 056705 (2009)

  140. 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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Authors and Affiliations

  1. Huaiyin Institute of Technology, Huaiyin, 223000, China

    Yu-Yu Ma

  2. School of Art, Ningbo Polytechnic, Ningbo, 315800, China

    Chia-Liang Lin

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

    T. E. Simos

  4. Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Dongtong Road 705, Neijiang, 641100, China

    T. E. Simos

  5. Group of Modern Computational Methods, Ural Federal University, 19 Mira Street, Ekaterinburg, Russian Federation, 620002

    T. E. Simos

  6. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece

    T. E. Simos

  7. Athens, Greece

    T. E. Simos

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  1. Yu-Yu Ma
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  2. Chia-Liang Lin
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  3. T. E. Simos
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T. E. Simos: Highly Cited Researcher (https://clarivate.com/hcr/), 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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Ma, YY., Lin, CL. & Simos, T.E. 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

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