Trigonometrically Fitted Runge-Kutta Methods for the Numerical Solution of the Oscillatory Problems
DOI:
https://doi.org/10.29304/jqcm.2021.13.3.835Keywords:
Explicit Runge-Kutta methods, oscillating problems, trigonometrically fittedAbstract
In this paper, two trigonometrically methods were established based on classical Runge-Kutta methods of the fourth and fifth-order respectively. The new methods will be applied for the numerical integration of oscillatory problems and have high effectiveness as the results demonstrate. Numerical results show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientific literature.
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References
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[2] Ch. Tsitouras, and T.E. Simos, "Optimized Runge–Kutta pairs for problems with oscillating solutions," Journal of Computational and Applied Mathematics, 147(2), (2002), pp.397-409.
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[13] S. Liu; J. Zheng; and Y. Fang, "A new modified embedded 5 (4) pair of explicit Runge–Kutta methods for the numerical solution of the Schrödinger equation," Journal of Mathematical Chemistry, 51(3), (2013), pp.937-953.
[14] J.R. Dormand, Numerical methods for differential equations: a computational approach, CRC Press, New York, (1996).
[15] F.A. Fawzi, N. Senu, F. Ismail, and Z.A. Majid, "New phase-fitted and amplification-fitted modified Runge-Kutta method for solving oscillatory problems," Global Journal of Pure and Applied Mathematics, 12(2), (2016), pp.1229-1242.
[16] D.P. Sakas, and T.E. Simos, "A fifth algebraic order trigonometrically-fitted modified Runge-Kutta Zonneveld method for the numerical solution of orbital problems," Mathematical and Computer Modelling, 42(7-8), (2005), pp.903-920.
[17] T.E. Simos, and J.V. Aguiar, "A modified Runge–Kutta method with phase-lag of order infinity for the numerical solution of the Schrödinger equation and related problems," Computers & Chemistry, 25(3), (2001), pp.275-281.
[18] Q. Ming, Y. Yang, and Y. Fang, "An optimized Runge-Kutta method for the numerical solution of the radial Schrödinger equation," Mathematical Problems in Engineering, 2012, Article ID 867948, (2012), pp.1-12.
[19] A.A. Kosti, Z. A. Anastassi, and 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," Journal of Mathematical Chemistry, 47(1), (2010), pp.315-330.
[20] M. Salih, F. Ismail, and N. Senu, "Phase fitted and amplification fitted of Runge-Kutta-Fehlberg method of order 4 (5) for solving oscillatory problems," Baghdad Science Journal, 17(2), (2020), pp.689-693.
[21] K. Hussain, F. Ismail, N. Senu, and F. Rabiei, "Optimized fourth-order Runge-Kutta method for solving oscillatory problems," In AIP Conference Proceedings, 1739(1), (2016), pp.1-7.
[22] K.A. Hussain, and Z.E. Abdulnaby, "A new two derivative FSAL Runge-Kutta method of order five in four stages," Baghdad Science Journal, 17(1), (2020), pp.161-171.
[23] K.A. Hussain, "Trigonometrically fitted fifth-order explicit two-derivative Runge-Kutta method with FSAL property," Journal of Physics: Conference Series. 1294(3), (2019), pp.1-10.
[2] Ch. Tsitouras, and T.E. Simos, "Optimized Runge–Kutta pairs for problems with oscillating solutions," Journal of Computational and Applied Mathematics, 147(2), (2002), pp.397-409.
[3] Z.A. Anastassi, and T.E. Simos, "Special optimized Runge–Kutta methods for IVPs with oscillating solutions," International Journal of Modern Physics C, 15(01), (2004), pp.1-15.
[4] W. Gautschi, "Numerical integration of ordinary differential equation based on trigonometric polynomials," Numerische Mathematik, 3(1), (1961), pp.381-397.
[5] T. Lyche, "Chebushevian multistep methods for ordinary differential equations," Numerische Mathematik, 19(1), (1972), pp.65–75..
[6] G.V. Berghe, H.D. Meyer, M.V. Daele, and T.V. Hecke, "Exponentially fitted Runge-Kutta methods," Journal of Computational and Applied Mathematics, 125(1-2), (2000), pp. 107–115.
[7] G.V. Berghe, H.D. Meyer, M.V. Daele, and T.V. Hecke, "Exponentially fitted Runge-Kutta methods," Computer Physics Communications, 123(1-3), (1999), pp.7–15.
[8] B. Paternoster, "Runge–Kutta(–Nystrom) methods for ODEs with periodic solutions based on trigonometric polynomials," Applied Numerical Mathematics, 28(2-4), (1998), pp.401-412.
[9] Z.A. Anastassi, and T.E. Simos, "Trigonometrically fitted fifth-order runge-kutta methods for the numerical solution of the Schrödinger equation," Mathematical and computer modelling, 42(7-8), (2005), pp.877-886.
[10] Z.A. Anastassi, and T.E. Simos, "Trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation," Journal of mathematical chemistry, 37(3), (2005), pp.281–293.
[11] Y. Zhang, H. Che, Y. Fang, and X. You, "A new trigonometrically fitted two-derivative Runge-Kutta method for the numerical solution of the Schrödinger equation and related problems," Journal of Applied Mathematics, 2013, Article ID 937858, (2013),pp.1–9.
[12] Y. Fang, X. You, and Q. Ming, "Trigonometrically fitted two-derivative Runge-Kutta methods for solving oscillatory differential equations," Numerical Algorithms, 65(3), (2014), pp.651–667.
[13] S. Liu; J. Zheng; and Y. Fang, "A new modified embedded 5 (4) pair of explicit Runge–Kutta methods for the numerical solution of the Schrödinger equation," Journal of Mathematical Chemistry, 51(3), (2013), pp.937-953.
[14] J.R. Dormand, Numerical methods for differential equations: a computational approach, CRC Press, New York, (1996).
[15] F.A. Fawzi, N. Senu, F. Ismail, and Z.A. Majid, "New phase-fitted and amplification-fitted modified Runge-Kutta method for solving oscillatory problems," Global Journal of Pure and Applied Mathematics, 12(2), (2016), pp.1229-1242.
[16] D.P. Sakas, and T.E. Simos, "A fifth algebraic order trigonometrically-fitted modified Runge-Kutta Zonneveld method for the numerical solution of orbital problems," Mathematical and Computer Modelling, 42(7-8), (2005), pp.903-920.
[17] T.E. Simos, and J.V. Aguiar, "A modified Runge–Kutta method with phase-lag of order infinity for the numerical solution of the Schrödinger equation and related problems," Computers & Chemistry, 25(3), (2001), pp.275-281.
[18] Q. Ming, Y. Yang, and Y. Fang, "An optimized Runge-Kutta method for the numerical solution of the radial Schrödinger equation," Mathematical Problems in Engineering, 2012, Article ID 867948, (2012), pp.1-12.
[19] A.A. Kosti, Z. A. Anastassi, and 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," Journal of Mathematical Chemistry, 47(1), (2010), pp.315-330.
[20] M. Salih, F. Ismail, and N. Senu, "Phase fitted and amplification fitted of Runge-Kutta-Fehlberg method of order 4 (5) for solving oscillatory problems," Baghdad Science Journal, 17(2), (2020), pp.689-693.
[21] K. Hussain, F. Ismail, N. Senu, and F. Rabiei, "Optimized fourth-order Runge-Kutta method for solving oscillatory problems," In AIP Conference Proceedings, 1739(1), (2016), pp.1-7.
[22] K.A. Hussain, and Z.E. Abdulnaby, "A new two derivative FSAL Runge-Kutta method of order five in four stages," Baghdad Science Journal, 17(1), (2020), pp.161-171.
[23] K.A. Hussain, "Trigonometrically fitted fifth-order explicit two-derivative Runge-Kutta method with FSAL property," Journal of Physics: Conference Series. 1294(3), (2019), pp.1-10.
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Published
2021-08-16
How to Cite
Ghazal, Z. K., & Hussain, K. A. (2021). Trigonometrically Fitted Runge-Kutta Methods for the Numerical Solution of the Oscillatory Problems. Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(3), Math Page 25– 33. https://doi.org/10.29304/jqcm.2021.13.3.835
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