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One Step Adapted Hybrid Second Derivative Block Method for Initial Value Problems with Oscillating Solutions

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Abstract

The purpose of this study is to develop a one-step block hybrid formula for solving a system of first-order differential equations with oscillating solutions. The formula is designed to avoid the problems that come with the conventional hybrid formulas. To increase accuracy, the formula incorporates the second order derivative into its construction. The construction of the formula is on the assumption that when the real solution is a linear combination of polynomial and sinusoidal functions that includes a parameter, it causes no errors. The formula is a sixth-order convergence, according to the analysis. The proposed formula’s performance is demonstrated using some typical numerical examples from recent literature whose solutions exhibit oscillatory behavior.

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References

  1. Abdulganiy, R.I.: Trigonometrically Fitted Block Backward Differentiation Methods for First Order Initial Value Problems with Periodic Solution. J. Adv. Math. Computer Sci. 28(5), 1–14 (2018). https://doi.org/10.9734/JAMCS/2018/42774

    Article  Google Scholar 

  2. Abdulganiy, R.I., Akinfenwa, O.A., Okunuga, S.A.: A family of \(L_0\) Stable Third Derivative Block Methods for Solving Systems Of First Order Initial Value Problems. J. Nigerian Association Math. Phys. 36(14), 47–54 (2016)

    Google Scholar 

  3. Abdulganiy, R.I., Akinfenwa, O.A., Okunuga, S.A.: Maximal Order Block Trigonometrically Fitted Scheme for the Numerical Treatment of Second Order Initial Value Problem with Oscillating Solutions. International J. Math. Anal. Optim. 2017, 168–186 (2017)

    Google Scholar 

  4. Abdulganiy, R.I., Akinfenwa, O.A., Okunuga, S.A., Oladimeji, G.O.: A Robust Block Hybrid Trigonometric Method for the Numerical Integration of Oscillatory Second Order Nonlinear Initial Value Problems. AMSE JOURNALS-AMSE IIETA publication-2017-Series: Advances A 54, 497–518 (2017)

    Google Scholar 

  5. Abdulganiy, R.I., Akinfenwa, O.A., Okunuga, S.A.: Construction of L Stable Second Derivative Trigonometrically Fitted Block Backward Differentiation Formula for the Solution of Oscillatory Initial Value Problems. Afr. J. Sci. Technol. Innov. Dev. 10(4), 411–419 (2018). https://doi.org/10.1080/20421338.2018.1467859

    Article  Google Scholar 

  6. Abdulganiy, R.I., Akinfenwa, O.A., Yusuff, O.A., Enobabor, O.E., Okunuga, S.A.: Block Third Derivative Trigonometrically-Fitted Methods for Stiff and Periodic Problems. J. Nigerian Society Phys. Sci. 2(1), 12–25 (2020)

    Article  Google Scholar 

  7. Akinfenwa, O.A., Jator, S.N., Yao, N.: On the Stability of Continuous Block Backward Differentiation Formula for Solving Stiff Ordinary Differential Equation. J. Mod. Meth. In Numer. Maths 3(2), 50–58 (2012)

    Article  Google Scholar 

  8. Akinfenwa, O.A., Abdulganiy, R.I., Akinnukawe, B.I., Okunuga, S.A.: Seventh Order Block Hybrid Method for solution of First Order Stiff Systems of Initial value Problems. J. Egyptian Math. Soc. 28(34), 11 (2020). https://doi.org/10.1186/s42787-020-00095-3

    Article  MATH  Google Scholar 

  9. And, Neta B., Ford, C.H.: Families of Methods for Ordinary Differential Equations Based on Trigonometric Polynomials. J. Comp. Appl. Math. 10, 33–38 (1984)

    Article  MathSciNet  Google Scholar 

  10. Dahlquist, G.G.: Numerical integration of ordinary differential equations. Math. Scand. 4, 69–86 (1956)

    Article  Google Scholar 

  11. Dhiman, N., Huntul, M.J., Tamsir, M.: A modified trigonometric cubic B-spline collocation technique for solving the time-fractional diffusion equation. Eng. Comput. 38(7), 2921–2936 (2021). https://doi.org/10.1108/EC-06-2020-0327

    Article  Google Scholar 

  12. Ehigie, J.O., Okunuga, S.A., Sofoluwe, A.B., Akanbi, M.A.: On generalized 2-step Continuous Linear Multistep Method of Hybrid Type for the Integration of Second Order Ordinary Differential Equation. Archives appl. sci. res. 26, 362–372 (2010)

    Google Scholar 

  13. Ehigie, J.O., Okunuga, S.A., Sofoluwe, A.B.: A Generalized 2-Step Continuous Implicit Linear Multistep Method of Hybrid Type. J. Inst. Math. Computer Sci. 21(1), 105–119 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Ehigie, J.O., Okunuga, S.A., Sofoluwe, A.B.: A class of 2-step continuous Hybrid implicit linear multistep methods of IVPs. J. Nigerian Math. Soc. 30, 145–162 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Ehigie, J.O., Jator, S.N., Okunuga, S.A.: A multi-point integrator with trigonometric coefficients for initial value problems with periodic solutions. Numer. Anal. Appl. 10(3), 329–344 (2017)

    Article  MathSciNet  Google Scholar 

  16. Fang, Y., Wu, X.: A trigonometrically fitted explicit Numerov-type method for second order initial value problems with oscillating solutions. Appl. Numer. Math. 58, 341–351 (2007)

    Article  MathSciNet  Google Scholar 

  17. Fang, Y., Song, Y., Wu, X.: A robust trigonometrically fitted embedded pair for perturbed oscillators. J. Comput. Appl. Math. 225, 347–355 (2009)

    Article  MathSciNet  Google Scholar 

  18. Fawzi, F. A., Senu, N., Ismail, F., Majid, Z. A.: A phase fitted and amplification-fitted modified Runge-Kutta method of fourth order for periodic initial value problems. Proceedings of the International Conference on Research and Education in Mathematics (ICREM7 ’15), 25-28 (2015)

  19. Fawzi, F.A., Senu, N., Ismail, F., Majid, Z.A.: A new efficient phase-fitted and amplification-fitted runge-kutta method for oscillatory problems. International J. Pure Appl. Math. 107, 69–86 (2016)

    Google Scholar 

  20. Franco, J.M.: An embedded pair of Exponentially-Fitted explicit Runge-Kutta methods. J. Comput. Appl. Math. 149, 407–414 (2002)

    Article  MathSciNet  Google Scholar 

  21. Franco, J.M.: Runge-Kutta-methods adapted to the numerical integration of oscillatory problems. Appl. Numer. Math. 50, 427–443 (2004)

    Article  MathSciNet  Google Scholar 

  22. Gautschi, W.: Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials. Numer. Math. 3, 381–397 (1961)

    Article  MathSciNet  Google Scholar 

  23. Gear, C.W.: Hybrid methods for initial value problems in ordinary differential equations. SIAM J. Numer. Anal. 2, 69–86 (1965)

    MathSciNet  MATH  Google Scholar 

  24. Gragg, W., Stetter, H.J.: Generalized multistep predictor-corrector methods. J. Assoc. Comput. Mach. 11, 188–209 (1964)

    Article  MathSciNet  Google Scholar 

  25. Gupta, G.K.: Implementing second-derivative multistep methods using Nordsieck polynomial representation. Math. Comput. 32, 13–18 (1978)

    Article  MathSciNet  Google Scholar 

  26. Ixaru, L Gr., Vanden Berghe, G., Van Daele, M.: Frequency evaluation in exponentially-fitted algorithms for ODEs. J. Comput. Appl. Math. 140, 423–434 (2002)

    Article  MathSciNet  Google Scholar 

  27. Jator, S.N., Swindell, S., French, R.D.: Trigonmetrically Fitted Block Numerov Type Method for \(y^{\prime \prime }=f(x, y, y^{\prime })\). Numer. Algor 62, 13–26 (2013)

    Article  Google Scholar 

  28. Jikantoro, Y.D., Ismail, F., Senu, N.: Zero-Dissipative Trigonometrically Fitted Hybrid Method for Numerical Solution of Oscillatory Problems. Sains Malaysiana 44(3), 473–482 (2015)

    Article  Google Scholar 

  29. Lambert, J.D.: Computational Methods in Ordinary Differential System, the Initial Value Problem. John Wiley & Sons, New York (1973)

    Google Scholar 

  30. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. John Wiley & Sons, New York (1991)

    MATH  Google Scholar 

  31. Li, J., Lu, M., Qi, X.: Trigonometrically fitted multi-step hybrid methods for oscillatory special second-order initial value problems. Int. J. Comput. Math. 95, 979–997 (2018). https://doi.org/10.1080/00207160.2017.1303138

    Article  MathSciNet  MATH  Google Scholar 

  32. Moo, K.W., Senu, N., Ismail, F., Suleiman, M.: New phase fitted and amplification-fitted fourth-order and fifth-order Runge-Kutta-Nystrom methods for oscillatory problems. Abstr. Appl. Anal. 2013, 9 (2013)

    Article  MathSciNet  Google Scholar 

  33. Ndukum, P.L., Biala, T.A., Jator, S.N., Adeniyi, R.B.: On a family of trigonometrically fitted extended backward differentiation formulas for stiff and oscillatory initial value problems. Numer. Algor 74(1), 267–287 (2017). https://doi.org/10.1007/s11075-016-0148-1

    Article  MathSciNet  MATH  Google Scholar 

  34. Neta, B.: Families of Backward Differentiation Methods Based on Trigonometric Polynomials. Int. J. Comput. Math. 20, 67–75 (1986)

    Article  Google Scholar 

  35. Ngwane, F.F., Jator, S.N.: A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value problems. Journal of Applied Mathematics 2015, 1–7 (2015). https://doi.org/10.1155/2015/343295

  36. Ngwane, F.F., Jator, S.N.: Solving Oscillatory Problems Using a Block Hybrid Trigonmetrically Fitted Method with Two Off-Step Points. Texas State University. San Marcos, Electronic Journal of Differential Equation. 20, 119–132 (2013)

    Google Scholar 

  37. Ngwane, F.F., Jator, S.N.: Trigonometrically-Fitted Second Derivative Method for Oscillatory Problems. Springer Plus. 3, 304 (2014)

    Article  Google Scholar 

  38. Ramos, H., Lorenzo, C.: Review of explicit Falkner methods and its modifications for solving special second-order I.V.P.s. Comput. Phys. Commun. 181(11), 1833–1841 (2010)

    Article  MathSciNet  Google Scholar 

  39. Ramos, H., Patricio, M.F.: Some new implicit two-step multiderivative methods for solving special second-order IVP’s. Appl. Math. Comput. 239, 227–241 (2014)

    MathSciNet  MATH  Google Scholar 

  40. Ramos, H., Singh, G.: A tenth order A-stable two-step hybrid block method for solving initial value problems of ODE. Appl. Math. Comput. 310, 75–88 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Ramos, H., Vigo-Aguiar, J.: Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.’s. J. Comput. Appl. Math. 204(1), 102–113 (2007)

    Article  MathSciNet  Google Scholar 

  42. Ramos, H., Vigo-Aguiar, J.: On the frequency choice in trigonometrically fitted methods. Appl. Math. Lett. 23, 1378–1381 (2010)

    Article  MathSciNet  Google Scholar 

  43. Ramos, H., Singh, G., Kanwar, V., Bhatia, S.: An efficient variable step-size rational Falkner-type method for solving the special second-order IVP. Appl. Math. Comput. 291, 39–51 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Ramos, H., Mehta, S., Vigo-Aguiar, J.: A unified approach for the development of \(k\)-step block Falkner-type methods for solving general second-order initial-value problems in ODEs. J. Comput. Appl. Math. 318, 550–564 (2017)

    Article  MathSciNet  Google Scholar 

  45. Samat, F., Ismail, E.S.: A Two-Step Modified Explicit Hybrid Method with Step-Size-Dependent Parameters for Oscillatory Problems. J. Math. Article ID 5108482, 7 (2020). https://doi.org/10.1155/2020/5108482

  46. Senu, N., Suleiman, M., Ismail, F.: An embedded explicit Runge-Kutta-Nyström method for solving oscillatory problems. Phys. Scr. 80(1), 015005 (2009)

    Article  Google Scholar 

  47. Senu, N., Kasim, I.A., Ismail, F., Bachok, N.: Zero-dissipative explicit Runge-Kutta method for periodic initial value problems. International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering. 8, 1226–1229 (2014)

    Google Scholar 

  48. Vigo-Aguiar, J., Ramos, H.: On the choice of the frequency in trigonometrically fitted methods for periodic problems. J. Comput. Appl. Math. 277, 94–105 (2015)

    Article  MathSciNet  Google Scholar 

  49. Yakubu, D.G., Aminu, M., Tumba, P., Abdulhameed, M.: An efficient family of second derivative runge-kutta collocation methods for oscillatory systems. J. Nigerian Math. Society. 37(2), 111–138 (2018)

    MathSciNet  MATH  Google Scholar 

  50. Zarina, B.I., Yatim, S.A.M, Othman, K.I, Suleiman, M.B.: Numerical Solution of Extended Block Backward Formulae for Solving Stiff ODEs. Proceeding of the world congress Engineering. 1 (2012)

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Acknowledgements

The authors are grateful to anonymous referees whose suggestions substantially enhanced the initial version of the paper, and especially to Professor Higinio Ramos for reading and editing the revised version.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Distance Learning institute, University of Lagos, Lagos, Nigeria

    R. I. Abdulganiy

  2. Department of Mathematics, University of Lagos, Lagos, Nigeria

    G. O. Inakoju, M. A. Gaffari & S. A. Okunuga

  3. Department of Mathematics/Computer Science, Ibrahim Badamasi Babangida University, Lapai, Nigeria

    Y. D. Jikantoro

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  1. R. I. Abdulganiy
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  2. G. O. Inakoju
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  3. M. A. Gaffari
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Correspondence to R. I. Abdulganiy.

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Abdulganiy, R.I., Inakoju, G.O., Gaffari, M.A. et al. One Step Adapted Hybrid Second Derivative Block Method for Initial Value Problems with Oscillating Solutions. Int. J. Appl. Comput. Math 8, 150 (2022). https://doi.org/10.1007/s40819-022-01358-6

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