Approximate solution of higher-order oscillatory differential equations via modified linear block techniques

Folashade Mistura  Jimoh; Adam Ajimoti  Ishaq; Kazeem Issa

doi:10.46481/asr.2025.4.2.275

Authors

Folashade Mistura Jimoh
Department of Physical Sciences, Al-Hikmah University, Ilorin, Nigeria
Adam Ajimoti Ishaq
Department of Physical Sciences, Al-Hikmah University, Ilorin, Nigeria
Kazeem Issa
[email protected]

Department of Mathematics and Statistics, Kwara State University, Ilorin, Nigeria

Keywords:

Ordinary differential equations, Higher-order oscillatory problems, Linear block method, Computational efficiency

Abstract

In many areas of science and engineering, problems modeled by ordinary differential equations (ODEs) often lack analytical solutions, requiring the use of numerical methods for approximation. The proposed method tackles the challenges of solving higher-order oscillatory differential equations and introduces a new linear block technique for directly solving these equations. This method improves accuracy, reduces computational effort, and simplifies coding complexity compared to previous approaches. A generalized algorithm is presented to derive the proposed method, which enhances existing techniques for second-order and higher-order oscillatory problems. The basic properties of the method were numerically analyzed, confirming its accuracy, stability, consistency, and convergence. The proposed method proves to be efficient and suitable for various test problems, including real-life problems, which demonstrate its accuracy as compared to the existing methods.

Dimensions

REFERENCES

[1] A. J. Adewale & J. Sabo, “Simulating the dynamics of oscillating differential equations of mass in motion”, International Journal of Development Mathematics 1 (2024) 54. https://doi.org/10.62054/ijdm/0101.07.

[2] J. Sabo, A. M. Ayinde, T. Oyedepo & A. A. Ishaq, “Simulation of two-step block approach for solving oscillatory differential equations”, Cumhuriyet Science Journal 45 (2024) 366. https://doi.org/10.17776/csj.1345303.

[3] Y. Skwame, D. J. Zirra & J. Sabo, “The numerical application of dynamic problems involving mass in motion governed by higher order oscillatory differential equations”, Physical Science International Journal 28 (2024) 8. https://doi.org/10.9734/psij/2024/v28i5845.

[4] A. M. Ayinde, S. Ibrahim, J. Sabo & D. Silas, “The physical application of motion using single step block method”, Journal of Material Science Research and Review 6 (2023) 708. https://www.sdiarticle5.com/review-history/103447.

[5] D. J. Zirra, Y. Skwame, J. Sabo, J. A. Kwanamu & S. Daniel, “On the numerical approximation of higher order differential equation”. Asian Journal of Research and Reviews in Physics 8 (2024) 26. https://doi.org/10.9734/ajr2p/2024/v8i1154.

[6] A. A. Ishaq, A. M. Ayinde, J. Sabo, S. Ibrahim & B. S. Jaiyeola, “Three-step collocation method for solution of third derivative initial value problem”, International Journal of Development Mathematics 1 (2024) 25. https://doi.org/10.62054/ijdm/0101.03.

[7] M. D. Aloko, A. M. Ayinde, A. D. Usman & J. Sabo, “An efficient scheme for direct simulation of third and fourth oscillatory differential equations”, International Journal of Development Mathematics 1 (2024) 72. https://doi.org/10.62054/ijdm/0101.06.

[8] J. Z. Donald, T. Y. Kyagya, A. A. Bambur & J. Sabo, J. “The effective use of block algorithm for mathematical treatment of some problematic system of order three”. FUW Trends in Science & Technology Journal 7 (2022) 413. https://www.ftstjournal.com/uploads/docs/73%20Article%2054%20pp.pdf.

[9] D. Raymond, T. Y. Kyagya, J. Sabo & A. Lydia, “Numerical application of higher order linear block for solving some fourth order initial value problems”, African Scientific Reports 2 (2023) 67. https://doi.org/10.46481/asr.2022.2.1.67.

[10] D. Raymond, T. P. Pantuvu, A. Lydia, J. Sabo & R. Ajia, “An optimized half-step scheme third derivative methods for testing higher order initial value problems”, African Scientific Reports 2 (2023) 76. https://doi.org/10.46481/asr.2023.2.1.76.

[11] O. M. Ogunlaran, M. A. Kehinde, M. A. Akanbi & E. I. Akinola, “A Chebyshev polynomial based block integrator for the direct numerical solution of fourth order ordinary differential equations,” Journal of the Nigerian Society of Physical Sciences 6 (2024) 1917. https://doi.org/10.46481/jnsps.2024.1917.

[12] J. Sabo, Y. Skwame & J. Z. Donald, “On the simulation of higher order linear block algorithm for modelling fourth order initial value problems”, Asian Research Journal of Mathematics 18 (2022) 22. https://doi.org/10.9734/arjom/2022/v18i1030415.

[13] E. O. Adeyefa & J. O. Kuboye, “Derivation of new numerical model capable of solving second and third order ordinary differential equations directly”, IAENG International Journal of Applied Mathematics 50 (2020) 2. https://www.iaeng.org/IJAM/issues_v50/issue_2/IJAM_50_2_02.pdf

[14] A. K. Jimoh, “Two-step hybrid block method for solving second order initial value problem of ordinary differential equations”, Earthline Journal of Mathematical Sciences. 14 (2024) 1047. https://doi.org/10.34198/ejms.14524.10471065.

[15] H. Ramos, S. N. Jator & M. I. Modebei, “Efficient k-step linear block methods to solve second-order initial value problems directly", Mathematics 8 (2020) 1752. https://doi.org/10.3390/math8101752.

[16] A. I. Muhammada, J. A. Kwanamu & J. Sunday", An algorithm for approximating third-order ordinary differential equations with applications to thin film flow and nonlinear genesio problems”, International Journal of Development Mathematics 1 (2024) 15. https://doi.org/10.62054/ijdm/0101.01.

[17] J. Sabo, T. Y. Kyagya & M. Solomon, “One-step hybrid block scheme for the numerical approximation for solution of third order initial value problems”, Journal of Scientific Research & Reports 27 (2021) 51. https://doi.org/10.9734/jsrr/2021/v27i1230476.

[18] S. A. A. Ahmedai Abd Allah, P. Sibanda, S. P. Goqo, U. O. Rufai, M. H. Sithole & O. A. I. Noreldin, “A block hybrid method with equally spaced grid points for third-order initial value problems”, AppliedMath 4 (2024) 320. https://doi.org/10.3390/appliedmath4010017.

[19] V. O. Atabo & S. O. Adee, “A new special 15-step block method for solving general fourth-order ordinary differential equations”, Journal of the Nigerian Society Physical Sciences 3 (2021) 308. https://doi.org/10.46481/jnsps.2021.337.

[20] E. A. Areo & M. A. Rufai, “An efficient one-eighth step hybrid block method for solving second-order initial value problems of ODEs”. International Journal of DifferentialEquations and Application 15 (2016) 117. http://dx.doi.org/10.12732/ijdea.v15i2.3352.

[21] Y. Skwame, P. I. Dalatu, J. Sabo & M. Mathew, “Numerical application of third derivative hybrid block methods on third order initial value problem of ordinary differential equations”, International Journal of Statistics and Applied Mathematics 4 (2019) 90. https://tinyurl.com/mu596c6m.

[22] J. Sabo, Y. Skwame, T. Y. Kyagya & J. A. Kwanamu, “The direct simulation of third order linear problems on single step block method”, Asian Journal of Research in Computer Science 12 (2021) 8. https://doi.org/10.9734/AJRCOS/2021/v12i230277.

[23] A. D. Familua & E. O. Omole, “Five points mono hybrid point linear multistep method for solving nth order ordinary differential equations using power series function”. Asian Research Journal of Mathematics 3 (2017) 17. https://doi.org/10.9734/ARJOM/2017/31190.

[24] L. O. Adoghe & E. O. Omole, “A two-step hybrid block method for the numerical integration of higher order initial value problem of ordinary differential equations”, World Science News, An International Scientific Journal 118 (2019) 236. https://tinyurl.com/bd32jzux.

[25] R. Allogmany, I. Fudziah, A. M. Zanariah & B. I. Zarina, “Implicit two-point block method for solving fourth-order initial value problem directly with application”, Mathematical Problems in Engineering 2020 (2020) 351279. https://doi.org/10.1155/2020/6351279.

[26] M. D. Aloko, M. A. Ayinde, A. D. Usman & J. Sabo, “An efficient scheme for direct simulation of third and fourth oscillatory differential equations”, International Journal of Development Mathematics 1 (2024) 72. https://doi.org/10.62054/ijdm/0101.06.

[27] O. Adeyeye & Z. Omar, “Implicit five-step block method with generalised equidistant points for solving fourth order linear and non-linear initial value problems”, Ain Shams Engineering Journal 10 (2019) 881. https://doi.org/10.1016/j.asej.2017.11.011.