One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

Authors

  • B. I. Akinnukawe Department of Mathematics, University of Lagos, Lagos, Nigeria
  • S. A. Okunuga Department of Mathematics, University of Lagos, Lagos, Nigeria

Keywords:

Collocation techniques, Hybrid points, Numerical integration, Optimized scheme

Abstract

In this paper, a one-step block of optimized hybrid schemes for the numerical integration of second-order initial value problems (IVP) of ordinary differential equations (ODE) is constructed via collocation techniques. The developed scheme is obtained by considering two intra-step nodal points as hybrid points, which are chosen in order to achieve optimized errors of the main formulae approximating the solution such that 0 < v1 < v 2 < 1 where v1 and v2 are defined as hybrid points. The characteristics of the developed scheme are analyzed. Application of the new scheme on some second-order IVPs shows the accuracy and effectiveness of the scheme compared with some existing methods.

Dimensions

Y. A. Abdullahi, O. Zumi & J. O. Kuboye, “Derivation of block predictor - block borrector method for direct solution of third-order ordinary differential equations”, Global Journal of Pure and Applied Mathematics 12 (2016) 343. https://www.ripublication.com/gjpam16/gjpamv12n1_30.pdf.

B. I. Akinnukawe, “One-step block hybrid integrator for the numerical solution of semi-explicit index-1 DAEs systems”, International Journal of Mathematical Analysis and Optimization: Theory and Application 7 (2021) 67. https://doi.org/10.6084/m9.figshare.14829315.

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://api.semanticscholar.org/corpusID:221609820.

M. O. Alabi, K. S., Adewoye & O. Z. Babatunde, “Integrator block off-grid points collocation method for direct solution of second order ordi-nary differential equations using Chebyshev as basis polynimials, Inter-national Journal of Mathematics and Statistics Studies 7 (2019) 12. https://www.eajournals.org/wp-content/uploads/Integrator-Block-Off.pdf.

M. A. Rufai & H. Ramos, “Numerical solution of second-order singular problems arising from astrophysics combining a pair of one step hybrid block Nystrom methods”, Astrophysics Space Science 365 (2020) 96. https://doi.org/10.1007/s10509-020-03811-8.

B. I. Akinnukawe & M. R. Odekunle, “Block Bi-basis collocation method for direct approximation of fourth-order initial value problems”, Journal of the Nigerian Mathematical Society 42 (2023) 1. https://ojs.ictp.it/jnms/index.php/jnms/article/view/888.

B. I. Akinnukawe & K. O. Muka, “L-stable block hybrid numerical algorithm for first-order ordinary differential equations”, Journal of the Nigerian Society of Physical Sciences 2 (2020) 160. https://doi.org/10.46481/jnsps.2020.108.

R. P. K. Chan, P. Leone & A. Tsai, “Order conditions and symmetry for two-step hybrid methods”, International Journal of Computer Mathematics 81 (2004) 1519. https://doi.org/10.1080/03057920412331272180.

S. Gholamtabar, N. Parandin, “Numerical solution of second-order differential equation by adams bashforth method”, American Journal of Engerineering Research 3 (2014) 318. https://www.ajer.org/papers/v3(6)/AN36318322.pdf.

O. A. Akinfenwa, S. A. Okunuga, B. I. Akinnukawe, U. O. Rufai & R. I. Abdulganiy, “Multi-derivative hybrid implicit runge-kutta method for solving stiff system of a first order differential equation”, Far East Journal of Mathematical Sciences 106 (2018) 543. http://dx.doi.org/10.17654/MS106020543.

E. O. Adeyafa, “Orthogonal-based hybrid block method for solving general second-order initial value problems”, Italian Journal of Pure and Applied Mathematics 37 (2017) 659. https://www.researchgate.net/publication/319091860.

T. A. Anake, Continuous implicit hybrid one-step block method for solution of initial value problems of general second-order ordinary differential equations, Ph.D. Thesis, Covenant University Ota, Nigerian, 2011. http://eprints.covenantuniversity.edu.ng/id/eprint/7856.

F. O. Obarhua & O. J. Adegboro, “An order four continuous numerical method for solving general second-order ordinary differential equations”, Journal of Nigerian Society of Physical Sciences 3 (2021) 42. https://doi.org/10.46481/jnsps.2021.150.

A. Bilesanmi, A. S. Wusu & A. L. Olutimo, “Solution of second-order ordinary differential equations via simulation annealing”, Open Journal of Optimization 8 (2019) 32. https://doi.org/10.4236/ojop.2019.81003.

J. O. Kuboye, Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach, Ph.D Thesis, Universiti Utara Malaysia, 2015. http://malrep.uum. edu.my/rep/Record/my.uum.etd.5789.

U. Mohammed, “A class of implicit five-step block method for general second-order ordinary differential equations”, Journal of the Nigerian Mathematical Society 30 (2011) 25. http://repository.futminna.edu.ng:8080/jspui/handle/123456789/573.

U. Mohammed & R. B. Adeniyi, “Derivation of five-step block hybrid bdf through the continuous multi-step collocation for solving second-order differential equations”, Pacific Journal of Science and Technology 15 (2014) 89. http://repository.futminna.edu.ng:8080/jspui/handle/123456789/425.

J. D. Lambert, Computational methods for ordinary differential equations, John Wiley, New York, 1973. https://www.amazon.com/Computational-Methods-Ordinary-Differential-Equations/dp/B002BEWLGO.

P. Henrici, Discrete variable methods in ordinary differential equations, John Wiley, New York, 1962. http://dx.doi.org/10.1002/zamm.19660460521.

Published

2024-04-23

How to Cite

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations. (2024). Journal of the Nigerian Society of Physical Sciences, 6(2), 1827. https://doi.org/10.46481/jnsps.2024.1827

Issue

Volume 6, Issue 2, May 2024

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2024 B. I. Akinnukawe, S. A. Okunuga

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations. (2024). Journal of the Nigerian Society of Physical Sciences, 6(2), 1827. https://doi.org/10.46481/jnsps.2024.1827