DEVELOPMENT OF A CONTINUOUS BACKWARD DIFFERENTIATION FORMULAE FOR SOLVING FIRST-ORDER AND SECOND-ORDER INITIAL VALUE PROBLEM
Abstract
Numerical methods for solving ordinary differential equations (ODEs) are essential in modeling dynamical systems across science and engineering. While specialized methods exist for first-order and second-order ODEs, developing a unified approach that efficiently handles both classes remain an active area of research. In this paper, we present a novel two-step hybrid block method based on the backward differentiation formula (BDF), capable of approximating solutions for both first- and second-order ODEs without requiring separate derivations. The method is constructed using interpolation and collocation techniques, and its numerical analysis confirms consistency, zero-stability, and convergence. Furthermore, stability analysis via the general linear method demonstrates that the scheme is A-stable, making it suitable for stiff systems. Numerical experiments including applications to the SIR epidemic model, Riccati differential equations, nonlinear stiff chemical systems, and second-order nonlinear ODEs—validate the method’s accuracy and computational efficiency. Comparative results with existing methods in the literature highlight its superior performance in terms of error reduction and stability. This work contributes a versatile, high-precision tool for ODE solutions, bridging gaps in the adaptability of traditional BDF-based approaches.
References
Abdelrahim R. & Omar Z. (2016). Direct solution of second-order ordinary differential equation sing a single-step hybrid block method of order five. Mathematical and Computational Applications, 21(12), 1 7. https://doi.org/10.3390/mca21020012 DOI: https://doi.org/10.3390/mca21020012
Adamu, M. A., Suleman, T., Dahiru, U.& Aisha I. A. (2019) Continuous implicit one step four-point hybrid block method for the solution of initial value problem for third order ordinary differential equations. International Journal of Science for Global Sustainability. 5(2), 56 -98. https://doi.org/10.11648/j.acm.20190803.11 DOI: https://doi.org/10.11648/j.acm.20190803.11
Adeniyi R. B. & Adeyefa E. O. (2013). Chebyshev collocation approach for a continuous formulation of implicit hybrid methods for IVPs in Second Order Odes. IOSR J. Maths., 6(4), 09-12. https://doi.org/10.9790/5728-0640912 DOI: https://doi.org/10.9790/5728-0640912
Ajileye G., Okedayo G. T. & Aboiyar T. (2017). Laguerre collocation approach for continuous hybrid block method for solving initial value problems of second order ordinary differential equations. FUW Trends in Science & Technology Journal, 2(1b), 379 383. https://doi.org/10.9734/jamcs/2018/42357 DOI: https://doi.org/10.9734/JAMCS/2018/42357
Akinfenwa, O.A., Jator, S.N. and Yao, N.M. (2011), A self- starting block Adams methods for solving stiff ordinary differential equations. The 14th IEEE International Conference on Computational Science and Engineering. https://doi.org/10.1109/cse.2011.34 DOI: https://doi.org/10.1109/CSE.2011.34
Akinfenwa, O. A., Abdulganiy, R. I., Okunuga, S. A. & Irechukwu, V. (2017). Simpsons 3/8 type block method for stiff systems of ordinary differential equations. Journal of the Nigerian Mathematical Society, 36(3), 503-514. https://doi.org/10.1016/j.aml.2016.08.012 DOI: https://doi.org/10.1016/j.aml.2016.08.012
Areo, E.A. and Adeniyi, R.B., (2014), A self-starting linear multistep method for direct solution of initial value problem of second order ordinary differential equations. International Journal of Pure and Applied Mathematics, 82: 345-364. https://doi.org/10.1155/2015/893763 DOI: https://doi.org/10.1155/2015/893763
Guler, C., Kaya, S. O. & Sezer, M. (2019). Numerical solution of a class of nonlinear ordinary differential equations in Hermite series. Thermal Science: International Scientific Journal, 23, 12051210. https://doi.org/10.2298/tsci181215047g DOI: https://doi.org/10.2298/TSCI181215047G
Hussain K., Ismail F., & Senu N. (2016). Solving Directly Special Fourth-Order Ordinary Differential Equations Using Runge-Kutta Type Method. J. Comput. Appl. Math., https://doi.org/10.1155/2015/893763 DOI: https://doi.org/10.1016/j.cam.2016.04.002
Jator S. (2015). Numerical integrators for fourth order initial and boundary value problems. Int. J. Pure Appl. Math., 8(2),
Jikantoro Y. D., Aliyu Y. B., Maali A. I., Abubakar A., & Musa I. (2019a). A numerical integrator for oscillator problems. Asian Research Journal of Mathematics, 14(1), 1 10. 1-10. https://doi.org/10.9734/arjom/2019/v14i130119 DOI: https://doi.org/10.9734/arjom/2019/v14i130119
Jikantoro Y. D., Ismail F., Senu N. & Ibrahim Z. B. (2018). Hybrid methods for direct integration of special third order ordinary differential equations. Applied Mathematics and Computation, 320(1), 452 463. https://doi.org/10.1016/j.amc.2017.10.003 DOI: https://doi.org/10.1016/j.amc.2017.10.003
Jikantoro Y. D., Ismail F., Senu N., Ibrahim Z. B. & Aliyu Y. B. (2019b). Hybrid method for solving special fourth order ordinary differential equations. Malaysian Journal of Mathematics Sciences, 13(1), 27 40. https://doi.org/10.1063/1.4801239 DOI: https://doi.org/10.1063/1.4801239
Kashkari B. S. H. & Syam M. I. (2019). Optimization of one step block method with three hybrid points for solving first-order ordinary differential equations. Results in Physics, 12, 592596. https://doi.org/10.1016/j.rinp.2018.12.015 DOI: https://doi.org/10.1016/j.rinp.2018.12.015
Kuboye, J. O., & Omar, Z. (2015). Derivation of a six-step block method for direct solutions of second order ordinary differential equations. Mathematical and Computational Applications, 20(3), 151-159. https://doi.org/10.19029/mca-2015-013 DOI: https://doi.org/10.3390/mca20010159
Lambert J. D. (1991). Numerical methods for ordinary differential systems. John Wiley, New York.
Lambert, J.D., (1973), Computational methods in ordinary differential equations. 1st Edition, Wiley & Son, Chichester New York, Brisbane, Toronto.
Ishaku Maal, A., Mohammed, U., Ben, A. G., & Jikantoro, Y. D. (2022). Three-step numerical approximant based on block hybrid backward differentiation formula for stiff system of ordinary differential equations. European Journal of Mathematics and Statistics, 3(3), 48-53. https://doi.org/10.24018/ejmath.2022.3.3.120
Mehdizadeh K., Nasehi N. and Hojjati G. (2012), A Class of Second Derivative Multistep Methods for Sti Systems. Acta Universitatis Apulensis, 30, 171188.
Mohammed U., Maali A. I., Garba J & Akintububo B. G. (2022). An order 2k hybrid backward differentiation formula for stiff system of ordinary differential equations using Legendre polynomial as basis function. Abacus (Mathematics Science Series) 49(2), 173 202. https://doi.org/10.24018/ejmath.2022.3.3.120 DOI: https://doi.org/10.24018/ejmath.2022.3.3.120
Ogunlaran O. M. & Kehinde M. A. (2022). A new block integrator for second order initial value problems. Neuroquantology,20(17), 1976 1980.
Omar, Z. and Kuboye, J.O. (2015). Computation of an accurate implicit block method for solving third order ordinary differential equations directly. Global Journal of Pure Applied Mathematics, 11(1), 177-186.https://doi.org/10.17654/fjmsoct2015_315_332 DOI: https://doi.org/10.17654/FJMSOct2015_315_332
Onumanyi, P. Awoyemi, D.O. Jator, S.N. and Sirisena, U.W. (1994). New linear multistep methods with continuous coecients for rst order initial value problems, J. Nig. Math. Soc., 13, 37-51.
Raft, A., Nizar A., Firas, S. & Mohammad, H. (2020). Multistep collocation hybrid block method for direct solution of high order ordinary differential equations. International Journal of Advanced Science and Technology, 29(3), 3367- 3381.https://doi.org/10.12691/tjant-8-2-1 DOI: https://doi.org/10.12691/tjant-8-2-1
Sahi1, R.K., Jator, S.N. and Khan, N.A., (2012). A Simpsons-Type Second Derivative Method For Sti Systems, International Journal of Pure and Applied Mathematics, 81(4), 619-633 https://doi.org/10.1016/j.matcom.2019.03.003 DOI: https://doi.org/10.1016/j.matcom.2019.03.003
Yakubu D.G., Kumlen, G.M. and Markus, S. (2017), Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems. Journal of Modern Methods in Numerical Mathematics. 8:1-2(2017), 118-138.https://doi.org/10.20454/jmmnm.2017.1349 DOI: https://doi.org/10.20454/jmmnm.2017.1349
Yakusak N. S. & Owolanke A. O. (2018). A class of linear multi-step method for direct solution of second order initial value problems in ordinary differential equation by collocation method. Journal of Advances in Mathematics and Computer Science, 26(1), 1 11. https://doi.org/10.9734/jamcs/2018/34424 DOI: https://doi.org/10.9734/JAMCS/2018/34424
Copyright (c) 2025 FUDMA JOURNAL OF SCIENCES
This work is licensed under a Creative Commons Attribution 4.0 International License.
FUDMA Journal of Sciences
