Two Step Block Method for the Solution of Nonlinear Second Order Singular Boundary Value Problem of Ordinary Differential Equations
DOI:
https://doi.org/10.62054/ijdm/0204.11Abstract
This work presents a two-step second-derivative block method for solving nonlinear singular boundary value problems. The method is formulated using interpolation and collocation at selected grid points to obtain a continuous linear multistep scheme, which is then discretized for block implementation. The proposed approach is proven to be convergent and A-stable. Numerical experiments confirm its effectiveness and reliability, showing improved accuracy compared to existing methods in the literature.
References
Abba, I. B. (2019). Hybrid block extended second derivative backward differentiation formulae for the solution of stiff ordinary differential equations PhD Thesis Adamawa State University, Mubi, Nigeria.
Abdelrahim, R. & Omar, Z. (2016). A four-step implicit block method with three generalized o -step points for solving fourth order initial value problem directly, Journal of King Saud University Science 29, 401.
Adamu, S., Bitrus, K. & Buhari, H. L. (2019). One step second derivative method using Chebyshev collocation point for the solution of ordinary differential equations. Journal of the Nigerian Association of Mathematical Physics, 51(May), 47-54. http://nampjournals.org/publications-download/vol51/.
Adamu, S., Danhausa, A. A., Stephen, L. &Williams B. (2020). Two Hybrid Points Block Methods for the Solution of Initial Value Problems. Journal of the Nigerian Association of Mathematical Physics, Volume 54, January issue, 7-12. http://nampjournals.org/publications-download/vol54/.
Adamu S., Favour A., Bukar, H., Aduroja, O. O. & Tahir, A. (2025). A Study on Some Numerical Methods for Simulating Mathematical Models of Ordinary Differential Equations. UMYU Scientifica, 4(2), 007–015. DOI: https://doi.org/10.56919/usci.2542.002.
Adegboye, Z. A. & Ahmed, U. I. (2014). Modification of Simpson's Block Hybrid Multistep Method for General Second Order ODEs. International Journal of Science and Technology, 3(1), 21-32.
Adesanya, A. O., Odekunle, M. R.& Adeyeye, A. O. (2012). Continuous block hybrid predictor corrector method for the solution of . International Journal of Mathematics and Soft Computing 2, 35.
Adesanya, A. O., Abdulqadri, B. & Ibrahim, Y. S. (2014). Hybrid one step block method for the solution of third order initial value problems of ordinary differential equations, International Journal of Applied Mathematics and Computation 6 (2014) 10.
Anake,T. A., Awoyemi, D.O., & Adesanya, A. A. (2012). One-step implicit hybrid block method for the direct solution of general second order ordinary differential equations. IAENG International Journal of Applied Mathematics. 42(4).
Biala, T. A., Jator, S. N., Adeniyi, R. B. & Ndukum, P. L. (2015). Block Hybrid Simpson's Method with Two off-grid Points for Stiff Systems. International Journal of Nonlinear Science, 20(1), 3-10.
Chakravarthy, P. P,. Phaneendra, K,. & Reddy, Y. N. (2007). A seventh order numerical method for singular perturbation problems. Applied Mathematics and Computation 186. 860-871.
Fotta, A., U., Alabi, T. J. & Abdulqadir, B. (2015). Block Method with one Hybrid point for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. International Journal of Pure and applied Mathematics, 103(3), 511-521.
Hasan, Y. Q. & Zhu, L. M. (2007). Solving Singular Initial Value Problems in the Second-order Ordinary Differential Equations. Journal of Applied Sciences. 7(17), 2505-2508.
Jiang, J., Liu, W. & Wang, H. (2018). Wang, Positive solutions to singular Dirichlet-type boundary value problems of nonlinear fractional differential equations. Advances in Difference Equations. Springer open Journal. https://doi.org/10.1186/s13662-018-1627-6.
Joshua, S. (2022). Optimized two-step second derivative methods for the solutions of stiff systems, Journal of Physics Communications. 6, 055016O.
Khalsaraei, M. M., Oskuyi, N. N. & Hojjati, G. (2012). A Class of Second Derivative Multistep Methods for stiff systems. Acta Universitatis Apulensis, 30, (2012), 171-188.
Kida, M, Adamu, S., Aduroja, O. O., Pantuvo, T. P. (2022), Numerical Solution of Stiff and Oscillatory Problems using Third Derivative Trigonometrically Fitted Block Method, J. Nig. Soc. Phys. Sci. 4 (1)34-48. DOI: https://doi.org/10.46481/jnsps.2022.271.
Kumar, N., Sinha, K. R. & Ranjan, R. (2023). Singularly perturbed two-point boundary value problem by applying exponential fitted finite difference method. Iranian Journal of Numerical Analysis and Optimization.13 (4), 711--727. https://doi.org/10.22067/ijnao.2023.83070.1283.
Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations, 1st edn, (New York, USA: Willey)
Ngwane, F. F. & Jator, S. N. (2012). Block hybrid second derivative method for stiff systems. International Journal of Pure and Applied Mathematics 80 (4), 543-59.
Ogunniran, M. O., Haruna, Y Adeniyi. R. B. & Olayiwola, M. O. (2021). Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations, Cankaya University Journal of Science and Engineering.
Olabode, B. T., Kayode,S. J., Odeniyan-Fakuade, F. H.,& Momoh, A. L. (2024). Numerical Solution to Singular Boundary Value Problems (SBVPS) using Modified Linear Multistep Formulas (LMF), International Journal of Mathematical Sciences and Optimization: Theory and Applications. 10(1), , 34 - 52.
Omar, A. A., Zaer, A. H., Shaher, M. & Nabil,Solving, S. (2012). Singular Two-Point Boundary Value Problems Using Continuous Genetic Algorithm. Hindawi Publishing Corporation Abstract and Applied Analysis. doi:10.1155/2012/205391.
Ramos, H. (2020). Comments on the use of block methods for solving singular boundary value problems, Third ICAMNM, ITM Web of Conferences 34,01005, Third ICAMNM. https://doi.org/10.1051/itmconf/20203401005.
Widlund, O. (1967). A note on unconditionally stable linear multistep methods BIT 7, 65-70.
Yahaya, H., Raphael, B. A., & Muideen, O. O. (2021). A Two-step Hybrid Block Method with Four O -step Points on Singular Initial and Boundary Value Problems, Turk. J. Math. Comput. Sci.13(2), 248-260.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Musa Kida, Mathew, R. Odekunle, Solomon, O. Adee, Sunday Samuel (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors are solely responsible for obtaining permission to reproduce any copyrighted material contained in the manuscript as submitted. Any instance of possible prior publication in any form must be disclosed at the time the manuscript is submitted and a
copy or link to the publication must be provided.
The Journal articles are open access and are distributed under the terms of the Creative
Commons Attribution-NonCommercial-NoDerivs 4.0 IGO License, which permits use,
distribution, and reproduction in any medium, provided the original work is properly cited.
No modifications or commercial use of the articles are permitted.
