Block Optimized Hybrid Methods for Integrating Singular Second Order Ordinary Differential Equations

In this work, a block methods with characteristics of LMF are derived, analyzed and numerically applied to solve singular Initial/Boundary value problems. It was done by applying shift operator to two linear multi-step formula and combined with Optimize hybrid set of formula which are developed at the the first sub-interval to circumvent the singularity at the left end of the integration interval.The mathematical derivation of the proposed methods is based on method of undetermined coefficients where the coefficient in our Linear Multi-step Formulas (LMF) are determined. The fundamental properties of the proposed scheme are analyzed. Finally, the numerical implementation of the method are done on some singular I/B value problem which demonstrate the accuracy and validity of the suggested technique when compared to various strategies available in the current literature.

One-block methods; shift operator, undetermined coefficients

1. Asifa, T., Sania, Q., Amanullah , S., Evren , H., & Asif, A. (2022). A new continuous hybrid block method with one optimal intra step point through interpolation and collocation Fixed Point . Theory Algorithms Sci Eng , 22. [Google Scholar] [Crossref]

2. Ajie, I. J., Durojaye, M.O, Utalor, I. K., & Onumanyi, P. A. (2020). Construction of a family of stable one- Block methods Using Linear Multi- Step Quadruple. Journal of Mathjematics., e-ISSN: 2278-5728 [Google Scholar] [Crossref]

3. Ajie, I. J., Utalor, I. K., & Onumanyi, P. A. (2019). A family of high order one-block methods for the solution of stiff initial value problems. Journal of Advances in Mathjematics and Computer Science., 31(6), 1–14. [Google Scholar] [Crossref]

4. Anake , T. A., Awoyemi, D. O., & Adesanya , A. O. (2012). One-step implicit hybrid block method for the direct solution of general second order ordinary differential equations,. Int. J. App. Math. , 42(4), 224–228. [Google Scholar] [Crossref]

5. Allouche, H.; Tazdayte, A. (2017). Numerical solution of singular boundary value problems with logarithmic singularities by Padè approximation and collocation methods. J. Comput. Appl. Math. , 311(324–341.). [Google Scholar] [Crossref]

6. Aydinlik, S.; Kiris, A. (2018). A high-order numerical method for solving nonlinear Lane-Emden-type equations arising in astrophysics. Astrophys. Space Sci., 363, 264. [Google Scholar] [Crossref]

7. Brugnano, L., Trigiante, D. (1998). Solving Differential Problems by Multistep Initial and Boundary Value Methods. Amsterdam: Gordon and Breach Science Publishers. [Google Scholar] [Crossref]

8. Caglar, H.; Caglar, N.; Ozer, M. (2009). B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Solitons Fract., 39(1232–1237.). [Google Scholar] [Crossref]

9. Duggan R & Goodman A. (2017). Pointwise bounds for a nonlinear heat conduction model of the human head. Bulletin of Mathematical Biology,Springer Science and Business Media LLC, 48(2), 229-236. DOI:10.1016/s0092-8240(86)80009-x [Google Scholar] [Crossref]

10. Gumgum, S. (2020). Taylor wavelet solution of linear and non-linear Lane-Emden equations. Appl. Numer. Math,158, 44–53. [Google Scholar] [Crossref]

11. Jator, S.N.: On a class of hybrid methods for y = f (x,y,y ). Int. J. Pure Appl. Math. 59(4), 381– 395 (2010) [Google Scholar] [Crossref]

12. Kadalbajoo, M.K.; Kumar, V. (2007). B-Spline method for a class of singular two-point boundary value problems using optimal grid. Appl. Maths. Comput. , 188(1856–1869.). [Google Scholar] [Crossref]

13. Kumar, M. A . (2003). difference scheme based on non-uniform mesh for singular two-point boundary value problems. Appl. Math. Comput. , 136(281–288.). [Google Scholar] [Crossref]

14. Lambert J. D.(1973), Computational methods in Ordinary Differential Equations, John Wiley and sons, New York. [Google Scholar] [Crossref]

15. Malele, J.; Dlamini, P.; Simelane, S. (2023). Solving Lane–Emden equations with boundary conditions of various types using high-order compact finite differences. Appl. Math. Sci. Eng. , 31, 2214303. [Google Scholar] [Crossref]

16. Pandey, R.K. (1997). A finite difference methods for a class of singular two point boundary value problems arising in physiology. Int. J. Comput. Math., 65(131–140.). [Google Scholar] [Crossref]

17. Ramos, Z. Kalogiratou, Th. Monovasilis and T. E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems, Numerical Algorithm, Springer 2015. DOI 10.1007/s11075-015- 0081-8. [Google Scholar] [Crossref]

18. 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. [Google Scholar] [Crossref]

19. Rajat, S., Gurjinder, S., Higinio , R., & Kanwar , V. (2023). An efficient optimized adaptive step-size hybrid block method for integrating w′′ = f (t,w,w′) directly. Journal of Computational and Applied Mathematics, 114838, 420. [Google Scholar] [Crossref]

20. Rufai, M. A., and Ramos, H. (2021). Numerical Solution for Singular Boundary Value Problems Using a Pair of Hybrid Nyström Techniques.Axioms 2021, 10, 202. Axioms , 10, 202. [Google Scholar] [Crossref]

21. Thula, K.; Roul, P. A High-Order B-Spline Collocation Method for Solving Nonlinear Singular Boundary Value Problems Arising in Engineering and Applied Science. Mediterr. J. Math. 2018, 15, 176. [Google Scholar] [Crossref]

22. Umesh M, & Kumar, M. (2021). Numerical solution of singular boundary value problems using advanced Adomian decomposition method. Eng. Comput., 37, 2853–2863. [Google Scholar] [Crossref]

23. Utalor, I. K., Ajie, I. J., Durojaye, M.O, (2025). Construction of Block Hybrid Methods for the Solution of Singular Second Order Ordinary Differential Equation. Asian Research Journal of Mathematics,21(5),23–37. https://doi.org/10.9734/arjom/2025/v21i5923 [Google Scholar] [Crossref]

24. Zhang, Y. (1993). Existence of solutions of a kind of singular boundary value problem. Nonlinear Anal. Theory Methods Appl. , 21(153–159.). [Google Scholar] [Crossref]

25. Henrici, P.(1962). Discrete Variable Methods in ODE. New York: John Wiley & Sons. [Google Scholar] [Crossref]

• Interplay of Students’ Emotional Intelligence and Attitude toward Mathematics on Performance in Grade 10 Algebra
• Numerical Simulation of Fitzhugh-Nagumo Dynamics Using a Finite Difference-Based Method of Lines
• Fixed Point Theorem in Controlled Metric Spaces
• Usage of Moving Average to Heart Rate, Blood Pressure and Blood Sugar
• Exploring Algebraic Topology and Homotopy Theory: Methods, Empirical Data, and Numerical Examples