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Articles

CJPLS: VOL. 14, NO. 1, JUNE 2026

Generalized Third-Derivative Block Hybrid Method for Solving Third-Order Differential Equations

Submitted
February 16, 2026
Published
2026-03-30

Abstract

In many scientific and engineering applications, real-world phenomena are modeled using differential equations in order to describe, analyze and interpret physical processes. This study is designed to formulate the development of third-derivative block hybrid method for the direct numerical solution of third-order initial value problems. The method is formulated using interpolation and collocation techniques based on power series expansion, resulting in a block scheme that incorporates four off-step points for improved accuracy and computational efficiency. The mathematical properties of the proposed method including order, error constant, consistency, zero-stability, convergence and region of absolute stability are thoroughly investigated to ensure its reliability. Stability analysis using the Boundary Locus Method confirms that the new method possesses a satisfactory region of absolute stability suitable for stiff and non-stiff problems. The effectiveness of the method is demonstrated through four numerical experiments involving both highly stiff and non-stiff third-order linear problems. Comparative analysis with existing methods reveals that the proposed block hybrid method consistently produces numerical approximations that closely match the exact solution, outperforming several classical approaches in terms of accuracy and stability. The results confirm that the new method is robust, computationally efficient, and suitable for solving a wide class of higher-order ordinary differential equations.

References

  1. Omale D., Ojih, P. B. & Ogwo M. O. (2014) Mathematical Analysis of stiff and non-stiff initial value problems of Ordinary Differential Equation using MATLAB. International Journal of Scientific & Engineering Research, 3 (9), 49 – 59.
  2. Fatunla, S.O. (1991). Block methods for second order ODEs. International Journal of Computer Mathematics, 41(1-2), 55-63.
  3. Sabo, J. (2021). Single-step block hybrid methods for direct solution of higher order initial value problems. M.Sc. Dissertation, Adamawa State Univeristy, Mubi, Adamawa State. 33-40. (Unpublished).
  4. Abdelraim, R., Omar, Z., Ala’yed, O, & Batiha, B. (2019). Hybrid third derivative block method for the solution of general second order initial value problems with generalized one step point. Eur. J. of Pure and App. Mat. 12(3), 1199-1214.
  5. Skwame, Y., Sabo, J. & Mathew, M. (2019).The treatment of second order ordinary differential equations using equidistant one-step block hybrid. Asian Journal of Probability and Statistics. 5(3), 1-9.
  6. Adoghe, L.O & Omole E.O. (2019). A fifth-fourth continuous block implicit hybrid method for the solution of third order initial value problems in ordinary differential equations. Applied and Computational Mathematics. 8(3), 50-57. http://www.sciencepublishinggroup.com/j/acm
  7. Kuboye, J. O. & Omar, Z. (2015). Derivation of a six step block method for direct solution of second order ordinary differential equations. Math. Comput. Appl. 20, 151-159.
  8. Adeyeye, O. & Omar, Z. (2018). New self-starting approach for solving special third order initial value problems, Int. J. Pure and Appl. Math. 118 (3), 511-517.
  9. Duromola, M. K., Momoh, A. L. & Adeleke, J. M. (2022). One-step Hybrid block method for directly solving fifth-order initial value problems of ordinary differential equations. Asian Research Journal of Mathematics, 18(1), 53- 64.
  10. Mechee, M. S., & Fawzi, F. A. (2021). Generalized Runge-Kutta integrators for solving fifth-order ordinary differential equations. Italian Journal of Pure and Applied Mathematics, 45, 600–610.
  11. Aliyu B. K., Osheku C. A., Funmilayo A. A. & Musa J. I. (2014) Identifying stiff ordinary differential equations and Problem solving Environments (PSGs). Journal of Scientific Research & Reports, 3 (11),1439 – 1448.
  12. Kuboye, J.O. & Omar, Z., Abolarin, O.E. & Abdelrahim, O.E. (2018). Generalized hybrid block method for solving second order ordinary differential equations directly. Research and Reports on Mathematics. 2, 1-7.
  13. Lawrence, O.A., Ogunware, B.G. & Ezekiel, O.O. (2016). A family of symmetric implicit higher order methods for the solution of third order initial value problems in ordinary differential equations Theoretical Mathematics & Applications, vol. 6(3), 67-84.
  14. Omole E.A. & Ogunware B.G. (2018). 3-point single hybrid block method (3PSHBM) for direct solution of general second order initial value problem of ordinary differential equations. Journal of Scientific Research & Reports. 20(3), 1-11.
  15. Raymond, D., Skwame, Y. & Sunday, J. (2018). An Implicit Two-Step One Off-Grid Point Third Derivative Hybrid Block Method for the Direct Solution of Second-Order Ordinary Differential Equations. Academic Journal of Applied Mathematical Sciences. 4(2), 8-14.
  16. Skwame, Y., Dalatu, P.I., Sabo, J. & Mathew, M. (2019). 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(6), 90-100.
  17. Skwame, Y., Sabo, J. & Mathew, M. (2019). The treatment of second order ordinary differential equations using equidistant one-step block hybrid. Asian Journal of Probability and Statistics. 5(3), 1-9.
  18. Sunday J. (2018). On the oscillation criteria and computation of third order oscillatory differential equations. Communication in Mathematics and Applications. 6(4), 615-625.
  19. Taparki, R.M., Gurah, D. & Simon, S. (2010). An implicit Runge-Kutta method for solution of third order initial value problem in ODE, Int. J. Numer. Math. 6, 174-189.
  20. Aibiremhon, A. A. and Omole, E. O. (2020). A four-step collocation procedure by means of perturbation term with application to third-order ordinary differential equation. International journal of computer Applications. 175(24), 25-36.
  21. Omar, Z. and Abdelrahim, R. (2016). Application of single step with three generalized hybrid points block method for solving third order ordinary differential equations. J. Nonlinear Sci. Appl. 9, 2705-2717.
  22. Kayode, S. J. and Obaruha, F. O. (2017). Symmetric 2-step 4-point hybrid method for the solution of general third order differential equations. Journal of Applied and Computational Mathematics. 6(2). 1-4.
  23. Adeyeye, O. and Omar, Z. (2019). Solving third order ordinary differential equation using one-step block method with four equidistance generalized hybrid points. International Journal of Applied Mathematics. 49(2), 1-9.
  24. Areo, E. A. and Omojola, M. T. (2017). One-twelveth step continuous block method for the solution of y′′′ = f(x, y, y′, y′′). International Journal of Pure and Applied Mathematics. 114(2), 165-178.
  25. Sunday, J., Joshua, K. V. and Danladi, N. (2019). On the derivation and implementation of a one-step algorithm for third order oscillatory problems. Adamawa State University, Journal of Scientific Research (ADSUJSR). 7(1), 8-20.
  26. Omar, Z., Abdullahi, Y. A. and Kuboye, J. O. (2016). Predictor-Corrector block method of order seven for solving third order ordinary differential equations. International Journal of Mathematical Analysis. 10(5), 223-235.