Adaptive Block Hybrid Method with Quasilinearization for the Accurate Solution of General Third-Order IVPs
DOI:
https://doi.org/10.52737/18291163-2026.18.09-1-29Keywords:
Block Hybrid Method, Quasilinearization, Embedded Procedure, Lagrange InterpolationAbstract
This paper introduces a novel variable-step block hybrid method for the direct solution of general third-order initial value problems. The proposed approach combines quasilinearization with an embedded procedure to enhance accuracy and computational efficiency. Rigorous analysis establishes the convergence and $A$-stability of the proposed method, providing a reliable numerical framework for a broad class of third-order initial value problems. Numerical experiments on a range of benchmark test problems demonstrate that the proposed method achieves high accuracy and competitive computational efficiency compared with several existing approaches. These results highlight the effectiveness and potential of the proposed method as a robust tool for the direct numerical solution of third-order differential equations arising in engineering and related applications.
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O. Adeyeye and Z. Omar, Solving third order ordinary differential equations using one-step block method with four equidistant generalized hybrid points. IAENG Int. J. Appl. Math., 49 (2019), no. 2, pp. 2-15.
L.O. Adoghe, B.G. Ogunware, B. Nikouravan and E.O. Omole, A family of symmetric implicit higher order methods for the solution of third order initial value problems in ordinary differential equations. Theor. Math. Appl., 6 (2016), no. 3, pp. 67-84.
M. Alkasassbeh and Z. Omar, Hybrid one-step block fourth derivative method for the direct solution of third order initial value problems of ordinary differential equations. Int. J. Pure Appl. Math., 119 (2018), no. 1, pp. 207-224.
R. Allogmany and F. Ismail, Implicit three-point block numerical algorithm for solving third order initial value problem directly with applications. Mathematics, 8 (2020), no. 10, article 1771.
A.R. Appadu, Optimized low dispersion and low dissipation Runge-Kutta algorithms in computational aeroacoustics. Appl. Math. Inf. Sci., 8 (2014), no. 1, pp. 57-68.
A.R. Appadu, and A.A.I. Peer, Optimized weighted essentially nonoscillatory third-order schemes for hyperbolic conservation laws. J. Appl. Math., 2013 (2013), article ID 428681, pp. 1-12.
D.O. Awoyemi, A P-stable linear multistep method for solving general third order ordinary differential equations. Int. J. Comput. Math., 80 (2003), no. 8, pp. 985-991.
J.C. Butcher, Numerical Methods for Ordinary Differential Equations, 3rd ed., John Wiley & Sons, Chichester, 2016.
M.K. Duromola and A.L. Momoh, Hybrid numerical method with block extension for direct solution of third order ordinary differential equations. Am. J. Comput. Math., 9 (2019), no. 2, pp. 68-80.
S. Jator, T. Okunlola, T. Biala and R. Adeniyi, Direct integrators for the general third-order ordinary differential equations with an application to the Korteweg-de Vries equation. Int. J. Appl. Comput. Math., 4 (2018), article number 110.
Y.D. Jikantoro, F. Ismail, N. Senu and Z.B. Ibrahim, A new integrator for special third order differential equations with application to thin film flow problem. Indian J. Pure Appl. Math., 49 (2018), no. 1, pp. 151-167.
L.V. Kantorovich, On Newton's method for functional equations. Doklady Akademii Nauk SSSR, 59 (1948), pp. 1237-1240.
L.V. Kantorovich and G.P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982.
J.D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley & Sons, London, 1973.
J.O. Kuboye, O.F. Quadri and O.R. Elusakin, Solving third order ordinary differential equations directly using hybrid numerical models. J. Niger. Soc. Phys. Sci., 2 (2020), pp. 69-76.
K.O. Lawal, Y.A. Yahaya and S.D. Yakubu, Four-step block method for solving third order ordinary differential equation. International Journal of Mathematics Trends and Technology, 57 (2018), no. 5, pp. 331-344.
Z.A. Majid, N.A. Azmi, M. Suleiman and Z.B. Ibrahim, Solving directly general third order ordinary differential equations using two-point four step block method. Sains Malaysiana, 41 (2012), no. 5, pp. 623-632.
M. Mechee, N. Senu, F. Ismail, B. Nikouravan and Z. Siri, A three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film flow problem. Math. Probl. Eng., 2013 (2013), article ID 795397, pp. 1-7.
S. Mehrkanoon, A direct variable step block multistep method for solving general third-order ODEs. Numer. Algor., 57 (2011), no. 1, pp. 53-66.
A. Olagunju and E. Adeyefa, Hybrid block method for direct integration of first, second and third order IVPs. Cankaya University Journal of Science and Engineering, 18 (2021), no. 1, pp. 1-8.
Z. Omar and R. Abdelrahim, Application of single step with three generalized hybrid points block method for solving third order ordinary differential equations. J. Nonlinear Sci. Appl., 9 (2016), pp. 2705-2717.
Z. Omar and J.O. Kuboye, Developing block method of order seven for solving third order ordinary differential equations directly using multistep collocation approach. Int. J. Appl. Math. Stat., 53 (2015), no. 3, pp. 165-173.
A.L. Osa and O.E. Olaoluwa, A fifth-fourth continuous block implicit hybrid method for the solution of third order initial value problems in ordinary differential equations. Appl. Comput. Math., 8 (2019), no. 3, pp. 50-57.
M.A. Rufai and H. Ramos, A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications. Int. J. Comput. Math., 99 (2022), no. 2, pp. 292-308.
S.D. Yakubu, Family of one-step A-stable optimized third derivative hybrid block methods for solving general second-order IVPs. AL-Rafidain Journal of Computer Sciences and Mathematics, 17 (2023), no. 2, pp. 125-140.
S.D. Yakubu and P. Sibanda, A novel one-step optimized hybrid block method for solving general second-order ordinary differential equations. Recent Advances in Natural Sciences, 3 (2025), no. 1, article ID 135, pp. 1-12.
S.D. Yakubu and P. Sibanda, Implicit one-step optimized fourth-derivative hybrid block method for directly solving general third-order IVPs. Recent Advances in Natural Sciences, 3 (2025), no. 2, article ID 142, pp. 1-12.
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