Hybrid Block Methods with Constructed Orthogonal Basis for Solution of Third-Order Ordinary Differential Equations




Hybrid block, Collocation, Interpolation, Third-order ODE, Integration scheme


In this work, an orthogonal polynomial with weight function w(x) =x2 + x + 1 in the interval [-1, 1] was constructed and used as the basis function to develop block methods, using collocation and interpolation approach. An efficient class of continuous and discrete numerical integration schemes of implicit hybrid form for third-order problems were developed and successfully implemented. Three different problems were solved with these schemes and they performed favourably. The investigation, using the appropriate existing theorems, shows that the methods are consistent, zero-stable and hence, convergent.


R. A. Bun & Y. D. Vasil’yev, “A numerical method for solving diferential equations of any orders”, Journal of Computer Mathematical Physics 32 (1992) 3.

F. L. Joseph, R. B. Adeniyi & E. O. Adeyefa, “A Collocation Technique for Hybrid Block Methods with a Constructed Orthogonal basis for Second Order Ordinary Differential Equations”, Global Journal of Pure and Applied Mathematics 14 (2018) 4.

E. O. Adeyefa, “A Model for Solving First, Second and Third Order IVPs Directly”, Int. J. Appl. Comput. Math. 7 (2021) 131. https://doi.org/10.1007/s40819-021-01075-6 DOI: https://doi.org/10.1007/s40819-021-01075-6

O. E. Abolarin, J. O. Kuboye, E. O. Adeyefa & B. O. Ogunware “New efficient numerical model for solving second , third and fourth order ordinary differential equations directly”, Gazi University Journal of Science 33 (2020) 4. DOI: https://doi.org/10.35378/gujs.627677

J. Sunday, G. M. Kumleng, N. M. Kamoh, J. A. Kwanamu, Y. Skwame & O. Sarjiyus, “Implicit Four-Point Hybrid Block Integrator for the Simulations of Stiff Models”, J. Nig. Soc. Phys. Sci. 4 (2022) 287. https://doi.org/10.46481/jnsps.2022.777 DOI: https://doi.org/10.46481/jnsps.2022.777

E. O. Adeyefa, Y. Haruna, R. O. Ajewole & R. I. Ndu, “On polynomials construction”, International Journal of Mathematical Analysis12 (2018) 6. DOI: https://doi.org/10.12988/ijma.2018.8423

R. B. Adeniyi & H. Yahaya, “A collocation technique based on orthogonal polynomial for construction of continuous hybrid methods”, Ilorin Journal of science 2 (2014) 1. https://doi.org/10.54908/iljs.2014.01.02.014. DOI: https://doi.org/10.54908/iljs.2014.01.02.014

R. B. Adeniyi & M. O. Bamgbola, “Formulations of Discrete and continuous Hybrid method, using orthogonal polynomials as basis function”, Journal of Nigerian Association of Mathematical Physics 29 (2015) 491.

F. Ekundayo & R. B. Adeniyi, “A numerical integration scheme with certain orthogonal polynomial in a collocation approximation technique for ODEs”, Journal of the Nigerian Association of Mathematical Physics 28 (2014) 2.

R. O. Folaranmi, R. B. Adeniyi & E. O. Adeyefa “An orthogonal based block method for the solution of ODEs”, Pacific Journal of Science and Technology 17 (2017) 2.

E. Momoniat & F. M. Mohammed, “Symmetry reduction and numerical solution of a third-order ODE from thin ln flow”, Mathematical and Computational Applications 15 (2010) 4. DOI: https://doi.org/10.3390/mca15040709

R. Genesio & A. Tesi, “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems”, Automatica 28 (1992) (3) 531. http://dx.doi.org/10.1016/0005-1098(92)90177-H. DOI: https://doi.org/10.1016/0005-1098(92)90177-H

T. A. Anake, A. O. Adesanya, G. J. Oghonyon & M. C. Agarana, “Block algorithm for general third order ordinary differential equations”, ICASTOR Journal of Mathematical Sciences 7 (2013) 2.

E. O. Adeyefa & J. O. Kuboye, “Derivation of New Numerical Model Capable of Solving Second and Third Order Ordinary Di erential Equations Directly”, IAENG International Journal of Applied Mathematics 50 (2020) 2.

J. O. Kuboye & E. O Adeyefa, “New development numerical formula for solution of first and higher order ordinary differential equations”, Journal of interdisciplinary mathematics 25 (2022) 2. https://doi.org/10.1080/09720502.2021.1925453. DOI: https://doi.org/10.1080/09720502.2021.1925453

S. O. Fatunla, Numerical methods for initial value problems in ordinary differential equations, Academic press Incorporation, Harcourt Brace, Jovanovich Publishers, New York (1988). DOI: https://doi.org/10.1016/B978-0-12-249930-2.50012-6

G. Dahlquist, Some properties of linear multistep and one leg method for ordinary di erential equations. Department of Computer Science, Royal Institute of Technology, Stockholm (1979).

J. D. Lambert, Computational methods in ordinary differential equation, John Wiley and Sons, New York (1973).

https://doi.org/10.1002/zamm.19740540726 DOI: https://doi.org/10.1002/zamm.19740540726

K. Lee, I. Fudziah & S. Norazak, “An accurate block hybrid collocation method for third order ordinary differential equations”, Journal of Applied Mathematics, Hindawi Publishing Corporation 549597 (2014) 9. https://doi.org/10.1155/2014/549597 DOI: https://doi.org/10.1155/2014/549597

G. Dahlquist, “On one-leg multistep method”, SIAM Journal on Numerical Analysis, 20 (1983) 1130. DOI: https://doi.org/10.1137/0720082

P. Henrichi, Discrete variable methods in ODE, John Wiley and Sons, New York (1962).

L. F. Shampine & H. A. Watts, “Block implicit one-step methods”, Journal of Numerical Mathematics 23 (1972) 731. https://doi.org/10.1007/BF01932819 DOI: https://doi.org/10.1090/S0025-5718-1969-0264854-5

A. S. Olagunju & F. L. Joseph, “Construction of Functions for Fractional Derivatives using Matlab”, Journal of Advances in Mathematics & Computer Science 36 (2021) 6. https://doi.org/10.9734/jamcs/2021/v36i630368. DOI: https://doi.org/10.9734/jamcs/2021/v36i630368



How to Cite

Joseph, F. L. ., Olagunju, A. S., Adeyefa, E. O., & James, A. A. (2022). Hybrid Block Methods with Constructed Orthogonal Basis for Solution of Third-Order Ordinary Differential Equations. Journal of the Nigerian Society of Physical Sciences, 5(1), 865. https://doi.org/10.46481/jnsps.2023.865



Original Research