Direct Numerical Algorithm as First, Second and Third-Order IVPS Solver

Authors

  • Emmanuel Oluseye Adeyefa Department of Mathematics, Federal University Oye-Ekiti, Ekiti State, NIGERIA. https://orcid.org/0000-0003-0942-6430
  • Victoria Oluwatoyin Ojo Department of Statistics, Oyo State College of Agriculture and Technology, Igbo-Ora, Oyo State, NIGERIA.
  • Rotimi O. Folaranmi Department of Mathematical and Computing Sciences, Thomas Adewumi University, Oko, Kwara State, NIGERIA. https://orcid.org/0000-0002-9344-4430
  • Adebisi A. Ibrahim Department of Mathematical Sciences, Oduduwa University, Ipetumodu, Osun State, NIGERIA.

DOI:

https://doi.org/10.22452/

Keywords:

block method, consistency, convergence, collocation, zero-stability

Abstract

This paper presents a novel numerical model for accurately integrating initial value problems of multi-order ordinary differential equations (ODEs) of first, second, and third orders. Chebyshev polynomials are employed as the basis functions, and the collocation technique is used to develop continuous schemes that are evaluated at selected points to formulate the proposed multi-order ODE solver, which is applied in a block-by-block manner. The convergence analysis is carried out to establish the zero-stability and consistency of the method. Comparisons with existing methods show the superior performance of the proposed method. The results indicate its ability to solve multi-order ODEs more effectively while reducing the computational cost. This work represents a significant advance in numerical integration for ODEs, providing improved accuracy and efficiency in solving a wide range of multi-order ODE problems.

Author Biographies

  • Emmanuel Oluseye Adeyefa, Department of Mathematics, Federal University Oye-Ekiti, Ekiti State, NIGERIA.

    Professor, Department Of Mathematics

  • Victoria Oluwatoyin Ojo, Department of Statistics, Oyo State College of Agriculture and Technology, Igbo-Ora, Oyo State, NIGERIA.

    Department of Statistics, Oyo State College of Agriculture and Technology, Igbo-Ora, Oyo State, Nigeria

    Lecturer

  • Rotimi O. Folaranmi, Department of Mathematical and Computing Sciences, Thomas Adewumi University, Oko, Kwara State, NIGERIA.

    Department of Mathematical and Computing Science, Thomas Adewumi University, Oko, Kwara State, Nigeria

     

References

Adeyefa, E.O., Olajide, O.A., Akinola, L.S., Abolarin, O.E., Ibrahim, A.A. & Haruna, Y. (2020), On direct integration of second and third order ordinary differential equations, J. Eng. Appl. Sci. 15 1972-1976.

Ajileye, G., Amoo, S.A. & Ogwumu, O.D. (2018), Hybrid block method algorithms for solution of first order initial value problems in ordinary differential equation, J. Appl. Comput. Math. 7 1-4. https://doi.org/10.4172/2168-9679.1000390.

Allogmany, R. & Ismail F. (2020), Implicit three-point block numerical algorithm for solving third order initial value problem directly with applications, Math. 8 1771. doi:10.3390/math8101771.

Awoyemi, D.O. (1999), A class of continuous linear multistep methods for general second-order initial value problems in ordinary differential equations, Intern. J. Comput. Math. 72 29-37.

Dahlquist, G. (1979), Some properties of linear multistep and one leg method for ordinary differential equations. Department of comput. sci. Royal institute of tech., Stockholm

Fatunla, S.O. (1991), Block method for second-order initial value problem, Intern. J. Compt. Math. 41 55-63.

Folaranmi, R.O. Adeniyi, R.B. & Adeyefa, E.O. (2016), An orthogonal based self-starting numerical integrator for third order IVPs in ordinary differential equations, The Pacific J. Sci. Tech. 17 73 – 86.

Henrichi, P. (1962), Discrete variable methods in ordinary differential equations, {it John Wiley and Sons}, New York

Hussain, K.A. Ismail, F. Senu, N. & Rabiei, F. (2017), Fourth-order improved runge–kutta method for directly solving special third order ordinary differential equations, Iranian J. Sci. Tech. 41 429-437. DOI 10.1007/s40995-017-0258-1

Ismail, F. Ken, Y.L. & Othman, M. (2009), Explicit and implicit 3-point block methods for solving special second order ordinary differential equations directly, Intern. J. Math. Analy. 3 239-254.

Jator, S.N. (2010), Solving second order initial value problems by a hybrid multistep method without predictors, Appl. Math. Comput. 217 4036–4046

Kayode, S.J. & Adegboro, J.O. (2018), Predictor-corrector linear multistep method for direct solution of initial value problems of second order ordinary differential equations, Asian J. Phys. Chem. Sci. 6 1-9.

Kuboye, J.O. (2015), Block methods for direct solution of higher-order ordinary differential equations using interpolation and collocation approach, Ph.D. thesis, Universiti Utara Malaysia.. http://etd.uum.edu.my/id/eprint/5789.

Lambert, J.D. (1991), Numerical methods for ordinary differential systems, John Willey and Sons New York.

Lambert, J.D. (1973), Computational methods for ordinary differential equations, John Willey and Sons New York.

Mohammed, U. & Adeniyi, R.B. (2014), Derivation of five-step block hybrid backward differential formulas through the continuous multi-step collocation for solving second order differential equation, Pacific J. Sci. Tech. 15 89 – 95.

Olabode, B.T. (2009), An accurate scheme by block method for the third-order ordinary differential equation, Pacific J. Sci. Tech. 10 136 – 142.

Ramos, H., Mehta, S. Vigo-Aguiar, J. A unified approach for the development of k-Step block Falkner-type methods for solving general second-order initial-value problems in ordinary differential equations, J. Comput. Appl. Math. Article in Press.

Ramos, H.& Rufai, M.A. (2021) An adaptive one-point second derivative lobatto-type hybrid method for solving efficiently differential systems, Intern. J. Comput. Math. 1-19. DOI: 10.1080/00207160.2021.1999429.

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Published

30-06-2026

Issue

Vol 45 No 2 (2026): June 2026

Section

Original Articles

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