This paper introduces a novel unified numerical approach to obtain explicit approximations of initial value formulations (IVFs) encompassing ODEs ranging from first- to third-order. The proposed technique leverages Chebyshev polynomials as basic functions and is developed using continuous schemes formulated through both collocation and interpolation strategies. It operates on a block-by-block basis, providing an efficient framework for numerically solving ODEs of multiple orders. The convergence properties of this method are thoroughly examined through the lens of zero-stability and consistency. In-depth discussions unfold, shedding light on the efficacy of this approach in addressing first, second, and third-order ODEs. Through comparative analyses against existing methods, it is distinctly evident that the proposed model surpasses its counterparts in terms of accuracy, marking a significant advancement in the numerical treatment of IVPs. This model not only introduces a unified approach for diverse ODE orders but also stands as a testament to its superior performance, establishing itself as a noteworthy contribution to the realm of numerical integration methodologies.

Keywords: Unified Numerical Approach, Initial Value Formulations (IVFs), Ordinary Differential Equations (ODEs), Chebyshev Polynomials, Collocation, Interpolation, Block Method, Zero-Stability, Consistency, Convergence Analysis, Numerical Integration, Accuracy.

[1] Adeniyi, R.B., Joseph, F.L., & Adeyefa, E.O. (2011). A collocation method for direct numerical integration of initial value problems in higher-order ordinary differential equations. Analele Stiintifice Ale Universitatii AL. I. Cuza Din Iasi (SN), Matematica, Tomul., 2: 311–321.

[2] Adeyefa, E.O., & Kuboye, J.O. (2020). Derivation of new numerical model capable of solving second and third-order ordinary differential equations directly. International Journal of Applied Mathematics.

[3] Ajileye, G., Amoo, S.A., & Ogwumu, O.D. (2018). Hybrid block method algorithms for the solution of first-order initial value problems in ordinary differential equations. Journal of Applied and Computational Mathematics, 7(2): 1–4.

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

[5] Fatunla, S.O. (1988). Numerical methods for initial value problems in ordinary differential equations. Academic Press Inc. Harcourt Brace, Jovanovich Publishers, New York.

[6] Adesanya, A.O., & Adewale, A. (2014). A note on the construction of constant order predictor corrector algorithm for the solution of ''. British Journal of Mathematics and Computer Science, 4(6): 886–895. Retrieved from www.sciencedomain.org.

[7] Kayode, S.J., & Adegboro, J.O. (2018). Predator-corrector linear multistep method for the direct solution of initial value problems of second-order ordinary differential equations. Asian Journal of Physical and Chemical Sciences, 6: 1–9.

[8] 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.

[9] Lambert, J.D. (1991). Numerical methods for ordinary differential systems. John Wiley and Sons, New York.

[10] James, A.A., Ajileye, G., Ayinde, A.M., & Dunama, W. (2022). Hybrid-block method for the solution of second order non-linear differential equations. Journal of Advances in Mathematics and Computer Science, 37(12): 156–169.

[11] Lambert, J.D. (1973). Computational methods for ordinary differential equations. John Wiley, New York.

[12] Mohammed, U., & Adeniyi, R.B. (2014). Derivation of five-step block hybrid backward differential formulas (HBDF) through continuous multi-step collocation for solving second-order differential equations. Pacific Journal of Science and Technology, 15: 89–95.

[13] Olabode, B.T. (2009). An accurate scheme by block method for the third-order ordinary differential equation. Pacific Journal of Science and Technology, 10: 136–142.

[14] Ramos, H., Mehta, S., & Vigo-Aguiar, J.A. (2016). Unified approach for the development of k-step block Falkner-type methods for solving general second-order initial-value problems in ODEs. Journal of Computational and Applied Mathematics, Article in Press.

[15] 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 ODEs. Journal of Engineering and Applied Sciences, 15: 1972–1976.

[16] Sunday, J., Odekunle, M.R., & Adesanya, A.O. (2013). Order six-block integrator for the solution of first-order ordinary differential equations. International Journal of Mathematics and Computer Applications Research, 3(4): 45–56.

[17] Isah, I.O., Salawu, A.S., Olayemi, K.S., & Enesi, L.O. (2020). An efficient 4-step block method for solution of first order initial value problems via shifted Chebyshev polynomial. Tropical Journal of Science and Technology, 1(2): 25–36. http://doi.org/10.47524/tjst.v1i2.5.

Source of Funding:

This study received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Competing Interests Statement:

The authors affirm that they have no competing interests that may have affected the findings or interpretations outlined in this study.

Consent for publication:

The authors declare that they consented to the publication of this study.

Authors' contributions:

All the authors made an equal contribution in the Conception and design of the work, Data collection, Drafting the article, and Critical revision of the article. All the authors have read and approved the final copy of the manuscript.

Institutional Review Board Statement:

Not applicable for this study.