MODELLING OF DYNAMICAL SYSTEMS: HYBRID BLOCK METHODS FOR STIFF INITIAL VALUE PROBLEMS

Uwem Akai; Jackson Ante; Emmanuel Asuk; Samuel Essang; Runyi Francis; Augustine Otobi

doi:10.4314/njt.2026.4845

Authors

U. P. Akai Department of Mathematics Topfaith University, Mkpatak Nigeria.
J. E. Ante Department of Mathematics Topfaith University, Mkpatak, Nigeria
E. E. Asuk Department of Mathematics University of Cross River State, Calabar, Nigeria
S. O. Essang Department of Mathematics and Computer Science, Arthur Jarvis University, Akpabuyo, Nigeria
R. E. Francis Department of Statistics, Federal Polytechnic, Ugep Cross River, NIgeria.
A. O. Otobi Department of Computer science, University of Calabar, Calabar, Nigeria

DOI:

https://doi.org/10.4314/njt.2026.4845

Keywords:

A-stability, Block methods, Dynamical systems, Hybrid methods, Interpolation and Collocation, Stiff IVPs

Abstract

Stiffness is a common phenomenon in differential equations that describe models of dynamical systems. Stiff initial value problems (IVPs) in ordinary differential equations (ODEs) pose challenges in their computation due to rapidly varying timescales in their solution components. In this study, a family of high order hybrid block methods with enhanced stability for stiff IVPs is developed. The technique of interpolation and collocation is used to determine the parameters of the hybrid block methods. The family of methods is shown to attain A-stability of order p ≤ 14. Numerical experiments conducted on stiff IVPs show that the proposed family of methods has superior accuracy when compared to existing methods in the literature.

Author Biographies

J. E. Ante, Department of Mathematics Topfaith University, Mkpatak, Nigeria

Department of Mathematics
E. E. Asuk, Department of Mathematics University of Cross River State, Calabar, Nigeria

Department of Mathematics
S. O. Essang, Department of Mathematics and Computer Science, Arthur Jarvis University, Akpabuyo, Nigeria

Department of Mathematics and Computer Science
R. E. Francis , Department of Statistics, Federal Polytechnic, Ugep Cross River, NIgeria.

Department of Statistics
A. O. Otobi, Department of Computer science, University of Calabar, Calabar, Nigeria

Department of Computer science

References

[1] Qureshi, S., Akanbi, M. A., Shaikh, A. A., Wusu, A. S., Ogunlaran, O. M., Mahmoud, W. and Osman, M. S. “A new adaptive nonlinear numerical method for singular and stiff differential problems”, Alexandria Engineering Journal, 74, pp. 585–597, 2023. doi: 10.1016/j.aej.2023.05.055.

[2] Ijam, H. M., Ibrahim, Z. B., Majid, Z. A. and Senu, N. “Stability analysis of a diagonally implicit scheme of block backward differentiation formula for stiff pharmacokinetics models”, Advances in Difference Equations, 2020(400), pp. 1-22, 2020. doi: 10.1186/s13662-020-02846-z.

[3] Eghbaljoo, M., Hojjati, G. and Abdi, A. “Adaptive second derivative multistep methods for solving stiff chemical problems”, Journal of Mathematical Chemistry, 62, pp. 1114–1133, 2024. doi: https://doi.org/10.1007/s10910-024-01582-z.

[4] Sagir, A. M. and Abdullahi, M. “A Robust Diagonally Implicit Block Method for Solving First Order Stiff IVP of ODEs”, Applied Mathematics and Computational Intelligence, 11(1), pp. 252–273. 2022.

[5] Adee, S. O. and Yunusa, S. “Some new hybrid block methods for solving non-stiff initial value problems of ordinary differential equations”, Nigerian Annals of Pure & Applied Sciences, 5, (10), pp. 263–277, 2022. doi: 10.5281/zenodo.7135518.

[6] Sharifi, M., Abdi, A., Braś, M., and Hojjati, G. “High Order Second Derivative Diagonally Implicit Multistage Integration Methods for ODES”, Mathematical Modelling and Analysis, 28(1), pp. 53–70, 2023. doi: 10.3846/mma.2023.16102.

[7] Kida, M., Adamu, S., Aduroja, O. O. and Pantuvo, P. T. “Numerical Solution of Stiff and Oscillatory Problems using Third Derivative Trigonometrically Fitted Block Method”, Journal of the Nigerian Society of Physical Sciences, 4(1) pp. 34–48, 2022. doi: 10.46481/jnsps.2022.271.

[8] Matthew, D. A. “Families of Adam’s type block schemes for stiff systems of ordinary differential equations”, M Sc Thesis, Department of Mathematics, University of Benin, Benin City, 2023.

[9] Khalsaraei, M. M., Shokri, A., and Molayi, M. “The new class of multistep multiderivative hybrid methods for the numerical solution of chemical stiff systems of first order IVPs”, Journal of Mathematical Chemistry, 58(9), pp. 1987–2012, 2020. doi: 10.1007/s10910-020-01160-z.

[10] Ebadi, M., Maleki, I. M. and Ebadian, A. “New class of hybrid explicit methods for numerical solution of optimal control problems”, Iranian Journal of Numerical Analysis and Optimization, 11(2), pp. 283–304, 2021. doi: 10.22067/ijnao.2021.67961.1005.

[11] Tassaddiq, A., Qureshi, S., Soomro, A., Hincal, E. and Shaikh, A. A. “A new continuous hybrid block method with one optimal intrastep point through interpolation and collocation”, Fixed Point Theory Algorithms for Sciences and Engineering, 2022(22), pp. 1-17, 2022. doi: 10.1186/s13663-022-00733-8.

[12] Rufai, M. A., Carpentieri, B. and Ramos, H. “A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering”, Fractal and Fractional, 7(703), pp. 1-12, 2023. doi: 10.3390/fractalfract7100703.

[13] Akai, U. P. and Muka, K. O. “Second derivative off-node block adaptive backward differentiation formulae for stiff initial value problems”, Computing and Applied Sciences Impact, 2(1), pp. 13–26, 2025.

[14] Yakubu, S. D. and Sibanda, P. “One-step family of three optimized second-derivative hybrid block methods for solving first-order stiff problems”, Journal of Applied Mathematics, 2024, pp. 1–18, 2024.

[15] Abdulganiy, R., Akinfenwa, O., Yusuff, O., Enobabor, O. and Okunuga, S. “Block Third Derivative Trigonometrically-Fitted Methods for Stiff and Periodic Problems”, Journal of the Nigerian Society of Physical Sciences, 2(1), pp. 12–25, 2020. doi: 10.46481/jnsps.2020.33.

[16] Akinfenwa, O. A., Abdulganiy, R. I., Akinnukawe, B. I. and Okunuga, S. A. “Seventh order hybrid block method for solution of first order stiff systems of initial value problems”, Journal of the Egyptian Mathematical Society, 28(34), pp. 1-11, 2020. doi: 10.1186/s42787-020-00095-3.

[17] Akinnukawe, B. I. and Muka, K. O. “L-stable block hybrid numerical algorithm for first-order ordinary differential equations”, Journal of the Nigerian Society of Physical Sciences, 2(3) pp. 160–165, 2020. doi: 10.46481/jnsps.2020.108.

[18] Kona, J. E. and Muka, K. O. “Hybrid block formulae whose eigenvalues of Jacobian matrices are close to the imaginary axis of the complex plane”, Recent Advances in Natural Sciences, 2(74), pp. 1-11, 2024. doi: 10.61298/rans.2024.2.2.74.

[19] Okor, T. and Nwachukwu, G. C. “Generalized Cash-type second derivative extended backward differentiation formulas for stiff systems of ODEs”, Journal of the Nigerian Mathematical Society, 41(2), pp. 163–191, 2022.

[20] Areo, E. A. and Edwin, O. A. “Multi-derivative multistep method for initial value problems using boundary value technique”, Open Access Library Journal, 7(e6063), pp. 1-17, 2020. doi: https://doi.org/10.4236/oalib.1106063.

[21] Sunday, J. O., Odekunle, M. R. and Adesanya, A. O. “Order Six Block Integrator for the solution of first-order ordinary differential equations”, International Journal of Mathematics and Soft Computing, 3(1), pp. 87–96, 2013 doi: https://doi.org/10.26708/IJMSC.2013.1.3.10.

[22] Cash, J. R. “Second derivative extended backward differentiation formulas for the numerical integration of stiff systems”, SIAM Journal on Numerical Analysis, 18(1), pp. 21–36, 1981. doi: 10.1137/0718003.

[23] Adamou, A. “Free variation analysis of thin rectangular plates using piecewise shape functions in Ritz procedure”, Nigerian Journal of Technology, 43(4), pp. 646–654, 2024. doi: 10.4314/njt.v43i4.5.

[24] Soomro, H., Zainuddin, N., Daud, H., Sunday, J., Jamaludin, N., Abdullah, A., Mulono, A. and Kadir, E. A. “3-point block backward differentiation formula with an off-step point for the solutions of stiff chemical reaction problems”, Journal of Mathematical Chemistry, 61(1), pp. 75–97, 2023. doi: 10.1007/s10910-022-01402-2.