Computational method for Volterra integro-differential equations of the second kind

Authors

  • Abdullahi Muhammed Ayinde
    Department of Mathematics, University of Abuja, FCT, 900211, Nigeria
  • Adam Ajimoti Ishaq
    Department of Physical Sciences, Al-Hikmah University, Ilorin, Nigeria
  • Lukman Olalekan Ahmed
    Department of Mathematics, Baze University, Abuja, Nigeria
  • Emmanuel Jacob
    Department of Mathematical Sciences, Prince Abubakar Audu University, Anyigba, Nigeria

Keywords:

Volterra integro-differential equations, Interpolation technique, Stability analysis, Collocation technique

Abstract

This study presents a computational method (CM) for solving second-kind Volterra integro-differential equations (VIDEs), which are widely used in engineering, physics, biology, and control systems. The method combines interpolation and collocation techniques to produce a continuous one-step computational scheme that accurately approximates the solution while maintaining stability and convergence properties. Consistency, zero-stability, order of accuracy, convergence, and the region of absolute stability are examined to confirm the suitability of the method for both linear and nonlinear problems. Numerical experiments with selected VIDEs compare the CM with the fifth--order Adams–Bashforth–Moulton predictor--corrector method, the two-point three-step block method, the trigonometrically fitted block method, the Haar wavelet method, and trapezoidal schemes. Across the tested examples, the CM provides accurate, stable, and efficient approximations, making it a useful tool for solving complex integro-differential systems in practical applications.

Dimensions

[1] A. Adu, N. C. Ine & C. A. John, ``Three-step three off-grid points for solving Volterra integral equation of the second kind'', Asian Journal of Pure and Applied Mathematics 5 (2023) 512. https://jofmath.com/index.php/AJPAM/article/view/4.

[2] J. C. Ahmadu, F. Ogunfiditimi & I. Sale, ``Derivation of continuous linear multistep hybrid block method for the solution of Volterra integral equation of second kind'', Asian Basic and Applied Research Journal 6 (2024) 59. https://jofresearch.com/index.php/ABAARJ/article/view/140.

[3] O. O. Aduroja, L. Adiku & O. Agbolade, ``Collocation approximation method for the solution of Volterra integro-differential equations'', FUOYE Journal of Management, Innovation and Entrepreneurship 2 (2023) 240. https://fjmie.fuoye.edu.ng/index.php/public_html/article/view/53.

[4] G. Agarwala, M. Mohana, A. M. Menon, A. Sharma, T. C. Dakal & S. D. Purohit, ``Analysis of the Adomian decomposition method to estimate the COVID-19 pandemic'', in Methods of Mathematical Modeling, H. Singh, H. M. Srivastava, and D. Baleanu (Eds.), Academic Press, 2022, pp. 173–187. https://doi.org/10.1016/B978-0-323-99888-8.00011-5.

[5] B. C. Agbata, B. N. Ani, M. M. Shior, I. G. Ezugorie, R. V. Paul & P. K. Meseda, ``Analysis of Adomian decomposition method and its application for solving linear and nonlinear differential equations'', Asian Research Journal of Mathematics 18 (2022) 56. https://www.researchgate.net/publication/359326653_Analysis_of_Adomian_Decomposition_Method_and_Its_Application_for_Solving_Linear_and_Nonlinear_Differential_Equations.

[6] H. Brunner, Volterra Integral Equations: An Introduction to Theory and Applications, Cambridge University Press, Cambridge, United Kingdom, 2017. https://doi.org/10.1017/9781107341661.

[7] A. Ajileye, A. O. Olamipos, I. A. Grace & A. M. Ajiley, ``Numerical method for the solution of linear Volterra integral equations'', International Journal of Development Mathematics 1 (2024) 37. https://doi.org/10.62054/ijdm/0101.04.

[8] G. Ajileye & S. A. Amoo, ``Numerical solution to Volterra integro-differential equations using collocation approximation'', Mathematics and Computational Sciences 4 (2023) 1. https://www.researchgate.net/publication/369295547_Numerical_solution_to_Volterra_integro-differential_equations_using_collocation_approximation.

[9] N. M. Kamoh, T. Aboiyar & E. S. Onah, ``On investigating some quadrature rules for the solution of second-order Volterra integro-differential equations'', IOSR Journal of Mathematics 13 (2017) 45. https://iosrjournals.org/iosr-jm/papers/Vol13-issue5/Version-3/H1305034550.pdf.

[10] E. Cimen & S. Yatar, ``Numerical solution of Volterra integro-differential equation with delay'', Journal of Mathematics and Computer Science 20 (2020) 255. https://doi.org/10.22436/jmcs.020.03.08.

[11] D. U. Islam & A. Srivastava, ``Comparative analysis of Adomian decomposition method and variational iteration method for solving nonlinear integral equations'', JuniKhyat, UGC Care Group I Listed Journal 14 (2024) 129. https://www.researchgate.net/publication/386336425_COMPARATIVE_ANALYSIS_OF_ADOMIAN_DECOMPOSITION_METHOD_AND_VARIATIONAL_ITERATION_METHOD_FOR_SOLVING_NONLINEAR_INTEGRAL_EQUATIONS.

[12] R. A. Olowe, O. A. Akinfenwa, R. I. Abdulganiy & A. Okunuga, ``A trigonometrically fitted method for solving Volterra integro-differential equations'', Journal of the Nigerian Mathematical Society 43 (2024) 203. https://www.researchgate.net/publication/381742554_A_TRIGONOMETRICALLY_FITTED_METHOD_FOR_SOLVING_VOLTERRA_INTEGRO-DIFFERENTIAL_EQUATIONS.

[13] D. A. Maturi, ``The variational iteration method for solving Volterra integro-differential equation using Maple'', Applied Mathematical Sciences 13 (2019) 897. https://doi.org/10.12988/ams.2019.98120.

[14] O. A. Uwaheren, A. F. Adebisi, C. Y. Ishola, M. T. Raji, A. O. Yekeen & O. J. Peter, ``Numerical solution of Volterra integro-differential equations by Akbari-Ganji's method'', BAREKENG: Journal of Mathematics and Its Applications 16 (2022) 1123. https://www.researchgate.net/publication/364151036_NUMERICAL_SOLUTION_OF_VOLTERRA_INTEGRO-DIFFERENTIAL_EQUATIONS_BY_AKBARI-GANJI'S_METHOD.

[15] F. M. Alharbi & S. S. Althubiti, ``Numerical simulation of initial value problem of integro-differential equation models'', European Journal of Pure and Applied Mathematics 17 (2024) 286. https://doi.org/10.29020/nybg.ejpam.v17i1.5024.

[16] B. Ghosh & J. Mohapatra, ``Numerical solution of time-fractional Volterra integro-differential equations'', Miskolc Mathematical Notes 26 (2025) 395. https://doi.org/10.18514/MMN.2025.4723.

[17] I. J. Otaide & M. O. Oluwayemi, ``Numerical treatment of linear Volterra integro-differential equations using variational iteration algorithm with collocation'', Partial Differential Equations in Applied Mathematics 10 (2024) 100693. https://doi.org/10.1016/j.padiff.2024.100693.

[18] C. Ouaret & Z. Malendas, Numerical solution of some Volterra nonlinear integral equations by using spline iterative collocation method, Master's thesis, Department of Mathematics, Centre Universitaire Abdelhafid Boussouf Mila, Mila, Algeria, 2023. https://theses-algerie.com/2452229097228842/memoire-de-master/centre-universitaire-abdel-hafid-boussouf---mila/www.centre-univ-mila.dz/?lang=en.

[19] Y. Zhu, T. Dong, X. Zheng, Y. Li, S. Jia & X. Guo, ``Numerical approximation and simulation of a Volterra integro-differential equation with a peridynamic differential operator'', Zeitschrift für Angewandte Mathematik und Mechanik 105 (2025) e202400619. https://doi.org/10.1002/zamm.202400619.

[20] J. Sabo, A. M. Ayinde, T. Oyedepo & A. A. Ishaq, ``Simulation of two-step block approach for solving oscillatory differential equations'', Cumhuriyet Science Journal 45 (2024) 366. https://doi.org/10.17776/csj.1345303.

[21] P. Tumba, Y. Skwame & D. Raymond, ``Half-step implicit linear hybrid block approach of order four for solving third-order ordinary differential equations'', Dutse Journal of Pure and Applied Sciences 7 (2021) 124. https://globaljournals.org/GJSFR_Volume21/5-Half-Step-Implicit-Linear.pdf.

[22] Z. A. Majid & N. A. Mohamed, ``Fifth-order multistep block method for solving Volterra integro-differential equations of second kind'', Sains Malaysiana 48 (2019) 677. https://doi.org/10.17576/jsm-2019-4803-22.

[23] I. Singh & S. Kumar, ``Haar wavelet method for some nonlinear Volterra integral equations of the first kind'', Journal of Computational and Applied Mathematics 292 (2016) 541. https://doi.org/10.1016/j.cam.2015.07.022.

[24] F. Ishak & S. N. Ahmad, ``Development of extended trapezoidal method for numerical solution of Volterra integro-differential equations'', International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 10 (2016) 579. https://doi.org/10.5281/zenodo.1127402.

[25] I. Fuziyah & N. F. S. Muhammad, ``New development of extended trapezoidal method for solving first-order linear Volterra integro-differential equations'', ASM Science Journal 13 (2020) 1. https://doi.org/10.32802/asmscj.2020.sm26(5.24)}.

fig 1

Published

2026-07-15

How to Cite

Computational method for Volterra integro-differential equations of the second kind. (2026). African Scientific Reports, 5(2), 476. https://doi.org/10.46481/asr.2026.5.2.476

Issue

Volume 5, Issue 2, August 2026

Section

SPECIAL ISSUE IN HONOUR OF PROF. OLABODE MATTHIAS BAMIGBOLA
Copyright & Licensing

Copyright (c) 2026 Abdullahi Muhammed Ayinde, Adam Ajimoti Ishaq, Lukman Olalekan Ahmed, Emmanuel Jacob

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Computational method for Volterra integro-differential equations of the second kind. (2026). African Scientific Reports, 5(2), 476. https://doi.org/10.46481/asr.2026.5.2.476

Similar Articles

61-70 of 79

You may also start an advanced similarity search for this article.