This study presented a New Numerical Scheme (NNS) using a linear block algorithm for solving Volterra integro-differential equations of the second kind. The NNS was derived using a linear block algorithm and analyzed with respect to key numerical properties, including order and error constant, consistency, zero-stability, convergence, and region of absolute stability. Theoretical analysis confirmed that the new numerical scheme is of uniformly orders, consistent, zero-stable, and possesses an A-stability region. To validate its performance, the NNS is applied to both linear and nonlinear Volterra integro-differential equations of the second kind and compared with existing methods such as the Adams-Bashforth-Moulton method, trapezoidal-based approaches, and block methods. The numerical results reveal that the NNS consistently produced solutions close to the exact values, outperforming other established methods across various step sizes. These findings demonstrate that the NNS is not only mathematically reliable but also computationally efficient, making it a robust tool for solving a wide range of problems involving Volterra integro-differential equations of the second kind.
Research Article | Open Access | Download Full Text
Volume 3 | Issue 4 | Year 2025 | Article Id: MS-V3I4P102 DOI: https://doi.org/10.59232/MS-V3I4P102
New Numerical Algorithm for Integrations of Volterra Integral Equations of Second Kind
John Sabo, Skwame Yusuf, Donald, John Zirra
| Received | Revised | Accepted | Published |
|---|---|---|---|
| 23 Oct 2025 | 25 Nov 2025 | 10 Dec 2025 | 30 Dec 2025 |
Citation
John Sabo, Skwame Yusuf, Donald, John Zirra. “New Numerical Algorithm for Integrations of Volterra Integral Equations of Second Kind.” DS Journal of Modeling and Simulation, vol. 3, no. 4, pp. 18-33, 2025.
Abstract
Keywords
Volterra Integro-Differential Equations, New Numerical Scheme, Convergence, Numerical integration, Linear and nonlinear problems.
References
[1] Abdul-Majid Wazwaz, A First Course in Integral Equations, 2nd ed., World Scientific Connect, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[2] Mustapha Yahaya, and Sirajo Lawan Bichi, “An Approximate Solution to Volterra Integral Equation of Second Kind with Quadrature Rule,” Dutse Journal of Pure and Applied Sciences (DUJOPAS), vol. 9, no. 2b, pp. 349-353, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[3] Adnan A. Jalal, Nejmaddin A. Sleman, and Azad I. Amen, “Numerical Methods for Solving the System of Volterra-Fredholm Integro-Differential Equations,” ZANCO Journal of Pure and Applied Sciences, vol. 31, no. 2, pp. 25-30, 2019.
[Google Scholar]
[4] M. Matinfar, and A. Riahifar, “Analytical-Approximate Solution for Nonlinear Volterra Integro-Differential Equations,” Journal of Linear Topological Algebra, vol. 4, no. 3, pp. 217-228, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[5] Ojo Olamiposi Aduroja, Lydia ADIKU, and Olumuyiwa AGBOLADE, “Collocation Approximation Method for the Solution of Volterra Integro-Differential Equations,” FUOYE Journal of Management, Innovation and Entrepreneurship, vol. 2, no. 1, pp. 240-245, 2023.
[Google Scholar] [Publisher Link]
[6] A.O. Adesanya et al., “Numerical Solution of Linear Integral and Integro-Differential Equations using Boubakar Collocation Method,” International Journal of Mathematical Analysis and Optimization: Theory and Applications, vol. 5, no. 2, pp. 592-598, 2023.
[Google Scholar] [Publisher Link]
[7] Adhraa M. Muhammad, and A.M. Ayal, “Numerical Solution of Linear Volterra Integral Equations with Delay using Bernstein Polynomial,” International Electronic Journal of Mathematics Education, vol. 14, no. 3, pp. 735–740, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[8] H.O. Bakodah et al., “Laplace Discrete Adomian Decomposition for Solving Nonlinear Integro-Differential Equations,” Journal of Applied Mathematics and Physics, vol. 7, no. 6, pp. 1388-1407, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[9] Faranak Rabiei et al., “Numerical Solution of Volterra Integro-Differential Equations using General Linear Method,” Numerical Algebra, Control and Optimization, vol. 9, no. 4, pp. 433-444, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[10] Yavuz Ugurlu, Dogan Kaya, and Ibrahim E. Inan, “Comparison of Three Semi-Analytical Methods for Solving (1+1)-Dimensional Dispersive Long Wave Equations,” Computers and Mathematics with Applications, vol. 61, no. 5, pp. 1278-1290, 2011.
[CrossRef] [Google Scholar] [Publisher Link]
[11] D. Rani, and D. Mishra, “Solution of Volterra Integral and Integro-Differential Equations using Modified Laplace Adomian Decomposition Methods,” Journal of Applied Mathematics, Statistics and Informatics, vol. 15, no. 1, pp. 5-18, 2019.
[CrossRef] [Google Scholar] [Publisher Link]
[12] Nurul Atikah binti Mohamed, and Zanariah Abdul Majid, “One-Step Block Method for Solving Volterra Integro-Differential Equations,” AIP Conference Proceedings, vol. 1682, no. 1, 2015.
[CrossRef] [Google Scholar] [Publisher Link]
[13] E.S. Shoukralla, and B.M. Ahmed, “Numerical Solution of Volterra Integral Equation of the Second Kind using Lagrange Interpolation Via the Vandermonde Matrix,” Journal of Physics: Conference Series, vol. 1447, no. 1, pp. 1-8, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[14] Ganiyu Ajileye, and Sikiru A. Amoo, “Numerical Solution to Volterra Integro-Differential Equations using Collocation Approximation,” Mathematics and Computational Sciences, vol. 4, no. 1, pp. 1-8, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[15] Jaysmita Patra, “Some Problems on Variational Iteration Method,” M.Sc. Thesis, National Institute of Technology, Rourkela, pp. 1-35, 2015.
[Google Scholar] [Publisher Link]
[16] Muhammad Memon, Khuda Bux Amur, and Wajid A. Shaikh, “Combined Variational Iteration Method with Chebyshev Wavelet for the Solution of Convection-Diffusion-Reaction Problem,” Mehran University Research Journal of Engineering and Technology, vol. 42, no. 2, pp. 93-107, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[17] M.D. Aloko, “New Approximate Methods for Solving Nonlinear Volterra-Fredholm Integro-Differential Equations of the Second Kind,” Ph.D. Thesis, University of Lagos, Nigeria, 2018.
[18] Fuziyah Ishak, and Muhammad Nur Firdaus Selamat, “New Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations,” International Journal of Mathematics, Computational, Physical, Electrical and Computer Engineering, vol. 10, no. 11, 2016.
[Google Scholar]
[19] Oluwaseun Adeyeye, and Zurni Omar, “A New Algorithm for Developing Block Methods for Solving Fourth Order Ordinary Differential Equations,” Global Journal of Pure and Applied Mathematics, vol. 12, no. 2, pp. 1465-1471, 2016.
[Google Scholar] [Publisher Link]
[20] Adewale A. James, and Sabo John, “Simulating the Dynamics of Oscillating Differential Equations of Mass in Motion,” International Journal of Development Mathematics (IJDM), vol. 1, no. 1, pp. 54-69, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[21] Yusuf Skwame, Donald J. Zirra, and Sabo John, “The Numerical Application of Dynamic Problems Involving Mass in Motion Governed by Higher Order Oscillatory Differential Equations,” Physical Science International Journal, vol. 28, no. 5, pp. 8-31, 2024.
[CrossRef] [Google Scholar] [Publisher Link]
[22] Fuziyah Ishak, and Muhammad Nur Firdaus Selamat, “New Development of Extended Trapezoidal Method for Solving First Order Linear Volterra Integro-Differential Equations,” ASM Science Journal, vol. 13, pp. 1-10, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[23] Zanariah Abdul Majid, and Nurul Atikah Mohamed, “Fifth Order Multistep Block Method for Solving Volterra Integro-Differential Equations of Second Kind,” Sains Malaysiana, vol. 48, no. 3, pp. 677-684, 2019.
[CrossRef] [Google Scholar]
[24] R.A. Olowe et al., “Trigonometrically-Fitted Simpson’s Method for Solving Volterra Integro-Differential Equations,” International Journal of Mathematical Sciences and Optimization: Theory and Applications, vol. 8, no. 2, pp. 68-78, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[25] N.M. Kamoh, T. Aboiyar, and A.R. Kimbir, “Continuous Multistep Methods for Volterra Integro-Differential Equation of the Second-Order,” Science World Journal, vol. 12, no. 3, pp. 11-14, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[26] R.I. Abdulganiy, O.A. Akinfenwa, and S.A. Okunuga, “Construction of L Stable Second Derivative Trigonometrically Fitted Block Backward Differentiation Formula for the Solution of Oscillatory Initial Value Problems,” African Journal of Science, Technology, Innovation and Development, vol. 10, no. 4, pp. 411-419, 2018.
[CrossRef] [Google Scholar] [Publisher Link]
[27] N.M. Kamoh, T. Aboiyar, and E.S. Onah, “On Investigating some Quadrature Rules for the Solution of Second-Order Volterra Integro-Differential Equation,” IOSR Journal of Mathematics, vol. 13, no. 5, pp. 45-50, 2017.
[CrossRef] [Google Scholar]