SOLVING OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS OFTHE SECOND KIND USING BLOCK HYBRID METHOD

Authors

  • Ezekiel Olaoluwa OMOLE
    Federal University of Technology and Environmental Sciences, Iyin-Ekiti 362005, Ekiti State, Nigeria
  • Taiwo Stephen Fayose
    Federal Polytechnic Ado-Ekiti, Ekiti State, Nigeria
    Federal University of Technology and Environmental Sciences, Iyin-Ekiti 362005, Ekiti State, Nigeria
  • Folasade Helen Odeniyan-Fakuade
    Osun State University, Osogbo, Osun State, Nigeria
  • Bamikole Gbenga Ogunware
    Adekunle Ajasin University, Akungba-Akoko, Ondo State, Nigeria
  • Olanrewaju Thomas Olotu
    University of Ilorin, Kwara State, Nigeria
  • Taiwo Aanu Ogunlusi
    Federal University Oye-Ekiti, Ekiti State, Nigeria
  • Taiwo Eniola Fayode
    Bamidele Olumilua University of Education, Science and Technology, Ikere-Ekiti, Ekiti State, Nigeria

Keywords:

BHM , VIDEs , Second kind , LBA , Consistency , Numerical simulation , Power series polynomial , Convergence analysis

Abstract

This study presented a Block Hybrid Method (BHM) for numerical integration of linear and nonlinear Volterra Integro-Differential Equations (VIDEs) of the second kind. The BHM was derived using the linear multistep method with the Linear Block Algorithm (LBA) to obtain a new block algorithm and its higher derivatives. Numerically, the analysis of the basic properties of the BHM was studied and found to be of uniform order seven, consistency, zerostability, and convergence, with an A-stable region of absolute stability. The numerical simulation of BHM was carried out by integrating the BHM on linear and nonlinear Volterra integro-differential equations of second kind. The results obtained were compared with existing methods and presented in tables and graphically shown. The numerical comparisons show that the BHM produces highly accurate approximations and minimize the truncation error over the existing methods. The study shows that the BHM is efficient, stable and suitable for solving Volterra integro-differential equations of the second kind.

Author Biographies

Ezekiel Olaoluwa OMOLE

Department of Mathematics

Taiwo Stephen Fayose

Department of Statistics
Department of Mathematical Sciences

Folasade Helen Odeniyan-Fakuade

Department of Mathematics

Bamikole Gbenga Ogunware

Department of Mathematical Sciences

Olanrewaju Thomas Olotu

Department of Mathematics

Taiwo Aanu Ogunlusi

Department of Mathematics

Taiwo Eniola Fayode

Department of Mathematical Sciences

Dimensions

Abdulganiy, R. I., Akinfenwa, O. A. and Okunuga, S. A. (2018). 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, 10(4), 411–419. https://doi.org/10.1080/20421338.2018.1467859

Adeyeye, O. and Omar, Z. (2016). A new algorithm for developing block methods for solving fourth order ordinary differential equations. Global J. of Pure and Applied Mathematics, 12(2), 1465–1471. http://www.ripublication.com/gjpam.htm

Adeyeye, O. and Omar, Z. (2019). Direct solution of initial and boundary value problems of third order ODEs using maximal-order fourth-derivative block method. AIP Conference Proceedings, 2164, 020014. https://doi.org/10.1063/1.5121039

Aduroja, O. O., Ajileye, A. M., Ajileye, G. and Oyedepo, T. (2025). Numerical solution of Volterra integro-differential equations of fractional order with initial conditions using collation approach. J. of Fractional Calculus and Application, 16(1), 1-10.

Agarwala, G., Mohana, M., Menon, A. M., Sharma, A., Dakal, T. C. and Purohit, S. D. (2022). Analysis of the Adomian decomposition method to estimate the COVID-19 pandemic. In: Methods of Mathematical Modeling (pp. 173-187). Elsevier. https://doi.org/10.1016/B978-0-323-99888-8.00011-5

Agbata, B. C., Ani, B. N., Shior, M. M., Ezugorie, I. G., Paul, R. V. and Meseda, P. K. (2022). Analysis of Adomian decomposition method and its application for solving linear and nonlinear differential equations. Asian Research Journal of Mathematics, 18(2), 56-70. https://doi.org/10.9734/ARJOM/2022/v18i230368

Aggarwal, S., Kumar, R. and Chandel, J. (2023). Solution of linear Volterra integral equation of second kind via Rishi transform. Journal of Scientific Research, 15(1), 111-119. https://doi.org/10.3329/jsr.v15i1.60337

Faires, D. and Burden, R. L. (2015). Numerical Analysis. Belmont, CA: Thomson Brooks/Cole.

Fuziyah, I. and Muhammad, N. F. S. (2020). New development of extended trapezoidal method for solving first order linear Volterra integro-differential equations. ASM Science Journal, 13, 1-6. https://doi.org/10.9734/psij/2024/v28i5845

Ishak, F. and Ahmad, S. N. (2016). Development of extended trapezoidal method for numerical solution of Volterra integro-differential equations. Int. J. of Mathem., Computational, Physical, Electrical and Computer Engineering, 10(11), 524–527.

Ishak, F. and Selamat, M. N. F. (2020). New development of extended trapezoidal method for solving first order linear Volterra integro-differential equations. ASM Science Journal, 13, https://doi.org/10.32802/asmscj.2020.sm26(5.24)

Kamoh, N. M., Aboiyar, T. and Kimbir, A. R. (2017). Continuous multistep methods for Volterra integro-differential equation of the second-order. Science World Journal, 12(3), 11–14.

Liu, C. S., Li, B. and Kuo, C. L. (2025). Variational iteration and linearized Liapunov methods for seeking the analytic solutions of nonlinear boundary value problems. Mathematics, 13(354), 1-38. https://doi.org/10.3390/math13030354

Majid, Z. A. and Mohamed, N. A. (2019). Fifth order multistep block method for solving Volterra integro-differential equations of second kind. Sains Malaysiana, 48(3), 677–684. https://doi.org/10.17576/jsm-2019-4803-22.

Mak, M. K., Leung, C. S. and Harko, T. (2021). A brief introduction to the Adomian decomposition method, with applications in astronomy and astrophysics. Romanian Astronomical Journal, 1(1), 1-41. http://arxiv.org/abs/2102.10511v1

Maturi, D. A. and Malaikah, H. M. (2021). The Adomian decomposition method for solving nonlinear partial differential equation using maple. Advances in Pure Mathematics, 11, 595-603. https://doi.org/10.4236/apm.2021.116038

Memon, M., Amur, K. B. and Shaikh, W. A. (2023). Combined variational iteration method with Chebyshev wavelet for the solution of convection-diffusion-reaction problem. Mehran University Res. J. of Engr. and Techn., 42(2), 93-107. https://doi.org/10.22581/muet1982.2302.11

Olowe, R. A. Akinfenwa, O. A., Abdulganiy, R. I. and Okunuga, A. (2024). A trigonometrically fitted method for solving volterra integro-differential equations. Journal of Nigerian Mathematical Society, 43(2), 203-220.

Olowe, R. A., Akinfenwa, O. A., Abdulganiy, R. I. and Okunuga, S. A. (2023). Trigonometrically-fitted simpson’s method for solving volterra integro-differential equations. Int. J. of Mathem. Sci. and Optimization: Theory and Applic., 8(2), 68-78. https://doi.org/10.6084/m9.figshare.22015772

Omole, E. O., Adeyefa, E. O., Apanpa, K. I., Ayodele, V. I., Amoyedo, F. E. and Emadifar, H. (2024). Unveiling the power of an implicit six-point block scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems. PLoS ONE, 19, e0301505. https://doi.org/10.1371/journal.pone.0301505

Otaide, I. J. and Oluwayemi, M. O. (2024). Numerical treatment of linear Volterra integro differential equations using variational iteration algorithm with collocation. Partial Differential Equations in Applied Mathematics, 10, 1-7. https://doi.org/10.1016/j.padiff.2024.100693

Oyedepo, T., Ajileye, G. and Ayoade, A. A. (2024a). Bernstein computational algorithm forbintegro-differential equations. Partial Differential Equations in Applied Mathematics, 11, 1-7. https://doi.org/10.1016/j.padiff.2024.100896

Oyedepo, T., Ayinde, A. M. and Didigwu, E. N. (2024b). Vieta-lucas polynomial computational technique for volterra integro-differential equations. Ellectronic Journal of Mathematical Analysis and Applications, 12(1), 1-8.

Patra, J. (2015). Some problems on variational iteration method. Master’s thesis, National Institute of Technology, Rourkela, 1-35p.

Rabiel, F., Hamid, F. A., Majid, Z. A. and Ismail, F. (2019). Numerical solutions of Volterra integro-differential equations using general linear method. Numerical Algebra, Control and Optimization, 9(4), 433-444. https://doi.org/10.3934/naco.2019042

Rama, E., Somaiah, K. and Sambaiah, K. (2021). A study of variational iteration method for solving various types of problems. Malaya J. of Matematik, 9, 701-708. https://doi.org/10.26637/MJM0901/0123

Raymond, D., Kyagya, T. Y., Sabo, J. and Lydia, A. (2023). Numerical application of higher order linear block method for solving some fourth order initial value problems. African Scientific Reports, 2(1), 67. https://doi.org/10.46481/asr.2022.2.1.67

Rufai, M. A., Shokri, A. and Omole, E. O. (2023). A one-point third-derivative hybrid multistep technique for solving second-order oscillatory and periodic problems. Journal of Mathematics, 2343215. https://doi.org/10.1155/2023/2343215

Sabo, J. and Adeyeye, O. (2025). Analysis of a block method developed with six generalised grid points for solving fourth order initial value problems. J. of Mathematical Anal. & Modeling, 6(1), 155-173. DOI: 10.48185/jmam.v6i1.1361

Singh, I. and Kumar, S. (2016). Haar wavelet method for some Volterra integral equations of first kind. Journal of Computational and Applied Mathematics, 292, 541–552. https://doi.org/10.1016/j.cam.2015.07.022

Skwame, Y., Zirra, D. J. and John, S. (2024). The numerical application of dynamic problems involving mass in motion governed by higher order oscillatory differential equations. Physical Science International Journal, 28(5), 8–31.

Sunday, J., Mahmud, M. S., Udo, M. O., Amoyedo, F. E., Ajewole, K. P., Olanrewaju, A. F., Omole, E. O. and Famutimi, O. F. (2025). An efficient intervalwise partitioning technique for the solution of ordinary differential systems. Nigerian Journal of Science and Technology Research (NIPES), 7(2), 2869–2878. https://doi.org/10.37933/nipes/7.4.2025.SI342

Sunday, J., Omole, E. O., Ogunrinde, R. B., Kumleng, G. M., Bamisile, O. O. and Kayode, O. O. (2026). Mathematical formulation of a symmetry-compact three-step algorithm for computing the spatio-temporal generalized FitzHugh–Nagumo equations. Symmetry, 18(2), 324. https://doi.org/10.3390/sym18020324

Tumba, P., Skwame, Y. and Raymond, D. (2021). Half-step implicit linear hybrid block approach of order four for solving third order ordinary differential equations. Dutse Journal of Pure and Applied Sciences (DUJOPAS), 7(2b), 124–133.

Wadi, A. M. and Sulaiman, N. A. (2025). Using W Transform for solving Volterra integro-differential equations. Communications on Applied Nonlinear Analysis, 32(7s), 675–683. https://internationalpubls.com

Published

30-06-2026

How to Cite

SOLVING OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS OFTHE SECOND KIND USING BLOCK HYBRID METHOD. (2026). FULafia Journal of Science and Technology , 10(2), 154-164. https://doi.org/10.62050/fjst2026.v10n2.796

Issue

Vol. 10 No. 2 (2026): Fulafia Journal of Science and Technology (FJST)

Section

Physical Sciences
Copyright & Licensing

How to Cite

SOLVING OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS OFTHE SECOND KIND USING BLOCK HYBRID METHOD. (2026). FULafia Journal of Science and Technology , 10(2), 154-164. https://doi.org/10.62050/fjst2026.v10n2.796

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