Three-step Method of Volterra Integral Equation of the Second Kind


Published: 2023-07-14

Page: 264-273

D. Raymond

Department of Mathematics and Statistics, Federal University Wukari, Nigeria.

A. Adu *

Department of Mathematics and Statistics, Federal University Wukari, Nigeria.

P. O. Olanrewaju

Department of Mathematics and Statistics, Federal University Wukari, Nigeria.

R. Ajia

Department of Mathematics and Statistics, College of Agriculture, Science and Technology Jalingo, Nigeria.

*Author to whom correspondence should be addressed.


The interpolation and collocation method has been used in this study to build a class of three step implicit second order derivative hybrid block methods for the solutions of the second kind of Volterra integral problems. After the continuous block techniques were evaluated at each step point, the discrete block methods could be reconstructed. Each discrete scheme obtained from the simultaneous solution of the block using the block methods confirmed that it had the same level of accuracy as the main endless approach. The new family of k-step procedures provides a high order of accuracy with exceedingly low error constants and stable intervals of absolute stability. The characteristics of the approaches revealed that they were convergent, zero-stable, and consistent. Results from two Volterra integral equations of the second kind showed that the new methods in the study achieved better than those we compared with in the literature.

Keywords: Volterra integral, trigonometric, hybrid point, off-grid, power series

How to Cite

Raymond , D., Adu , A., Olanrewaju , P. O., & Ajia , R. (2023). Three-step Method of Volterra Integral Equation of the Second Kind. Asian Journal of Pure and Applied Mathematics, 5(1), 264–273. Retrieved from


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Baker CTH. A perspective on the numerical treatment of Volterra equations, Journal of Computer and Applied Mathematics. 2000;125(1):217–249.

Wazwaz AM. Linear and Nonlinear Integral Equations: Methods and Applications, Springer; 2011.

Ibrahimov P, Imanova M. Multistep methods of the hybrid type and their application to solve the second kind volterra integral equation. Symmetry. 2021;13:1087. Available:

Shihab SN, Abdulrahman AA, Ali MM. Collocation orthonormal Bernstein polynomials method for solving integral equation. Eng. and Tech. Journal. 2015;33(8):1493-1502

Nouri M, Maleknejad K. Numerical Solution of delay integral equations by using block pulse functions arises in biological sciences. International Journal of Mathematics Modelling and Computation. 2016;6(3):221-23.

Alturk A. Application of the Bernstein polynomial for solving Volterra Integral Equations with Convolution kernels. Filomant. 2016;30(4):1045-1052.

Hong Z, Fang X, Yan Z, Hao H. On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method. J. Appl. Math. Phys. 2016;4:1315–1320.

Mechee MS, Al-Ramahi AM, Kadum RM. Variation iteration method for solving a class of volterra integral equations. Journals of University of Babylon. 2016;24(9).

Mohammad A, Omar Z. Generalized Two-Hybrid One-Step Implicit Third derivative Block Method for the Direct Solution of Second Order Ordinary differential Equations. International Journal of Pure and Applied Mathematics. 2017;112(3):497-517.

Draidi W, Qatanani N. Numerical Schemes for solving Volterra integral equations with Carleman kernel. International Journal of Appl. Math. 2018;31:647–669.

Aggarwal S, Chauhan R, Sharma N. Application of Kamal transform for solving linear Volterra integral equations of first kind. International Journal of Research and Advance Technology. 2018;6(8):2081-2088.

Muhammad AM, Ayal AM. Numerical Solution of Linear Volterra Integral Equation with Delay using Bernstein Polynomial. IEJME. 2019;14:735–740.

Majid ZA, Mohamed NA. Fifth order multistep block method for solving volterra integro differential equation of second kind. Sains Malaysiana. 2019;48(3):677-684.

Assari P, Dehghan MA. Mesh less local Galerkin method for solving Volterra integral equations deduced from nonlinear fractional differential equations using the moving least squares technique. Applied Numerical Mathematics. 2019;143:276–299.

Areo EA, Omojola MO. A new one-twelfth step continuous Block method. International journal of pure and applied mathematics. 2015;114(2):165-178.

Shoukralla ES, Ahmed BM. Numerical solutions of volterra integral equations of the second using Lagrange interpolation via thee vandermonde matrix. In Journal of Physics: Conference Series (2020;1447(1):o12003). IOP publish.

Rouibah K, Bellour A, lima P, Rawashden E. Iterative continuous collocation for solving nonlinear volterra integral equations. Kragujevac Journal of Mathematics. 2022;46(4):635-645.