# Numerical solution to Volterra integro-differential equations using collocation approximation

Document Type : Original Article

Authors

1 Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria

2 Department of Mathematics and Statistics, Federal University Wukari, Taraba State

Abstract

This paper considers the collocation method for the numerical solution of the Volterra integro- differential equation using polynomial basis functions. The modeled equation was converted into a linear algebraic system of equations and matrix inversion was employed to solve the algebraic equation. We substitute the result of algebraic into the approximate solution to obtain the numerical result. Some numerical problems are solved to show the method's efficiency and consistency.

Keywords

Main Subjects

#### References

[1] A.O. Adesanya, Y.A.Yahaya, B. Ahmed, R.O. Onsachi, Numerical Solution of Linear integral and Integro-Differential Equations Using Boubakar Collocation Method. International Journal of Mathe-matical Analysis and Optimization: Theory and Application, 2019(2) 2019, 592-598.
[2] A.O. Agbolade, T.A. Anake, Solution of  rst order volterra linear integro differential equations by collocation method. J. Appl. Math., Article ID 1510267, 2017.
[3] G. Ajileye, F.A. Aminu, Approximate Solution to First-Order Integro-differential Equations Using Polynomial Collocation Approach. J Appl Computat Math., 11 2022, 486.
[4] G. Ajileye, A.A. James, A.M. Ayinde, T. Oyedepo, Collocation Approach for the Computational Solution Of Fredholm-Volterra Fractional Order of Integro-Differential Equations. J. Nig. Soc. Phys. Sci. 4 2022, 834.
[5] A.H. Bhraway, E. Tohidi, F. Soleymani, A new Bernoulli matrix method for solving high order linear and nonlinear Fredholm integro-differential equations with piecewise interval. Appl. Math. Comput. 219(2) 2012, 482-497.
[6] C. Ercan, T. Kharerah, Solving a class of Volterra integral system by the differential transform method. Int. J. Nonlinear Sci. 16(1) 2013, 87-91.
[7] M. El-kady, M. Biomy, Efficient Legendre pseudospectral method for solving integral and integro differential equation. Commom Nonlinear Sci. Numer Simulat, 15(7) 2010, 1724-1739.
[8] S.E. Fadugba, Solution of Fractional Order Equations in the Domain of the Mellin Transform. Journal of the Nigerian Society of Physical Sciences, 2019, 138-142.
[9] D.A. Gegele, O.P. Evans, D. Akoh, Numerical solution of higher order linear Fredholm integro-differential equations. American Journal of Engineering Research, 3(1) 2014, 243-247.
[10] K. Issa, F. Saleh, Approximate solution of pertubed Volterra Fredholm integro differential equation by Chebyshev-Galerkin method. Journal of Mathematics, 2017.
[11] N. Irfan, S. Kumar, S. Kapoor, Bernstein Operational Matrix Approach for Integro-Differential Equation Arising in Control theory, Nonlinear Engineering, 3(2) 2014, 117-123.
[12] R.H. Khan,H.O. Bakodah, Adomian decomposition method and its modi cation for nonlinear Abel's integral equations. Computers and Mathematics with Applications, 7(45-48) 2013, 2349-2358.
[13] G. Mehdiyeva, V. Ibrahimov, M Imanova, On the Construction of the Multistep Methods to Solving the Initial-Value Problem for ODE and the Volterra Integro-Differential Equations, IAPE, Oxford, United Kingdom 2019.
[14] G. Mehdiyera, M. Imanova, V. Ibrahim, Solving Volterra integro differential equation by second derivative methods, 43rd Appl. Math. Inf. Sci., 9(5) 2015, 2521-2527.
[15] R.C. Mittal, R. Nigam, Solution of fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. Mech, 4(2) 2008, 87-94.
[16] Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations, Computers and Mathematics with Applications,61(8) 2011, 2330-2341.
[17] S. Nemati, P. Lima, Y. Ordokhani, Numerical method for the mixed Volterra-Fredholm integral equations using hybrid Legendre function, Conference Application of Mathematics, 2015, 184-192.
[18] M.O. Olayiwola, A.F. Adebisi, Y.S. Arowolo, Application of Legendre Polynomial Basis Function on the Solution of Volterra Integro-differential Equations Using Collocation Method, Cankaya University Journal of Science and Engineering, 17(1) 2020, 041-051.

### History

• Receive Date: 04 December 2022
• Revise Date: 22 January 2023
• Accept Date: 09 March 2023