# Application of Chebyshev polynomial basis function on the solution of Volterra integro-differential equations using Galerkin method

Document Type : Original Article

Authors

1 Osun State University, Osogbo, Nigeria.

2 University of Ilorin Nigeria

Abstract

In this paper, the Galerkin Method (GM) is employed for finding the solution of Volterra integro-differential equations with the aid of Chebyshev polynomial basis function. Some examples are presented to demonstrate the effectiveness of the method and its reliability. The approximate solutions are in agreement with the exact solutions and are compared favorably with an existing method available in the literature.

Keywords

#### References

[1] S. Abbasbany, Application of He’s Homotopy perturbation method for functional integral equations, Chaos Soliton and fractals, 31(5) 2017, 1245-1247.
[2] S. Abbasbany, Application of He’s Homotopy perturbation method for Laplace transform, Chaos Soliton and fractals, 30(5) 2006, 1206-1212.
[3] T.A. Abassy, Improved Adomian decomposition method, Computer and mathematics with Applications, 59(1) 2010, 42-54.
[4] A.F. Adebisi, O.A. Uwaheren, O.E. Abolarin, R.M. Tayo, J.A. Adedeji, O.J. Peter, Solution of Typhoid Fever Model by Adomian Decomposition Method, Journal of Mathematical and Computer Science, 11(2) 2021, 1242-1255.
[5] A.A. Ayoade, O.J. Peter, S. Amadiegwu, A.I. Abioye, A.A. Victor, A.F. Adebisi, Solution of Cholera Disease Model by Parameter Expansion Method, The Pacific Journal of Science and Technology, 19(2) 2018, 36-50.
[6] A. Boutiara, S. Etemad, A. Hussain, S.H. Rezapour, The generalized U–H and U–H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving φ-Caputo fractional operators, Advances in Difference Equations 2021(1) 2021, 1-21
[7] A. Cardone, D.D. Conte, R. Ambrusio, Collocation method for Volterra integral and integro-differential equations: A review, Axioms, 7(3) 2018, 45-61.
[8] F. Fakhar-Izadi, M. Dehghan, An efficient pesudo-spectral Legendre Galerkin method for solving a nonlinear partial integro-differential equation arising in population dynamics, Mathematical Method in the Applied Sciences, 36(12) 2012, 1485-1511.
[9] 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, Journal of science and engineering, 17(1) 2020, 41-51.
[10] T. Oyedepo, A.F. Adebisi, R.M. Tayo, J.A. Adedeji, M.A. Ayinde, O.J. Peter, Perturbed least squares technique for solving volterra fractional integro-differential equations based on constructed orthogonal polynomials, Journal of Mathematical and Computer Science, 11(1) 2021, 203-218.
[11] O.J. Peter, A. Yusuf, K. Oshinubi F.A, Oguntolu, A.A. Ibrahim, A.I. Abioye, T.A. Ayoola, Fractional Order of Pneumococcal Pneumonia Infection Model with Caputo Fabrizio Operator, Results in Physics, 29 2021, 104581.
[12] O.J. Peter, A.S. Shaikh, M.O. Ibrahim, K.S. Nisar, D. Baleanu, I. Khan, A.I. Abioye, Analysis and Dynamics of Fractional Order Mathematical Model of COVID-19 in Nigeria Using Atangana-Baleanu Operator. Computers, Materials and Continua, 66(2) 2021, 1823-1848.
[13] O.J. Peter, Transmission Dynamics of Fractional Order Brucellosis Model Using Caputo–Fabrizio Operator, International Journal of Differential Equations, 2020 2020, 1-11.
[14] O.J. Peter, A.F Adebisi, F.A. Oguntolu, S. Bitrus, C.E. Akpan, Multi-step Homotopy Analysis Method for Solving Malaria Model, Malaysian Journal of Applied Sciences, 3(2) 2018, 34-45.
[15] D. Rani, V. Mishra, solution of volterra integral and integro-differential equations using modified Laplace Adomian Decomposition Method, Journal of Applied Mathematics, Statistics and Informatics, 15(1) 2019, 5-18.
[16] SH. Rezapour, S.K. Ntouyas, M.Q. Iqbal, A. Hussain, S. Etemad, J. Tariboon. An Analytical Survey on the Solutions of the Generalized Double-Order –Integro-differentialEquation, Journal of Function Spaces, 2021.
[17] SH. Rezapour. Shahram, S.K. Ntouyas, A. Amara, S. Etemad, J. Tariboon, Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality, Mathematics, 9(11) 2021, 1165.
[18] O.A. Taiwo and A.F. Adebisi, Multiple perturbed collocation Tau method for special class of higher order linear fredholm and volterra integro- differential equations, Projournal of physical science Research, 2(3) 2014,3-22.
[19] A.M. Wazwaz, The combined Laplace transfom-Adomain decomposition method for handling nonlinear Volterra integro-differential equations, Applied Mathematics and Computation, 216(4) 2010, 1304-1309.
[20] A. Wazwaz. Two methods of solving integral equations, Appl.math. compu, 77 1996, 79-89.
[21] A. Wazwaz Linear and Nonlinear integral equation methods and Applications, Higher education press, 2011, springer.

### History

• Receive Date: 23 October 2021
• Revise Date: 12 November 2021
• Accept Date: 12 November 2021
• First Publish Date: 12 November 2021