# NUMERICAL SOLUTION OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY AKBARI-GANJI’S METHOD

• O. A. Uwaheren Department of Mathematics, University of Ilorin, Kwara State, Nigeria
• A. F. Adebisi Department of Mathematical Sciences, Osun State University, Osogbo, Nigeria
• C. Y. Ishola Department of Mathematics, National Open University of Nigeria, Jabi Abuja, Nigeria
• M. T. Raji Department of Mathematics, Federal University of Agriculture, Abeokuta, Ogun State, Nigeria
• A. O. Yekeen Department of Mathematics, University of Ibadan, Oyo State, Nigeria
• O. J. Peter Department of Epidemiology and Biostatistics, School of Public Health, University of Medical Sciences, Ondo City, Ondo State, Nigeria
Keywords: Akbari-Ganji Method, Volterra Integro-differential and Volterra Inttegro-differential Difference equations

### Abstract

In this study, Akbari-Ganji’s Method (AGM) was applied to solve Volterra Integro-Differential Difference Equations (VIDDE) using Legendre polynomials as basis functions. Here, a trial solution function of unknown constants that conform with the differential equations together with the initial conditions were assumed and substituted into the equations under consideration. The unknown coefficients are solved for using the new proposed approach, AGM which principally involves the application of the boundary conditions on successive derivatives and integrals of the problem to obtain a system of equations. The system of equation is solved using any appropriate computer software, Maple 18. Some examples were solved and the results compared to the exact solutions.

### References

Yin Yang: Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method. Applied Mathematics, (2013), 4, 113-118.

Md. Mamun-Ur-Rashid Khan and Goutam Saha: Analytical Solution of LiÃ©nard differential equation using Homotopy Perturbation Method. GANIT Journal Bangladesh Mathematical Society 39 (2019) 87-100

F. A. Hendi 1 and M. M. Al-Qarni: An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two-Dimensional Volterra-Fredholm Integro-differential Equations. Advances in Mathematical Physics. (2017), Article ID 9385040, 1-8

Behrouz Raftari: Numerical Solutions of the Linear Volterra Integro-differential Equations: Homotopy Perturbation Method and Finite Difference Method. World Applied Sciences Journal 9 (Special Issue of Applied Math): (2010) 07-12 ISSN 1818-4952

Abdulla-Al-Mamun, Weidong Tao and Md. Asaduzzaman (2019) Solution of Volterra Integro-Differential Equations by Using Variational Iteration Method doi:10.20944/preprints201905.0164.v1

A. K. Hussain, F. S. Fadhel, Z. R. Yahya and N. Rusli : Iterative Method (VIM) for solving partial integro-differential equations. Journal of Theoretical and Applied Information Technology. (2016) 88 (2) 366-374

www.jatit.org

J. Biazar,M. G. Porshokouhi and B. Ghanbari, Numerical solution of functional integral equations by the Variational iteration method, Journal of Computational and Applied Mathematics, (2011) 235, 2581-2585,

E. U. Agom1, F. O. Ogunfiditimi and A. Tahir: Numerical Solution of Fourth Order Linear Differential Equations by Adomian Decomposition Method. British Journal of Mathematics and Computer Science (2016) 17(4): 1-8.

DOI: 10.9734/BJMCS/2016/24714

W. Li and Y. Pang: Application of Adomian Decomposition Method to nonlinear systems. Advances in Difference Equations. (2020) 2020, 67

A. Arikoglu and I. Ozkol: Solutions of integral and integro-differential equation systems by using differential transform method. Computers and Mathematics with Applications 56 (2008) 2411-2417

A. Verma and M. Kumar. Numerical Solution of Lane-Emden type equation using Multilayer Perceptron Neural Network Method, International Journal of Applied and Computational Mathematics. 5 (2019) 141. link.springer.com

O.A. Uwaheren , A.F. Adebisi and O.A. Taiwo: Perturbed Collocation Method For Solving Singular Multi-order Fractional Differential Equations of Lane-Emden Type. Journal of Nigerian Society of Physical Sciences. 2(2020), 141-148.

Akbari M.R, Ganji D.D, Majidian A, Ahmadi A.R: Dynamic Vibration Analysis for Non-linear Partial Differential Equation of the Beam; Columns with Shear Deformation and Rotary Inertia by AGM. Development and Applications of Oceanic Engineering (2014) 3:22-31.

Akbari, M.R., Ganji, D.D., Nimafar, M., and Ahmadi, A.R: Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach. Frontiers of Mechanical Engineering (2014) 9 (4): 390-401.

M. Nimafar, M.R. Akbari, D.D. Ganji, R. Azadi: Akbari-Ganji method for vibration under external harmonic loads, Nonlinear Sci. Lett. A, (2017) 8(4),416-437.

H. Mirgolbabaee, S.T. Ledari, D. D. Ganji and E. K. Valujai Analyzing the nonlinear heat transfer equation by AGM. New Trends in Mathematical Sciences. (2017) 5(1) 51-58

A. F. Adebisi, T. A. Ojurongbe, K. A. Okunlola, O. J. Peter. Application of Chebyshev polynomial basis function on the solution of volterra integro-differential equations using Galerkin method Mathematics and Computational Sciences, 2(1), 2021: 41-51

O. A. Uwaheren, A. F. Adebisi, O. T. Olotu, M. O. Etuk, O. J. Peter. Legendre Galerkin Method for Solving Fractional Integro-Differential Equations of Fredholm Type. The Aligarh Bulletin of Mathematics. 40(1), (2021), 1-13.

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(2021), 2, 1242-1255

A. F. Adebisi, K.A. Okunola, M. T. Raji, J. A. Adedeji1 & O. J. Peter. Galerkin and perturbed collocation methods for solving a class of linear fractional integro-differential equations. The Aligarh Bulletin of Mathematics. 40, (2) (2021), 45-57

Oyedepo, T., Uwaheren, O. A., Okperhie E. P. and O. J. Peter. Solution of Fractional Integro-Differential Equation Using Modified Homotopy Perturbation Technique and Constructed Orthogonal Polynomials as Basis Functions. Journal of Science Technology and Education 7(3), 157-164 (2019).

C. Y. Ishola, O. A. Taiwo. A.F, Adebisi, O. J. Peter. Numerical solution of two- dimensional Fredholm integro-differential equations by Chebyshev integral operational matrix method. Journal of Applied Mathematics and Computational Mechanics. 21(1), 29-40, (2022).

Published
2022-09-01
How to Cite
[1]
O. Uwaheren, A. Adebisi, C. Ishola, M. Raji, A. Yekeen, and O. Peter, “NUMERICAL SOLUTION OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY AKBARI-GANJI’S METHOD”, BAREKENG: J. Math. & App., vol. 16, no. 3, pp. 1123-1130, Sep. 2022.
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Articles