Keywords:-
Keywords:
Collocation method; Volterra; Integro-differential equations; Approximate solution
Article Content:-
Abstract
In this study, we develop and implement a numerical approach for solving first-order Volterra integro- differential equations. We derive the integral form of the problem, which is then transformed into an algebraic equation system using standard collocation points. We established the approach's uniqueness as well as its convergence and numerical examples were used to test the method's efficiency which shows that the method competes favourably with existing methods.
References:-
References
1. V. Volterra, "Theory of Functional and of Integral and Integro−differential Equations," Dover Publications. (2005).
2. R. H. Khan and H. O. Bakodah, "Adomian decomposition method and its modification for nonlinear Abel’s integral equations," Computers and Mathematics with Applications, vol.7, pp. 2349-2358, 2013.
3. R. C. Mittal and R. Nigam, "Solution of fractional integro−differential equations by Adomian decomposition method," The International Journal of Applied Mathematics and Mechanics, vol.2, pp. 87-94, 2008.
4. G. Ajileye and F. A. Aminu, "A Numerical Method using collocation approach for the solution of Volterra−Fredholm Integro−differential Equations," African Scientific Reports 1, pp. 205–211, 2022.
5. A. O. Agbolade and T. A. Anake, "Solution of first order Volterra linear integro differential equations by collocation method," J. Appl. Math., 2017.
6. S. Nemati, P. Lima and Y. Ordokhani, "Numerical method for the mixed Volterra−Fredholm integral equations using hybrid Legendre function," Conference Application of Mathematics,, vol. 4, pp.184-192, 2015.
7. G. Ajileye and F. A. Aminu, "Approximate Solution to First−Order Integro−differential Equations Using Polynomial Collocation Approach," J Appl Computat Math., vol.1, pp. 486, 2022.
8. G, Mehdiyera, M. Imanova and V. Ibrahim, "Solving Volterra integro differential equation by second derivative methods," 43rd Appl. Math. Inf. Sci. vol. 9, pp. 2521-2527, 2015.
9. G. Mehdiyeva, V. Ibrahimov and 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. ISBN: 978-1-912532-05-6.
10. K.Issa and F. Saleh, "Approximate solution of perturbed Volterra Fredholm integro differential equation by Ghebyshev−Galerkin method," Journal of Mathematics, 2017. doi:10,1155/2017/8213932.
11. A. H. Bhraway, E. Tohidi and F. Soleymani, "A new Bernoulli matrix method for solving high order linear and nonlinear Fredholm integro−differential equations with piecewise interval," Appl. Math. Comput, vol. 219, pp.482-497, 2012.
12. C. Ercan and T. Kharerah, "Solving a class of Volterra integral system by the differential transform method," Int. J. Nonlinear Sci., vol.16, pp.87-91, 2013.
13. A. Shahsavaran and A. Shahsavaran, "Application of Lagrange Interpolation for Nonlinear Integro Differential Equations," Applied Mathematical Sciences, vol.6 no.18, pp.887 - 892, 2012.
14. N. Irfan, S. Kumar and S. Kapoor, "Bernstein Operational Matrix Approach for Integro−Differential Equation Arising in Control theory," Nonlinear Engineering vol.3, pp.117-123, 2014.
15. P. Darania and A. Ebadian, "A method for the numerical solution of the integro−differential equations," Applied Mathematics and Computation, vol.188, pp.657–668, 2007.
16. A. Maadadi and A. Rahmoune, "Numerical solution of nonlinear Fredholm integro-differential equations using Chebyshev polynomials," International Journal of Advanced Scientific and Technical Research , vol.8, no.4, 2018. https://dx.doi.org/10.26808/rs.st.i8v4.09
17. B. H. Garba and S. L. Bichi, "A hybrid method for solution of linear Volterra integro−differential equations (GVIDES) via finite difference and Simpsons' numerical methods (FDSM)," Open J. Math Anal. vol.5, no.1, 2021.
18. L. Zada, M. Al-Hamami, R. Nawaz, S. Jehanzeb, A. Morsy, A. Abdel-Aty and K.S. Nisar, "A New Approach for Solving Fredholm Integro−Differential Equations," Information Sciences Letters. vol.10, no .2 , 2021.
19. L. Rahmani, B. Rahimi and M. Mordad, "Numerical Solution of Volterra−Fredholm Integro−Differential Equation by Block Pulse Functions and Operational Matrices," Gen. Math. Notes, vol.4, no.2, pp. 37-48, 2011.
20. Z. P. Atabakan, A. K. Nasab, A. Kiliçman and Z. K. Eshkuvatov, "Numerical Solution of Nonlinear Fredholm Integro−Differential Equations Using Spectral Homotopy Analysis Method," Mathematical Problems in Engineering, vol. 9 no. 7, 2013. http://dx.doi.org/10.1155/2013/674364
21. G. Ajileye and S. A Amoo, "Numerical solution to Volterra integro−differential equations using collocation approximation. Mathematics and Computational Sciences. 4(1) 2023. 1-8
22. 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., vol.4, pp.834, 2022.
2. R. H. Khan and H. O. Bakodah, "Adomian decomposition method and its modification for nonlinear Abel’s integral equations," Computers and Mathematics with Applications, vol.7, pp. 2349-2358, 2013.
3. R. C. Mittal and R. Nigam, "Solution of fractional integro−differential equations by Adomian decomposition method," The International Journal of Applied Mathematics and Mechanics, vol.2, pp. 87-94, 2008.
4. G. Ajileye and F. A. Aminu, "A Numerical Method using collocation approach for the solution of Volterra−Fredholm Integro−differential Equations," African Scientific Reports 1, pp. 205–211, 2022.
5. A. O. Agbolade and T. A. Anake, "Solution of first order Volterra linear integro differential equations by collocation method," J. Appl. Math., 2017.
6. S. Nemati, P. Lima and Y. Ordokhani, "Numerical method for the mixed Volterra−Fredholm integral equations using hybrid Legendre function," Conference Application of Mathematics,, vol. 4, pp.184-192, 2015.
7. G. Ajileye and F. A. Aminu, "Approximate Solution to First−Order Integro−differential Equations Using Polynomial Collocation Approach," J Appl Computat Math., vol.1, pp. 486, 2022.
8. G, Mehdiyera, M. Imanova and V. Ibrahim, "Solving Volterra integro differential equation by second derivative methods," 43rd Appl. Math. Inf. Sci. vol. 9, pp. 2521-2527, 2015.
9. G. Mehdiyeva, V. Ibrahimov and 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. ISBN: 978-1-912532-05-6.
10. K.Issa and F. Saleh, "Approximate solution of perturbed Volterra Fredholm integro differential equation by Ghebyshev−Galerkin method," Journal of Mathematics, 2017. doi:10,1155/2017/8213932.
11. A. H. Bhraway, E. Tohidi and F. Soleymani, "A new Bernoulli matrix method for solving high order linear and nonlinear Fredholm integro−differential equations with piecewise interval," Appl. Math. Comput, vol. 219, pp.482-497, 2012.
12. C. Ercan and T. Kharerah, "Solving a class of Volterra integral system by the differential transform method," Int. J. Nonlinear Sci., vol.16, pp.87-91, 2013.
13. A. Shahsavaran and A. Shahsavaran, "Application of Lagrange Interpolation for Nonlinear Integro Differential Equations," Applied Mathematical Sciences, vol.6 no.18, pp.887 - 892, 2012.
14. N. Irfan, S. Kumar and S. Kapoor, "Bernstein Operational Matrix Approach for Integro−Differential Equation Arising in Control theory," Nonlinear Engineering vol.3, pp.117-123, 2014.
15. P. Darania and A. Ebadian, "A method for the numerical solution of the integro−differential equations," Applied Mathematics and Computation, vol.188, pp.657–668, 2007.
16. A. Maadadi and A. Rahmoune, "Numerical solution of nonlinear Fredholm integro-differential equations using Chebyshev polynomials," International Journal of Advanced Scientific and Technical Research , vol.8, no.4, 2018. https://dx.doi.org/10.26808/rs.st.i8v4.09
17. B. H. Garba and S. L. Bichi, "A hybrid method for solution of linear Volterra integro−differential equations (GVIDES) via finite difference and Simpsons' numerical methods (FDSM)," Open J. Math Anal. vol.5, no.1, 2021.
18. L. Zada, M. Al-Hamami, R. Nawaz, S. Jehanzeb, A. Morsy, A. Abdel-Aty and K.S. Nisar, "A New Approach for Solving Fredholm Integro−Differential Equations," Information Sciences Letters. vol.10, no .2 , 2021.
19. L. Rahmani, B. Rahimi and M. Mordad, "Numerical Solution of Volterra−Fredholm Integro−Differential Equation by Block Pulse Functions and Operational Matrices," Gen. Math. Notes, vol.4, no.2, pp. 37-48, 2011.
20. Z. P. Atabakan, A. K. Nasab, A. Kiliçman and Z. K. Eshkuvatov, "Numerical Solution of Nonlinear Fredholm Integro−Differential Equations Using Spectral Homotopy Analysis Method," Mathematical Problems in Engineering, vol. 9 no. 7, 2013. http://dx.doi.org/10.1155/2013/674364
21. G. Ajileye and S. A Amoo, "Numerical solution to Volterra integro−differential equations using collocation approximation. Mathematics and Computational Sciences. 4(1) 2023. 1-8
22. 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., vol.4, pp.834, 2022.
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Ajileye, G., Aduroja, O., Adiku, L., & Salihu, I. (2023). A Polynomial Approximation for the Numerical Solution of First Order Volterra Integro-Differential Equations. International Journal Of Mathematics And Computer Research, 11(9), 3716-3720. https://doi.org/10.47191/ijmcr/v11i9.03