Solving FIDEs by Using Semi-Analytical Techniques

Year 2019, Volume: 2 Issue: 3, 192 - 198, 30.09.2019

Ahmed A. Hamoud , Kirtiwant P. Ghadle Nedal M. Mohammed

https://doi.org/10.33434/cams.559717

Abstract

This paper mainly focuses on the recent advances in the semi-analytical approximated methods for solving Fredholm Integro-Differential Equations  (FIDEs) of the second kind by using  Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM) and Direct Homotopy Analysis Method (DHAM). Convergence analysis of the exact solution of the proposed methods is established. Moreover, we proved the uniqueness of the solution.   To illustrate the methods, an example is presented.

Keywords

Fredholm integro-differential equation, variational iteration method, homotopy perturbation method, direct homotopy analysis method

References

• [1] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech., 35(1) (2000), 37-43.
• [2] J.H. He, The homotopy perturbation method for non-linear oscillators with discontinuities, Appl. Math. Comput., 151(1) (2004), 287-292.
• [3] S. Abbasbandy, Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Appl. Math. Comput., 175(1) (2006), 581-589.
• [4] P. Ariel, T. Hayat, S. Asghar, Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int. J. Nonlinear Sci. Numer. Simul., 7(4) (2006), 399-406.
• [5] M. Rafei, D.D. Ganji, Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul., 7(3) (2006), 321-328.
• [6] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A, 352(4-5) (2006), 404-410.
• [7] A. Hamoud, K. Ghadle, A study of some reliable methods for solving fuzzy Volterra-Fredholm integral equations, Acta Univ. Apul., 53 (2018), 65-92.
• [8] A. Hamoud, K. Ghadle, S. Atshan The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method, Khayyam J. Math., 5(1) (2019), 21-39.
• [9] M. Javidi, A Golbabai, A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method, Appl. Math. Comput., 189(2) (2007), 1921-1928.
• [10] S. Alao, F. Akinboro, F. Akinpelu, R. Oderinu, Numerical solution of integro-differential equation using Adomian decomposition and variational iteration methods, IOSR Journal of Mathematics, 10(4) (2014), 18-22.
• [11] A. Hamoud, K. Ghadle, The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations, Korean J. Math. 25(3) (2017), 323-334.
• [12] A. Hamoud, K. Ghadle, On the numerical solution of nonlinear Volterra-Fredholm integral equations by variational iteration method, Int. J. Adv. Sci. Tech. Research, 3 (2016), 45-51.
• [13] A. Hamoud, K. Ghadle, The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integro-differential equations, J. Korean Soc. Ind. Appl. Math. 21 (2017), 17-28.
• [14] A. Hamoud, K. Ghadle, Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations, J. Indian Math. Soc. 85(1-2) (2018), 52-69.
• [15] A. Hamoud, K. Ghadle, Recent advances on reliable methods for solving Volterra-Fredholm integral and integro-differential equations, Asian Journal of Mathematics and Computer Research, 24(3) (2018), 128-157.
• [16] A. Hamoud, K. Ghadle, M. Bani Issa, Giniswamy, Existence and uniqueness theorems for fractional Volterra-Fredholm integro-differential equations, Int. J. Appl. Math., 31(3) (2018), 333-348.
• [17] A. Hamoud, K. Ghadle, Existence and uniqueness of the solution for Volterra-Fredholm integro-differential equations, Journal of Siberian Federal University. Mathematics & Physics, 11(6) (2018), 692-701.
• [18] A. Hamoud, A. Azeez, K. Ghadle, A study of some iterative methods for solving fuzzy Volterra-Fredholm integral equations, Indonesian J. Elec. Eng. & Comp. Sci., 11(3) (2018), 1228-1235.
• [19] A. Hamoud, K. Ghadle, Homotopy analysis method for the first order fuzzy Volterra-Fredholm integro-differential equations, Indonesian J. Elec. Eng. & Comp. Sci., 11(3) (2018), 857-867.
• [20] A. Hamoud, K. Ghadle, Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integrodifferential equation of the second kind, Tamkang Journal of Mathematics, 49(4) (2018), 301-315.
• [21] A. Hamoud, M. Bani Issa, K. Ghadle, M. Abdulghani, Existence and convergence results for Caputo fractional Volterra integro-differential equations, Journal of Mathematics and Applications, 41 (2018), 109-122.
• [22] A. Hamoud, K. Ghadle, Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations, Indian J. Math., 60(3) (2018), 375-395.
• [23] A. Hamoud, K. Ghadle, Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations, Journal of Mathematical Modeling, 6(1) (2018), 91-104.