Usage of the Variational Iteration Technique for Solving Fredholm Integro-Differential Equations

Document Type: Research Paper


1 Department of Mathematics, Faculty of Education and Science, Taiz University, Taiz, Yemen

2 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004 India


Integral and integro-differential equations are one of the most useful mathematical tools in both pure and applied mathematics.
In this article, we present a variational iteration method for solving Fredholm integro-differential equations. This study provides an analytical approximation to determine the behavior of the solution. To show the efficiency of the present method for our problems in comparison with the exact solution we report the absolute error. From the computational viewpoint, the variational iteration method is more efficient, convenient and easy to use. The method is very powerful and efficient in nding analytical as well as numerical solutions for wide classes of linear and nonlinear
Fredholm integro-differential equations. Moreover, It proves the existence and uniqueness results and convergence of the solution of Fredholm integro-differential equations. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique. The convergence theorem and the numerical results establish the precision and efficiency of the proposed technique.


Main Subjects

[1]     Abbaoui, K. and Cherruault, Y. Convergence of Adomian’s method applied to nonlinear equations, Mtath. Comput. Modelling, 20(9) (1994), 69–73.

[2]     Abbasbandy, S. and Elyas, S. Application of variational iteration method for system of nonlinear Volterra integro-differential equations, Mathematics and Computational Applications, 2(14) (2009), 147–158.

[3]     Adomian, G. A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135(2) (1988), 501–544.

[4]     Alao, S. Akinboro1, F. Akinpelu, F. and Oderinu, R. Numerical solution of integro-differential equation using Adomian decomposition and variational iteration methods, IOSR Journal of Mathematics, 10(4) (2014), 18–22.

[5]     Behzadi, S. Abbasbandy, S. Allahviranloo, T. and Yildirim, A. Application of homotopy analysis method for solving a class of nonlinear Volterra-Fredholm integro-differential equations, J. Appl. Anal. Comput. 2(2) (2012), 127–136.

[6]     Hamoud, A.A. and Ghadle, K.P. 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.

[7]     Asemi, R., Mohammadi, A. and Farajpour, A.  A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures11(9), (2014), 1515–1540.

[8]     Safarabadi, M., Mohammadi, M., Farajpour, A. and Goodarzi, M.   Effect of surface energy on the vibration analysis of rotating nanobeam, Journal of Solid Mechanics, 7(3), (2015), 299–311.

[9]     Mohammadi, M.,   Ghayour, M. and   Farajpour, A.  Analysis of free vibration sector plate based on elastic medium by using new version of differential quadrature method, Journal of Solid Mechanics in Engineering, 3(2), (2011), 47–56.

[10]  Goodarzi, M., Mohammadi, M., Farajpour, A. and Khooran, M.   Investigation of the effect of pre-stressed on vibration frequency of rectangular nanoplate based on a visco-Pasternak foundation, Journal of Solid Mechanics, 6 (1), (2014), 98–121.

[11]  Mohammadi, M., Ghayour, M. and Farajpour, A.   Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B: Engineering45(1), (2013), 32–42.

[12]  Mittal, R. and Nigam, R. Solution of fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. Mech., 4(2) (2008), 87–94.

[13]  Yang, C. and Hou, J. Numerical solution of integro-differential equations of fractional order by Laplace decomposition method, Wseas Trans. Math., 12(12) (2013), 1173–1183.

[14]  Hamoud, A.A. and Ghadle, K.P. Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind, Tamkang Journal of Mathematics, 49(4),   (2018), 301–315.

[15]  Ghadle, K.P., Hamoud, A.A. and Bani Issa M.SH.  A comparative study of variational iteration and Adomian decomposition techniques for solving Volterra integro-differential equations, International Journal of Mathematics Trends and Technology, ICETST. (2018), 16–21.

[16]  Farajpour, A., Danesh, M. and Mohammadi, M. Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E: Low-dimensional Systems and Nanostructures, 44(3), (2011), 719–727.

[17]  Hamoud, A.A. and Ghadle, K.P. The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Probl. Anal. Issues Anal., 7(25) (2018), 41–58.

[18]  Burton, T.A. Integro-differential equations, compact maps, positive kernels, and Schaefer’s fixed point theorem, Nonlinear Dyn. Syst. Theory, 17(1) (2017), 19–28.

[19]  Burton, T.A. Existence and uniqueness results by progressive contractions for integro-differential equations, Nonlinear Dyn. Syst. Theory, 16(4) (2016), 366–371.

[20]  Hamoud, A.A. and Ghadle, K.P. The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations, Korean J. Math. 25(3) (2017), 323–334.

[21]  Hamoud, A.A. and Ghadle, K.P. Existence and uniqueness theorems for fractional Volterra-Fredholm integro-differential equations, Int. J. Appl. Math. 31(3) (2018), 333–348.

[22]  Hamoud, A.A. and Ghadle, K.P. Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations, J. Indian Math. Soc. 85(1-2) (2018), 52–69.

[23]  He, J.H. A variational approach to nonlinear problems and its application, Mech. Applic. 20(1) (1998), 30–34.

[24]  He, J.H. and Wang, S.Q. Variational iteration method for solving integro-differential equations, Phys. Lett. A367 (2007), 188–191.

[25]  Wazwaz, A.M. A comparison between variational iteration method and Adomian decomposition method, Journal of Computational and Applied Mathematics, 207 (2007), 129–136.

[26]  Wazwaz, A.M. The variational iteration method for solving linear and non-linear Volterra integral and integro-differential equations, Int. J. Comput. Math. 87(5) (2010), 1131–1141.

[27]  Wazwaz, A.M. Linear and Nonlinear Integral Equations Methods and Applications, Springer Heidelberg Dordrecht London New York, 2011.

Volume 50, Issue 2
December 2019
Pages 303-307
  • Receive Date: 11 February 2019
  • Revise Date: 28 February 2019
  • Accept Date: 28 February 2019