A new numerical scheme for solving systems of integro-differential equations

Document Type : Research Paper


Shiraz University of Technology


This paper has been devoted to apply the Reconstruction of Variational Iteration Method (RVIM) to handle the systems of integro-differential equations. RVIM has been induced with Laplace transform from the variational iteration method (VIM) which was developed from the Inokuti method. Actually, RVIM overcome to shortcoming of VIM method to determine the Lagrange multiplier. So that, RVIM method provides rapidly convergent successive approximations to the exact solution. The advantage of the RVIM in comparison with other methods is the simplicity of the computation without any restrictive assumptions. Numerical examples are presented to illustrate the procedure. Comparison with the homotopy perturbation method has also been pointed out.


[1] S. Abbasbandy, E. Shivanian, Application of variational iteration method for n-th order
integro-di erential equations, Verlag der Zeitschrift furnatur for Schung, 46a (2009),
[2] J. Biazar, E. Babolian and R. Islam, Solution of the system of Volterra integral equations
of the rst kind by Adomian decomposition method, Appl. Math. Comput., 139 (2003),
[3] A. Bratsos, M. Ehrhardt and T. h. Famelis, A discrete Adomian decomposition method
for discrete nonlinear Schrodinger equations, Appl. Math. Comput. , 197 (2008), 190-
[4] Y. S. Choi, R. Lui, An integro-di erential equation arising from an electrochemistry
model, Quart. Appl. Math. 4 (1997) 677686.
[5] J. A. Cuminato, A. D. Fitt, M. J. S. Mphaka, A. Nagamine, A singular integro-
di erential equation model for dryout in LMFBR boiler tubes, IMA J. Appl. Math.
75 (2009) 269290.
[6] C. M. Cushing, Integro-di erential Equations and Delay Models in Population Dynam-
ics, in: Lecture Notes in Biomathematics, vol. 20, Springer, NewYork, 1977.
[7] D. D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation meth-
ods in heat radiation equations, Int. Commun., Heat and Mass Transfer 33 (3) (2006)
[8] D. D. Ganji, A. Sadighi, Application of Hes homotopy-perturbation method to nonlinear
coupled systems of reaction-di usion equations, Int. J. Nonlinear Sci. Numer. Simul.,
7 (4) (2006) 411418.
[9] A. Golbabai and M. Javidi, Application of He's homotopy perturbation method for n-th
order integro-di erential equations, Appl. Math. Comput., 190 (2007), 1409-1416.
[10] J. H. He, Homotopy perturbation technique, Comput. Math. Appl. Mech. Eng., 178
(1999), 257-262.
[11] E. Hesameddini and H. Lati zadeh, Reconstruction of variational iteration algorithm
using the Laplace transform, Int. J. of Non. Sci. and Numer. Sim., 10 (2009), 1365-
[12] A. J. Jerri, Introduction to integral equations with applications, Seconded, Wiley Inter-
science, 1999.
[13] K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear
Volterra-Fredholm integro-di erential equations, Appl. Math. Comput., 145 (2003), 641-
[14] H. Thieme, A model for the spatial spread of an epidemic, J. Math. Biol., 4 (1977)
[15] S. Q. Wang and J. H. He, Variational iteration method for solving integro-di erential
equations, Phys. Lett., 367 (2007), 188-191.
[16] A. M. Wazwaz, A reliable modi cation of Adomaion's decomposition method, Appl.
Math. Comput., 102 (1999), 77-86.
[17] X. Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics,
vol. 16, Springer, 2003.