
Research Article


Solution of Delay Volterra Integral Equations Using the Variational Iteration Method 

M. Avaji,
J.S. Hafshejani,
S.S. Dehcheshmeh
and
D.F. Ghahfarokhi



ABSTRACT

In this study, the wellknown Variational Iteration Method (VIM) is implemented for finding the solution of linear and nonlinear Delay VolterraIintegral Equations (DVIEs). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory and the initial approximations can be freely chosen with unknown constants. The proposed method is shown to be highly accurate and yields the closed form series solutions of the exact solution. Several illustrative linear and nonlinear experiments are included to show the validity and capability of the presented method.





Received: October 30, 2011;
Accepted: December 26, 2011;
Published: January 18, 2012


INTRODUCTION
The Variational Iteration Method (VIM) was proposed by Hu
(1999) and He (1999a, b)
and has been proved by many authors (Marinca, 2002;
Abdou and Soliman, 2005) to be a powerful mathematical
tool for solving various types of linear and nonlinear problems arising in different
fields. For instance, ordinary differential equations (He,
2000; Momani et al., 2006; Ramos,
2008), Integral Equations (IEs) (Xu, 2007; Yousefi
et al., 2009), Integro Differential Equations (IDEs) (Shang
and Han, 2010; Nawaz, 2011), Fractional Differential
Equations (FDEs) (Odibat and Momani, 2006; Momani
and Odibat, 2007), Partial Differential Equations (PDEs) (Bildik
and Konuralp, 2006; Dehghan and Shakeri, 2008; Sweilam
and Khader, 2007) and etc. (Shakeri and Dehghan, 2007;
Ozis and Yildirim, 2007; Shakeri
and Dehghan, 2008; Dehghan and Tatari, 2006). In
this study, we are interested in extending the VIM for solving DVIEs.
DVIEs are used extensively in the applied and mathematical sciences for modeling
various phenomenon. For instance, medical science, biomathematics and biological
models (Baker and Derakhshan, 1993; Hu,
1999; Precup, 1995), population growth (Canada
and Zertiti, 1994), infectious diseases and epidemics (Williams
and Leggett, 1982), physics, physical models and dynamical systems (Brunner,
1994; Cahlon and Schimidt, 1997; Alnasr,
2004), the influence of noise (Ashwin et al., 2001)
and etc. (Brunner and Hu, 2005; Zhang
and Brunner, 1998; Vanani et al., 2011a,
2011b).
An ordinary form of DVIEs is given as (Alnasr, 2004):
wher g, ψ and K are given smooth functions and β is a constant delay. We consider a more general form of DVIE (1) as follows: where τ(s) represents a general delay function. HE’S VARIATIONAL ITERATION METHOD
Variational Iteration method was first proposed by He (2006a,
b) and has been successfully used by many researchers
to solve various linear and nonlinear models (He and Wu,
2006). The idea of the method is based on constructing a correction functional
by a general Lagrange multiplier and the multiplier is chosen in such a way
that its correction solution is improved with respect to the initial approximation
or to the trial function.
To illustrate the basic concept of the method, we consider the following general nonlinear differential equation given in the form: where, L is a linear operator, N is a nonlinear operator and g (t) is a known analytical function. We can construct a correction functional according to the variational method as:
where, λ(s) is a general Lagrange multiplier which can be identified optimally
via variational theory, the subscript n denotes the n th approximation and
is considered as a restricted variation, namely .
Successive approximations, u_{n+1}(t) will be obtained by applying the
obtained Lagrange multiplier and a properly chosen initial approximation u_{0}(t).
APPLICATION Here three experiments of nonlinear DVIEs are given to illustrate the efficiency and validity of the method. In all experiment, the Taylor expansion series of each iteration are used to overcome the difficulty of computations of complicated integrals arising in computations. We show that Taylor expansion series reduces the volume of computations and runtime of the method. The computations associated with the experiments discussed below were performed in Maple 14 on a PC with a CPU of 2.4 GHz. Experiment 1: Consider the following nonlinear DVIE: The exact solution is u(t) = e^{t}. The correspondent ODE for (5) is as: The Lagrange multiplier can be readily identified as λ = 1. As a result, we obtain the iteration formula
We have solved this problem using the iteration formula (7) for n = 5. The
sequence of approximate solution is obtained as follows:
Table 1: 
Maximum absolute error and runtime of the method for different
n of Experiment 1 

Therefore, we conclude that:
This has the closed form u(t) = e^{t} which is the exact solution of the problem. Also, we test the runtime of the method for different n. Table 1, shows the results including the maximum absolute error and runtime of the method for different n. From the numerical results in Table 1, it is easy to conclude that obtained results by VIM are in good agreement with the exact solution. Also, the runtime of the proposed algorithm illustrate the method as a fast and powerful tool. Experiment 2: Consider the following nonlinear DVIE: The exact solution is u(t) = e^{t}. The correspondent ODE for (8) is as: The iteration formula can be obtained as:
Table 2: 
Maximum absolute error and runtime of the method for different
n of Experiment 2 

We have solved this problem using the iteration formula (10) for n = 5. The
sequence of approximate solution is obtained as follows:
Therefore, we conclude that:
This has the closed form u(t) = e^{t} which is the exact solution of the problem. Table 2, shows the results including the maximum absolute error and runtime of the method for different n. Experiment 3: Consider the following nonlinear DVIE: The exact solution u(t) = cosht. The correspondent ODE for (12) is as: The obtained iteration formula is as:
Table 3: 
Maximum absolute error and runtime of the method for different
n of Experiment 3 

We have solved this problem using the iteration formula (14) for n = 6. The sequence of approximate solution is obtained as follows: Thus, we conclude that:
This has the closed form u(t) = cosht which is the exact solution of the problem. The obtained results including the maximum absolute error and runtime of the method for different n are given in Table 3. Results presented here, agree well with the exact solution. Also, the method yield the desired accuracy only in a few terms in a short time. CONCLUSION In this study, the Variational Iteration Method (VIM) has been successfully employed to obtain the approximate the exact solutions of linear and nonlinear DVIEs. Several illustrative linear and nonlinear experiment were solved and some results are obtained. The results show that the method is simple, easy to use and is very accurate for DVIEs. It was shown that the method is reliable, efficient and requires less computations.

REFERENCES 
1: He, J.H., 1999. Approximate analytical solution of Blasius equation. Commu. Nonlinear Sci. Numer. Simul., 4: 7578. CrossRef 
2: He, J.H., 1999. Variational iteration methoda kind of nonlinear analytical technique: Some examples. Int. J. Nonlinear Mech., 34: 699708. CrossRef  Direct Link 
3: Marinca, V., 2002. An approximate solution for onedimensional weakly nonlinear oscillations. Int. J. Nonlinear Sci. Numer. Simul., 3: 107120. Direct Link 
4: He, J.H., 2000. Variational iteration method for autonomous ordinary differential systems. Applied Math. Comput., 114: 115123. CrossRef  Direct Link 
5: Momani, S., S. Abuasad and Z. Odibat, 2006. Variational iteration method for solving nonlinear boundary value problems. Applied Math. Comput., 183: 13511358.
6: Ramos, J.I., 2008. On the variational iteration method and other iterative techniques for nonlinear differential equations. Applied Math. Comput., 199: 3969. CrossRef 
7: Xu, L., 2007. Variational iteration method for solving integral equations. Comput. Mathe. Appl., 54: 10711078. CrossRef 
8: Yousefi, S.A., A. Lotfi and M. Dehghan, 2009. He's variational iteration method for solving nonlinear mixed VolterraFredholm integral equations. Comput. Math. Applied, 58: 21722176. CrossRef 
9: Shang, X. and D. Han, 2010. Application of the variational iteration method for solving nthorder integrodifferential equations. J. Comput. Applied Math., 234: 14421447. CrossRef 
10: Nawaz, Y., 2011. Variational iteration method and homotopy perturbation method for fourthorder fractional integrodifferential equations. Comput. Math. Appl., 61: 23302341. CrossRef 
11: Odibat, Z.M. and S. Momani, 2006. Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul., 7: 2734. CrossRef  Direct Link 
12: Momani, S. and Z. Odibat, 2007. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals, 31: 12481255. CrossRef 
13: Bildik, N. and A. Konuralp, 2006. The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations. Int. J. Nonlinear Sci. Numer. Simul., 7: 6570. CrossRef  Direct Link 
14: Dehghan, M. and F. Shakeri, 2008. Application of He's variational iteration method for solving the Cauchy reactiondiffusion problem. J. Comput. Applied Math., 214: 435446. CrossRef 
15: Sweilam, N.H. and M.M. Khader, 2007. Variational iteration method for one dimensional nonlinear thermoelasticity. Chaos Solitons Fract., 32: 145149. CrossRef  Direct Link 
16: Shakeri, F. and M. Dehghan, 2007. Numerical solution of a biological population model using He's variational iteration method. Comput. Math. Appl., 54: 11971209. CrossRef 
17: Ozis, T. and A. Yildirim, 2007. A study of nonlinear oscillators with u^{1/3} force by He's variational iteration method. J. Sound Vib., 306: 372376. CrossRef 
18: Shakeri, F. and M. Dehghan, 2008. Solution of a model describing biological species living together using the variational iteration method. Math. Comput. Model., 48: 685699. CrossRef 
19: Dehghan, M. and M. Tatari, 2006. The use of He's variational iteration method for solving a FokkerPlanck equation. Phys. Scr., 74: 310316.
20: Baker, C.T.H. and M.S. Derakhshan, 1993. Convergence and stability of quadrature methods applied to volterra equations with delay. IMA J. Numer. Anal., 13: 6791. CrossRef  Direct Link 
21: Hu, Q., 1999. Multilevel correction for discrete collocation solutions of volterra integral equations with delay arguments. Applied Numer. Math., 31: 159171. CrossRef 
22: Precup, R., 1995. Monotone technique to the initial values problem for a delay integral equation from biomathematics. Studia Univ. BabesBolyai Math., 40: 6373.
23: Canada, A. and A. Zertiti, 1994. Method of upper and lower solutions for nonlinear delay integral equations modelling epidemics and population growth. Math. Models Methods Applied Sci., 4: 107119. CrossRef 
24: Williams, L.R. and R.W. Leggett, 1982. Nonzero solutions of nonlinear integral equations modeling infectious disease. Soc. Ind. Applied Math. J. Math. Anal., 13: 112121. Direct Link 
25: Brunner, H., 1994. Collocation and continuous implicit RungeKutta methods for a class of delay volterra integral equations. J. Comput. Applied Math., 53: 6172. CrossRef 
26: Cahlon, B. and D. Schimidt, 1997. Stability criteria for certain delay integral equations of volterra type. J. Comput. Applied Math., 84: 161188. CrossRef 
27: Alnasr, M.H., 2004. The numerical stability of multistep methods for volterra Integral equations with many delays. Int. J. Comput. Math., 81: 12571263. CrossRef  Direct Link 
28: Ashwin, P., E. Covas and R. Tavakol, 2001. Influence of noise on scalings for inout intermittence. Phys. Rev. E, Vol. 64. 10.1103/PhysRevE.64.066204
29: Brunner, H. and Q. Hu, 2005. Super convergence of iterated collocation solutions for Volterra integral equations with variable delays. Soc. Ind. Applied Math. J. Numer. Anal., 43: 19431949. Direct Link 
30: Zhang, W. and H. Brunner, 1998. Collocation approximations for secondorder differential equations and volterra integrodifferential equations with variable delays. Can. Applied Math. Q., 6: 269285. Direct Link 
31: He, J.H., 2006. New Interpretation of homotopy perturbation method. Int. J. Mod. Phys. B, 20: 25612568. Direct Link 
32: He, J.H., 2006. NonPerturbative Methods for Strongly Nonlinear Problems. Dissertation, deVerlag im Internet GmbH, Berlin.
33: He, J.H. and X.H. Wu, 2006. Construction of solitary solution and compactonlike solution by variational iteration method. Chaos, Solitons Fractals, 29: 108113. CrossRef 
34: Vanani, S.K., S. Heidari and M. Avaji, 2011. A lowcost numerical algorithm for the solution of nonlinear delay boundary integral equations. J. Applied Sci., 11: 35043509. CrossRef  Direct Link 
35: Vanani, S.K., S. Heidari and F.K. Haghani, 2011. A numerical algorithm for solving delay boundary integral equations. Int. J. Acad. Res., 3: 318326. Direct Link 
36: Abdou, M.A. and A.A. Soliman, 2005. New applications of variational iteration method. Phys. D: Nonlinear Phenomena, 211: 18. CrossRef  Direct Link 



