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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
 
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ABSTRACT

In this study, the well-known 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.

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  How to cite this article:

M. Avaji, J.S. Hafshejani, S.S. Dehcheshmeh and D.F. Ghahfarokhi, 2012. Solution of Delay Volterra Integral Equations Using the Variational Iteration Method. Journal of Applied Sciences, 12: 196-200.

DOI: 10.3923/jas.2012.196.200

URL: https://scialert.net/abstract/?doi=jas.2012.196.200
 
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):

(1)

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:

(2)

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:

(3)

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:

(4)

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, un+1(t) will be obtained by applying the obtained Lagrange multiplier and a properly chosen initial approximation u0(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:

(5)

The exact solution is u(t) = et. The correspondent ODE for (5) is as:

(6)

The Lagrange multiplier can be readily identified as λ = -1. As a result, we obtain the iteration formula

(7)

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) = et 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:

(8)

The exact solution is u(t) = e-t. The correspondent ODE for (8) is as:

(9)

The iteration formula can be obtained as:

(10)


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:

(11)

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:

(12)

The exact solution u(t) = cosht. The correspondent ODE for (12) is as:

(13)

The obtained iteration formula is as:

(14)

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:

(15)

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.

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