APPLICATION OF THE VARIATIONAL ITERATION METHOD FOR SYSTEM OF NONLINEAR VOLTERRA’S INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, the variational iteration method is proposed to solve system of nonlinear Volterra's integro-differential equations. Four numerical examples are illustrated by this method. The results reveal that this method is very effective and highly promising in comparison with other numerical methods. Key WordsVariational iteration method, System of nonlinear Volterra's integrodifferential equations 1.INTRODUCTION The variational iteration method [1, 2], which is a modified general Lagrange multiplier method [3] has been shown to solve effectively, easily and accurately, a large class of nonlinear problems with approximations which converge quickly to accurate solutions. It was successfully applied to autonomous ordinary differential equation [4], to nonlinear partial differential equations with variable coefficients [5], to SchrödingerKDV, generalized KDV and shallow water equations [6], to Burgers' and coupled Burgers' equations [7], to the linear Helmoltz partial differential equation [8] and recently to nonlinear fractional differential equations with Caputo differential derivative [9], and other fields [10-12, 28-31]. On the other hand, one of the interesting topics among researchers is solving integro-differential equations. In fact, integro-differential equations arise in many physical processes, such as glass-forming process [13], nanohydrodynamics [14], drop wise condensation [15], and wind ripple in the desert [16]. There are various numerical and analytical methods to solve such problems, for example, the homotopy perturbation method [17], the Adomian decomposition method [18], but each method limits to a special class of integro-differential equations. J.H. He used the variational iteration method for solving some integro-differential equations [19]. This Chinese mathematician chooses [19] initial approximate solution in the form of exact solution with unknown constants. M. Ghasemi et al solved the nonlinear Volterra's integro-differential equations [26] by using homotopy perturbation method. In [21], the variational iteration method was applied to solve the system of linear integro-differential equations. Also, J. Biazar et al solved systems of integro-differential equations by He's homotopy perturbation method [22]. S. Abbasbandy and E. Shivanian 148 The purpose of this paper is to extend the analysis of the variational iteration method to solve the system of general nonlinear Volterra's integro-differential equations which is as follows: ( ) ( ) ( ) ( ) 1 1 2 2 3 3 x ( ) ( ) 1 1 1 0 ( ) ( ) ( ) 2 2 1 1 3 3 ( ) ( , ( ),..., ( ), ( ),..., ( ),..., ( ),..., ( )) + K ( , , ( ),..., ( ),..., ( ),..., ( )) d , ( ) ( , ( ),..., ( ), ( ),..., ( ),. m m m m n n m m n n m m m u x H x u x u x u x u x u x u x x t u t u t u t u t t u x H x u x u x u x u x =

The purpose of this paper is to extend the analysis of the variational iteration method to solve the system of general nonlinear Volterra's integro-differential equations which is as follows: In system (1), m is order of derivatives and the continuous several variables functions i H and i K , 1, 2,..., i n = are given, the solutions to be determined are ( ) i u x , 1, 2,..., i n = . In Section 2, the basic ideas of variational iteration method are stated.Three examples are given in Section 3 also; we compare our results with another numerical method in this section.It is shown that this method is very simple and effective.Finally conclusions are stated in Section 4.

BASIC IDEAS OF VARIATIONAL ITERATION METHOD
To illustrate the basic concept of variational iteration method, we consider the following general nonlinear system where L is a linear operator, N is a nonlinear operator and ( ) x ψ is a given continuous function.According to the variational iteration method [10,[23][24][25], we can construct a correction functional in the form ∫ where 0 ( ) u x is an initial approximation with possible unknowns, λ is a Lagrange multiplier which can be identified optimally via variational theory, the subscript k denotes the th k approximation, and % k u is considered as a restricted variation [10,23], i.e. % 0 k u δ = .It is shown this method is very effective and easy for linear problem, its exact solution can be obtained by only one iteration, because λ can be exactly identified.
For solving (1) by the variational iteration method, for simplicity, we consider all of terms as restricted variation except ( ) . According to the variational iteration method, we derive correction functional as follow:  and the stationary conditions of the above correction functional can be expressed as follows: The Lagrange multiplier, therefore, can be identified as follows: As a result, we obtain the following iteration formulas

APPLICATIONS
In this section, we present some examples to show efficiency and high accuracy of the variational iteration method for solving system of nonlinear Volterra's integrodifferential equations (1).Example 3.1: [22] Let us first consider the system of nonlinear integro-differential equations as follow: with the exact solutions ( ) , ( ) .
According to (2) we have the following iteration formulas Now, we choose initial approximations 0 ( ) 2 1 u x x = + and 0 ( ) 1 v x = which satisfy initial conditions then, we obtain It is obvious that the iterations converge to exact solutions.This example has been solved by the homotopy perturbation method in [22].In order to show the efficiency and high accuracy of the present method, in Figure (1), we plotted the error functions, i.e.As we can see, this method is very effective and it is applied very convenient.Also, the obtained solutions with 6 iterations are showed in Figure ( 2) graphically.
x x u x x e v x x e = + = − Above system was solved in [26] and [27] that results are the same.Using the variational iteration method (2) with the initial approximations 0 ( ) Then, we have the following primary approximations It is obvious that the iterations converge to expand of exact solutions.In order to show the efficiency and high accuracy of the present method we compared results on interval [ 2, 2] − with homotopy perturbation method and Adomian decomposition method graphically in Figure (6).Also, the obtained solutions with 10 iterations are showed in Figure (7)

CONCLUSIONS
In this paper, we have studied system of nonlinear integro-differential equations with the variational iteration method.The initial approximation was selected arbitrary not in form of the exact solution with unknown constants.The results showed that the variational iteration method is remarkably effective and performing is very easy.In addition, it has more accuracy than homotopy perturbation method and Adomian decomposition method for this kind of problems.

Fig. 1 :
Fig. 1: (a): Solid line: The error of VIM with third iteration for ( ) u x ; dashed line: The error of homotopy method with 3 terms for ( ) u x ; (b): Solid line: The error of VIM with third iteration for ( ) v x ; dashed line: The error of homotopy method with 3 terms for ( ) v x .

Fig. 2 :
Fig. 2: (a): The error of VIM with sixth iteration for ( ) u x (b): The error of VIM with sixth iteration for ( ) v x .
It is obvious that the iterations converge to expand of exact solutions.Also, as the same of Example 1, obtained results by VIM and homotopy perturbation method on interval [ 1,1] − are showed in Figures(3) and (

Fig. 3 :
Fig. 3: (a): Solid line: The error of VIM with only second iteration for ( ) u x ; dashed line: The error of homotopy method with 2 terms for ( ) u x ; (b): Solid line: The error of VIM with second iteration for ( ) v x ; dashed line: The error of homotopy method with 2 terms for ( ) v x .

Fig. 4 :
Fig. 4: (a): The error of VIM with fifth iteration for ( ) u x (b): The error of VIM with fifth iteration for ( ) v x . .

Fig. 6 :
Fig. 6: (a): Solid line: The error of VIM with seventh iteration for ( ) u x ; dashed line: The error of homotopy method with 7 terms for ( ) u x ; (b): Solid line: The error of VIM with seventh iteration for ( ) v x ; dashed line: The error of homotopy method with 7 terms for ( ) v x .

Fig. 7 :
Fig. 7: (a): The error of VIM with tenth iteration for ( ) u x (b): The error of VIM with tenth iteration for ( ) v x .