LOCAL FRACTIONAL VARIATIONAL ITERATION METHOD FOR SOLVING VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITHIN LOCAL FRACTIONAL OPERATORS

The paper uses the Local fractional variational Ite ration Method for solving the second kind Volterra integro-differential equations within the local fra ctional integral operators. The analytical solution s within the non-differential terms are discussed. So me illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems for the integral equations.

Recently, the local fractional variational iteration method (Yang and Baleanu, 2012) is derived from local fractional operators (Yang, 2011a;2011b;2011c;2012a;2012b;2012c;2012d;2012e;Zhong et al., 2012;Zhong and Gao, 2011). The method, which accurately computes the solutions in a local fractional series form or in an exact form, presents interest to applied sciences for problems where the other methods cannot be applied properly.
This study is organized as follows. In section 2, the basic mathematical tools are reviewed. Section 3 presents the local fractional variational iteration method based on local fractional operator. Illustrative examples is shown in section 4. Conclusions are in section 5.

Definition 1
Suppose that there is the relation Equation 2.1:

Definition 2
Suppose that the function ( ) f x satisfies condition (2.1), for ; it is so called local fractional continuous on the interval ( , ) a b , denoted by

Definition 3
In fractal space, let ( ) Local fractional derivative of high order is written in the form Equation 2.3:

Definition 4
A partition of the interval [ , ] a b is denoted as Note: If the functions are local fractional continuous then the local fractional derivatives and integrals exist. Some properties of local fractional derivative and integrals are given in (Yang, 2012f).

Definition 5
In fractal space, the Mittage Leffler function, sine function and cosine function are, respectively Equation 2.5 to 2.7:

ANALYSIS OF THE METHOD
The standard ka order local fractional Volterra integrodifferential equation of the second kind is given by: ar e the initial conditions. According to the rule of local fractional variational iteration method, the correction local fractional functional for Equation 3.1 is given by Equation 3.2: Here, we can construct a correction functional as follows Equation 3.3: where, n u ɶ is considered as a restricted local fractional variation; that is, 0, n u α δ = ɶ we obtain the following fractal Lagrange multiplier Equation 3.4: Therefore Equation 3.5 and 3.6: Finally, the solution is Equation 3

ILLUSTRATIVE EXAMPLES
In this section three examples for the local fractional Volterra integro-differential equation is presented in order to demonstrate the simplicity and the efficiency of the above method.

Example 1
We consider the local fractional Volterra integrodifferential Equation 4.1: The correction functional for this Equation 4.2 is given by: where, we used for first-order integrodifferential equation as shown in (3.4).
We can use the initial condition to select u 0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations Equation 4.3 to 4.7:

Example 2
We consider the local fractional Volterra integrodifferential Equation 4.9: (2 ) The correction functional for this Equation 4.10 is given by: where, we used for second-order integro-differential equation as shown in Equation 3.4. We can use the initial condition to select . Using this selection into the correction functional gives the following successive approximations: Equation 4.11 to 4.15: (2 ) 3 2 2 2 0 0 2 4 6 8 10 12 And so on:

Example 3
We consider the local fractional Volterra integro-differential Equation 4.17 and 4.18: The correction functional for this equation is given by:

CONCLUSION
In this study the Volterra integro-differential equations within the local fractional differential operator had been analyzed using the local fractional variational iteration method. The non-differentiable solutions are obtained. The present method is a powerful tool for solving many integral equations within the local fractional derivatives.

ACKNOWLEDGMENT
The researchers are grateful to the referees for their invaluable suggestions and comments for the improvement of the paper.