Numerical Solutions for Quadratic Integro-Differential Equations of Fractional Orders

In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) solve the nonlinear initial value problems of first-order fractional quadratic integro-differential equations (FQIDEs). We use the Caputo sense in this article to describe the fractional derivatives. The solutions of the problems are derived by infinite convergent series, and the results show that both methods are most convenient and effective.


Introduction
The fractional calculus has appeared in many areas during the recent decades.Some scientists use approximation and numerical methods because there are almost no exact solutions of the fractional differential equations.He has proposed the VIM and HPM to solve the problems of linear and nonlinear [1] [2] [3] [4].VIM is based on Lagrange multiplier.The another method is HPM which defines as a coupling of the traditional perturbation method and homotopy in topology.Many authors successfully apply these methods to find the solutions of functional equations which arise in scientific and engineering problems [1] [5] [6] [7] [8] [9].The Adomian decomposition method presents solution of functional equations but perhaps we find some difficulties that will arise during the computation of Adomian polynomials, the VIM and HPM overcome it is difficulties [7].Fractional differential equations have diverse applications of physical phenomena [10] [11] [12] [13], for instance, acoustics, electromag-netism, control theory, robotics, viscoelastic materials, diffusion, edge detection, turbulence, signal processing, anomalous diffusion and fractured media [14].In literature, Momani and Noor [15] used the Adomian decomposition method for solving the fourth order fractional integro-differential Equation.Elbeleze et al. [16] [17], Kadem and Kilicman [18] applied the HPM and VIM methods for integro-differential Equation of fractional order with initial-boundary conditions.Recently, Gaafar [19] studied the existence and nondecreasing solution for the initial value problem of a quadratic integro-differential equations.However, there is little work on nonlinear fractional quadratic integro-differential equations.
Our goal for this article is extending the analysis of VIM and HPM to construct the approximate solutions of the following nonlinear initial value problems for first-order fractional quadratic integro-differential equations.
subject to the following initial condition: D α is the fractional derivative in the caputo sense, ( x is any nonlinear function, 0 γ is real constant and g is given and can be approximated by taylor polynomials.

Basic Definitions
In this section, we intend to present some basic definitions and properties of fractional calculus theory which are further used in this article.
Definition 1. Areal function The Riemann-Liouville fractional integral operator of order Some properties of the operator α  can be found in [11], which are needed here, as follows: for , 1, , 0 ( ) ( ), The fractional derivative of ( ) f t in the caputo sense is defined as ( ) ( ),

Analysis of VIM
The basic concept of the VIM is constructing the correction functional for the fractional quadratic integro-differential equation sees Equation (1) with initial conditions, ) taking the variation of Equation ( 7) to the independent variable y k we find ( ) ( ) ( ) ( ) to make the previous equation stationary, we gain the following stationary conditions: finally, the Lagrange multiplier is: We achieve the following iteration formula by substitution of (10) into the functional (6) the initial approximation ( ) 0 y x can be selected by the following way which satisfies initial conditions ( ) ( ) , where y 0 .

Analysis of HPM
The main concept of the HPM is constructing the homotopy for fractional quadratic integro-differential equation sees Equation ( 1), [ ]  1) can be considered as a power series in p which is the basic assumption of HPM : when 16) the approximate solution of Equation ( 1) can be as following First, substitute the relation (16) in the Equation (15).Second, equate the terms which have the same power's of p which yield to the following series of equations: ( ) and so on, the functions 1 2 , , F F  satisfy the following condition: ( )

Applications
n this section, we apply VIM and HPM to first-order nonlinear (FQIDEs).
Consider the following nonlinear first-order (FQIDEs): subject to the following initial condition ( ) According to VIM, the expression of the iteration formula (12) for Equation (21) can be observed in the following form: To avoid the difficulty of fractional integration, for the exponential term we take the truncated taylor expansion in (23), e.g., first-order approximation takes the following form by using iteration Formula (23): Table 1 and Figure 1 presents the approximate solution for the different values of α, we have noticed that the accuracy is improving.First, by computing more terms of the approximate solutions.The second way is taking more terms in the taylor expansion of the exponential term.
According to HPM, we build the following homotopy: and so on, apply the operator α  to the previous equations, and use the initial condition (22), to gain the following equations: The two terms approximation are formed as the following Equation Table 2 and Figure 2 shows the approximate solutions of (33) for 0 and for some values of ( ] 0,1 α ∈  According to VIM, the expression of the iteration Formula (12) for Equation (34) can be observed in the following form: To avoid the difficulty of fractional integration, for the exponential term we  first-order approximation takes the following form by using iteration Formula (35): Table 3 and Figure 4 presents the approximate solution for the different values of α, we have noticed that the accuracy is improving.First, by computing more terms of the approximate solutions.The second way is taking more terms in the taylor expansion of the exponential term.
According to HPM, we build the following homotopy: First, substitute the relation (16) in the Equation (37).
Second, equate the terms which have the same power's of p which yield to the following series of Equations: ( ) and so on, by taking the truncated taylor expansions for the exponential term in (40, 41): e.g., the two terms approximation are formed as the following Equation ) Table 4 and Figure 5 shows the approximate solutions of (34) for 0 and for some values of ( ] 0,1 α ∈ and so on, we obtain

Conclusion
In this paper, we have applied the VIM and HPM to find the solution of nonlinear initial value problem of fractional quadratic integro-differential equations for the first order.The methods do not require any linearization, perturbation or restrictive assumptions, we have observed that the VIM and HPM is a very powerful and effective tool for finding the solutions of the fractional quadratic integro-differential Equation.We use the Maple package (2015) in calculations.

β
is the Rieman-Liouville fractional integral operator of order then Equation(15) becomes to be the original problem.The solution of Equation (

Figure 1 .
Figure 1.Approximate solution for Equation (21) is obtained by VIM with different values of α.

Figure 2 .
Figure 2. Approximate solution for Equation (21) is obtained by HPM with different values of α.

Figure 3 .
Figure 3.Comparison of approximate solution by using HPM and VIM at 1 α = .
so on, applying the operator α  to the previous Equations, and use the initial condition (34), to gain the following Equations:

Figure 4 .
Figure 4. Approximate solution for Equation (34) is obtained by VIM with different values of α.

Figure 6
Figure 6 represent a comparison between two approximate solutions by using VIM and HPM methods.

Figure 5 .
Figure 5. Approximate solution for Equation (34) is obtained by HPM with different values of α.

Figure 6 .
Figure 6.Comparison of approximate solution by using HPM and VIM at 1 α = .

Table 4 .
Approximate solution for Equation (34) at different values of α.

Table 1 .
Approximate solution for Equation (21) at different values of α.

Table 2 .
Approximate solution for Equation (21) at different values of α.

Table 3 .
Approximate solution for Equation (34) at different values of α.