Homotopy Perturbation and Adomian Decomposition Methods for a Quadratic Integral Equations with Erdelyi-Kober Fractional Operator

In this paper, we investigated the existence, uniqueness of the solution and convergence for NFQIE (1), using two methods; HPM and ADM. The homotopy perturbation method (HPM) was suggested by Ji-Huan [10-15] in 1999. In this method, the solution can be expressed by an infinite series, which commonly converges fast to the exact solution. It is a coupling of the traditional perturbation method and homotopy in topology, which is solved differential and integral equations, linear and nonlinear. The HPM does not require a small parameter in equations. Also, it has an important advantage which enlarges the application of nonlinear problem in applied science.


Introduction
It is well-known that the theory of integral equations has many applications in describing numerous events and problems of the real world. Nonlinear quadratic integral equations (NQIE)are also often encountered in the theories of radiative transfer and neutron transport [1,2].
Hashem [9], studied the existence of maximal and minimal at least one continuous solution for NQIE of Erdelyi-kober type In this paper, we investigated the existence, uniqueness of the solution and convergence for NFQIE (1), using two methods; HPM and ADM. The homotopy perturbation method (HPM) was suggested by Ji-Huan [10][11][12][13][14][15] in 1999. In this method, the solution can be expressed by an infinite series, which commonly converges fast to the exact solution. It is a coupling of the traditional perturbation method and homotopy in topology, which is solved differential and integral equations, linear and nonlinear. The HPM does not require a small parameter in equations. Also, it has an important advantage which enlarges the application of nonlinear problem in applied science.
The Adomian decomposition method (ADM) solves many of functional equations, for example, differential, integro-differential, differential-delay, and partial differential equations. The solution usually appears in a series form, this method has many significant advantages, it does not require linearization, perturbation and other restrictive methods. Also, it might change the problem to a solved one [16][17][18][19]. It is worth mentioning that our results are motiviated by the generalization of the work.
Then the nonLinear fractional quadratic integral (Theorem 1) has a unique positive solution x∈C.   Where the H n are the so-called He's polynomials [22] which can be calculated by using the formula

Adomian Decomposition Method (ADM)
The ADM suggest the solution x(t) be decomposed by infinite series solution and the nonlinear functions g(t,x(t)) and f(t,x(t)) of Equation (2), represented by Adomian polynomials as follows substituting (9) and (10) into (2) gives the following recursive scheme ( , ) = , ( , ) = , Let S p and s q be two arbitrary partial sums with p>q Now, We are going to prove that {S p } is a cauchy Sequence in the Banach Space E [23][24][25].
By (H4), The operator T is a contraction map from S into S, hence the conclusion of the theorem follows.

Main Results
In this section, we prove the existence and uniqueness of continuous solutions and the convergence for Equation 1 1 we denote by C=C(I) the space of all real-valued functions which are continuous on I=[0,1].We can transform (2) into an equivalent fixed point problem Tx=x, where the operator T:C→C is defined by Observe that the existence of a fixed point for the operator T implies the existence of a solution for the (2). Now define a subset S of C as Then operator T maps S into S, since for x∈S

Homotopy Perturbation Method
The He's homotopy perturbation technique [10,11] defines the homotopy ( , ) : Where t∈Ω and p∈[0,1] is an impeding parameter, u 0 is an initial approximation which satisfies the boundary conditions, we can define H(u,p) by where F(u) is an integral operator such that F(u)=u(t)-a(t), and L(u) has the form, and continuously trace an implicitly defined curve from a starting point H(u 0 ,0) to a solution function H(x,t). The embedding parameter p monotonically increases from zero to one as the trivial problem F(u)=0 is continuously deformed to the original problem L(u)=0.
The embedding parameter p∈(0,1] can be considered as an expanding parameter [20]. when p→1, (6) corresponds to (4) and give an approximation to the solution of (2) as follows, The series (7) converges in most cases, and the rate of convergence depends on L(u) [21].
We substitute (6) into (4) and equate the terms with identical powers of p, obtaining

Numerical Example
In this section, We shall study some numerical examples and applying HPM and ADM methods, then comparing the result [26][27][28].
and has the exact solution x(t)=t 2 . First applying homotopy perturbation method .

Case 2:
We can be constructed a distinct convex homotopy as follows It can continuously trace an implicity defined curve from a starting point H(u,0) to a solution function H(u,1), and equating the coefficients of the same powers of p, we obtain   Table 1: Comparison betwee n the absolute error of (HPM) (when n=1) and (ADM) solutions (when q=1). Table 2 shows a comparison between the absolute error of (HPM) (when n=1) and (ADM) solutions (when q=1), (Figure 2).

Example 2
( ) Case 1: First applying homotopy perturbation method, we can be constructed a homotopy as follows substituting (6) into (16), and equating the same powers of p     and so on. Table 4 shows a co.parison between the absolute error between (HPM) and (ADM) (when n=2, q=2), (Figure 4).