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Homotopy Perturbation and Adomian Decomposition Methods for a Quadratic Integral Equations with Erdelyi-Kober Fractional Operator | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Homotopy Perturbation and Adomian Decomposition Methods for a Quadratic Integral Equations with Erdelyi-Kober Fractional Operator

Hendi FA1*, Shammakh W1 and Al-badrani H2

1Department of Mathematics, King Abdulaziz University, Saudi Arabia

2Department of Mathematics, Taibah University,Saudi Arabia

*Corresponding Author:
Hendi FA
Department of Mathematics
King Abdulaziz University, Saudi Arabia
Tel: 25774662470
E-mail: [email protected]

Received Date: May 18, 2016; Accepted Date: May 25, 2016; Published Date: May 31, 2016

Citation: Hendi FA, Shammakh W, Al-badrani H (2016) Homotopy Perturbation and Adomian Decomposition Methods for a Quadratic Integral Equations with Erdelyi-Kober Fractional Operator. J Appl Computat Math 5:306. doi:10.4172/2168-9679.1000306

Copyright: © 2016 Hendi FA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This paper is devoted with two analytical methods; Homotopy perturbation method (HPM) and Adomian decomposition method(ADM). We display an efficient application of the ADM and HPM methods to the nonlinear fractional quadratic integral equations of Erdelyi-kober type. The existence and uniqueness of the solution and convergence will be discussed. In particular, the well-known Chandrasekhar integral equation also belong to this class, recent will be discussed. Finally, two numerical examples demonstrate the efficiency of the method.

Keywords

Adomian decomposition; Integral equations; Homotopy; Functional equations

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].

Many research about (QIE) appear in the literaturely, numerous research papers have appeared devoted to nonlinear fractional quadratic integral equation (NFQIE) [3-8]. However, there few work on NFQIE with Erdelyi-kober fractional operator.

Hashem [9], studied the existence of maximal and minimal at least one continuous solution for NQIE of Erdelyi-kober type

(1)

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.

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 [16-19]. It is worth mentioning that our results are motiviated by the generalization of the work.

Theorem 1: Assume that

image is acontinuous function on [0,1],

imageare continuous and bounded with

image

there exist constant L1 and L2 such that

image image

Then the nonLinear fractional quadratic integral (Theorem 1) has a unique positive solution x∈C.

proof: For x,y∈S and for each t∈[0,1], we obtain

image image image image image image image image image image image

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

image (2)

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

image

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

image

Then operator T maps S into S, since for x∈S

image

It is clear that S is a closed subset of C.

Homotopy Perturbation Method

The He’s homotopy perturbation technique [10,11] defines the homotopy image which satisfies

Where t∈Ω and p∈[0,1] is an impeding parameter, u0 is an initial approximation which satisfies the boundary conditions, we can define H(u,p) by

image

where F(u) is an integral operator such that F(u)=u(t)-a(t), and L(u) has the form,

image

and continuously trace an implicitly defined curve from a starting point H(u0,0) 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].

image (6)

when p→1, (6) corresponds to (4) and give an approximation to the solution of (2) as follows,

image (7)

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

image image image image image image image

Where the Hn are the so-called He’s polynomials [22] which can be calculated by using the formula

image

Adomian Decomposition Method (ADM)

The ADM suggest the solution x(t) be decomposed by infinite series solution

image (8)

and the nonlinear functions g(t,x(t)) and f(t,x(t)) of Equation (2), represented by Adomian polynomials as follows

image (9)

image (10)

substituting (9) and (10) into (2) gives the following recursive scheme

image image (11)

Theorem 2: Assume that the solution of the (2)exist . If | x1 (t) |< m,m is a positive constant, then the series solution (8) of the(2) converges.

Proof: Define the Sequence {Sp} such that image is the sequence of partial sums from the series solutionimage and we have

image

Let Sp and sq be two arbitrary partial sums with p>q Now, We are going to prove that {Sp} is a cauchy Sequence in the Banach Space E [23-25].

image image image image image image image image image image image image image image

image image image

Let p=q+1 then

image image image image image image image image image image

Numerical Example

In this section, We shall study some numerical examples and applying HPM and ADM methods, then comparing the result [26-28].

Example 1: Consider the following nonlinear (FQIE),

image (12)

and has the exact solution x(t)=t2. First applying homotopy perturbation method .

Case 1: We can be constructed a homotopy as follows

image (13)

substituting (6) into (13), and equating the same powers of p

image image image image

and so on. Then the approximate solution is

image

Second applying (ADM) to equation (12), We get

image image

Where Ai are Adomian polynomials of the nonlinear term x2, and the solution will be

image

Table 1 shows a comparison between the absolute error of (HPM) (when n=1) and (ADM) solutions (when q=1), (Figure 1).

t uHPM uADM uExact  |uHPM-uExact| |uADM-uExact|
0.1 0.01000000 0.01000000 0.01000000 0.00000000 0.00000000
0.2 0.04000002 0.04000002 0.04000000 0.00000002 0.00000002
0.3 0.09000065 0.09000065 0.09000000 0.00000065 0.00000065
0.4 0.16000868 0.16000868 0.16000000 0.00000868 0.00000868
0.5 0.25006406 0.25006406 0.25000000 0.00006406 0.00006406
0.6 0.36032343 0.36032343 0.36000000 0.00032343 0.00032343
0.7 0.49125223 0.49125223 0.49000000 0.00125223 0.00125223
0.8 0.64396212 0.64396212 0.64000000 0.00396212 0.00396212
0.9 0.82057130 0.82057130 0.81000000 0.01057130 0.01057130
1 1.02378744 1.02378744 1.00000000 0.02378744 0.02378744

Table 1: Comparison betwee n the absolute error of (HPM) (when n=1) and (ADM) solutions (when q=1).

applied-computational-mathematics-exact-approximate-solutions

Figure 1: The difference between exact and approximate solutions by (HPM)and (ADM).

Case 2: We can be constructed a distinct convex homotopy as follows

image image

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

image image image image imageimage image image imageimage

and so on.

Table 2 shows a comparison between the absolute error of (HPM) (when n=1) and (ADM) solutions (when q=1), (Figure 2).

t uHPM uADM uExact |uHPM-uExact| |uADM-uExact|
0.1 0.01000000 0.01000000 0.01000000 0.00000000 0.00000000
0.2 0.04000000 0.04000002 0.04000000 0.00000000 0.00000002
0.3 0.08999983 0.09000065 0.09000000 0.00000017 0.00000065
0.4 0.15999700 0.16000868 0.16000000 0.00000300 0.00000868
0.5 0.24997210 0.25006406 0.25000000 0.00002790 0.00006406
0.6 0.35982724 0.36032343 0.36000000 0.00017276 0.00032343
0.7 0.48919293 0.49125223 0.49000000 0.00080707 0.00125223
0.8 0.63693217 0.64396212 0.64000000 0.00306783 0.00396212
0.9 0.80003776 0.82057130 0.81000000 0.00996224 0.01057130
1 0.97142857 1.02378744 1.00000000 0.02857143 0.02378744

Table 2: Comparison between the absolute error of (HPM) (when n=1) and (ADM) solutions (when q=1).

applied-computational-mathematics-exact-approximate-solutions

Figure 2: The difference between exact and approximate solutions by (HPM) and (ADM).

Example 2

image

Case 1: First applying homotopy perturbation method, we can be constructed a homotopy as follows

image image

substituting (6) into (16), and equating the same powers of p

image image image image

and so on. Then the approximate solution is

image

Second applying (ADM) to equation (15), we get

image image

Where Ai are Adomian polynomials of the nonlinear term cos(x21+x2) and the solution will be

image

Table 3 shows a comparisons between (HPM) and (ADM) solutions (when n=2, q=2), (Figure 3).

t uHPM uADM
0.1 0.29428565 0.28924531
0.2 0.32447042 0.31590677
0.3 0.37627732 0.36286938
0.4 0.46222603 0.44089584
0.5 0.59623287 0.56123678
0.6 0.79544168 0.73670859
0.7 1.08184953 0.98233639
0.8 1.48424305 1.31593221
0.9  2.04062381 1.75867138
 1  2.80125802  2.33568188

Table 3: Comparisons between (HPM) and (ADM) solutions (when n=2, q=2).

applied-computational-mathematics-approximate-solutions

Figure 3: Approximate solution for (HPM)and (ADM).

Case 2: We can be constructed a distinct convex homotopy as follows

image image

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

image image image image image image

and so on.

Table 4 shows a co.parison between the absolute error between (HPM) and (ADM) (when n=2, q=2), (Figure 4).

t uHPM uADM
0.1 0.28925 0.28924531
0.2 0.31591 0.31590677
0.3 0.36287 0.36286938
0.4 0.4409 0.44089584
0.5 0.56124 0.56123678
0.6 0.73671 0.73670859
0.7 0.98234 0.98233639
0.8 1.31593 1.31593221
0.9 1.75867 1.75867138
 1 2.33568 2.33568188

Table 4: Comparison between the absolute error between (HPM) and (ADM) (when n=2, q=2).

applied-computational-mathematics-approximate-solutions

Figure 4: Approximate solution for (HPM) and (ADM).

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