Reach Us
+44-1474-556909

^{1}Department of Mathematics, King Abdulaziz University, Saudi Arabia

^{2}Department 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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

**Adomian decomposition**; Integral equations; Homotopy; Functional equations

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

is acontinuous function on [0,1],

are continuous and bounded with

there exist constant L_{1} and L_{2} such that

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

By (H4), The operator T is a contraction map from S into S, hence the conclusion of the theorem follows.

In this section, we prove the existence and uniqueness of continuous solutions and the convergence for Equation

(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

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

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

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

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(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].

(6)

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

(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

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

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

(8)

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

(9)

(10)

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

(11)

**Theorem 2:** Assume that the solution of the (2)exist . If | x_{1} (t) |< m,m is a positive constant, then the series solution (8) of the(2) converges.

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

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].

Let p=q+1 then

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),

(12)

and has the exact solution x(t)=t^{2}. First applying homotopy perturbation method .

**Case 1:** We can be constructed a homotopy as follows

(13)

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

and so on. Then the approximate solution is

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

Where Ai are Adomian polynomials of the nonlinear term x^{2}, and the solution will be

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

t | u_{HPM} |
u_{ADM} |
u_{Exact} |
|u-_{HPM}u|_{Exact} |
|u-_{ADM}u|_{Exact} |
---|---|---|---|---|---|

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).

**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

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 | u_{HPM} |
u_{ADM} |
u_{Exact} |
|u-_{HPM}u|_{Exact} |
|u-_{ADM}u|_{Exact} |
---|---|---|---|---|---|

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).

**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. Then the approximate solution is

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

Where A_{i} are Adomian polynomials of the nonlinear term cos(x^{2}1+x^{2}) and the solution will be

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

t | u_{HPM} |
u_{ADM} |
---|---|---|

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).

**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

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 | u_{HPM} |
u_{ADM} |
---|---|---|

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).

- Baruch C, Eskin M (1981) Existence theorems for an integral equation of the Chandrasekhar H-equation with perturbation. Journal of Mathematical Analysis and Applications 83: 159-171.
- Hashem HHG (2015) On successive approximation method for coupled systems of Chandrasekhar quadratic integral equations. Journal of the Egyptian Mathematical Society 23: 108-112.
- Banas J, Caballero J, Rocha J, Sadarangani K (2005) Monotonic solutions of a class of quadratic integral equations of Volterra type. Computers Mathematics with Applications 49: 943-952.
- Banas J, Martinon A (2004) Monotonic solutions of a quadratic integral equation of Volterra type. Computers Mathematics with Applications 47: 271-279.
- Banas J, Martin JR, Sadarangani K (2006) On solutions of a quadratic integral equation of Hammerstein type. Mathematical and Computer Modelling 43: 97-104.
- Caballero J, Darwish MA, Sadarangani K (2014) Solvability of a quadratic integral equation of Fredholm type in Holder spaces. Electronic Journal of Differential Equations 31: 1-10.
- Shou-Zhong F, Zhong W, Jun-Sheng D (2013) Solution of quadratic integral equations by the Adomian decomposition method. CMES-Comput Model Eng Sci 92: 369-385.
- Ziada AA (2013) Adomian solution of a nonlinear quadratic integral equation. Journal of the Egyptian Mathematical Society 21: 52-56.
- Hashem HH, Zaki MS (2013) Caratheodory theorem for quadratic integral equations of Erdelyi-Kober type. Journal of Fractional Calculus and Applications 4: 56-72.
- He JH (1999) Homotopy perturbation technique. Computer methods in applied mechanics and engineering 178: 257-262.
- He JH (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics 35: 37-43.
- He JH (2003) Determination of limit cycles for strongly non-linear oscillators. Phys Rev Lett 90: 174301.
- He JH (2003) Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and computation 135: 73-79.
- He JH (2004) Asymptotology by homotopy perturbation method. Applied Mathematics and Computation 156: 591-596.
- He JH (2006) Homotopy perturbation method for solving boundary value problems. Physics letters A 350: 87-88.
- Abbaoui K, Cherruault Y (1994) Convergence of Adomian’s method applied to differential equations. Comput Math 28: 103-109.
- Adomian G (1983) Stochastic system. Academic press, New York.
- Adomian G (1995) Solving frontier problems of physics: the decomposition method. Kluwer, Dordercht.
- Adomian G, Rach R, Mayer R (1992) Modified decomposition. J Math Comput 23: 17-23.
- He JH (2006) New interpretation of homotopy perturbation method. Internat J Modern Phys B 20: 2561-2568.
- Liao SJ (1997) Boundary element method for general nonlinear differential operators. Engineering Analysis with Boundary Elements 20: 91-99.
- Ghorbani A (2009) Beyond Adomian polynomials: he polynomials. Chaos, Solitons Fractals 39: 1486-1492.
- Banas J, O’Regan D (2008) On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order. Journal of Mathematical Analysis and Applications 345: 573-582.
- Darwish MA, Beata R (2014) Asymptotically stable solutions of a generalized fractional quadratic functional-integral equation of Erdélyi-Kober type. J Funct Spaces.
- Darwish MA, Sadarangani K (2015) On a quadratic integral equation with supremum involving Erdélyi-Kober fractional order. Mathematische Nachrichten 288: 566-576.
- El-Sayed AM, Hashem HHG, Ziada EAA (2010) Picard and Adomian methods for quadratic integral equation. Comp Appl Math 29: 447-463.
- El-Sayed AM, Hashem HHG, Ziada EAA (2014) Picard and Adomian decomposition methods for a quadratic integral equation of fractionl order. Computational and Applied Mathematics 33: 95-109.
- Wang J, Dong X, Yong Z (2012) Analysis of nonlinear integral equations with Erdélyi–Kober fractional operator. Communications in Nonlinear Science and Numerical Simulation 17: 3129-3139.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebraic Geometry
- Analytical Geometry
- Applied Mathematics
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Complex Analysis
- Computational Model
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Differential Equations
- Differential Transform Method
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Hamilton Mechanics
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Physical Mathematics
- Quantum Mechanics
- Quantum electrodynamics
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Riemannian Geometry
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topology
- mirror symmetry
- vector bundle

- Total views:
**10301** - [From(publication date):

May-2016 - Jul 22, 2019] - Breakdown by view type
- HTML page views :
**10112** - PDF downloads :
**189**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2019-20