COMPARISON OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD WITH ADOMIAN DECOMPOSITION AND HOMOTOPYPERTURBATION METHODSTO SOLVE WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS OF ABEL TYPE

COMPARISON OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD

WITH ADOMIAN DECOMPOSITION AND HOMOTOPYPERTURBATION METHODSTO SOLVE WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS OF ABEL TYPE

Hamed Daei Kasmaei

Department of Mathematics and Statistics, Faculty of Science, Central Tehran Branch, Islamic Azad University ,Tehran, Iran . Cell phone :+989123937613,
[email protected]

ABSTRACT

In the present paper , we obtain analytical-approximate solution of Abel Volterra integral equations by using Optimal Homotopy Asymptotic method (OHAM).This approach has been compared with some other powerful and efficient methods such as He’s homotopy perturbation method (HPM) and Adomian decomposition method (ADM). This method uses simple computations with quite acceptable approximate solutions, which has close agreement

with exact solutions. The accuracy and efficiency of OHAM approach is compared and illustrated by presenting four test examples that satisfy the power of OHAM compared to ADM and HPM methods.

Keywords: Nonlinear singular Volterra integral equations of Abel type, Optimal Homotopy

Asymptotic method , Least square method , Homotopy Perturbation method, Asymptotic behavior, Adomian Decomposition method.

2000 AMS Classification: 65R20, 45E10, 45D05

1 INTRODUCTION

 Many problems in science and engineering such as solid state physics, plasma physics, fluid mechanics, chemical kinetics and mathematical biology lead to nonlinear singular Volterra integral equations of Abel type as:

​​ 

in which ​​ is leading term, ​​ is known function , is unknown nonlinear functional and

 In recent years researchers have turned their attention towards solving Volterra integral equations with ​​ and have represented different methods [10, 11, 30,31].
Also, many powerful and applicable methods have been proposed and applied successfully to approximate many types of non-linear singular integral equations with a wide range of applications [19, 20, 42, 44].  Also, the generalized Abel integral equation on a finite interval was studied by zeilon [53].

 Abel Volterra integral equations have been proposed in first and second types. This equation was applied by Niels Abel in 1823 to describe a sliding point mass in a vertical plane on a unknown curve under gravitational force. The point mass starts its motion without initial velocity from a point which has a vertical distance x from the lowest point of the curve [45]. Using the work-energy theorem, the equation of the unknown curve that obtained is the well-known Abel integral equation

where is a given function and ​​ ​​ is an unknown function.

 In this paper, we articulate the concept of OHAM to express a reasonable and reliable method to solve weakly singular integral equations of Abel type. This approach was established by Marinca and Herisanu [22, 32]. Afterwards, they published some papers presented in [33, 34, 35, 36] to show the ability of OHAM to expand their ideas in order to implement it to solve a vast domain of non linear problems. The advantage of OHAM is built in convergence criteria which are similar to HAM but more flexible. Also ,a series of papers by Iqbal et al [23] Iqbal and Javed [24] and Haq [15] have proved the effectiveness, generalization and reliability of this method and obtained solutions of currently important application in science and engineering. In order to explain reliability of the method, we deal with different examples in the subsequent section.

 Finally, numerical comparison between OHAM and other existing methods shows the efficiency of OHAM. Comparison graphs of exact solutions and approximate solutions are also plotted to visualize the performance of OHAM .Since there is a paucity of exact solution of a non linear problems, we may go for approximate analytic solutions.

 Many asymptotic techniques are used for solving non linear problems. So by keeping this fact in mind, we have presented a powerful technique OHAM which is generalized form of HPM and HAM. OHAM is simple, straightforward technique and does not require the existence of any small or large parameter as do traditional perturbation methods. OHAM has successfully applied to a number of non-linear problems arising in the science and engineering by various researchers. This proves the validity and acceptability of OHAM as a useful technique [25, 38, 39, 43].

 In this paper , we propose Semi-Analytical methods respectively as follows:

 (1) Adomian decomposition method(ADM).

 (2) Homotopy perturbation method(HPM).

 (3) Optimal Homotopy Asymptotic method (OHAM).

 We compare Optimal Asymptotic Homotopy Perturbation method to two other different methods namely, He’s homotopy perturbation method (HPM) and Adomian Decomposition method (ADM) to solve weakly singular Abel Volterra integral equations of the second kind.

 Then , we present four different test examples to show the ability of OHAM method rather ​​ to classic ADM and HPM. Also, the results have been compared to each other to show the power of OHAM rather to other two compared methods. Its noticed that using modified versions of ADM and HPM are not possible to be used all the time. However, it seems that for some special cases, these methods give us the closed form of the equation in just one or two iterations. But, it is not applicable to all types of these equations.

 To empower ADM and HPM , researchers usually use a combination of Classic Semi-Analytical methods along with some tools such as pad´e approximant, Laplace transformations and so on in order to reach to the best approximation just by modifying them in one or two iterations. But, OHAM uses a direct method in two steps by adding optimization parameters to homotopy equation that enables us to use it for all types of linear and non linear problems too.

2 ADOMIAN DECOMPOSITION METHOD

 The Adomian decomposition method (ADM) [46, 47, 48] is a well-known systematic method for solution of linear or non-linear and deterministic or stochastic operator equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), integral equations, integro-differential equations, etc. The ADM is a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. The accuracy of the analytic approximate solutions obtained by ADM, can be verified by direct substitution. Advantages of the ADM over Picard’s iterated method were demonstrated in [41]. More advantages of the ADM over the variational iteration method were presented in [49, 50]. Adomian and co-workers have solved nonlinear differential equations for a wide class of nonlinearities, including product [1], polynomial [2], exponential [3], trigonometric [4], hyperbolic [5], composite [6], negative-power [7], radical [8] and even decimal-power nonlinearities [9].

 We find that the ADM solves nonlinear operator equations for any analytic nonlinearity, providing us with an easily computable and rapidly convergent sequence of analytic approximate functions.

 In Adomian decomposition method, we consider the functional equation of Abel integral equation of the form

where is a nonlinear operator and is a given function. We assume the solution as infinite series for the unknown function, given by

and then we decompose the non linear terminto a series

where the , depending on ​​ are called the Adomian polynomials and are obtained for the nonlinearity ​​ by the definitional formula:

 We list the formulas of the first several Adomian polynomials for the one-variable simple analytic nonlinearity ​​ from through, inclusively, for convenient reference as

and so on.Substituting Eq.4 and Eq.5 into Abel’s integral equation of the form Eq.3 , we get:

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ The components ​​ are usually determined by using the recurrence relation:

Having determined the components , the solution ​​ of Eq.4 is determined in the form of a rapid convergent power series by substituting the derived components ​​ in Eq.8. Thus in order to implement ADM on Abel integral equation, weuse this form of equation :

Substituting Eq.4 ​​ into Eq.9, results:

Then, we can use the following recursive relation to evaluate the various iterations  ​​​​ as follows :

Here, we assume that the kernel ​​ to be Abel’s kernel i.e.

3 BASIC IDEA OF HOMOTOPY PERTURBATION METHOD

 In this method, using the homotopy technique of topology, a homotopy is constructed

with an embedding parameter p ∈[0,1], which is considered as a small parameter.

 This method became very popular among the scientists and engineers, even though it involves continuous deformation of a simple problem into a more difficult problem under consideration. Most of the perturbation methods depend on the existence of a small perturbation parameter but many nonlinear problems have no small perturbation parameter at all. Many new methods have been proposed in the late nineties to solve such nonlinear equation devoid of such small parameters, Dehghan and Shakeri [12, 13], Ganji and Rajabi [14], He [16, 17], Liao [27, 28]. The homotopy perturbation method is considered as a combination of the classical perturbation technique and the homotopy (whose origin is in the topology), but not restricted to small parameters as occur with traditional perturbation methods. This method can be done in few iterations to obtain highly accurate solutions. When the homotopy theory is coupled with perturbation theory ,it provides a powerful mathematical tool to solve non linear problems. A review of recently developed methods of nonlinear analysis can be found in He [18]. To figure out the basic concept of HPM, consider the following nonlinear functional equation

with the following boundary conditions:

whereis a general differential operator, is a boundary operator, ​​ a known analytic

function, and is the domain boundary for .can be divided into two operators L and N, where L is linear and is non linear so that Eq.13 can be rewritten as

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ Generally, a homotopy function can be constructed as

or

where p is a homotopy parameter, whose values are within range of 0 and 1, and ​​ is the first approximation for the solution of Eq.13 that satisfies the boundary conditions.

 Assuming that solution for Eq.13 or Eq.15 can be written as a power series of

Substituting Eq.18 into Eq.16 or Eq.17 and equating identical powers of term, there can be found values for the sequence