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Necdet Bildik, Mustafa Inc, "A Comparison between Adomian Decomposition and Tau Methods", Abstract and Applied Analysis, vol. 2013, Article ID 621019, 5 pages, 2013. https://doi.org/10.1155/2013/621019
A Comparison between Adomian Decomposition and Tau Methods
Abstract
We present a comparison between Adomian decomposition method (ADM) and Tau method (TM) for the integrodifferential equations with the initial or the boundary conditions. The problem is solved quickly, easily, and elegantly by ADM. The numerical results on the examples are shown to validate the proposed ADM as an effective numerical method to solve the integrodifferential equations. The numerical results show that ADM method is very effective and convenient for solving differential equations than Tao method.
1. Introduction
The decomposition method was introduced by Adomian in [1–3] in the 1980s in order to solve linear and nonlinear functional equations (algebraic, differential, partial differential equations and systems, integral, delay, integrodifferential equations, etc.) [1–10]. This method leads to computable, accurate, approximate convergence solutions to linear and nonlinear deterministic and stochastic operator equations. The solution can verify any stage of approximation. The convergence of this method was proved by Cherruault and coauthors in [11–13].
In this paper we will be concerned with approximate solutions of the linear or nonlinear Volterra integrodifferential equations. Firstly, this type of equation was introduced by Volterra [14] in the early 1900s. These equations can be found in physics, biology, and engineering applications such as heat transfer, diffusion process in general, and neutron diffusion [4].
Many authors have compared the ADM with some existing methods in solving different linear or nonlinear evolution equations, integral and integrodifferential equations. Bellomo and Monaco [15] compared the ADM and the perturbation techniques. Advantages of the ADM over Picard’s method have been shown by Rach [16]. Edwards et al. [17] compared the ADM and the RungeKutta methods for approximate solutions of some predatorprey models. Additionally, Wazwaz [18] presented a comparison between the ADM and the Taylor series methods. He showed that the ADM minimizes the computational difficulties of the Taylor series in that the components of the solution were determined elegantly by using simple integrals. More recently, ElSayed and AbdelAziz [19] introduced a comparison of the ADM and the WaveletGalerkin method for the solution of integrodifferential equations. They showed that the ADM was simple and easy to use.
In [20], Hosseini and Shahmorad employed Tau method to obtain a numerical solution to the integrodifferential equations given by (1). Batiha et al. [21] presented the variational iteration method (VIM) and the ADM for solving nonlinear integrodifferential equations. Fariborzi Araghi and Sadigh Behzadi [22–24] solved nonlinear VolterraFredholm integrodifferential equations by using the modified ADM, the VIM, and the homotopy analysis method, respectively. Borhanifar and Abazari [25] implemented the differential transform method for solving nonlinear integrodifferential equations with the kernel functions including derivative type of unknown solution. Ben Zitoun and Cherruault [26] presented a method for solving nonlinear integrodifferential equations with constant or variable coefficients with initial or boundary conditions. ElKalla [27] introduced a new technique for solving a class of quadratic integral and integrodifferential equations.
In this work, we will describe and adapt Adomian’s decomposition method to obtain an approximate solution for (1). As we will see, the method converges rapidly. The balance of this paper is as follows: in Section 2, we will give analysis of ADM for the problem; in Section 3 we will give three examples to demonstrate the method. Concluding remarks are given in the last section.
2. Analysis
We consider the nonlinear Volterra integrodifferential equations of the form in [4] as follows: where indicates the th derivative of with respect to , constants that define the initial conditions, and is nonlinear operator. In this work we take equal to or . Thus, applying the inverse operator to (1) yields where is obtained by using the initial conditions in [4] and is fold integration operator; that is, We obtain the zeroth component which is defined by all terms that arise from the initial conditions and from integrating the source terms. Then, decomposing the unknown function gives a sum of the component defined by the decomposition series Since the nonlinear terms or , then it can be expressed as where appropriate Adomian is polynomial which is generated form of the following formula [1–3, 6]:
Substituting (5) and (6) into (2) yields, The components are completely determined by using the recurrent formula for . It is useful to note that the recursive formula is constructed on the basis that the zeroth component is defined by all terms that arise from the initial conditions and from integrating the source terms. The remaining components , can be completely determined such that each term is computed by using the previous term. As a result, the components are identified, and the series solutions are thus entirely determined.
The term approximation is defined by which can be used for numerical approximation.
3. Test Problems
In this section, we report on numerical results of some examples, selected through integral and integrodifferential equations, solved by ADM. These examples can be solved analytically by reducing them to differential equations, and they are also solved numerically by Tau method in [20]. Here the aim is to solve these examples using the ADM given Section 2 and compare these results with the presented results in [20].
Problem 1. We mainly present the method using the algorithm given in Section 2. As a first example, consider the equation In order to illustrate the proposed method, we get zeroth component and obtain by using (9) to determine the other individual terms of the decomposition series. Thus and so on. Consequently, the series solution is obtained as so that the closed form of the solution is
Problem 2. We consider Fredholm integrodifferential equation which is given as follows [20]:
Proceeding as before, we obtain
Consequently, the series solution is found as In Table 1, ADM and TM values are presented which correspond to the various values of . As it is seen in this table, the values obtained by [20] and the results we obtained which are close and but present method better accuracy and easy to use than the TM. It is to be noted that only few iterations were needed to obtain the accuracy for approximate solutions. The overall errors can be made even much smaller by adding new terms of the decomposition. Thus the convergence would be seen more rapidly.

The numerical solutions showed that ADM is a very convenient method for such linear and nonlinear integral and integrodifferential equations. By using this method, it is possible to obtain more precise results than the traditional methods, with less calculations and consuming the less time.
Problem 3. In [4, 28], Wazwaz proposed that the construction of the zeroth component of the decomposition series can be defined in a slightly different way. In [4, 28], he assumed that if the zeroth component is and the function is possible to be divided into two parts such as and , then one can formulate the recursive algorithm in a form of a modified recursive scheme as follows:
We finally consider the Volterra integral equation in the following form [20]: Using the modified decomposition method, we first decompose the function into two parts as and , namely, Consequently, we obtain Other components for . Therefore, the exact solution follows immediately. It is clear that two components are calculated to determine the exact solution.
4. Concluding Remarks
In this paper, we calculated the approximate solutions of the integral and Volterra integrodifferential equations by using Adomian decomposition method. We demonstrated that the decomposition procedure is quite efficient in order to determine the solution in closed form by using initial and boundary conditions. Our present method avoids the tedious work needed by traditional techniques. In the studies by Hosseini and Shahmorad in [20], they spent more time, and boring operations were done to get approximate solutions by using TM. In our study, however, we got more accurate approximate solutions by using the initial condition in this method; Hosseini and Shahmorad in [20] obtained the approximate solutions for Problems 1 and 3, such that, the Tauerror is for and in Problem 1. Moreover, the exact solutions are obtained by our present method for Problems 1 and 3. Our method avoids the difficulties and massive computational work that usually arise from WaveletGalerkin, Tau, and finite difference methods.
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Copyright © 2013 Necdet Bildik and Mustafa Inc. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.