A Comparison between Adomian Decomposition and Tau Methods

Abstract

We present a comparison between Adomian decomposition method (ADM) and Tau method (TM) for the integro-differential 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 integro-differential 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, integro-differential 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 integro-differential 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 integro-differential 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 Runge-Kutta methods for approximate solutions of some predator-prey 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, El-Sayed and Abdel-Aziz [19] introduced a comparison of the ADM and the Wavelet-Galerkin method for the solution of integro-differential 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 integro-differential equations given by (1). Batiha et al. [21] presented the variational iteration method (VIM) and the ADM for solving nonlinear integro-differential equations. Fariborzi Araghi and Sadigh Behzadi [22–24] solved nonlinear Volterra-Fredholm integro-differential 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 integro-differential equations with the kernel functions including derivative type of unknown solution. Ben Zitoun and Cherruault [26] presented a method for solving nonlinear integro-differential equations with constant or variable coefficients with initial or boundary conditions. El-Kalla [27] introduced a new technique for solving a class of quadratic integral and integro-differential 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 integro-differential equations of the form in [4] as follows: 𝑢^(𝑛)(𝑥)=𝑓(𝑥)+^𝑥₀𝐾()(𝑡)𝑢𝑥,𝑡𝑁𝑢𝑑𝑡,_(𝑚)(0)=𝑐_𝑚,0≤𝑚≤(𝑛−1), (1) where 𝑢^(𝑛)(𝑥) indicates the 𝑛th derivative of 𝑢(𝑥) with respect to 𝑥, 𝑐_𝑚 constants that define the initial conditions, and 𝑁𝑢 is nonlinear operator. In this work we take 𝑁𝑢 equal to 𝑢² or 𝑢𝑢_𝑥. Thus, applying the inverse operator 𝐿⁻¹ to (1) yields 𝑢(𝑥)=^𝑛−1_𝑘=01𝑐𝑘!^𝑚𝑥^𝑘+𝐿⁻¹𝑓(𝑥)+𝐿^−1𝑥₀𝐾(𝑥,𝑡)𝑁𝑢(𝑡)𝑑𝑡, (2) where ∑^𝑛−1_𝑘=0(1/𝑘!)𝑐_𝑚𝑥^𝑘 is obtained by using the initial conditions in [4] and 𝐿⁻¹ is 𝑛-fold integration operator; that is, 𝐿⁻¹(⋅)=^𝑥₀⋯^𝑥_0𝑛-times(⋅)𝑑𝑥⋯𝑑𝑥_𝑛-times. (3) We obtain the zeroth component 𝑢₀(𝑥)=^𝑛−1_𝑘=01𝑐𝑘!^𝑚𝑥^𝑘+𝐿⁻¹𝑓(𝑥), (4) which is defined by all terms that arise from the initial conditions and from integrating the source terms. Then, decomposing the unknown function 𝑢(𝑥) gives a sum of the component defined by the decomposition series 𝑢(𝑥)=^∞_𝑛=0𝑢_𝑛(𝑥). (5) Since the nonlinear terms 𝑁𝑢=𝑢² or 𝑁𝑢=𝑢𝑢_𝑥, then it can be expressed as 𝐹(𝑢)=𝑁𝑢=^∞_𝑛=0𝐴_𝑛, (6) where 𝐴_𝑛 appropriate Adomian is polynomial which is generated form of the following formula [1–3, 6]: 𝐴₀=𝐹𝑢₀,𝐴₁=𝑢₁𝐹^′𝑢₀,𝐴₂=𝑢₂𝐹^′𝑢₀+12𝑢²₁𝐹^′′𝑢₀,𝐴₃=𝑢₃𝐹^′𝑢₀+𝑢₁𝑢₂𝐹^′′𝑢₀+1𝑢3!³¹𝐹^′′′𝑢₀,⋮ (7)

Substituting (5) and (6) into (2) yields, ^∞_𝑛=0𝑢_𝑛(𝑥)=^𝑛−1_𝑘=01𝑐𝑘!^𝑚𝑥^𝑘+𝐿⁻¹𝑓(𝑥)+𝐿^−1𝑥₀𝐾(𝑥,𝑡)^∞_𝑛=0𝐴_𝑛(𝑡)𝑑𝑡. (8) The components 𝑢₁(𝑥),𝑢₂(𝑥),… are completely determined by using the recurrent formula 𝑢₁(𝑥)=𝐿^−1𝑥₀𝐾(𝑥,𝑡)𝐴₀(𝑡),⋮𝑢𝑑𝑡_𝑛(𝑥)=𝐿^−1𝑥₀𝐾(𝑥,𝑡)𝐴_𝑛−1(𝑡)𝑑𝑡, (9) for 𝑛≥0. It is useful to note that the recursive formula is constructed on the basis that the zeroth component 𝑢₀(𝑥) is defined by all terms that arise from the initial conditions and from integrating the source terms. The remaining components 𝑢_𝑛(𝑥),𝑛≥0, can be completely determined such that each term is computed by using the previous term. As a result, the components 𝑢₀(𝑥),𝑢₁(𝑥),𝑢₂(𝑥),… are identified, and the series solutions are thus entirely determined.

The 𝑛-term approximation 𝜙_𝑛 is defined by 𝜙_𝑛=^𝑛−1_𝑘=0𝑢_𝑘(𝑥), (10) 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 integro-differential 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 𝑦(𝑠)−^𝑠₀𝑦(𝑡)𝑑𝑡=1,0≤𝑠≤1. (11) In order to illustrate the proposed method, we get zeroth component 𝑦₀=1 (12) and obtain 𝑦₁(𝑥),𝑦₂(𝑥),… by using (9) to determine the other individual terms of the decomposition series. Thus 𝑦₁=^𝑠₀𝑦₀(𝑡)𝑦𝑑𝑡=𝑠,₂=^𝑠₀𝑦₁(𝑡)1𝑑𝑡=𝑠2!²,𝑦₃=^𝑠₀𝑦₂(𝑡)1𝑑𝑡=𝑠3!³,⋮ (13) and so on. Consequently, the series solution is obtained as 𝑦(𝑠)=^∞_𝑛=0𝑦_𝑛(𝑠)=𝑦₀+𝑦₁+𝑦₂+𝑦₃1+⋯=1+𝑠+𝑠2!²+1𝑠3!³+⋯, (14) so that the closed form of the solution is 𝑦(𝑠)=𝑒^𝑠. (15)

Problem 2. We consider Fredholm integro-differential equation which is given as follows [20]: 𝑦^′′(𝑠)(𝑠)+1−𝑦20¹₀𝑡³⁹𝑦(𝑡)𝑑𝑡=−𝑠²−2𝑠+2111𝑦(0)344400,0≤𝑠≤1,−𝑦^′(0)=0,𝑦(1)+𝑦^′(1)=9. (16)

Proceeding as before, we obtain 𝑦₀=𝑦(0)+𝑠𝑦^′(0)+𝐿⁻¹−𝑠²−2𝑠+2111344400=2+2𝑠+2111𝑠688800²−13𝑠³−1𝑠12⁴,𝑦₁=^𝑠₀^𝑠₀𝑦₀(𝑠)−1𝑑𝑠𝑑𝑠20^𝑠₀^𝑠₀¹₀𝑡³⁹𝑦₀(𝑡)𝑑𝑡𝑑𝑠𝑑𝑠=𝑠²−29772923𝑠13349952000²+13𝑠³+2111𝑠8265600⁴−1𝑠60⁵−1𝑠360⁵,𝑦₂=^𝑠₀^𝑠₀𝑦₁(𝑠)−1𝑑𝑠𝑑𝑠20^𝑠₀^𝑠₀¹₀𝑡³⁹𝑦₁(𝑡)=𝑑𝑡𝑑𝑠𝑑𝑠−5478328678327𝑠7049842652160000²[phone]𝑠160199424000⁴+1𝑠60⁵+2111𝑠247968000⁶−1𝑠2520⁷−1𝑠201160⁸,⋮ (17)

Consequently, the series solution is found as 𝑦(𝑥)=^∞_𝑛=0𝑦_𝑛(𝑥)=𝑦₀+𝑦₁+𝑦₂+⋯=2+2𝑠+2111𝑠688800²−13𝑠³−1𝑠12⁴+𝑠²−29772923𝑠13349952000²+13𝑠³+⋯. (18) In Table 1, ADM and TM values are presented which correspond to the various values of 𝑠. 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.

Table 1 

Numerical results for Problem 2.

The numerical solutions showed that ADM is a very convenient method for such linear and nonlinear integral and integro-differential 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 𝑦₀=𝑓 and the function 𝑓 is possible to be divided into two parts such as 𝑓₁ and 𝑓₂, then one can formulate the recursive algorithm in a form of a modified recursive scheme as follows: 𝑦₀(𝑠)=𝑓₁,𝑦₁(𝑠)=𝑓₂+𝐿⁻¹𝑓(𝑥)+𝐿^−1𝑥₀𝐾(𝑥,𝑡)𝑦₁(𝑡),𝑦𝑑𝑡_𝑛+1(𝑠)=𝐿^−1𝑥₀𝐾(𝑥,𝑡)𝑦_𝑛(𝑡)𝑑𝑡,𝑛≥1. (19)

We finally consider the Volterra integral equation in the following form [20]: 𝑦(𝑠)=1+120𝑠−1001−𝑒^−𝑠+^𝑠₀100𝑒^𝑡−𝑠−120𝑦(𝑡)𝑑𝑡,0≤𝑠≤20. (20) Using the modified decomposition method, we first decompose the function 𝑓(𝑠) into two parts as 𝑓₁ and 𝑓₂, namely, 𝑓₁(𝑠)𝑓=1,₂(𝑠)=120𝑠−1001−𝑒^−𝑠. (21) Consequently, we obtain 𝑦₀(𝑠)𝑦=1,₁(𝑠)=120𝑠−1001−𝑒^−𝑠+^𝑠₀100𝑒^𝑡−𝑠−120𝑦₀(𝑡)𝑑𝑡=0. (22) Other components 𝑦_𝑛(𝑠)=0 for 𝑛≥2. Therefore, the exact solution 𝑦(𝑠)=1 (23) 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 integro-differential 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 Tau-error is 2.73127𝑒−08 for 𝑛=10 and 𝑠=1.00 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 Wavelet-Galerkin, Tau, and finite difference methods.