A New Modification of Adomian Decomposition Method for Volterra Integral Equations of the Second Kind

Abstract

We propose a new modification of the Adomian decomposition method for Volterra integral equations of the second kind. By the Taylor expansion of the components apart from the zeroth term of the Adomian series solution, this new technology overcomes the problems arising from the previous decomposition method. The validity and applicability of the new technique are illustrated through several linear and nonlinear equations by comparing with the standard decomposition method and the modified decomposition method. The results obtained indicate that the new modification is effective and promising.

1. Introduction

The Adomian decomposition method (ADM) [1, 2] has been efficiently used to solve linear and nonlinear problems such as differential equations and integral equations. The method provides the solution as an infinite series in which each term can be easily determined. The rapid convergence of the series obtained by this method is thoroughly discussed by Cherruault et al. [3].

Recently, Wazwaz [4] proposed a reliable modified technique of ADM that accelerates the rapid convergence of decomposition series solution. The modified decomposition needs only a slight variation from the standard decomposition method. Although the modified decomposition method may provide the exact solution by using two iterations only and sometimes without using the so-called Adomian polynomials, its effectiveness is based on the assumption that the function 𝑓 can be divided into two parts, and thus the success of the modified method depends on the proper choice of 𝑓₁ and 𝑓₂.

More recently, another new modification of decomposition method was proposed by Wazwaz and El-Sayed [5]. In the new modification, the process of dividing 𝑓 into two components was replaced by a Taylor series of infinite components. The ideas have been shown to be computationally efficient in applying the proposed technique in several differential and integral equations. However, as will be seen from the examples below, the modified decomposition method will sometimes encounter obstacles to obtain the exact solution. After that, several authors further proposed expressing the function 𝑓 by the orthogonal polynomial series, such as Chebyshev series [6], Legendre series [7, 8], and Jacobi and Gegenbauer series [9].

In this paper, we introduce a new modification of decomposition method and make further progress beyond the achievements made so far in this regard. Several examples concerning integral equations are tested, and the results suggest that this new idea proposes a promising tool for the computation of integral equations both linear and nonlinear.

2. Description of the New Modification

Since the beginning of the 1980s, the Adomian decomposition method has been applied to a wide class of integral equations. To illustrate the procedure, consider the following Volterra integral equations of the second kind given by 𝑢(𝑥)=𝑓(𝑥)+𝜆^𝑥_𝑎𝐾(𝑥,𝑡)(𝐿(𝑢(𝑡))+𝑁(𝑢(𝑡)))d𝑡,𝜆≠0, (1) where the kernel 𝐾(𝑥,𝑡) and the function 𝑓(𝑥) are given real-valued functions, 𝜆 is a parameter, 𝐿(𝑢(𝑥)) and 𝑁(𝑢(𝑥)) are linear and nonlinear operators of 𝑢(𝑥), respectively, and the unknown function 𝑢(𝑥) is the solution to be determined.

In the decomposition method, we usually express the solution of (1) in a series form defined by 𝑢(𝑥)=^+∞_𝑘=0𝑢_𝑘(𝑥). (2) Moreover, the decomposition method identifies the nonlinear term 𝑁(𝑢(𝑥)) by the decomposition series 𝑁(𝑢(𝑥))=^+∞_𝑘=0𝐴_𝑘(𝑥), (3) where 𝐴_𝑘(𝑥) is the so-called Adomian polynomials, which can be evaluated by the following formula [10, 11]: 𝐴_𝑛=1d𝑛!d𝜆^𝑛𝑛𝑁^𝑛_𝑖=0𝜆^𝑖𝑢_𝑖,𝑛=0,1,2,…. (4) Substituting (2) and (3) into both sides of (1) gives ^+∞_𝑘=0𝑢_𝑘(𝑥)=𝑓(𝑥)+𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿^+∞_𝑘=0𝑢_𝑘(𝑡)+^+∞_𝑘=0𝐴_𝑘(𝑡)d𝑡. (5) In the following we outline the basic feature of the standard decomposition method [1, 2] and the modified decomposition method [4, 5].

The Standard Adomian Decomposition Method (SADM). By the standard decomposition method, the components 𝑢₀(𝑥), 𝑢₁(𝑥), 𝑢₂(𝑥),… of the solution 𝑢(𝑥) of (1), are completely determined in the following recurrence manner:𝑢₀(𝑥)(𝑥),𝑢=𝑓_𝑘+1(𝑥)=𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢_𝑘+𝐴_𝑘d𝑡,𝑘≥0. (6a)(6b)

Having determined the components 𝑢₀(𝑥), 𝑢₁(𝑥), 𝑢₂(𝑥),…, the solution 𝑢(𝑥) in a series form defined by (2) follows immediately.

The Modified Decomposition Method 1 (MADM1). The standard decomposition method by Adomian [1, 2] was modified by Wazwaz [4]. The modification is based on the assumption that the function 𝑓(𝑥) can be divided into two parts, namely, 𝑓₁(𝑥) and 𝑓₂(𝑥). Under this assumption we set 𝑓(𝑥)=𝑓₁(𝑥)+𝑓₂(𝑥). (7) Accordingly, a slight variation was proposed only for the components 𝑢₀(𝑥) and 𝑢₁(𝑥). The suggestion was that only the part 𝑓₁(𝑥) will be assigned to the zeroth component 𝑢₀(𝑥), whereas the remaining part 𝑓₂(𝑥) will be combined with the other terms given into (6b) to define 𝑢₁(𝑥). Consequently, the modified recursive relation 𝑢₀(𝑥)=𝑓₁(𝑥),𝑢₁(𝑥)=𝑓₂(𝑥)+𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢₀+𝐴₀𝑢d𝑡,_𝑘+1(𝑥)=𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢_𝑘+𝐴_𝑘d𝑡,𝑘≥1, (8) was developed.

It is important to note here that there were some conclusions made in [4]. First, the slight variation in reducing the number of terms of 𝑢₀ will result in a reduction of the computational work and will accelerate the convergence. Second, this slight variation in the definition of the components 𝑢₀ and 𝑢₁ may provide the solution by using two iterations only. Third, there is no need sometimes to evaluate the so-called Adomian polynomials required for the nonlinear equations.

The Modified Decomposition Method 2 (MADM2). In [5], it is pointed that sometimes it may be useful to express 𝑓(𝑥) in Taylor series as follows: 𝑓(𝑥)=^+∞_𝑖=0𝑓_𝑖(𝑥). (9) Meanwhile, a new recursive relationship in the following form can be obtained: 𝑢₀(𝑥)=𝑓₀(𝑥),𝑢₁(𝑥)=𝑓₁(𝑥)+𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢₀+𝐴₀𝑢d𝑡,_𝑘+1(𝑥)=𝑓_𝑘+1(𝑥)+𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢_𝑘+𝐴_𝑘d𝑡,𝑘≥1. (10) According to (10), the terms 𝑢₀(𝑥),𝑢₁(𝑥),𝑢₂(𝑥),… of the solution 𝑢(𝑥) follow immediately, and the solution 𝑢(𝑥) can be obtained using (2).

It is evident that algorithm (10) reduces the number of terms involved in each component, and hence the size of calculations is minimized compared to the standard Adomian decomposition method. Moreover, this reduction of terms in each component facilitates the construction of Adomian polynomials for nonlinear operators [5].

The New Modified Decomposition Method. As indicated earlier, MADM1 may provide the exact solution by using two iterations only and sometimes without any need for Adomian polynomials, but its effectiveness depends on the proper choice of 𝑓₁(𝑥) and 𝑓₂(𝑥), which may need quite a little computational work. Regarding SADM and MADM2, sometimes one will inevitably encounter the difficulties as will be seen from the examples below. In this new modification, we obtain the first two terms by𝑢₀(𝑥)(𝑥),𝑢=𝑓₁(𝑥)=𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢₀+𝐴₀d𝑡, (11a)(11b)

which are the same as the results by SADM. However, sometimes 𝑢₁(𝑥) is too complicated to continue or is analytically impossible. Here, we suggest that 𝑢₁(𝑥) may be expressed in Taylor series form as 𝑢₁(𝑥)=^+∞_𝑖=0𝑢_1𝑖(𝑥), (12) next, we set 𝑢₂(𝑥)=𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢₁+𝐴₁d𝑡, (13) where the Adomian polynomial 𝐴₁ can be evaluated by (4) with 𝑢₀(𝑥) and 𝑢₁(𝑥), which will be obtained by (11a) and (12), respectively. Also, we suggest that the same series expansion can be acted upon 𝑢₂(𝑥), that is, 𝑢₂(𝑥)=^+∞_𝑖=0𝑢_2𝑖(𝑥). (14) Similarly, we set 𝑢_𝑘(𝑥)=𝜆^𝑥_𝑎𝐾(𝑥,𝑡)𝐿𝑢_𝑘+𝐴_𝑘d𝑡,𝑘≥3, (15) and their Taylor expansions given by 𝑢_𝑘(𝑥)=^+∞_𝑖=0𝑢_𝑘𝑖(𝑥),𝑘≥3. (16) By combining (11a), (12), (14), and (16), the solution 𝑢(𝑥) can be obtained using the series (2).

In practice, for some problems, we have to determine a few terms in the series such as 𝑢(𝑥)=∑^𝑀_𝑖=0𝑢_𝑖(𝑥) by truncating the series at certain term, regardless of the standard decomposition method and the modified decomposition method. Because of the uniformly convergence property of the infinite series, a few terms will attain the maximum accuracy.

To give a clear overview of this study, we have chosen several linear and nonlinear Volterra integral equations to testify the validity and applicability of the new modification. For comparison, examples will also be discussed by using SADM, MADM1, and MADM2.

3. Application for Linear Integral Equations

Example 1. Consider the linear Volterra integral equation [12] 𝑢(𝑥)=sin𝑥+𝑒−𝑒^cos𝑥−^𝑥₀𝑒^cos𝑡𝑢(𝑡)d𝑡. (17)
Following suggestions (6a) and (6b) of SADM, the second component 𝑢₁(𝑥) of the solution 𝑢(𝑥) cannot be obtained because of the impossible integration, which also appears by using MADM2. If we turn to MADM1, the six different choices of 𝑓₁(𝑥) and 𝑓₂(𝑥) need quite a little computational work only if we set 𝑓₁(𝑥)=sin𝑥 and 𝑓₂(𝑥)=𝑒−𝑒^cos𝑥 directly. In the following part, we solve (17) by the new modification with the help of computer algebraic system Maple 12.
According to suggestions (11a) and (11b), we obtain 𝑢₀(𝑥)=sin𝑥+𝑒−𝑒^cos𝑥,𝑢₁(𝑥)=−^𝑥₀𝑒^cos𝑡𝑢₀(𝑡)d𝑡. (18) We then set the Taylor expression for 𝑢₁(𝑥) by 𝑢₁(𝑥)1=−2𝑒𝑥²−16𝑒²𝑥³+16𝑒𝑥⁴+1𝑒12²𝑥⁵−31720𝑒𝑥⁶−⋯. (19) And also, we set the Taylor expansion for 𝑢_𝑖(𝑥), 𝑖=2,3,4,… because the evaluation of integrals, obtained by applying (13) and (15) in this example, is analytically impossible. In this way, we have 𝑢₂(𝑥)=−^𝑥₀𝑒^cos𝑡𝑢₁(𝑡)=1d𝑡6𝑒²𝑥³+1𝑒24³𝑥⁴−1𝑒12²𝑥⁵−1𝑒36³𝑥⁶𝑢+⋯,₃(𝑥)=−^𝑥₀𝑒^cos𝑡𝑢₂(𝑡)1d𝑡=−𝑒24³𝑥⁴−1𝑒120⁴𝑥⁵+1𝑒36³𝑥⁶𝑢+⋯,₄(𝑥)=−^𝑥₀𝑒^cos𝑡𝑢₃(𝑡)=1d𝑡𝑒120⁴𝑥⁵+1𝑒720⁵𝑥⁶𝑢−⋯,₅(𝑥)=−^𝑥₀𝑒^cos𝑡𝑢₄(𝑡)1d𝑡=−𝑒720⁵𝑥⁶⋮−⋯, (20) Thus, the solution in a series form is given by 𝑢(𝑥)=sin𝑥+𝑒−𝑒^cos𝑥−1+𝑒2𝑥²+16𝑥⁴−31𝑥720⁶+⋯. (21) Combining (21) with the Taylor expansion of 𝑒−𝑒^cos𝑥1=𝑒2𝑥²−16𝑥⁴+31𝑥720⁶−⋯, (22) the solution in a closed form is given by 𝑢(𝑥)=sin𝑥. (23)

Example 2. Consider the linear Volterra integral equation [5, 13] 𝑢(𝑥)=1+sinh𝑥−cosh𝑥+^𝑥₀𝑢(𝑡)d𝑡. (24)
In this example, both SADM and MADM2 can fulfil the exact solution effectively. Whereas, MADM1 needs quite a little computational work to obtain the exact solution only if we set 𝑓₁(𝑥)=sinh𝑥 and 𝑓₂(𝑥)=1−cosh𝑥 directly. In the following part, we solve (24) by the new modification.
Proceeding as before, we set 𝑢₀(𝑥)𝑢=1+sinh𝑥−cosh𝑥,₁(𝑥)=^𝑥₀𝑢₀(𝑡)d𝑡=cosh𝑥−sinh𝑥+𝑥−1. (25) We then set the Taylor expression for 𝑢₁(𝑥) by 𝑢₁(𝑥)=1𝑥2!²−1𝑥3!³+1𝑥4!⁴−1𝑥5!⁵+1𝑥6!⁶−⋯, (26) and also 𝑢₂(𝑥)=^𝑥₀𝑢₁(𝑡)=1d𝑡𝑥3!³−1𝑥4!⁴+1𝑥5!⁵−1𝑥6!⁶+1𝑥7!⁷𝑢−⋯,₃(𝑥)=^𝑥₀𝑢₂(𝑡)=1d𝑡𝑥4!⁴−1𝑥5!⁵+1𝑥6!⁶−1𝑥7!⁷+1𝑥8!⁸𝑢−⋯,₄(𝑥)=^𝑥₀𝑢₃(𝑡)=1d𝑡𝑥5!⁵−1𝑥6!⁶+1𝑥7!⁷−1𝑥8!⁸+1𝑥9!⁹𝑢−⋯,₅(𝑥)=^𝑥₀𝑢₄(𝑡)=1d𝑡𝑥6!⁶−1𝑥7!⁷+1𝑥8!⁸−1𝑥9!⁹⋮+⋯, (27) The solution in a series form is given by 𝑢(𝑥)1=sinh𝑥−cosh𝑥+1+𝑥2!²+1𝑥4!⁴+1𝑥6!⁶+⋯, (28) and in a closed form by 𝑢(𝑥)=sinh𝑥. (29)

Example 3. Consider the system of linear Volterra integral equations [12] 𝑢(𝑥)+=cos𝑥−𝑥sin𝑥_𝑥0(()𝑢(𝑡)()𝑣(𝑡𝑣(𝑥)+sin𝑥−𝑡+cos𝑥−𝑡))d𝑡,=sin𝑥−𝑥cos𝑥_𝑥0(cos(𝑥−𝑡)𝑢(𝑡)−sin(𝑥−𝑡)𝑣(𝑡))d𝑡. (30) In this example, it is both difficult to obtain the exact solution by using SADM and MADM2. And also, like the problem mentioned in preceding examples, MADM1 needs much computational work to obtain the exact solution only if we set 𝑓₁(𝑥)=cos𝑥 and 𝑓₂(𝑥)=−𝑥sin𝑥 for the first equation and 𝑓₁(𝑥)=sin𝑥 and 𝑓₂(𝑥)=−𝑥cos𝑥 for the second equation, directly. In the following parts, we solve (30) by the new modification.
Proceeding as before, the system (30) is transformed into a set of recursive relations given by 𝑢₀(𝑥)𝑣=cos𝑥−𝑥sin𝑥,₀(𝑥)𝑢=sin𝑥−𝑥cos𝑥,₁(𝑥)=^𝑥₀sin(𝑥−𝑡)𝑢₀(𝑡)+cos(𝑥−𝑡)𝑣₀(𝑡)𝑣d𝑡,₁(𝑥)=^𝑥₀cos(𝑥−𝑡)𝑢₀(𝑡)−sin(𝑥−𝑡)𝑣₀(𝑡)d𝑡. (31) We then set the Taylor expansion for 𝑢₁(𝑥) and 𝑣₁(𝑥) by 𝑢₁(𝑥)=12𝑥²−1𝑥12⁴+1𝑥240⁶𝑣−⋯,₁(𝑥)2=𝑥−3𝑥³+7𝑥120⁵−⋯. (32) Similarly, 𝑢₂(𝑥)=^𝑥₀sin(𝑥−𝑡)𝑢₁(𝑡)+cos(𝑥−𝑡)𝑣₁(𝑡)=1d𝑡2𝑥²−16𝑥⁴+1𝑥80⁶𝑣−⋯,₂(𝑥)=^𝑥₀cos(𝑥−𝑡)𝑢₁(𝑡)−sin(𝑥−𝑡)𝑣₁(𝑡)=1d𝑡𝑥60⁵−1𝑥840⁷𝑢+⋯,₃(𝑥)=^𝑥₀sin(𝑥−𝑡)𝑢₂(𝑡)+cos(𝑥−𝑡)𝑣₂(𝑡)=1d𝑡𝑥24⁴−1𝑥240⁶𝑣+⋯,₃(𝑥)=^𝑥₀cos(𝑥−𝑡)𝑢₂(𝑡)−sin(𝑥−𝑡)𝑣₂(𝑡)=1d𝑡6𝑥³−1𝑥24⁵+1𝑥420⁷𝑢−⋯,₄(𝑥)=^𝑥₀sin(𝑥−𝑡)𝑢₃(𝑡)+cos(𝑥−𝑡)𝑣₃(𝑡=1)d𝑡𝑥24⁴−1𝑥144⁶𝑣+⋯,₄(𝑥)=^𝑥₀cos(𝑥−𝑡)𝑢₃(𝑡)−sin(𝑥−𝑡)𝑣₃(𝑡)=1d𝑡𝑥2520⁷−1𝑥45360⁹𝑢+⋯,₅(𝑥)=^𝑥₀sin(𝑥−𝑡)𝑢₄(𝑡)+cos(𝑥−𝑡)𝑣₄(𝑡)=1d𝑡𝑥720⁶−1𝑥10080⁸𝑣+⋯,₅(𝑥)=^𝑥₀cos(𝑥−𝑡)𝑢₄(𝑡)−sin(𝑥−𝑡)𝑣₄(𝑡=1)d𝑡𝑥120⁵−1𝑥840⁷𝑢+⋯,₆(𝑥)=^𝑥₀sin(𝑥−𝑡)𝑢₅(𝑡)+cos(𝑥−𝑡)𝑣₅(𝑡)=1d𝑡𝑥720⁶−1𝑥6720⁸⋮+⋯, (33) The solution in a series form is given by 𝑢(𝑥)1=cos𝑥−𝑥sin𝑥+𝑥𝑥−6𝑥³+1𝑥120⁵,𝑣(𝑥)1−⋯=sin𝑥−𝑥cos𝑥+𝑥1−2𝑥²+1𝑥24⁴−⋯, (34) and in a closed form by (𝑢(𝑥),𝑣(𝑥))=(cos𝑥,sin𝑥). (35)

4. Application for Nonlinear Integral Equations

Example 4. Consider the nonlinear Volterra integral equation [12, 14] 𝑢(𝑥)=𝑒^𝑥−13𝑥𝑒^3𝑥+13𝑥+^𝑥₀𝑥𝑢³(𝑡)d𝑡. (36)
Following suggestions (6a) and (6b) of 𝑆𝐴𝐷𝑀, the second component 𝑢₁(𝑥) described via the sum of twenty-seven terms is too complicated for no confidence to continue. Meanwhile, there does not exist noise terms [15], which leads to the result quickly. By MADM1, the six different choices of 𝑓₁(𝑥) and 𝑓₂(𝑥) need quite a little computational work as the examples given before, unless we set 𝑓₁(𝑥)=𝑒^𝑥 and 𝑓₂(𝑥)=−(1/3)𝑥𝑒^3𝑥+(1/3)𝑥 directly. The solution can be obtained by MADM2 exactly. In the following part, we solve (36) by the new modification.
According to suggestions (11a) and (11b), we obtain 𝑢₀(𝑥)=𝑒^𝑥−13𝑥𝑒^3𝑥+13𝑢𝑥,₁(𝑥)=^𝑥₀𝑥𝑢³₀(𝑡)d𝑡. (37) We then set the Taylor expression for 𝑢₁(𝑥) by 𝑢₁(𝑥)=𝑥²+32𝑥³+12𝑥⁴−32𝑥⁵−218𝑥⁶−227𝑥120⁷+⋯. (38) And also, we set the Taylor expansion for 𝑢_𝑖(𝑥), 𝑖=2,3,4,… because of the complicated integration. In this way, we have 𝑢₂(𝑥)=3^𝑥₀𝑥𝑢²₀(𝑡)𝑢₁(𝑡)d𝑡=𝑥⁴+218𝑥⁵+21𝑥10⁶−25𝑥12⁷𝑢−⋯,₃(𝑥)=3^𝑥₀𝑥𝑢²₀(𝑡)𝑢₂(𝑡)+𝑢₀(𝑡)𝑢²₁(𝑡)=6d𝑡5𝑥⁶+69𝑥16⁷⋮+⋯, (39) where the Adomian polynomials for nonlinear term 𝑢³ were calculated by using (4). The solution in a series form is given by 𝑢(𝑥)=𝑒^𝑥−13𝑥𝑒^3𝑥+13𝑥+𝑥²+32𝑥³+32𝑥⁴+98𝑥⁵+27𝑥40⁶+27𝑥80⁷+⋯. (40) Combining (40) with the Taylor expansion of −13𝑥𝑒^3𝑥+13𝑥=−𝑥²−32𝑥³−32𝑥⁴−98𝑥⁵−27𝑥40⁶−27𝑥80⁷−⋯, (41) the solution in a closed form is given by 𝑢(𝑥)=𝑒^𝑥. (42)

Example 5. Consider the nonlinear Volterra integrodifferential equation of the first kind [12] ^𝑥₀(𝑥−𝑡)𝑢²(𝑡)d𝑡+^𝑥₀𝑒^𝑥−𝑡𝑢^′(𝑡)d𝑡=𝑥𝑒^𝑥+14𝑒^2𝑥−14−12𝑢(0)𝑥,=1. (43)
By integrating the second integral by parts and using the initial conditions, (43) can be converted to Volterra integral equation of the second kind given by 𝑢(𝑥)=𝑒^𝑥+𝑥𝑒^𝑥+14𝑒^2𝑥−14−12𝑥−^𝑥₀(𝑥−𝑡)𝑢²(𝑡)+𝑒^𝑥−𝑡𝑢(𝑡)d𝑡. (44)

Following suggestions (6a) and (6b) of SADM, 𝑢₁(𝑥) consists of fourteen terms and 𝑢₂(𝑥) twenty-nine terms. It is so complicated that we can give up. The similar obstacle derived from the selections of 𝑓₁(𝑥) and 𝑓₂(𝑥) will be encountered if we use MADM1. By using MADM2, it is of certain difficulty to obtain the exact solution through the first finite terms of the series solution. In the following part, we solve (43) by the new modification.

Following as before, we set 𝑢₀(𝑥)=𝑒^𝑥+𝑥𝑒^𝑥+14𝑒^2𝑥−14−12𝑢𝑥,₁(𝑥)=−^𝑥₀(𝑥−𝑡)𝑢²₀(𝑡)+𝑒^𝑥−𝑡𝑢₀(𝑡)d𝑡=−𝑥−2𝑥²−116𝑥³−29𝑥24⁴−41𝑥60⁵𝑢−⋯,₂(𝑥)=−^𝑥₀2(𝑥−𝑡)𝑢₀(𝑡)𝑢₁(𝑡)+𝑒^𝑥−𝑡𝑢₁(𝑡)=1d𝑡2𝑥²+76𝑥³+43𝑥⁴𝑢+⋯,₃(𝑥)=−^𝑥₀(𝑥−𝑡)2𝑢₀(𝑡)𝑢₂(𝑡)+𝑢²₁(𝑡)+𝑒^𝑥−𝑡𝑢₂(𝑡)1d𝑡=−6𝑥³−12𝑥⁴𝑢−⋯,₄(𝑥)=−^𝑥₀2(𝑥−𝑡)𝑢₀(𝑡)𝑢₃(𝑡)+𝑢₁(𝑡)𝑢₂(𝑡)+𝑒^𝑥−𝑡𝑢₃(𝑡)=1d𝑡𝑥24⁴+⋯, (45) where Adomian polynomials for nonlinear term 𝑢² were calculated by using (4). The solution in a series form is given by 𝑢(𝑥)=𝑒^𝑥+𝑥𝑒^𝑥+14𝑒^2𝑥+−14−323𝑥−2𝑥²−56𝑥³−13𝑥⁴−⋯. (46) Combining (46) with the Taylor expansion of 𝑥𝑒^𝑥+14𝑒^2𝑥=14+323𝑥+2𝑥²+56𝑥³+13𝑥⁴+⋯, (47) the solution in a closed form is given by 𝑢(𝑥)=𝑒^𝑥. (48)

5. Discussion and Conclusion

In this work, we have carefully developed a new modification of Adomian decomposition method based on the Taylor expansion of the components apart from the zeroth term for Volterra integral equations. The new technology overcomes such obstacles as the complicated integrals and the impossible integrals analytically, which arise from either SADM or MADM2. If the exact solution exists in the zeroth component, MADM1 needs extensive workload for the suitable choice of 𝑓₁(𝑥) and 𝑓₂(𝑥), while the new modification can proceed in a fixed manner. Therefore, it is appropriate for one to implement through the mathematical software Maple without manual tedious calculations.

The idea has been shown to be computationally efficient in applying the proposed technique to several linear and nonlinear Volterra integral equations and integrodifferential equations. Also, through the examples, it is seen that the new modification suits for those integral equations where the exact solution appears as part of the zeroth term. It is also interesting to point out that one can obtain the approximate solution of integral equations as mentioned in [16, 17] simply by slightly varying the new modification of this work.

Acknowledgment

The work was supported by the Scientific Research Fund of Zhejiang Provincial Education Department of China (no. Y201120655), K.C. Wong Magna Fund in Ningbo University.