#### Abstract

We introduce an efficient recursive scheme based on Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems. This approach is based on a modification of the ADM; here we use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components. In fact, we develop the recursive scheme without any undetermined coefficients while computing the solution components. Unlike the classical ADM, the proposed method avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. The uniqueness of the solution is discussed. The convergence and error analysis of the proposed method are also established. The accuracy and reliability of the proposed method are examined by four numerical examples.

#### 1. Introduction

In this paper we propose an efficient recursive scheme based on Adomian decomposition method (ADM) for solving a class of nonlinear singular boundary value problems (SBVPs) that arising in physiology [1–5]: Here , , and are any finite real constants. We assume that for , the functions and are continuous and . Singular boundary value problems frequently arise in the modeling of many problems in biological, physical, and engineering sciences. For example, it arises in the study of steady-state oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics [6] with and , where and are positive real constants. In heat conduction model in human head [7, 8] with , , where and are positive reals.

There is considerable literature on the numerical treatment of singular boundary value problems [1–11] and many of the references therein. The main difficulty of the problem (1) is that the singularity behavior occurs at . Various efficient numerical techniques have been used to deal with such SBVPs, such as finite difference method (FDM) [9], cubic spline method (CSM) [1], and B-spline method (BSM) [2]. Although, these techniques are efficient and have many advantages, a huge amount of computational work is needed which combines some root-finding methods to obtain accurate numerical solution especially for nonlinear SBVPs. Recently, some newly developed approximate methods were also applied to deal with such SBVPs, such as variational iteration method (VIM) [5, 11], Adomian decomposition method (ADM) [3], and modified Adomian decomposition method (MADM) [4]. It is well known that solving nonlinear two-point boundary value problems (BVPs) using ADM/MADM is always a computationally involved task. Since it requires computation of unknown constants in a sequence of nonlinear or more difficult transcendental equations. Moreover, in some cases these unknown constants may not be uniquely determined and this may be the major disadvantage of ADM/MADM for solving nonlinear BVPs.

The aim of this work is to introduce an efficient recursive scheme to overcome the difficulties that occur in the ADM or MADM for solving nonlinear SBVPs (1). This approach is based on a modification of the ADM; here we use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components. In fact, we develop the recursive scheme without any undetermined coefficients while computing the solution components. Unlike the ADM, the proposed scheme avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximations of the solution are obtained in the form of series with easily computable components. The sufficient condition that guarantees the existence of a unique solution of the problem (1) is proved. The convergence analysis and the error estimation are also discussed. Finally, the accuracy and reliability of the proposed method are examined by four numerical examples.

##### 1.1. The Review of ADM

In this subsection, we briefly describe the ADM for solving SBVPs (1). Recently, many researchers [3, 4, 12–23] have applied the ADM to deal with many different scientific models. According to the ADM we rewrite (1) in a operator form as where is a second-order linear differential operator. In [24] Wazwaz defines the inverse operator as

Operating on both sides of (2) and using the condition , we have

The main idea of the ADM depends on decomposing the solution and the nonlinear function by an infinite series as where are Adomian’s polynomials which can be constructed for various classes of nonlinear functions with the formula given in [13] as Substituting the series (5) into (4), we obtain Upon matching both sides of (7), the ADM is given by where is unknown constant to be determined. Having determined the components , the series solution of follows immediately with the undetermined . Hence, the -term truncated series solution is given as It should be noted that the approximate solution depends on the unknown constant . This constant can be obtained approximately by imposing the boundary condition at on , which leads to a sequence of transcendental equations . For example, consider According to the ADM (8), with , we obtain the components as Consequently, the -term approximate series solution can be obtained as By imposing the boundary condition at on , we obtain a sequence of transcendental equations , as follows

However, solving such transcendental equations for requires additional computational work, and may not be uniquely determined. This may be the major disadvantage of the ADM for solving two-point boundary value problems.

#### 2. The Proposed Recursive Scheme

In this section, we propose a new efficient recursive scheme based on the ADM for solving nonlinear SBVPs (1). To overcome the singular behavior at , we rewrite SBVPs (1) in operator form as subject to the boundary conditions where is a second-order linear singular differential operator. Twofold integral operator regarded as the inverse operator of is proposed as In order to establish the new efficient recursive scheme, we operate on left hand side of (14) and impose the boundary condition ; we obtain Thus we have where is unknown constant to be determined.

We now operate on both sides of (14) and use (18); we get To eliminate unknown constant , we impose the boundary condition on (19) which leads to Substituting the value of into (19), we obtain Note that (21) does not involve any unknown constants to be determined.

We now decompose the solution by the series as where are the Adomian's polynomials [13]. In 2010, Duan [25, 26] reported several new efficient algorithms for rapid computer generation of the Adomian polynomials. Recently, El-Kalla [27] suggested another programmable formula for Adomian polynomials: where is partial sum of the series solution .

Substituting the series (22) into (21), we obtain Comparing both sides of (24), the proposed scheme is given by the following recursive scheme:

The recursive scheme (25) provides complete determination of solution components of the solution . Hence, the truncated -term series solution can be obtained as

Unlike ADM or MADM, the proposed recursive scheme (25) does not require any computation of unknown constants.

#### 3. Convergence Analysis

In this section, we will give the sufficient condition that guarantees existence of a unique solution of (28) in Theorem 1. Then we discuss the convergence analysis in Theorem 2 and the error analysis in Theorem 3 of the proposed scheme (25) for SBVPs (1). Note that many researcher [28–31] have also established the convergence of the ADM for solving differential as well as integral equations. Let be a Banach space with the norm Note that (21) can be written in operator form as where is given by

Theorem 1. *Let be a Banach space with the norm given by (27). Assuming that the nonlinear function satisfies the Lipschitz condition; that is, . Further, let be a constant defined as
**
If , then (28) has a unique solution in .*

*Proof. *For any , we have
Using Lipschitz continuity of , we obtain
Hence, we have
If , then is contraction mapping and hence by the Banach contraction mapping theorem, (28) has a unique solution in .

We now prove the convergence of the proposed scheme (25). Let be a sequence of partial sums of the series solution . Using (25) and (26), we have Using (23) in (34), it follows that which is equivalent to the following operator equation:

Note that the formulation (36) is used to prove Theorems 2 and 3.

Theorem 2. *Let be the nonlinear operator defined by (28) as contractive; that is,
**
Then the sequence of partial sums defined by (26) converges to the exact solution .*

*Proof. *Using the relation (36) and the estimate (33), we have
Thus we have
We now show that the sequence is convergent. Now for all , , with , we consider
Since , , and , it follows
Taking limit as , we obtain
Hence is cauchy sequence in . Hence there exits in such that . Note that is the exact solution of (28) as
This completes the proof.

Theorem 3. *Let be the exact solution of the operator equation (28). Let be the sequence of partial sums of series solution defined by (26). Then there holds
**
where .*

*Proof. *For any and using the estimate (41), we have
Since , fixing and letting in above estimate, we obtain
Since, we have
so
Combining the estimates (46) and (48) and using , we obtain
This completes the proof.

#### 4. Numerical Examples

In this section, we demonstrate the accuracy and reliability of the proposed scheme (25) by implementing it to four SBVPs, arising various physical models. All the numerical results obtained by the proposed (25) are compared with the results obtained by various numerical methods.

*Example 1. *Consider the following nonlinear SBVPs:
where and are positive constants involving the reaction rate and Michaelis constants. Thus we take and as used in [1, 3, 5].

According to the proposed method (25) with , , and . Consequently, we have the following recursive scheme: Using the formula (6), Adomian’s polynomials for with are obtained as

*Case (**).* Using (51) and (52), the components of the solution are obtained as
The 6-term approximate series solution is given by

*Case (**).* In a similar manner, we obtain the components of solution as
Hence, we obtain the 6-term approximate series solution as

*Case (**).* The components of the solution are obtained as
The series solution is given as follows:
Comparison of the numerical results obtained by the proposed method (25), the VIM used in [5], and the CSM used in [1] are shown in Tables 1, 2, and 3. These tables show that the numerical results obtained by the proposed scheme (25) are comparable with those in [1, 5]. In addition, we have also presented numerical results for in Tables 1 and 3.

*Example 2. *Consider the following nonlinear SBVPs [9]:
The analytical solution is .

According to the proposed method (25) with , , , and . Consequently, we have the following recursive scheme as Similar to the previous example, Adomian’s polynomials for are given as Using (60) and (61), we obtain the components of the solution as Hence, we obtain approximate series solution as For quantitative comparison, we now define the maximum absolute error by Here is the -term approximate series solution obtained by the proposed method (25) and is the analytical solution. Table 4 shows a comparison between the numerical results obtained by the proposed method (25) and the FDM used in [9]. The numerical results obtained by the proposed method (25) show good agreement with those in [9].

*Example 3. *Consider the following nonlinear SBVPs [3]:
The analytical solution is , where .

According to the proposed method (25) with , , , and , we have the following recursive scheme: Proceeding as before, Adomian's polynomials for with are given as Making use of (66) and (67), we can calculate the components of solution as Hence we obtain approximate series solution as

Table 5 shows a comparison of the numerical results obtained by the proposed method (25), the FDM used in [9], and the BSM used in [2]. Once again, the approximate series solutions using proposed method (25) are comparable with the results reported earlier in the literature [2, 9].

*Example 4. *Consider the following nonlinear SBVPs [8]:

According to the proposed method (25) with , , , and , we have the following scheme: Adomian’s polynomials, for with are given as Making use of (71) and (72), we obtain the component as follows: Hence, the -term approximate series solution is given by

Table 6 presents a comparison of the approximate solutions obtained by the proposed method (25), the FDM used in [32], and tangent chord technique used in [8]. We again obtain the numerical results using the proposed scheme (25) which are comparable with those in [8, 32].

#### 5. Conclusions

(i)In this work, the simplicity, efficiency, and reliability of the proposed method (25) have been tested by solving four nonlinear SBVPs that arise in physiology.(ii)The accuracy of the numerical results indicates that the proposed method is well suited for the solution of SBVPs. It has also been shown that only 6 terms are sufficient to obtain comparable solution with those in [2, 9].(iii)Unlike ADM or MDAM, the proposed method (25) does not require any computation of unknown constants. In fact, it provides a direct scheme to obtain approximations to the solution.(iv)Unlike the finite difference and the cubic spline methods, the proposed method does not require any linearization and discretization of the variables.(v)Uniqueness of the solution of (28) has been discussed in Theorem 1. The convergence analysis and the error estimation of the proposed scheme (25) have also been established in Theorems 2 and 3.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the anonymous referees for their useful comments and suggestions that led to improvement of the presentation and content of this paper. The authors thankfully acknowledge the financial assistance provided by the Council of Scientific and Industrial Research (CSIR), New Delhi, India.