Abstract

We introduce an effective methodology for solving a class of linear as well as nonlinear singular two-point boundary value problems. This methodology is based on a modification of Adomian decomposition method (ADM) and a new two fold integral operator. We use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components of solution. Thus, we develop modified recursive scheme without any undetermined coefficients while computing the successive solution components. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. However, most of earlier recursive schemes using ADM do require computation of undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the present technique. The results reveal that the method is very effective, straightforward, and simple.

1. Introduction

We consider the following class of singular two-point boundary value problems [1–5]: subject to the boundary conditions where , and   are any finite constants and . We assume that, for , the function and are continuous and . In particular, the problem (1) arises very frequently in applied sciences and in physiological studies, for example, in the study of steady-state oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics [6] and distribution of heat sources in the human head [7]. In particular, when and (1) is known as Thomas-Fermi equation [8], given the singular equation Recently, there has been much interest in the study of singular two-point boundary value problems of type (1), (see, e.g., [1–5, 8–15]) and many of the references therein. The main difficulty of problem (1) is that the singularity behavior occurs at . A lot of methods have been applied to tackle this singular boundary value problem. In [9], a standard three-point finite difference scheme was considered with uniform mesh for the solution of problem for . In [1, 5], the finite difference methods were used to obtain the numerical solutions. In [4], a numerical method based on Green’s function was used to obtain numerical solution of the same problem. A novel approach that combines a modified decomposition method with the cubic B-spline collocation technique is presented in [14] to obtain approximate solution with high accuracy. Recently, in [15], a new modified decomposition method was applied to tackle these problems. These numerical methods have many advantages, but a huge amount of computational work is required to obtain accurate numerical solution especially for nonlinear problems.

1.1. Adomian Decomposition Method (ADM)

In this subsection, we briefly describe standard ADM or MADM for nonlinear second-order equation.

Recently, many researchers [3, 8, 12–24] have shown interest in the study of ADM for different scientific models. Adomian [20] asserted that the ADM provides an efficient and computationally suitable method for generating approximate series solution for a large class of differential equations.

According to Wazwaz [24], (1) in operator form is given as where is a linear differential operator defined as The inverse operator is given as (see [24]) Operating the inverse linear operator on both sides of (4) yields Next, we decompose the solution and the nonlinear function by an infinite series as where are Adomian’s polynomials that can be constructed for various classes of nonlinear functions with the formula given in [17] as Substituting the series (8) into (7), we obtain Upon matching both sides of (10), the standard ADM is given by and modified ADM is given as Having determined the components , recurrently, the series solution of follows immediately with the undetermined coefficient , is as yet to be determined [12, 21, 22]. For numerical purpose, the -term truncated series may be used to give the approximate solution.

The decomposition method has been applied to solve nonlinear boundary value problems for ordinary differential equations by several researchers [3, 12–16, 18, 19, 21–23]. Solving such problems using standard ADM or MADM is always a computationally involved task as it requires the computation of undetermined coefficients in a sequence of nonlinear algebraic equations which increases the computational work (see [3, 14, 15, 21–23]). A major disadvantage of the earlier methods for solving nonlinear BVPs is that we need to solve a sequence of growingly higher-order polynomials or more difficult transcendental equations [3, 21, 23]. For example, consider nonlinear equation Applying the standard ADM (11) with initial guess , where to above problem (14), we obtain the solution components as and -term approximate solution as Note that the above calculations become more and more complicated and cannot lead to exact results for the other approximations. To obtain approximate solution, the boundary condition is imposed into , and solving for , we can obtain the approximation. However, solving this transcendental equation for requires additional computational work, and may not be uniquely determined (see [3, 15, 21, 23]).

In order to avoid solving such nonlinear algebraic or transcendental equations for two-point boundary value problems, in [16], extended ADM was introduced for nonsingular problems with Dirichlet boundary conditions. The method proposed in [13] was based on the new modification of ADM. Recently, a special modified inverse linear integral operators for higher-order boundary value problem was given in [25].

In this paper, a new modification of the ADM is proposed to overcome the difficulties occurred in the standard ADM or MADM for solving nonlinear singular boundary value problems (1). To obtain a new modified decomposition method, we operate our proposed twofold integral operator on both sides of (1) and use all the boundary conditions to establish the recursion scheme for the solution components of the solution. We propose a modified recursion scheme which does not require the computation of unknown constants, that is, without solving a sequence of growingly higher-order polynomial or difficult transcendental equations to obtain unknown constant [3, 14, 15, 21, 23]. The main advantage of our proposed method is that it provides a direct recursive scheme for solving the singular boundary value problem.

The rest of the paper is organized as follows. In Section 2, the description of the proposed recursive scheme based on new definition of differential operator and its inverse operator is given. In Section 3, we illustrate our method with numerical results along with the graphical representation. In Section 4, the conclusion is given.

2. New Modified Decomposition Method

In this section, we propose a new modified ADM based on two-fold integral operator for solving linear as well as nonlinear singular two-point boundary value problems.

To overcome the singular behavior at , we again rewrite (1) which can be rewritten in operator form as where is the linear differential operator.

Twofold integral operator regarded as the inverse operator of is proposed as To establish a new modified recursive scheme, we operate on the left-hand side of (17) and use the boundary condition , which yields where .

Now, we again operate the inverse operator on both sides of (17) and use (20), which gives Imposing the boundary condition into (21), we get For simplicity, we set .

Substituting the value of into (21), we obtain Note that the right-hand side of (23) does not involve any undetermined coefficients.

Next, we decompose the solution by series as and the nonlinear function is decomposed by series as where are Adomian’s polynomials [17].

Substituting the series (24) and (25) into (23), we obtain Upon matching both sides of (26), the new modified decomposition method is given by It should be noted that the new modified recursive scheme (27) requires no additional computational work as it does not involve any unknown constant. This slight change plays a major role in minimizing the computational work. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete the calculation of the solution by several earlier modified recursion schemes using the ADM or MADM [3, 14, 22, 23]. The above new modified scheme (27) gives the complete determination of solution components of solution , and hence, the approximate series solution can be obtained by adding the solution components . However, for numerical purpose, the truncated -term approximate series solution is given by and the limiting value of sequence converges to exact solution . We remark that the convergence of Adomian decomposition method for differential and integral equations has already been established by many authors [26–28].

3. Numerical Illustrations and Discussions

In this section, the new modified ADM (27) is implemented for tackling singular boundary value problems (1). We demonstrate the effectiveness of this method with five examples. All the numerical results obtained by the proposed method are compared with known results. Furthermore, the maximum error functions and approximate solutions are plotted.

Example 1. Consider the nonlinear singular two-point boundary value problem [10] with exact solution .
We apply scheme (27) to (29), where and , , , and , scheme (27) for (29) can be read as follows The Adomian’s polynomials for nonlinear function with are obtained by formula (9) given as For the demonstration purpose, we chose some specific values of and .
For , , by using (30) and (31), we obtain the successive solution components of solution as
For , , we again use (30) and (31) to obtain the successive solution components as

Remark 2. In a similar manner, we can calculate any solution components of solution for different values and . For simplicity, we have not listed all the solution components for different values of and , but we have calculated the first ten solution components.

Now, we define error functions as and the maximum absolute error is given as We plot approximate solutions , for , and exact solution in Figures 1 and 2 for different values of and . The error functions , for , are plotted in Figures 3 and 4. In addition, the maximum absolute errors , for , are listed in Tables 1 and 2.

Figure 1 

Comparison of exact solution and approximate solutions , of Example 1 when .

Figure 2 

Comparison of exact solution and approximate solutions , of Example 1 when .

Figure 3 

Error functions , , of Example 1 when .

Figure 4 

Error functions , of Example 1 when .

Example 3. Consider the nonlinear singular two-point boundary value problem with exact solution .
On applying scheme (27) to (35) where , , , and , scheme (27) for (35) becomes as follows: where .
As given above, the Adomian polynomials for nonlinear function with are given by Using (36) and (37), we have the successive solution components as In order to verify how close the approximate solutions are to the exact solution, we plot approximate solutions , and exact solution in Figure 5 and conclude that the approximate solution is very close to exact solution . In Figure 6, the error functions , for , are plotted. From Figure 6 we can see that when increases, the error decreases. In addition, the maximum absolute errors , for , are listed in Table 3.

Figure 5 

Comparison of exact solution and approximate solutions of Example 3.

Figure 6 

Error functions , , of Example 3.

Example 4. Consider the nonlinear singular two-point boundary value problem [15, 23] with exact solution .
We apply scheme (27) to (39), where and , , and , scheme (27) for (39) becomes Proceeding as before, the Adomian’s polynomials for with are given as Making use of (40) and (41), we have successive solution components of solution as To verify whether the proposed scheme (27) leads to accurate solution, we plot approximations , for , and exact in Figure 7. The figure shows good agreement with exact solution . In Figure 8 the error functions , for , are plotted. In addition, numerical results for maximum absolute error , for , are listed in Table 4.