1. Introduction

In this work, we consider the singular two-point boundary value problems (BVPs) of the type subject to the boundary conditions or where , , , and are continuous functions on , and the parameters , , are real constants.

Problems of form (1) and (2) have been studied in many areas of science and engineering, for example, fluid mechanics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion process, geophysics, and so forth. Exact/approximate solutions of these problems are of great importance due to their wide applications in scientific research. Singular BVPs have been studied by several authors. Bataineh et al. [1] used the modified homotopy analysis method (MHAM) to search for approximate solutions of a certain class of singular two-point BVPs. Ravi Kanth and Aruna. [2] and Lu [3] used differential transform method (DTM) and variational iteration method (VIM), respectively, for solving singular two-point boundary value problems. Abu-Zaid and Gebeily [4] provided a finite difference approximation to the solution of the above problems. Ravi Kanth and Reddy [5] presented a method based on cubic splines for solving a class of singular two-point BVPs. The existence of a unique solution of (1) and (2) was discussed in [4].

Legendre operational matrix of differentiation, first proposed by Saadatmandi and Dehghan [6], is a powerful method for solving linear and nonlinear problems. They extended the application of Legendre polynomials to solve fractional differential equations. Recently, Pandey et al. [7] employed the Legendre operational matrix of differentiation to solve Lane-Emden type equations. Most recently, Kazem et al. [8] constructed a general formulation for the fractional-order Legendre functions to obtain the solution of fractional-order differential equations. To the best of the authors' knowledge, the present work demonstrates for the first time the applicability of the method of Legendre operational matrix of differentiation for obtaining the exact/approximate solutions of the singular two-point BVPs of the type (1) and (2). Several examples are studied to demonstrate the capability of the method.

2. Legendre Polynomials and Operational Matrix of Differentiation

The th-order Legendre polynomials, , on the interval are defined as

These polynomials on the interval , so-called shifted Legendre polynomials, can be defined by introducing the change of variable . The shifted Legendre polynomials denoted by can be obtained as where and . The analytic form of the shifted Legendre polynomial of degree is given by Note that and satisfy the orthogonality condition

A function square integrable in may be expressed in terms of shifted Legendre polynomials as where the coefficients are given by

In practice, we consider the -term-shifted Legendre polynomial so that where the shifted Legendre coefficient vector and the shifted Legendre vector are given by

The derivative of the vector can be expressed as where is the operational matrix of derivative. A general method of constructing such operational matrix of derivative could be presented as follows.(1)Differentiate analytically some polynomials of first degree, (2)express these derivatives as a linear combination of polynomials of lower degree, and(3)find a general formula.

Now, the general formula of the operational matrix of derivative is given by

For example, for odd we have

3. Applications of the Operational Matrix of Derivative

To solve (1) and (2) by means of the operational matrix of derivative method [6], we approximate and by the shifted Legendre polynomials as where the vector represents the nonhomogenous term. By using (12), (15), and (16) we have

Employing (15)–(18), the residual for (1) can be written as

Now, finding the solution given in (10) can be divided into two cases: linear and nonlinear.

3.1. Linear Case

For , we generate linear equations as in a typical tau method [9] by applying Also, by substituting boundary conditions (2) and (3) into (15) and (18) we have or Equations (20)–(22) generate set of linear equations, respectively. These linear equations can be solved for unknown coefficients of the vector . Consequently, given in (15) can be easily calculated.

3.2. Nonlinear Case

For , we first collocate (19) at points. For suitable collocation points, we use the first shifted Legendre roots of . These equations together with (21) or (22) generate nonlinear equations which can be solved using Newton’s iterative method. Consequently, given in (10) can be calculated.

4. Numerical Experiments

To illustrate the effectiveness of the presented method, we will consider the following examples of singular two-point BVPs.

Example 1. We first consider the linear singular two-point BVP [1, 10], subject to the boundary conditions of the form (2) The exact solution of (23) subject to (24) in the case of is
To solve (23) and (24) we apply the technique described in Section 3.1. With , we approximate the solution as According to (14), we have
Therefore, using (20) we obtain
Now, from (21) we have
Solving the linear system (28)-(29) yields Thus, which is the exact solution (25).

Example 2. Consider the linear singular two-point BVPs [1], subject to the boundary conditions of the form (3)
The exact solution of (32) subject to (33) in the case is
By the same manipulations as in the previous example and assuming , we have
Thus, which is the exact solution (35).

Example 3. We next consider the linear singular two-point BVPs [11], subject to the boundary conditions
The exact solution of (38) subject to (39) in the case is
By using the technique described in Section 3.1, with and , the values of the unknown matrix are given in Table 1. Figure 1 shows the absolute errors of in the interval for different values of . Obviously, increasing the number of terms of the Legendre polynomials has the effect of increasing the solution accuracy.

Table 1: The values of the unknown matrix for and of Example 3.

Figure 1: Absolute errors of in the interval for different values of of Example 3.

Example 4. Finally, consider the nonlinear singular two-point BVP [12] subject to the boundary conditions
The exact solution of (42) subject to (43) in the case is
We approximate the solution as
Here, and are as given in (27). Using (19), we have
Now, we collocate (46) at the first nine roots of , that is

Also (21) gives

Equations (46) and (48) generate 11 nonlinear equations which can be solved using Newton's iterative method. The values of the unknown matrix for , , and are given in Table 2. Figure 2 shows the absolute errors of in the interval for different values of .

Table 2: The values of the unknown matrix for , and of Example 4.

Figure 2: Absolute errors of in the interval for different values of of Example 4.

5. Conclusions

In this paper, the Legendre operational matrix of derivative was applied to solve a class of linear and nonlinear singular two-point BVPs. Different from other numerical techniques, only a small size operational matrix is required to provide the solution at high accuracy. It can be clearly seen in the paper that the proposed method is working well even with a few number of terms of the Legendre polynomials.

Acknowledgment

This work was supported by the Universiti Kebangsaan Malaysia's Grant no. DIP-2012-12.