1 Introduction

Two-point singular boundary value problems for ordinary differential equations arise very frequently in many branches of applied mathematics and physics such as chemical reactions, atomic calculations, gas dynamics, nuclear physics, atomic structures, in the theory of shallow membrane caps, and study of positive radial solutions of nonlinear elliptic equations and physiological studies. In most of the cases, singular boundary value problems in general cannot be solved analytically. So these problems must be handled by various numerical techniques. However, the numerical treatment of the singular boundary value problems has always been far from trivial due to the singularity.

The objective of this paper is to propose an effective technique to solve a general nonlinear derivative-dependent singular boundary value problems (SBVPs). The proposed technique is based on the work of Singh et al. [1], where we transform the original nonlinear weakly singular boundary value problem with Dirichlet and Robin boundary conditions into an equivalent nonlinear Fredholm integral equation before establishing the recursive scheme for the solution. In this paper we consider the following class of derivative-dependent nonlinear singular boundary value [2–5]

(1.1)

where a>0, b and c are any finite real constants. The condition p(0)=0 says that the problem (1.1) is singular and if q is allowed to be discontinuous at x=0 then the problem (1.1) is called doubly singular [2]. Throughout the paper we assume the following conditions on p, q and f(x,y,py′):

(E 1):

p∈C[0,1]∩C 1(0,1] with p>0 in (0,1];

(E 2):

q>0 in (0,1], q∈L 1(0,1] and q is not identically zero;

(E 3):

\(\int _{0}^{1}\frac{1}{p(x)}\int _{0}^{x}q(s)ds dx<\infty\);

(F 1):

let f(x,y,py′) is continuous on \(D_{1}=\{{[0,1]\times \mathbb{R}\times \mathbb{R}}\}\) and is not identically zero;

(F 2):

let f(x,y,py′) satisfies the Lipschitz condition

(1.2)

where l 1 and l 2 are Lipschitz constants.

Past couple of decades, there has been much interest in the study of singular two-point boundary value problems, [1–4, 6–18] and many of the references therein. The main difficulty of (1.1) is that the singular behavior occurs at x=0. In [2, 3, 6], authors have discussed the existence and uniqueness of solution of the problem (1.1). A lot of numerical techniques have been applied to tackle the particular case of singular boundary value problem (1.1). For instant, the cubic spline and finite difference methods were carried out in [7–9]. Although, these numerical techniques have lot of advantages, but a huge amount of computational work is involved which combines some root-finding methods for obtaining accurate numerical solution especially for nonlinear singular boundary value problems.

Recently, some newly developed numerical-approximate methods have also been applied to handle some particular cases of (1.1). Such as, the ADM and MADM have been used in [10, 12]. The homotopy analysis method (HAM) was introduced in [13]. To solve equation (1.1) using ADM, MADM, and HAM is always a computationally involved task as it requires the computation of undetermined coefficients in a sequence of nonlinear algebraic or more difficult transcendental equations which increases the computational work (see [10, 12, 13]). Moreover, the undetermined coefficients may not be uniquely determined and this may be the main disadvantage of these methods for solving nonlinear two-point BVPs.

The variational iteration method (VIM) and its modified version for solving the particular cases of SBVP (1.1) have been employed in [15, 17, 19]. The VIM gives good approximations only when the problem is linear or nonlinear BVPs with nonlinearity of the form y n,yy′,y′n… etc., but the method suffers when the nonlinear function is of the form e y,ln(y),siny,sinhy… etc., this may be one of the major disadvantage of VIM for solving difficult nonlinear problems (see [17]).

1.1 Review of ADM

In this subsection, we shall briefly describe ADM for nonlinear second order differential equations.

In the recent past, a lot of researchers [1, 10–12, 14, 16, 17, 20–33] have expressed their interest in the study of ADM for various scientific models. Adomian [22] asserted that the ADM provides an efficient and computationally worthy method for generating approximate series solution for a large class of differential equations.

According to the ADM, the operator form of (1.1) can be written as

(1.3)

where \(\mathcal{L}\equiv\frac{d^{2}}{dx^{2}}\) is linear second-order differential operator, Ry=(−p′/p)y′ and Ny=(q/p)f(x,y,py′) represents the nonlinear term.

The inverse operator of \(\mathcal{L}\) can be defined as

$$\begin{aligned} \mathcal{L}^{-1}[\cdot]=\int _{0}^{x}\int _{0}^{x}[ \cdot] dx dx. \end{aligned}$$
(1.4)

Operating the inverse operator \(\mathcal{L}^{-1}[\cdot]\) on both sides of (1.3), we have

(1.5)

Then the solution y and the nonlinear function Ny are decomposed by infinite series

(1.6)

where A n are Adomian’s polynomials that can be constructed for various classes of nonlinear functions with the formula given by Adomian and Rach [23]

$$\begin{aligned} A_n=\frac{1}{n!}\frac{d^n}{d \lambda^n} \Biggl[N \Biggl(\sum_{k=0}^{\infty} y_k\lambda^k \Biggr) \Biggr]_{\lambda=0},\quad n=0,1,2, \ldots. \end{aligned}$$
(1.7)

New efficient algorithms and subroutines in ‘MATHEMATICA’ for rapid computer-generation of the Adomian polynomials to high orders are provided in [34–36]. In [36, 37], authors proposed a new modification of the ADM (MADM) based on a new formula for Adomian’s polynomials by rearranging Taylor series components of the analytic function N(y). By defining the series \(\psi_{n}=\sum_{j=0}^{n} y_{j}\), and according to new MADM, A k ’s are obtained as following:

$$\begin{aligned} A_k=N(\psi_{k})-\sum_{j=0}^{k-1}A_j. \end{aligned}$$
(1.8)

This formula enforces many additional terms to the calculation processes in (1.7), which implying faster convergence [36, 37].

Substituting the series (1.6) into (1.5), we obtain

(1.9)

On comparing both sides of (1.9), the ADM admits the following recursive scheme

$$\begin{aligned} y_0=y(0)+y'(0)x,\qquad y_{n+1}= \mathcal{L}^{-1} [Ry]+\mathcal{L}^{-1}[A_{n}],\quad n\geq0, \end{aligned}$$
(1.10)

that will lead to the complete determination of components y n , and the series solution of y follows immediately with the undetermined coefficient y(0), and the unknown constant y(0) will be determined later imposing the boundary conditions at x=1 (see [12, 17, 24, 25]). Then the n-term truncated approximate series solution is given by \(\psi_{n}(x)=\sum_{m=0}^{n} y_{m}(x)\).

In [11, 12, 16, 21, 24, 25], the researchers have used the ADM or MADM for solving nonlinear boundary value problems for ordinary differential equations. To solve two-point boundary value problems using 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 [10, 12, 16, 17, 21, 24, 25]). In other words, major disadvantage of the ADM for solving nonlinear boundary value problems is to solve a sequence of growingly higher order polynomials or more difficult transcendental equations for undetermined coefficients [21]. In order to avoid solving a sequence of difficult transcendental equations for a two-point boundary value problems. In [36, 38], authors proposed a new modified ADM for solving second order as well as higher order non-singular boundary value problems with Dirichlet as well as Robin boundary conditions, where they transformed the original nonlinear boundary value problem into an equivalent nonlinear Fredholm-Volterra integral equation for the solution before designing the recursion scheme. Later, in [1, 14, 20, 32], authors proposed some new modification of ADM for solving second-order singular boundary value problems with Dirichlet, Dirichlet and Robin, Neumann and Robin boundary conditions.

In this paper, we propose an efficient numerical algorithm to overcome the difficulties occur in ADM or MADM for solving nonlinear singular boundary value problems. To establish this algorithm, we first construct Green’s function to establish the recursion scheme for solution components. The proposed method provides a direct scheme for obtaining approximations for the solution of the singular boundary value problem without solving a sequence of difficult transcendental equations for unknown constant. In addition, the convergence analysis and error estimation of the proposed method is established. Finally, some numerical examples are included to demonstrate the accuracy of the proposed method.

2 Adomian decomposition method with Green’s function

In this section, we propose an efficient numerical algorithm, which is based on Green’s function and Adomian’s polynomial for solving a general nonlinear singular two-point boundary value problems of the form (1.1). To do this, we first consider the corresponding homogeneous problem of (1.1)

(2.1)

The unique solution of (2.1) can easily be found and given by

(2.2)

Now we again consider the SBVP (1.1) with homogeneous boundary conditions

(2.3)

The Green’s function of (2.3) can easily be constructed as

(2.4)

The derivation of Green’s function is provided in the appendix.

Now, using (2.2) and (2.4), the SBVP (1.1) can be converted into following integral equation as

(2.5)

In other words, the integral equation (2.5) is equivalent to SBVP (1.1). It should also be noted that the right hand side of (2.5) does not involve any undetermined coefficients.

We next decompose the solution y by an infinite series as:

$$\begin{aligned} y=\sum_{n=0}^{\infty}y_n, \end{aligned}$$
(2.6)

and the nonlinear function f(x,y,py′) by an infinite series

$$\begin{aligned} f\bigl(x,y,py'\bigr)=\sum _{n=0}^{\infty} A_n, \end{aligned}$$
(2.7)

where A n are Adomian’s polynomials [23].

Substituting the series (2.6) and (2.7) in (2.5), we obtain

$$\begin{aligned} \sum_{n=0}^{\infty}y_n= \frac{c}{a}+\int _{0}^{1} G(x,\xi)q(\xi) \sum _{n=0}^{\infty} A_n d\xi. \end{aligned}$$
(2.8)

Upon comparing both sides of (2.8), we obtain the following scheme as follows:

$$\begin{aligned} \left . \begin{array}{@{}l@{}} \displaystyle y_0=\frac{c}{a},\\ \displaystyle y_{n+1}=\int_{0}^{1} G(x,\xi) q(\xi) A_nd\xi,\quad n\geq0. \end{array} \right \} \end{aligned}$$
(2.9)

The solution components y n can be calculated using the scheme (2.9) and hence, the n-term approximate series solution is obtained as

$$\begin{aligned} \psi_n(x)=\sum_{j=0}^{n} y_j. \end{aligned}$$
(2.10)

Unlike existing ADM or MADM, the proposed scheme (2.9) avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients.

3 Convergence analysis

In this section, we shall suggest an alternative approach for proving the convergence analysis of proposed recursive scheme for singular boundary value problem (1.1). We remark that many authors [37, 39–42] have also established the convergence of ADM for differential and integral equations. To do this, let be a Banach space with the norm

(3.1)

where,

It is well known that is Banach space with the norm (3.1) (see, pp. 45, [2]).

Note that Eq. (2.5) can be written in the operator equation form as

$$\begin{aligned} y=\mathcal{N}y, \end{aligned}$$
(3.2)

where \(\mathcal{N}y\) is given by

(3.3)

We next discuss the existence of the unique solution of Eq. (3.2). To do this, we first prove the following Lemma.

Lemma 3.1

Let the assumptions (E 1)–(E 3) hold and the Green’s function of (2.3) is given by G(x,ξ), then

  1. (i)

    \(m_{1}:=\max_{0< x \leq1} |\int_{0}^{1} G(x,\xi)q(\xi)d\xi |<\infty\),

  2. (ii)

    \(m_{2}:=\max_{0< x \leq1} |\int_{0}^{1} p(x)G_{x}(x,\xi) q(\xi)d\xi |<\infty\), where \(G_{x}(x,\xi)=\frac{\partial G(x,\xi)}{\partial x}\).

Proof

(i) This is trivial, it follows from the assumptions (E 1)–(E 3) and from the Green’s function (2.4).

(ii) From (6.4), we see that

(3.4)

Hence, we obtain c 2=max0<x,ξ≤1|p(x)G x (x,ξ)|<∞.

Now again using the assumption (E 2), we have

(3.5)

Hence it follows that \(m_{2}=\max_{0< x \leq1} |\int_{0}^{1}p(x)G_{x}(x,\xi)q(\xi)d\xi |<\infty\). □

Theorem 3.1

Let be Banach space with norm given by (3.1). Also assume that the nonlinear function f(x,y,py′) satisfies the Lipschitz condition (F 2). Let m=max{m 1,m 2} and l=max{l 1,l 2}, where the constants m 1 and m 2 given as in Lemma 3.1 and l 1 and l 2 are Lipschitz constants. If δ=2lm<1, then Eq. (3.2) has a unique solution in  .

Proof

Using the Lemma 3.1, we have for any y 1, ,

Now using the Lipschitz continuity of f, we have

where l=max{l 1,l 2}. Thus, we have

(3.6)

Similarly, we have

Using the Lipschitz continuity of f, we obtain

Hence

(3.7)

Combining the estimates (3.6) and (3.7), we obtain

Thus, we have

(3.8)

where δ=2lm and m=max{m 1,m 2}. If δ<1, then is contraction mapping and hence by the Banach contraction mapping theorem, Eq. (3.2) has a unique solution in . □

In order to establish the convergence of proposed scheme (2.9), we write the scheme (2.9) in operator form as follows. For that, we define the sequence {ψ n } such that ψ n =y 0+y 1+⋯+y n , is a sequence of partial sums of the series solution \(\sum_{j=0}^{\infty}y_{j}\). Since

or as an approximation, we have

(3.9)

Now, by using (2.8) and (3.9), we have

Thus the operator form of the scheme can be written as

(3.10)

Note that the formulation (3.10) is used to prove the Theorems 3.2 and 3.3. Next, we give the convergence of the sequence {ψ n } to the exact solution y of (3.2).

Theorem 3.2

Let \(\mathcal{N}y\) be the nonlinear operator defined by (3.3) which satisfies the Lipschitz condition with Lipschitz constant δ<1. If ∥y 0∥<∞, then there holds ∥y k+1∥≤δ∥y k ∥, k=0,1,2,… and the sequence {ψ n } defined by (3.10) converges to the exact solution y.

Proof

Since

we see that y k+1=ψ k+1−ψ k , k=1,2,….

Now we show that the sequence {ψ n } is convergent sequence.

Using the estimate (3.8) and (3.10), we have

Hence, we obtain

Now for all \(n,m\in \mathbb{N}\), with n≥m, we have

$$\begin{aligned} \|\psi_{n}-\psi_{m}\| =&\bigl\|(\psi_n- \psi_{n-1})+(\psi_{n-1}-\psi_{n-2})+\cdots+( \psi_{m+1}-\psi_{m}) \bigr\| \\ \leq& \|\psi_n-\psi_{n-1}\|+\|\psi_{n-1}- \psi_{n-2}\|+\cdots+\|\psi_{m+1}-\psi_{m}\| \\ \leq& \delta^{n} \|y_{0} \|+ \delta^{n-1} \|y_{0} \|+\cdots+ \delta^{m+1}\| y_0\| \\ \leq& \delta^{m+1}\bigl( 1+\delta + \delta^{2} +\cdots+ \delta^{n-m-1} \bigr)\|y_0\| \\ \leq& \frac{\delta^{m+1}(1-\delta^{n-m})}{1-\delta}\|y_0\|. \end{aligned}$$

Since 0≤δ<1, implies (1−δ n−m)≤1 and since ∥y 0∥<∞, it follows that

$$\begin{aligned} \|\psi_{n}-\psi_{m}\|\leq \frac{\delta^{m+1}}{1-\delta}\|y_0\|, \end{aligned}$$
(3.11)

which converges to zero, that is, ∥ψ n −ψ m ∥→0, as m→∞. Hence {ψ n } is Cauchy sequence in . Since is Banach space, the sequence {ψ n } must be convergent in . Hence there exits ψ in such that lim n→∞ ψ n =ψ. But, we have \(y=\sum_{n=0}^{\infty}y_{n}=\lim_{n\rightarrow \infty} \psi_{n}\), that is, y=ψ which is exact solution of Eq. (3.2). □

Theorem 3.3

Let y be the exact solution of (3.2). Let ψ m be the sequence of approximate series solution obtained by (3.10). Then there holds

$$\max_{0\leq x\leq1}\Biggl \vert y-\sum_{j=0}^{m} y_j\Biggr \vert \leq\frac{\delta^{m+1}}{1-\delta}\|y_0\|.$$

Proof

Using the inequality (3.11) for \(n\geq m,~n,~m \in \mathbb{N}\), we have

Since lim n→∞ ψ n =y, fixing m and letting n→∞ in above estimate, we obtain

which completes the proof using \(\psi_{m}= \sum_{j=0}^{m}y_{j}\). □

4 Numerical illustrations

In this section, the proposed scheme (2.9) is applied to solve singular two point boundary value problems. In order to check the accuracy of the proposed method, we have consider three nonlinear singular examples. All the numerical results obtained by proposed method are compared with known results.

Example 4.1

Consider the following nonlinear SBVP

$$\begin{aligned} \left . \begin{array}{@{}l@{}} \displaystyle\bigl(x^3 y'(x)\bigr)'=x^3 \biggl(\frac{1}{8y^2(x)}-\frac{a_0}{y(x)}-b_0 x^{2\gamma-4} \biggr),\quad 0<x\leq1,\\ \displaystyle \lim_{x\rightarrow0+}x^3y'(x)=0,\quad y(1)=1, \end{array} \right \} \end{aligned}$$
(4.1)

where a 0≥0, b 0>0 and γ>1 are any real constants, arises in the theory of shallow membrane caps [3, 5, 15].

According to proposed scheme (2.9), we have p(x)=q(x)=x 3, a=1, b=0 and c=1. Consequently:

$$\begin{aligned} \left . \begin{array}{@{}l@{}} y_0=1,\\ \displaystyle y_{n+1}=\int_{0}^{1}G(x,\xi) \xi^{3} A_n(y_0,y_1,\ldots,y_{n}) d\xi,\quad n\geq0, \end{array} \right \} \end{aligned}$$
(4.2)

where the Green’s function is

(4.3)

The Adomian’s polynomials for \(f(y)=\frac{1}{8y^{2}(x)}-\frac{a_{0}}{y(x)}-b_{0} x^{2\gamma-4}\) about y 0, are given as:

(4.4)

Using (4.2), (4.3) and (4.4), for a 0=0, b 0=0.5 and γ=2, we obtain components as

In similar manner, for a 0=0.5, b 0=1 and γ=1.5, using (4.2), (4.7) and (4.4), we have

The comparison of approximate solution obtained by proposed recursive scheme and VIM used in [15] is presented in Table 1. From these results, we see that proposed recursive scheme provides good approximations which is comparable with those in [15].

Table 1 Numerical solution and residual error for Example 4.1, when a 0=0, b 0=0.5, γ=2
Full size table

Furthermore, since the exact solution of (4.1) is not known, we instead investigate the absolute residual error function, which is a measure of how well the approximation satisfies the original nonlinear differential equation as

$$R_n(x)= \biggl \vert \bigl(x^3 \psi_{n}'(x)\bigr)'-x^3 \biggl(\frac{1}{8\psi_{n}^2(x)}-\frac{a_0}{\psi_{n}(x)}-b_0 x^{2\gamma-4} \biggr)\biggr \vert , \quad 0<x\leq1 $$

where ψ n (x) is a sequence of approximate solution. Finally, the residual error R n , n=2,4,6 is presented in Tables 1 and 2 for various values of the parameters a 0,b 0 and γ. Also, in Figs. 1 and 2, we plot absolute residual error function R n , and observe that as the number of iterations increases the residual error decreases.

Fig. 1
figure 1

Residual error functions R 8(x) (dashed line), R 10(x) (dotted line) and R 12(x) (solid line) of Example 4.1, when a 0=0,b 0=0.5,γ=2

Full size image
Fig. 2
figure 2

Residual error functions R 8(x) (dashed line), R 10(x) (dotted line) and R 12(x) (solid line) of Example 4.1, when a 0=0.5,b 0=1,γ=1.5

Full size image
Table 2 Numerical solution and residual error for Example 4.1, when a 0=0.5, b 0=1, γ=1.5
Full size table

Example 4.2

Consider nonlinear derivative-dependent singular two-point boundary value problem

$$\begin{aligned} \left . \begin{array}{@{}l@{}} \displaystyle \bigl(x^{\alpha}y'\bigr)'= x^{\alpha+\beta-2} \beta e^{y} \bigl( -x y'-\alpha-\beta+1 \bigr),\quad 0<x\leq 1,\\ \displaystyle \lim_{x\rightarrow0+}x^\alpha y'(x)=0,\quad y(1)=-\ln (5 ), \end{array} \right \} \end{aligned}$$
(4.5)

with exact solution y(x)=−ln(4+x β), where α≥0 and β>0 are any constants.

Comparing with the proposed recursive scheme (2.9), we have p(x)=x α, q(x)=x α+β−2 and a=1, b=0, and c=−ln(5). Consequently, we have following scheme

(4.6)

where the Green’s function is

(4.7)

We now calculate the Adomian’s polynomials for nonlinear term f=−β(xe y y′+e y(α+β−1)) about y 0 using the formula (1.7) as:

(4.8)

For the demonstration purpose, we pick some specific values of α and β.

For α=2, β=1, using (4.6), (4.7) and (4.8), we obtain the components y n as

For α=2, β=3, making use of (4.6), (4.7) and (4.8), we have:

We now define error function as E n (x)=|ψ n (x)−y(x)| and the maximum absolute errors by

(4.9)

where y(x) is exact solution, and ψ n (x) is n-term approximate series solution.

In order to verify the efficiency of proposed recursive scheme, the maximum absolute error E (n), for n=3,4,…,10 are listed in Tables 3, 4, 5 and 6 for various values of α and β.

Table 3 Maximum absolute error of Example 4.2, when β=1
Full size table
Table 4 Maximum absolute error of Example 4.2, when β=1
Full size table
Table 5 Maximum absolute error of Example 4.2, when β=3
Full size table
Table 6 Maximum absolute error of Example 4.2, when β=3
Full size table

Example 4.3

Consider the following singular derivative-dependent two point boundary value problem

$$\begin{aligned} \left . \begin{array}{@{}l@{}} \displaystyle\bigl(x^{\alpha}y'\bigr)'=x^{\alpha+\beta-2} \beta \bigl( xy'+y(\alpha+\beta-1) \bigr),\quad 0<x\leq1, \\ \displaystyle \lim_{x\rightarrow0+}x^\alpha y'(x)=0,\quad y(1)=e, \end{array} \right \} \end{aligned}$$
(4.10)

with exact solution \(y(x)=e^{x^{\beta}}\).

According to proposed scheme (2.9), we have p(x)=x α, q(x)=x α+β−2, a=1, b=0, and c=e. Consequently

(4.11)

where,

(4.12)

and the Green’s function is

(4.13)

For α=2, β=1, we use (4.11), (4.13) and (4.12), to obtain the components y n :

The maximum absolute error E (n), for n=2,4,6,8,10,12,14,16 is listed in Tables 7, 8, 9 and 10 for various values of α and β. From these numerical results, we see that our approximate series solution converges to exact solution as the number of iterations increase.

Table 7 Maximum absolute error of Example 4.3, when β=1
Full size table
Table 8 Maximum absolute error of Example 4.3, when β=1
Full size table
Table 9 Maximum absolute error of Example 4.3, when β=4
Full size table
Table 10 Maximum absolute error of Example 4.3, when β=4
Full size table

For α=2, β=4, using (4.11), the components y n are obtained as:

5 Conclusion

In this work, we have shown the application of the proposed recursive scheme for solving nonlinear derivative-dependent singular boundary value problems. The accuracy of the computed numerical results measured using the maximum absolute error or absolute residual error shows that the proposed method is suitable for such singular boundary value problems. It provides a direct recursive scheme to obtain a sequence of approximate solutions whose limiting value is the exact solution of the problem. Unlike the existing methods such as ADM or MDAM, the proposed method does not require the computation of undermined coefficients. In addition, the proposed scheme does not require any linearization or discretization of variables. The proposed method is capable for solving a variety of nonlinear boundary value problems whereas the other methods like VIM suffers. The proposed method requires less computational work as compared to other existing methods for solving such equations. The approximate solution of the examples are presented and only a few terms are required to obtain accurate approximations for solutions. The convergence and error estimation of the proposed recursive scheme are also discussed. Finally, it is pointed out that the procedure described in this paper can be extended for higher order boundary value problems.