Analytic solution of system of singular nonlinear differential equations with Neumann-Robin boundary conditions arising in astrophysics

1 Introduction

We consider the following system of singular boundary value problems (SBVPs)

(1.1)

where $a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2}$ are real constants. Here, $p_{1} (x) = x^{k_{1}} g_{1} (x)$ , $p_{2} (x) = x^{k_{2}} g_{2} (x)$ , $g_{1} (0) \neq 0$ and $g_{2} (0) \neq 0$ with $p_{1} (0) = p_{2} (0) = 0$ .

We next consider the following system of Lane-Emden equations [2, 3, 4, 5, 6, 7, 1] a particular case of (1.1) with $p_{1} (x) = x^{k_{1}}$ and $p_{2} (x) = x^{k_{2}}$ as

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} (x^{k_{1}} y_{1}^{'} (x))^{'} = x^{k_{1}} f_{1} (x, y_{1} (x), y_{2} (x)), x \in (0, 1), (x^{k_{2}} y_{2}^{'} (x))^{'} = x^{k_{2}} f_{2} (x, y_{1} (x), y_{2} (x)), y_{1}^{'} (0) = 0, a_{1} y_{1} (1) + b_{1} y_{1}^{'} (1) = c_{1}, y_{2}^{'} (0) = 0, a_{2} y_{2} (1) + b_{2} y_{2}^{'} (1) = c_{2} . \end{matrix}

(1.2)

In recent years, the SBVPs have been studied extensively in [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and references therein. But as far as we know, we find only the following results on system of Lane-Emden equations. Recently, in [31, 5, 32] authors studied (1.2) with boundary conditions $y_{1}^{'} (0) = y_{2}^{'} (0) = 0, y_{1} (1) = y_{2} (1) = 1$ and shape factors $k_{1} = k_{2} = 2$ and $f_{1} = - b + a g_{1} (y_{1}, y_{2}) + c g_{2} (y_{1}, y_{2}), f_{2} = d g_{1} (y_{1}, y_{2}) + e g_{2} (y_{1}, y_{2}),$ that relates the concentration of the carbon substrate and the concentration of oxygen, the Michaelis-Menten functions

g_{i} (y_{1}, y_{2}) = M_{1} \times M_{2}, M_{1} = \frac{y_{1}}{l_{i} + y_{1}}, M_{2} = \frac{y_{2}}{m_{i} + y_{2}}, i = 1, 2,

where $M_{1}$ and $M_{2}$ are the respective Michaelis-Menten nonlinear operators.

In [3, 4, 33], authors considered the system of Lane–Emden equations (1.2) with boundary conditions $y_{1}^{'} (0) = y_{2}^{'} (0) = 0, y_{1} (1) = 1, y_{2} (1) = 2$ and shape factors $k_{1} = k_{2} = 2$ and $f_{1} = a y_{1}^{2} + b y_{1} y_{2}, f_{2} = c y_{1}^{2} - d y_{1} y_{2},$ occurs in catalytic diffusion reactions with the parameters $a, b, c$ , $d$ are actual chemical reactions.

In [4, 32], the ADM was applied to obtain a convergent analytic approximate solution of (1.2) with $k_{1} = k_{2} = 2$ . Later, in [34], the variational iteration method (VIM) was applied to obtain approximations to solutions of (1.2) for shape factors $k_{i} = 1, 2, 3, i = 1, 2$ . Most recently, a numerical procedure based on sinc-collocation method was developed in [35] to obtain the solution of (1.2). In [36] authors used the reproducing kernel Hilbert space method for solving to obtain the solution of (1.2). In [1], the ADM with Green’s function used to find numerical approximation of the solutions of (1.2).

The homotopy analysis method was developed and improved by S. Liao [37, 38, 39, 40] for solving a broad class of functional equations. Various modifications of the homotopy analysis method have also been elaborated, for example, the optimal homotopy asymptotic method was proposed by Marinca and Herisanu [41, 42], the optimal homotopy analysis method was introduced in [43, 44]. Recently, the homotopy analysis method with Green’s function was applied to solve singular boundary value problems in [45, 46, 47, 48, 49].

In this work, we present the homotopy analysis method combined with the Green’s function strategy for obtaining approximate solutions of coupled singular boundary value problems (1.1) and (1.2). Unlike standard HAM, our approach avoids solving the transcendental equations for the undetermined coefficients. Unlike the ADM and VIM, our proposed technique contains the convergence-control parameters which gives fast convergence of the series solution. Numerical results reveal that the present process provides better results as compared to some existing iterative methods [4, 5, 1].

2 Integral form of system of Lane-Emden-Fowler types equations

By following [20], we transform the following system of Lane-Emden-Fowler types equations with Neumann-Robin boundary conditions

(2.1)

into the equivalent system of integral equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} y_{1} (x) = \frac{c_{1}}{a_{1}} + 1 \int 0 G_{1} (x, s) p_{1} (s) f_{1} (s, y_{1} (s), y_{2} (s)) d s, y_{2} (x) = \frac{c_{2}}{a_{2}} + 1 \int 0 G_{2} (x, s) p_{2} (s) f_{2} (s, y_{1} (s), y_{2} (s)) d s, \end{matrix}

(2.2)

where $G_{1} (x, s)$ and $G_{2} (x, s)$ are given by

G_{1} (x, s) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} h_{1} (1) - h_{1} (x) + \frac{b_{1}}{a_{1}} h_{1}^{'} (1), & s \leq x, h_{1} (1) - h_{1} (s) + \frac{b_{1}}{a_{1}} h_{1}^{'} (1), & x \leq s, \end{matrix}

(2.3)

with $h_{1} (x) = x \int 0 \frac{d t}{p_{1} (t)}, h_{1}^{'} (1) = \frac{1}{p_{1} (1)}, h_{1} (1) - h_{1} (s) = 1 \int s \frac{d t}{p_{1} (t)}, h_{1} (1) - h_{1} (x) = 1 \int s \frac{d t}{p_{1} (t)} - x \int s \frac{d t}{p_{1} (t)}$ ,

G_{2} (x, s) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} h_{2} (1) - h_{2} (x) + \frac{b_{2}}{a_{2}} h_{2}^{'} (1), & s \leq x, h_{2} (1) - h_{2} (s) + \frac{b_{2}}{a_{2}} h_{2}^{'} (1), & x \leq s, \end{matrix}

(2.4)

with $h_{2} (x) = x \int 0 \frac{d t}{p_{2} (t)}, h_{2}^{'} (1) = \frac{1}{p_{2} (1)}, h_{2} (1) - h_{2} (s) = 1 \int s \frac{d t}{p_{2} (t)}, h_{2} (1) - h_{2} (x) = 1 \int s \frac{d t}{p_{2} (t)} - x \int s \frac{d t}{p_{2} (t)} .$

Similarly, by following [17, 1], we transform the following system of Lane-Emden types equations with Neumann-Robin boundary conditions

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} (x^{k_{1}} y_{1}^{'} (x))^{'} = x^{k_{1}} f_{1} (x, y_{1} (x), y_{2} (x)), x \in (0, 1), (x^{k_{2}} y_{2}^{'} (x))^{'} = x^{k_{2}} f_{2} (x, y_{1} (x), y_{2} (x)), y_{1}^{'} (0) = 0, a_{1} y_{1} (1) + b_{1} y_{1}^{'} (1) = c_{1}, y_{2}^{'} (0) = 0, a_{2} y_{2} (1) + b_{2} y_{2}^{'} (1) = c_{2}, \end{matrix}

(2.5)

into the equivalent system of integral equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} y_{1} (t) = \frac{c_{1}}{a_{1}} + 1 \int 0 G_{1} (x, s) s^{k_{1}} f_{1} (s, y_{1} (s), y_{2} (s)) d s, y_{2} (x) = \frac{c_{2}}{a_{2}} + 1 \int 0 G_{2} (x, s) s^{k_{2}} f_{2} (s, y_{1} (s), y_{2} (s)) d s, \end{matrix}

(2.6)

where $G_{1} (x, s)$ and $G_{2} (x, s)$ for $k_{1} = k_{2} = 1$ are

	$G_{1} (x, s) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} ln x - \frac{b_{1}}{a_{1}}, & s \leq x, ln s - \frac{b_{1}}{a_{1}}, & x \leq s, \end{matrix}$		(2.7)
	$G_{2} (x, s) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} ln x - \frac{b_{2}}{a_{2}}, & s \leq x, ln s - \frac{b_{2}}{a_{2}}, & x \leq s \end{matrix}$		(2.8)

and $G_{1} (x, s)$ and $G_{2} (x, s)$ for $k_{1} \neq 1, k_{2} \neq 1$ are

	$G_{1} (x, s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{x^{1 - k_{1}} - 1}{1 - k_{1}} - \frac{b_{1}}{a_{1}}, & s \leq x \frac{s^{1 - k_{1}} - 1}{1 - k_{1}} - \frac{b_{1}}{a_{1}}, & x \leq s, \end{matrix}$		(2.9)
	$G_{2} (x, s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{x^{1 - k_{2}} - 1}{1 - k_{2}} - \frac{b_{2}}{a_{2}}, & s \leq x, \frac{s^{1 - k_{2}} - 1}{1 - k_{2}} - \frac{b_{2}}{a_{2}}, & x \leq s . \end{matrix}$		(2.10)

3 Solution method

We consider the system of integral operator equation (2.2) as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} N_{1} [y_{1}] = y_{1} (x) - \frac{c_{1}}{a_{1}} - 1 \int 0 G_{1} (x, s) p_{1} (s) f_{1} (s, y_{1} (s), y_{2} (s)) d s = 0, N_{2} [y_{2}] = y_{2} (x) - \frac{c_{2}}{a_{2}} - 1 \int 0 G_{2} (x, s) p_{2} (s) f_{2} (s, y_{1} (s), y_{2} (s)) d s = 0. \end{matrix}

(3.1)

Note that the above system of integral equations reduce to (2.6) with $p_{1} (x) = x^{k_{1}}$ and $p_{2} (x) = x^{k_{2}}$ .

In order to apply the HAM with Green’s function [45, 46, 47, 48, 49], we construct the zeroth-order deformation equations for (3.1) as

{\begin{matrix} (1 - q) [φ_{1} (x, q) - y_{10} (x)] = q c_{10} N_{1} [φ_{1} (x, q)], (1 - q) [φ_{2} (x, q) - y_{20} (x)] = q c_{20} N_{2} [φ_{2} (x, q)], q \in [0, 1] \end{matrix}

(3.2)

where $q$ is an embedding parameter, $(y_{10} (x), y_{20} (x))$ are initial approximations, $(c_{10} \neq 0, c_{20} \neq 0)$ are convergence control parameters and ( $φ_{1} (x, q), φ_{2} (x, q)$ ) are unknown functions.

If $q = 0$ and $q = 1$ the homotopy equations (3.2) vary from . Now as $q$ varies from $0$ to $1$ , the solutions of (3.1) will vary from the initial guesses $(y_{10} (x), y_{20} (x))$ to the exact solutions $(y_{1} (x), y_{2} (x))$ of equations (3.1).

Expanding $φ_{1} (x, q)$ and $φ_{2} (x, q)$ as a Taylor series with respect to $q$ yields

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} φ_{1} (x, q) = \infty \sum k = 0 y_{1 k} (x) q^{k}, φ_{2} (x, q) = \infty \sum k = 0 y_{2 k} (x) q^{k}, \end{matrix}

(3.3)

where

y_{1 k} (x) = \frac{1}{k!} \frac{\partial^{k} φ_{1} (x, q)}{\partial q^{k}} {∣ ∣ ∣}_{q = 0}, y_{2 k} (x) = \frac{1}{k!} \frac{\partial^{k} φ_{2} (x, q)}{\partial q^{k}} {∣ ∣ ∣}_{q = 0} .

(3.4)

The series (3.3) converge for $q = 1$ if $c_{10} \neq 0, c_{20} \neq 0$ are chosen properly, and they take the form

(3.5)

which will be the solutions of (3.1).

For further analysis, we define the vectors as

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} {\to y}_{1 k} = {y_{10} (x), y_{11} (x), \dots, y_{1 k} (x)}, {\to y}_{2 k} = {y_{20} (x), y_{21} (x), \dots, y_{2 k} (x)} . \end{matrix}

(3.6)

Differentiating (3.2) $k$ -times with respect to $q$ , dividing them by $k!$ , setting $q = 0,$ we obtain the $k$ th-order deformation equations as

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} y_{1 k} (x) - χ_{k} y_{1 (k - 1)} (x) = c_{10} R_{1 k} [{\to y}_{1 (k - 1)}], y_{2 k} (x) - χ_{k} y_{2 (k - 1)} (x) = c_{20} R_{2 k} [\to y_{2 (k - 1)}], \end{matrix}

(3.7)

where $χ_{k} = {\begin{matrix} 0, & k \leq 1, 1, & k > 1, \end{matrix}$ and

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} R_{1 k} [{\to y}_{1 (k - 1)}] = \frac{1}{(k - 1)!} \frac{\partial^{k - 1}}{\partial q^{k - 1}} [φ_{1} (x, q) - \frac{c_{1}}{a_{1}} - 1 \int 0 G_{1} (x, s) p_{1} (s) f_{1} (s, φ_{1} (s, q), φ_{2} (s, q)) d s] {∣ ∣ ∣}_{q = 0} R_{2 k} [{\to x}_{2 (k - 1)}] = \frac{1}{(k - 1)!} \frac{\partial^{k - 1}}{\partial q^{k - 1}} [φ_{2} (x, q) - \frac{c_{2}}{a_{2}} - 1 \int 0 G_{2} (x, s) p_{2} (s) f_{2} (s, φ_{1} (s, q), φ_{2} (s, q)) d s] {∣ ∣ ∣}_{q = 0} \end{matrix}

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} = \frac{1}{(k - 1)!} \frac{\partial^{k - 1}}{\partial q^{k - 1}} [\infty \sum i = 0 y_{1 i} q^{i} - \frac{c_{1}}{a_{1}} - 1 \int 0 G_{1} (x, s) p_{1} (s) f_{1} (s, \infty \sum i = 0 y_{1 i} q^{i}, \infty \sum i = 0 y_{2 i} q^{i}) d s] {∣ ∣ ∣}_{q = 0}, = \frac{1}{(k - 1)!} \frac{\partial^{k - 1}}{\partial q^{k - 1}} [\infty \sum i = 0 y_{2 i} q^{i} - \frac{c_{2}}{a_{2}} - 1 \int 0 G_{2} (x, s) p_{2} (s) f_{2} (s, \infty \sum i = 0 y_{1 i} q^{i}, \infty \sum i = 0 y_{2 i} q^{i}) d s] {∣ ∣ ∣}_{q = 0} . \end{matrix}

On simplification we get

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} R_{1 k} [{\to y}_{1 (k - 1)}] = y_{1 (k - 1)} - (1 - χ_{k}) \frac{c_{1}}{a_{1}} - 1 \int 0 G_{1} (x, s) p_{1} (s) H_{1 (k - 1)} d s, R_{2 k} [{\to y}_{2 (k - 1)}] = y_{2 (k - 1)} - (1 - χ_{k}) \frac{c_{2}}{a_{2}} - 1 \int 0 G_{2} (x, s) p_{2} (s) H_{2 (k - 1)} d s, \end{matrix}

(3.8)

where $H_{1 k}$ and $H_{2 k}$ are given by

(3.9)

Making use of (3.8), the $k$ th-order deformation equations (3.7) are simplified as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} y_{1 k} - χ_{k} y_{1 (k - 1)} & = c_{10} [y_{1 (k - 1)} - (1 - χ_{k}) \frac{c_{1}}{a_{1}} - 1 \int 0 G_{1} (x, s) p_{1} (s) H_{1 (k - 1)} d s], y_{2 k} - χ_{k} y_{2 (k - 1)} & = c_{20} [y_{2 (k - 1)} - (1 - χ_{k}) \frac{c_{2}}{a_{2}} - 1 \int 0 G_{2} (x, s) p_{2} (s) H_{2 (k - 1)} d s] . \end{matrix}

(3.10)

By taking the initial approximations $y_{10} = \frac{c_{1}}{a_{1}}$ and $y_{20} = \frac{c_{2}}{a_{2}}$ , the solutions components are computed as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} y_{11} = c_{10} y_{10} - c_{10} \frac{c_{1}}{a_{1}} - c_{10} 1 \int 0 G_{1} (x, s) p_{1} (s) H_{10} d s, y_{21} = c_{20} y_{20} - c_{20} \frac{c_{2}}{a_{2}} - c_{20} 1 \int 0 G_{2} (x, s) p_{2} (s) H_{20} d s, ⋮ y_{1 k} = (1 + c_{10}) y_{1 (k - 1)} - c_{10} 1 \int 0 G_{1} (x, s) p_{1} (s) H_{1 (k - 1)} d s, y_{2 k} = (1 + c_{20}) y_{2 (k - 1)} - c_{20} 1 \int 0 G_{2} (x, s) p_{2} (s) H_{2 (k - 1)} d s, k \geq 2. \end{matrix}

(3.11)

The $n$ th-order approximations to solutions are obtained as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} ϕ_{1 n} (x, c_{10}, c_{20}) = n \sum k = 0 y_{1 k} (x, c_{10}, c_{20}), ϕ_{2 n} (x, c_{10}, c_{20}) = n \sum k = 0 y_{2 k} (x, c_{10}, c_{20}) . \end{matrix}

(3.12)

The unknown parameters $c_{10}, c_{20}$ have a significant influence on the convergence of the approximate solution. We next find the optimal values of $c_{10}, c_{20}$ by solving

\frac{\partial E_{1 n} (c_{10}, c_{20})}{\partial c_{10}} = 0, \frac{\partial E_{2 n} (c_{10}, c_{20})}{\partial c_{20}} = 0,

(3.13)

where $E_{1 n}$ and $E_{2 n}$ are given by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} E_{1 n} (c_{10}, c_{20}) = \frac{1}{N} N \sum k = 1 (N_{1} [ϕ_{1 n} (x_{k}, c_{10}, c_{20})])^{2}, E_{2 n} (c_{10}, c_{20}) = \frac{1}{N} N \sum k = 1 (N_{2} [ϕ_{2 n} (x_{k}, c_{10}, c_{20})])^{2}, x_{1}, x_{2}, \dots, x_{N} \in [0, 1], \end{matrix}

(3.14)

Then these optimal values of $c_{10}$ and $c_{20}$ will be substituting in (3.12) to obtain the optimal HAM-approximate solution

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} ϕ_{1 n} (x) = n \sum k = 0 y_{1 k} (x), ϕ_{2 n} (x) = n \sum k = 0 y_{2 k} (x) . \end{matrix}

(3.15)

Remark 3.1.

The ADM [32, 1] is a particular case of (3.11) when $c_{10} = c_{20} = - 1$ and given by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} y_{10} = \frac{c_{1}}{a_{1}}, y_{20} = \frac{c_{2}}{a_{2}}, y_{1 k} = 1 \int 0 G_{1} (x, s) p_{1} (s) H_{1 (k - 1)} d s, y_{2 k} = 1 \int 0 G_{2} (x, s) p_{2} (s) H_{2 (k - 1)} d s, k \geq 1. \end{matrix}

(3.16)

Hence, the ADM-approximate solutions are given by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} ψ_{1 n} (x) = n \sum k = 0 y_{1 k} (x), ψ_{2 n} (x) = n \sum k = 0 y_{2 k} (x) . \end{matrix}

(3.17)

4 Convergence analysis

Let $E = (C [0, 1], ∥ y ∥)$ be a Banach space with norm

∥ y ∥ = max {∥ y_{1} ∥, ∥ y_{2} ∥}, y \in E,

(4.1)

where, $∥ y_{1} ∥ = max x \in I = [0, 1] | y_{1} (x) |$ and $∥ y_{2} ∥ = max x \in I | y_{2} (x) |$ . We next discuss the convergence and error analysis of the proposed method. To do so, we first introduce the vector notations

	$y = (\begin{matrix} y_{1} y_{2} \end{matrix}), y_{k} = (\begin{matrix} y_{1 k} y_{2 k} \end{matrix}), c_{0} = (\begin{matrix} c_{10} c_{20} \end{matrix}), p (x) = (\begin{matrix} p_{1} (x) p_{2} (x) \end{matrix}),$
	$G (x, s) = (\begin{matrix} G_{1} (x, s) G_{2} (x, s) \end{matrix}), H_{k} = (\begin{matrix} H_{1 k} H_{2 k} \end{matrix}), f (x, y_{1}, y_{2}) = (\begin{matrix} f_{1} (x, y_{1}, y_{2}) f_{2} (x, y_{1}, y_{2}) \end{matrix}),$

$ϕ_{n} = (\begin{matrix} ϕ_{1 n} ϕ_{2 n} \end{matrix})$ and $M = max (\begin{matrix} {max}_{x \in I} ∣ ∣ 1 \int 0 G_{1} (x, s) p_{1} (s) d s ∣ ∣ {max}_{x \in I} ∣ ∣ 1 \int 0 G_{2} (x, s) p_{2} (s) d s ∣ ∣ \end{matrix}) .$

Theorem 4.1.

Suppose that the nonlinear function $f (x, z, w)$ satisfy Lipschitz condition: $| f (x, z, w) - f (x, z^{*}, w^{*}) | \leq l_{1} | z - z^{*} | + l_{2} | w - w^{*} |,$ where $l_{1}$ and $l_{2}$ are Lipschitz constants. If the parameter $c_{0}$ is chosen such that there exists constant $δ_{c_{0}} \in (0, 1)$ , then the series $ϕ_{n}$ defined by (3.15) is convergent in Banach space $E$ .

Proof.

From (3.10) and (3.12), we have

	$ϕ_{n}$
		$= (1 + c_{0}) ϕ_{n - 1} - c_{0} 1 \int 0 G (x, s) p (s) n \sum k = 1 H_{k - 1} d s .$		(4.2)

From (4), we consider for $n > m$ , for all $n, m \in N$ as

∥ ϕ_{n} - ϕ_{m} ∥

Using the inequality $\sum_{k = 0}^{n} H_{k} \leq f (s, ϕ_{1 n}, ϕ_{2 n})$ from ([50]) we have

	$∥ ϕ_{n} - ϕ_{m} ∥$	$\leq \| 1 + c_{0} \| max x \in I ∣ ∣ ϕ_{n - 1} - ϕ_{m - 1} ∣ ∣ + \| c_{0} \|$
		$max x \in I ∣ ∣ ∣ 1 \int 0 G (x, s) p (s) (f (s, ϕ_{1 (n - 1)}, ϕ_{2 (n - 1)}) - f (s, ϕ_{1 (m - 1)}, ϕ_{2 (m - 1)})) d s ∣ ∣ ∣ .$

Applying the Lipschitz condition, we get

	$∥ ϕ_{n} - ϕ_{m} ∥$
		$\leq \| 1 + c_{0} \| ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ + \| c_{0} \| max x \in I ∣ ∣ ∣ 1 \int 0 G (x, s) p (s) d s ∣ ∣ ∣$
		$\times 2 L max {∥ ∥ ∥ ϕ_{1 (n - 1)} - ϕ_{1 (m - 1)} ∥ ∥ ∥, ∥ ∥ ∥ ϕ_{2 (n - 1)} - ϕ_{2 (m - 1)} ∥ ∥ ∥}$
		$\leq \| 1 + c_{0} \| ∥ ∥ ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ ∥ ∥ + 2 L M \| c_{0} \| ∥ ∥ ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ ∥ ∥$
		$\leq (\| 1 + c_{0} \| + 2 L M \| c_{0} \|) ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ = δ_{c_{0}} ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥,$

Thus, we have

∥ ϕ_{n} - ϕ_{m} ∥ \leq δ_{c_{0}} ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ .

(4.3)

where $L = max {l_{1}, l_{2}}, δ_{c_{0}} = (| 1 + c_{0} | + 2 L M | c_{0} |) .$

Setting $n = m + 1$ in (4.3), we have

∥ ϕ_{m + 1} - ϕ_{m} ∥

\leq δ_{c_{0}} ∥ ϕ_{m} - ϕ_{m - 1} ∥ \leq δ_{c_{0}}^{2} ∥ ϕ_{m - 1} - ϕ_{m - 2} ∥ \leq \dots \leq δ_{c_{0}}^{m} ∥ ϕ_{1} - ϕ_{0} ∥ .

For all $n, m \in N$ , with $n > m$ , consider

	$∥ ϕ_{n} - ϕ_{m} ∥$	$= ∥ (ϕ_{n} - ϕ_{n - 1}) + (ϕ_{n - 1} - ϕ_{n - 1}) + \dots + (ϕ_{m + 1} - ϕ_{m}) ∥$
		$\leq ∥ ϕ_{n} - ϕ_{n - 1} ∥ + ∥ ϕ$

	$∥ ϕ_{n} - ϕ_{m} ∥$
		$\leq \| 1 + c_{0} \| ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ + \| c_{0} \| max x \in I ∣ ∣ ∣ 1 \int 0 G (x, s) p (s) d s ∣ ∣ ∣$
		$\times 2 L max {∥ ∥ ∥ ϕ_{1 (n - 1)} - ϕ_{1 (m - 1)} ∥ ∥ ∥, ∥ ∥ ∥ ϕ_{2 (n - 1)} - ϕ_{2 (m - 1)} ∥ ∥ ∥}$
		$\leq \| 1 + c_{0} \| ∥ ∥ ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ ∥ ∥ + 2 L M \| c_{0} \| ∥ ∥ ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ ∥ ∥$
		$\leq (\| 1 + c_{0} \| + 2 L M \| c_{0} \|) ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥ = δ_{c_{0}} ∥ ϕ_{n - 1} - ϕ_{m - 1} ∥,$

Analytic solution of system of singular nonlinear differential equations with Neumann-Robin boundary conditions arising in astrophysics

Solving coupled Lane-Emden equations by Green's function and decomposition technique

Approximation approach to the fractional BVP with the Dirichlet type boundary conditions

On boundary conditions parametrized by analytic functions

Wavenumber-explicit hp-FEM analysis for Maxwell's equations with impedance boundary conditions

Numerical Methods for a Diffusive Class Nonlocal Operators

Error estimates for a splitting integrator for semilinear boundary coupled systems

Dynamic Programming Method for Best Piecewise Linear Approximation for Vector Field of Nonlinear Boundary Value Problems on the Interval [0, 1]

1 Introduction

2 Integral form of system of Lane-Emden-Fowler types equations

3 Solution method

Remark 3.1.

4 Convergence analysis

Theorem 4.1.

Proof.

Analytic solution of system of singular nonlinear differential equations with Neumann-Robin boundary conditions arising in astrophysics

Related Research

Solving coupled Lane-Emden equations by Green's function and decomposition technique

Approximation approach to the fractional BVP with the Dirichlet type boundary conditions

On boundary conditions parametrized by analytic functions

Wavenumber-explicit hp-FEM analysis for Maxwell's equations with impedance boundary conditions

Numerical Methods for a Diffusive Class Nonlocal Operators

Error estimates for a splitting integrator for semilinear boundary coupled systems

Dynamic Programming Method for Best Piecewise Linear Approximation for Vector Field of Nonlinear Boundary Value Problems on the Interval [0, 1]

1 Introduction

2 Integral form of system of Lane-Emden-Fowler types equations

3 Solution method

Remark 3.1.

4 Convergence analysis

Theorem 4.1.

Proof.