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

05/14/2020 ∙ by Randhir Singh, et al. ∙ 0

In this paper, the Green's function and decomposition technique is proposed for solving the coupled Lane-Emden equations. This approach depends on constructing Green's function before establishing the recursive scheme for the series solution. Unlike, standard Adomian decomposition method, the present method avoids solving a sequence of transcendental equations for the undetermined coefficients. Convergence and error estimation is provided. Three examples of coupled Lane-Emden equations are considered to demonstrate the accuracy of the current algorithm.

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1 Introduction

This paper aims to extend the application of the Adomian decomposition method with Green’s function [1, 2] for solving the following coupled Lane-Emden boundary value problems

(1.1)

where

are real constants. In recent years, singular boundary value problems for ordinary differential equations have been studied extensively

[3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and references therein. However, we find only the following results on coupled Lane-Emden equations.

Recently, in [22, 23, 24] authors studied (1.1) with boundary conditions and that relates the concentration of the carbon substrate and the concentration of oxygen. In [25, 26], authors considered the coupled Lane–Emden equations (1.1) with boundary conditions , and occurs in catalytic diffusion reactions. In [26, 24], the Adomian decomposition method was applied to obtain a convergent analytic approximate solution of (1.1) with . Later, in [27], the variational iteration method was applied to obtain approximations to solutions of (1.1) for shape factors . In [28], the Sinc-collocation method was used to obtain the solution of (1.1). In [29] authors used the reproducing kernel Hilbert space method for solving to obtain the solution of (1.1).

2 Adomian decomposition method

Recently, many researchers [30, 31, 32, 33, 34, 35, 36] have applied the Adomian decomposition method to deal with many different scientific models. According to the Adomian decomposition method we rewrite (1.1) in a operator form as

(2.1)

where and are differential operators and their inverse integral operators are defined as

(2.2)

Operating on (2.1) and using , we get

(2.3)

According to the ADM, we decompose and as

(2.4)

where are Adomian’s polynomials [30] are given

(2.5)

Substituting (2.4) into (2.3), we get

(2.6)

Upon comparing both sides of (2.6), we have

(2.7)

where are unknown constants to be determined. The -term series solutions are given as

(2.8)

The unknown constants may be obtained by imposing boundary condition at on , that leads to

(2.9)

Solving above transcendental equations for require additional computational work, and may not be uniquely determined.

To avoid solving the above sequence of difficult transcendental equations, the Adomian decomposition method with Green’s function was introduced in [1, 2]. This technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. Unlike the standard Adomian decomposition method, this avoids solving a sequence of transcendental equations for the undetermined coefficients.

3 Green’s function and decomposition technique

In this section, we extend the application of the Adomian decomposition method with Green’s function [2, 12, 37, 38, 39], where we transformed the singular boundary value problem into the integral equation before establishing the recursive scheme for the approximate solution. To apply this technique to coupled Lane-Emden boundary value problems (1.1), we first consider the equivalent integral form of coupled Lane-Emden equation (1.1) as

(3.1)

where are given by

(3.2)

and

(3.3)

Substituting the series (2.4) into (3.1), we obtain

(3.4)

Comparing components from both sides of (3.4) we have the following recursive scheme

(3.5)

Then, we obtain the approximate series solutions as

(3.6)

Unlike ADM or MADM, the proposed recursive schemes (3.6) do not require any computation of unknown constants.

4 Convergence and error analysis

Let be a Banach space with norm

(4.1)

where, and .

From (3.5) and (3.6), we have

(4.2)

where

y
f
Definition 1.

The function satisfy Lipschitz condition as

(4.3)

where , and , are Lipschitz constants.

Theorem 4.1.

Suppose that the nonlinear function satisfy Lipschitz condition (4.3), then the series solution defined by (3.6) is convergent whenever .

Proof.

Define

For and using (4.2), we have

Using from ([40]), we have

Applying the Lipschitz condition, we get

where

(4.4)

Thus, we have

(4.5)

By taking in (4.5), we see that

For all , with , consider

It follows that as ,

(4.6)

Letting , we obtain Hence, is a Cauchy sequences in the Banach space . ∎

Theorem 4.2.

If the approximate solution converges to , then the maximum absolute truncated error is estimated

(4.7)
Proof.

From (4.6), we have

Since as , and the above inequality reduces to

(4.8)

From (3.5), we have , and we find

(4.9)

Combining (4.8) and (4.9), we obtain error estimate as

(4.10)

which completes the proof. ∎

5 Numerical Results

In this section, we consider three coupled Lane-Emden type boundary value problems to examine the accuracy of the present method. Since the exact solution of the problems is not known, we examine the accuracy and applicability of the present method by the absolute residual error

(5.1)

where is the absolute residual error and is the present approximate solution. The maximum residual errors are defined as

(5.2)

5.1 The catalytic diffusion reactions problem [25, 26]

Example 5.1.

Consider the coupled Lane-Emden equation occurs in catalytic diffusion reactions [25, 26] as

(5.3)

where the parameters and are the actual chemical reactions. Here , , and .

In Tables 1 and 3, we list the numerical results of the approximate solution and the absolute error obtained by the proposed method of Example 5.1 for () and (), respectively. We also compare the numerical results of the maximum residual error obtained by the present method and the results obtained by the modified ADM [26] in Table 2. In Table 4, we list the numerical results of the maximum residual error.

5.1.1 When

By applying the proposed scheme (3.5) with the initial guesses , we obtain the 5-terms series solutions as

0.1 0.7782151 1.6870682 7.00E-2 8.79E-2 0.7836523 1.6938487 5.51E-3 6.84E-3
0.2 0.7842287 1.6955995 6.55E-2 8.23E-2 0.7893632 1.7020042 5.10E-3 6.34E-3
0.3 0.7943299 1.7099257 5.85E-2 7.36E-2 0.7989874 1.7157379 4.48E-3 5.57E-3
0.4 0.8086434 1.7302167 4.97E-2 6.26E-2 0.8126872 1.7352661 3.71E-3 4.63E-3
0.5 0.8273577 1.7567278 3.98E-2 5.02E-2 0.8306972 1.7609008 2.89E-3 3.62E-3
0.6 0.8507387 1.7898165 2.97E-2 3.76E-2 0.8533324 1.7930602 2.10E-3 2.64E-3
0.7 0.8791464 1.8299638 2.02E-2 2.56E-2 0.8809992 1.8322829 1.39E-3 1.75E-3
0.8 0.9130553 1.8778003 1.18E-2 1.51E-2 0.9142115 1.8792486 7.90E-4 1.01E-3
0.9 0.9530791 1.9341363 5.09E-3 6.53E-3 0.9536120 1.9348042 3.30E-4 4.29E-4
Table 1: Numerical results of the approximate solution and the absolute error when of Example 5.1
Modified ADM [26]
2 8.72E-2 1.12E-1 4.13E-1 5.67E-1
3 3.95E-2 5.05E-2 2.36E-1 3.09E-1
4 2.00E-2 2.53E-2 6.43E-2 8.52E-2
5 1.07E-2 1.35E-2 4.79E-2 6.09E-2
6 6.01E-3 7.58E-3 2.09E-2 2.51E-2
7 3.47E-3 4.37E-3 1.09E-2 1.36E-2
8 2.06E-3 2.59E-3 6.21E-3 7.32E-3
9 1.24E-3 1.56E-4 3.35E-3 3.28E-3
10 7.60E-4 9.53E-4 1.79E-3 2.09E-3
11 5.91E-4 5.91E-4 9.61E-4 1.11E-3
Table 2: Comparison of the numerical results of the maximum residual error when of Example 5.1

5.1.2 When

On applying the proposed scheme (3.5) with the initial guesses , we obtain the 5-terms series solutions as

0.1 0.8054213 1.8054213 1.42E-2 1.42E-2 0.8065106 1.8065106 2.63E-4 2.63E-4
0.2 0.8107583 1.8107583 1.33E-2 1.33E-2 0.8117871 1.8117871 2.43E-4 2.43E-4
0.3 0.8197227 1.8197227 1.18E-2 1.18E-2 0.8206563 1.8206563 2.13E-4 2.13E-4
0.4 0.8324215 1.8324215 1.00E-2 1.00E-2 0.8332325 1.8332325 1.76E-4 1.76E-4
0.5 0.8490101 1.8490101 8.05E-3 8.05E-3 0.8496803 1.8496803 1.37E-4 1.37E-4
0.6 0.8696982 1.8696982 5.98E-3 5.98E-3 0.8702190 1.8702190 9.91E-5 9.93E-5
0.7 0.8947568 1.8947568 4.05E-3 4.05E-3 0.8951290 1.8951290 6.53E-5 6.53E-5
0.8 0.9245277 1.9245277 2.36E-3 2.36E-3 0.9247601 1.9247601 3.71E-5 3.71E-5
0.9 0.9594352 1.9594352 1.00E-3 1.01E-3 0.9595423 1.9595423 1.55E-5 1.55E-5
Table 3: Numerical results of the approximate solution and the absolute error when of Example 5.1
2 4.15E-2 4.15E-2
3 1.41E-2 1.41E-2
4 5.35E-3 5.35E-3
5 2.15E-3 2.15E-3
6 9.05E-4 9.05E-4
7 3.91E-4 3.91E-4
8 1.73E-4 1.73E-4
9 7.83E-5 7.83E-5
10 3.58E-5 3.58E-5
11 1.66E-5 1.66E-5
Table 4: The numerical results of the maximum residual error when of Example 5.1

5.2 The concentration of the carbon substrate and the concentration of oxygen problem [27]

Example 5.2.

Consider the coupled Lane-Emden equations, which was used to study the concentration of the carbon substrate and the concentration of oxygen, as

(5.4)

where the parameters are fixed as given in [27, 24]. Here, .

In Table 5, 6 and 7, we list the numerical results of the approximate solution and the absolute error obtained by the proposed method of Example 5.2 for ), and , respectively. We also compare the numerical results of the maximum residual error obtained by the present method and the results obtained by the modified ADM [24] in Table 8 for .

5.2.1 When the shape factors

By applying the proposed scheme (3.5) with the initial guesses , we obtain the 4-terms series solutions as

0.1 2.0145977 1.0371205 2.61E-4 7.68E-6 2.0145872 1.0371202 2.67E-4 7.87E-6
0.2 1.9838514 1.0359956 2.49E-4 7.33E-6 1.9838408 1.0359953 2.40E-4 7.07E-6
0.3