Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model

Mohamed A. Abdelkawy; Zulqurnain Sabir; Juan L. G. Guirao; Tareq Saeed

doi:10.1515/phys-2020-0185

Open Access Published by De Gruyter Open Access November 20, 2020

Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model

Mohamed A. Abdelkawy , Zulqurnain Sabir , Juan L. G. Guirao and Tareq Saeed

From the journal Open Physics

https://doi.org/10.1515/phys-2020-0185

Abstract

The present study aims to design a second-order nonlinear Lane–Emden coupled functional differential model and numerically investigate by using the famous spectral collocation method. For validation of the newly designed model, three dissimilar variants have been considered and formulated numerically by applying a famous spectral collocation method. Moreover, a comparison of the obtained results with the exact/true results endorses the effectiveness and competency of the newly designed model, as well as the present technique.

Keywords: coupled Lane–Emden; nonlinear; functional differential model; singular; spectral collocation scheme; exact solutions

1 Introduction

The present research work is related to the singular models for the second-order nonlinear system of functional differential (FD) equations. These FD equations have a huge variety of applications, e.g., the growth rate population model [1], electrodynamics [2], infection HIV-1 model [3], growth rate of tumor model [4], chemical kinetics model [5], hepatitis-B virus infection model [6], and gene regulatory system [7]. Few numerical techniques have been applied to solve these FD equations, e.g., Kadalbajoo and Sharma [8] presented a numerical scheme for solving the FD equations, Mirzaee and Hoseini [9] implemented a collocation technique, Xu and Jin [10] used fractional measures and boundary functions for presenting the solution of these equations, and Geng et al. [11] discussed a numerical approach for solving the singularly perturbed FD equations. Due to the singular point, these models have achieved the diverse attention of the research community. One of the significant and historical singular models is the Lane–Emden (LE) model introduced by Lane and further investigated by Emden, which has a wider range of applications in science, technology, and engineering. The LE model is used in the density field of the gaseous star [12], catalytic diffusion reactions [13], mathematical geometry and physics [14], isothermal and polytrophic gas spheres [15], the theory of electromagnetic [16], magnetic field oscillation [17], quantum and classical mechanics [18], isotropic continuous media [19], morphogenesis [20], and dusty fluid models [21]. To the solution of the LE model, many numerical and analytic techniques have been applied. Shawagfeh [22] applied the method of Adomian decomposition, Bender et al. [23] used the method of perturbation, Liao [24] proposed an analytic algorithm, Nouh [25] implemented the power series technique by using Pade approximation, Mandelzweig, and Tabakin [26] used method of Bellman and Kalabas quasi-linearization to solve the LE equation. Recently, some numerical techniques are also broadly implemented to solve the singular LE type of models [27,28,29,30,31,32,33].

The present research work is about to model the second kind of singular nonlinear coupled LE system of FD equations and its modeled form is written as follows [34]:

(1) Z ″ ( a x + α ) + η 1 x Z ′ ( b x + β ) + Q ( x ) Z ( c x + γ ) = ℱ ( x ) Q ″ ( a x + α ) + η 2 x Q ′ ( b x + β ) + Z ( x ) Q ( c x + γ ) = ℋ ( x ) , 0 ≤ x ≤ ℒ ,

with the following conditions:

(2) Z ( ℒ ) = ζ 1 , Z ′ ( 0 ) = ζ 2 , Q ( ℒ ) = ζ 3 , Q ′ ( 0 ) = ζ 4 ,

where a , b , c , α , β , γ , η 1 , η 2 , ζ 1 , ζ 2 , ζ 3 , and ζ 4 are given constants while ℱ ( x ) , ℋ ( x ) are given functions. The FD model is the extension of the research study of Sabir et al. [35,36,37], which is applied to solve the singular nonlinear FD equations. The designed model is verified by solving the three variants based on nonlinear LE second-order coupled FD equations using the numerical spectral collocation scheme. The novel features of the current work are briefly shortened as follows:

The mathematical model for the nonlinear LE second-order coupled FD equation is successfully presented and verified by solving the three variants of the models using the spectral collocation method.
The comparison is performed of the obtained numerical results from the spectral collocation method with the true results, which shows the correctness of the presented system, as well as, designed approach.
Manipulation of the present spectral collocation method by applying the designed model provided brilliance solutions with higher accuracy and greater dependability.
The consistency of the designed mathematical model is certified from the reliable absolute error of the proposed and exact outcomes.
The nonlinear LE second-order system of the FD model is not simple to handle numerically because of the singularity, harder in nature, and nonlinearity. The spectral collocation method is one of the best suggestions, as well as a great selection to tackle such kinds of complex systems.

A large amount of work to model the physical systems has been restricted to ordinary differential equations. Therefore, the urgent requirement to achieve the exact solutions or simply the approximate ones to these problems has emerged. Since the finding of the exact solutions is not possible for these fractional differential models mostly, numerical techniques have been implemented to find approximate solutions to solve them. Some local numerical techniques are introduced for solving such systems, and these methods may become computationally heavy. Moreover, the local schemes listed the approximate solution at particular points, whereas the global approaches provide the approximate results in the whole interval. Hence, the global behavior of the solution can be naturally taken into account. The spectral collocation technique is a global numerical method that is a particular kind of famous spectral method, which is widely applicable for almost every type of differential equations. Recently, there is more interest in appointing the spectral collocation method to treat with various types of integral and differential models [38,39], due to its importance to finite/infinite ranges [40,41]. The convergence speed is the major advantage of the spectral collocation method. This method has exponential convergence rates as well as a high accuracy level. The spectral method has been classified into four classes: collocation [42], tau [43], Galerkin [44], and Petrov Galerkin [45] method. The collocation approach is a particular kind of spectral technique, which is widely suitable for almost all kinds of differential systems.

The other parts of the paper are organized as follows: a few relevant properties of Jacobi shift polynomials, designed scheme, detailed result discussions, conclusions, and future research guidance are described in the remaining sections.

2 Shifted Jacobi polynomials

The Jacobi polynomials (JP) known as the eigen functions based on the singular form of the Sturm–Liouville equation. In view of this, many particular cases exist, such as Legendre, the four type of Gegenbauer and Chebyshev polynomials. Furthermore, the JP have been applied in extensive applications because of its wider ability to approximate the general categories of the functions. Few of them are the Gibbs’ phenomenon resolution, data compression electrocardiogram, and to solve differential models. For [ 0 , L ] interval, the shifted Jacobi polynomials (SJP) are indeed applied with the freedom to select the Jacobi indexes θ and ϑ ; the method can be calibrated for a wide variety of problems. To consider the SJP J k ( ρ , σ ) ( x ) , which satisfy the following properties:

J k + 1 ( ρ , σ ) ( x ) = ( a k ( ρ , σ ) x − b k ( ρ , σ ) ) J k ( ρ , σ ) ( x ) − c k ( ρ , σ ) J k − 1 ( ρ , σ ) ( x ) , k ≥ 1 , J 0 ( ρ , σ ) ( x ) = 1 , J 1 ( ρ , σ ) ( x ) = 1 2 ( ρ + σ + 2 ) x + 1 2 ( ρ − σ ) ,

(3) J k ( ρ , σ ) ( − x ) = ( − 1 ) k J k ( ρ , σ ) ( x ) , J k ( ρ , σ ) ( − 1 ) = ( − 1 ) k Γ ( k + σ + 1 ) k ! Γ ( σ + 1 ) ,

where ρ , σ > − 1 , x ∈ [ − 1 , 1 ] and

a k ( ρ , σ ) = ( 2 k + ρ + σ + 1 ) ( 2 k + ρ + σ + 2 ) 2 ( k + 1 ) ( k + ρ + σ + 1 ) , b k ( ρ , σ ) = ( σ 2 − ρ 2 ) ( 2 k + ρ + σ + 1 ) 2 ( k + 1 ) ( k + ρ + σ + 1 ) ( 2 k + ρ + σ ) , c k ( ρ , σ ) = ( k + ρ ) ( k + σ ) ( 2 k + ρ + σ + 2 ) ( k + 1 ) ( k + ρ + σ + 1 ) ( 2 k + ρ + σ ) .

Moreover, the rth derivative of J j ( ρ , σ ) ( x ) is formulated as follows:

(4) D r J j ( ρ , σ ) ( x ) = Γ ( j + ρ + σ + q + 1 ) 2 r Γ ( j + ρ + σ + 1 ) J j − r ( ρ + r , σ + r ) ( x ) ,

where r represents an integer value. For the SJP J ℒ , k ( ρ , σ ) ( x ) = J k ( ρ , σ ) ( 2 x ℒ − 1 ) , ℒ > 0 , the analytic explicit form is given as follows:

(5) P ℒ , k ( ρ , σ ) ( x ) = ∑ j = 0 k ( − 1 ) k − j Γ ( k + σ + 1 ) Γ ( j + k + ρ + σ + 1 ) Γ ( j + σ + 1 ) Γ ( k + ρ + σ + 1 ) ( k − j ) ! j ! ℒ j x j = ∑ j = 0 k Γ ( k + ρ + 1 ) Γ ( k + j + ρ + σ + 1 ) j ! ( k − j ) ! Γ ( j + ρ + 1 ) Γ ( k + ρ + σ + 1 ) ℒ j ( x − ℒ ) j .

To deduce the following:

(6) P ℒ , k ( ρ , σ ) ( 0 ) = ( − 1 ) k Γ ( k + σ + 1 ) Γ ( σ + 1 ) k ! , J ℒ , k ( ρ , σ ) ( ℒ ) = Γ ( k + ρ + 1 ) Γ ( ρ + 1 ) k ! ,

(7) D r J ℒ , k ( ρ , σ ) ( 0 ) = ( − 1 ) k − r Γ ( k + σ + 1 ) ( k + ρ + σ + 1 ) r L r Γ ( k − r + 1 ) Γ ( r + σ + 1 ) ,

(8) D r J ℒ , k ( ρ , σ ) ( ℒ ) = Γ ( k + ρ + 1 ) ( k + ρ + σ + 1 ) r L r Γ ( k − r + 1 ) Γ ( r + ρ + 1 ) ,

(9) D r J ℒ , k ( ρ , σ ) ( x ) = Γ ( r + k + ρ + σ + 1 ) ℒ r Γ ( k + ρ + σ + 1 ) J ℒ , k − r ( ρ + r , σ + r ) ( x ) .

3 Methodology of shifted Jacobi collocation method

The collocation technique is an easy weighted residuals approach. Lanczos [46] first time introduced the proper trial function form together with the collocation point distributions that are considered fundamental to the precision of the obtained outcomes. Furthermore, this research work is revived by Clenshaw et al. [47,48] and Wright [49]. These studies involve the applications of the expansions of the Chebyshev polynomial to the initial value problems. Here, in this section, a numerical method based on the shifted Jacobi collocation approach is presented to solve a new nonlinear singular second kind of coupled functional LE differential model, given as follows:

(10) Z ″ ( a x + α ) + η 1 x Z ′ ( b x + β ) + Q ( x ) Z ( c x + γ ) = ℱ ( x ) Q ″ ( a x + α ) + η 2 x Q ′ ( b x + β ) + Z ( x ) Q ( c x + γ ) = ℋ ( x ) , 0 ≤ x ≤ ℒ ,

with the following conditions:

(11) Z ( ℒ ) = ζ 1 , Z ′ ( 0 ) = ζ 2 , Q ( ℒ ) = ζ 3 , Q ′ ( 0 ) = ζ 4 ,

where a , b , c , α , β , γ , η 1 , η 2 , ζ 1 , ζ 2 , ζ 3 , and ζ 4 are given constants while ℱ ( x ) , ℋ ( x ) are given functions. The solution of equation (10) is approximated as follows:

(12) Z K ( x ) = ∑ j = 0 K ς j J ℒ , j ( ρ , σ ) ( x ) = Δ ℒ , K ( ρ , σ ) ( x ) , Q K ( x ) = ∑ j = 0 K ε j J ℒ , j ( ρ , σ ) ( x ) = Ω ℒ , K ( ρ , σ ) ( x ) .

We obtain the approximate independent variables by applying the shifted Jacobi collocation scheme at x ℒ , K , j ( ρ , σ ) grids. These grids are the point set in an indicated range, where the values of the dependent variable are estimated. Generally, the performance of the location of the node becomes optional using x ℒ , K , j ( ρ , σ ) as a Jacobi–Gauss–Lobatto nodes. Thus, we can approximate the functions Z ( c x + γ ) , Q ( c x + γ ) as follows:

(13) Z K ( c x + γ ) = ∑ j =0 K ς j J ℒ , j ( ρ , σ ) ( c x + γ ) = Δ ℒ , K ( ρ , σ ) ( c x + γ ) Q K ( c x + γ ) = ∑ j =0 K ε j J ℒ , j ( ρ , σ ) ( c x + γ ) = Ω ℒ , K ( ρ , σ ) ( c x + γ ) .

Thus, the required derivatives of first and second orders of the approximate solutions are then estimated as follows:

(14) Z K ′ ( x ) = ∑ j = 0 K ς j ( J ℒ , j ( ρ , σ ) ( x ) ) ′ = ∑ j = 0 K ς j j + ρ + σ + 1 ℒ P ℒ , j − 1 ( ρ + 1 , σ + 1 ) ( x ) = ℘ ℒ , K ( ρ , σ ) ( x ) ,

(15) Q K ′ ( x ) = ∑ j = 0 K ε j ( J ℒ , j ( ρ , σ ) ( x ) ) ′ = ∑ j = 0 K ε j j + ρ + σ + 1 ℒ P ℒ , j − 1 ( ρ + 1 , σ + 1 ) ( x ) = ℑ ℒ , K ( ρ , σ ) ( x ) ,

(16) Z K ′ ( b x + β ) = ∑ j = 0 K ς j ( J ℒ , j ( ρ , σ ) ( b x + β ) ) ′ = ∑ j = 0 K ς j b ( j + ρ + σ + 1 ) ℒ P ℒ , j − 1 ( ρ + 1 , σ + 1 ) ( b x + β ) = ϕ ℒ , K ( ρ , σ ) ( x ) ,

and

(17) Q K ′ ( b x + β ) = ∑ j = 0 K ε j ( J ℒ , j ( ρ , σ ) ( b x + β ) ) ′ = ∑ j = 0 K ε j b ( j + ρ + σ + 1 ) ℒ P ℒ , j − 1 ( ρ + 1 , σ + 1 ) ( b x + β ) = φ ℒ , K ( ρ , σ ) ( x ) .

Also, we get

(18) Z K ″ ( a x + α ) = ∑ j = 0 K ς j ( J ℒ , j ( ρ , σ ) ( a x + α ) ) ″ = ∑ j = 0 K ς j a 2 ( j + ρ + σ + 1 ) ( j + ρ + σ + 2 ) ℒ 2 × P ℒ , j − 2 ( ρ + 2 , σ + 2 ) ( a x + α ) = χ ℒ , K ( ρ , σ ) ( x )

and

(19) Q K ″ ( a x + α ) = ∑ j = 0 K ε j ( J ℒ , j ( ρ , σ ) ( a x + α ) ) ″ = ∑ j = 0 K ε j a 2 ( j + ρ + σ + 1 ) ( j + ρ + σ + 2 ) ℒ 2 × P ℒ , j − 2 ( ρ + 2 , σ + 2 ) ( a x + α ) = ψ ℒ , K ( ρ , σ ) ( x ) .

Then, we can estimate the residual of (10) as follows:

(20) χ ℒ , K ( ρ , σ ) ( x ) + η 1 x ϕ ℒ , K ( ρ , σ ) ( x ) + Ω ℒ , K ( ρ , σ ) ( x ) Δ ℒ , K ( ρ , σ ) ( c x + γ ) = ℱ ( x ) ψ ℒ , K ( ρ , σ ) ( x ) + η 2 x φ ℒ , K ( ρ , σ ) ( x ) + Δ ℒ , K ( ρ , σ ) ( x ) Ω ℒ , K ( ρ , σ ) ( c x + γ ) = ℋ ( x ) .

In the shifted Jacobi collocation method, the residual (20) is let to be zero at the K − 1 points,

(21) χ ℒ , K ( ρ , σ ) ( x ℒ , K , i ( ρ , σ ) ) + η 1 x ℒ , K , i ( ρ , σ ) ϕ ℒ , K ( ρ , σ ) ( x ℒ , K , i ( ρ , σ ) ) + Ω ℒ , K ( ρ , σ ) ( x ℒ , K , i ( ρ , σ ) ) Δ ℒ , K ( ρ , σ ) ( c x ℒ , K , i ( ρ , σ ) + γ ) = ℱ ( x ℒ , K , i ( ρ , σ ) ) , ψ ℒ , K ( ρ , σ ) ( x ℒ , K , i ( ρ , σ ) ) + η 2 x ℒ , K , i ( ρ , σ ) φ ℒ , K ( ρ , σ ) ( x ℒ , K , i ( ρ , σ ) ) + Δ ℒ , K ( ρ , σ ) ( x ℒ , K , i ( ρ , σ ) ) Ω ℒ , K ( ρ , σ ) ( c x ℒ , K , i ( ρ , σ ) + γ ) = ℋ ( x ℒ , K , i ( ρ , σ ) ) ,

where i = 1 , 2 , 3 … , K − 1 . So, the 2 K − 2 algebraic model for 2 K + 2 , the remaining unknown equations can be achieved from the conditions (11) as follows:

(22) Δ ℒ , K ( ρ , σ ) ( ℒ ) = ζ 1 , ℘ ℒ , K ( ρ , σ ) ( 0 ) = ζ 2 , Ω ℒ , K ( ρ , σ ) ( ℒ ) = ζ 3 , ℑ ℒ , K ( ρ , σ ) ( 0 ) = ζ .

Finally, from equations (21) and (22), the ( K + 1 ) nonlinear algebraic system can be implemented to the unidentified coefficients ς j , j = 0 , … , K .

4 Numerical results and comparisons

We shall use the algorithm presented in the last section, in particular the three numerical variants to solve the coupled FD model to show the high accuracy as well as precision of the proposed method.

4.1 Problem I

Consider the following nonlinear singular second-order coupled FD model of LE type:

(23) Z ″ ( 2 x − 1 ) + 3 x Z ′ ( 3 x ) + Q ( x ) Z ( x + 1 ) = ℱ ( x ) , Q ″ ( 2 x − 1 ) + 2 x Q ′ ( 3 x ) + Z ( x ) Q ( x + 1 ) = ℋ ( x ) , 0 ≤ x ≤ 1 , Z ( 1 ) = 2 , Z ′ ( 0 ) = 0 , Q ( 1 ) = 0 , Q ′ ( 0 ) = 0 ,

where ℱ ( x ) and ℋ ( x ) are selected for the exact solutions as follows:

Z ( x ) = 1 + x 3 , Q ( x ) = 1 − x 3 .

In Table 1, the numerical solutions are ( Z K and Q K ) of Problem I for different parameter values. The resulting values in Table 1 show more accurate results. The perfect matching of the obtained and exact solutions is observed in Figures 1 and 2. The curves of absolute error (AE) E Z and E Q for the Problem I are provided in (3) and (4) (Figure 3).

Table 1

Numerical solutions of Problem I

K	2	3
ρ = σ = 0
Z K	1.92174 x 2 − 2.2204 × 10 − 16 x + 0.0782624	x 3 + 2.2204 × 10 − 16 x 2 + 1.11022 × 10 − 16 x + 1
Q K	− 1.87192 x 2 + 2.2204 × 10 − 16 x + 1.87192	1 − 1.11022 × 10 − 16 x − 4.44089 × 10 − 16 x 2 − x 3
ρ = − 0 . 5 , σ = 0
Z K	− 0.463357 + 2.46336 x 2	1 + 2.22045 × 10 − 16 x 2 + x 3
Q K	2.48434 − 2.48434 x 2	1 − 4.44089 × 10 − 16 x − x 3
ρ = σ = 0 . 5
Z K	0.0782624 + 1.92174 x 2	1 − 1.11022 × 10 − 16 x + x 3
Q K	1.87192 − 1.87192 x 2	1 + 2.22045 × 10 − 16 x 2 − x 3
ρ = σ = − 0 . 5
Z K	0.0782624 + 1.92174 x 2	1 + 2.22045 × 10 − 16 x + x 3
Q K	1.87192 + 2.22045 × 10 − 16 x − 1.87192 x 2	1 + 1.11022 × 10 − 16 x − x 3

$Figure 1 Plots of the exact and numerical results ( Z {\mathcal{Z}} and Z K {{\mathcal{Z}}}_{{\mathcal{K}}} ) of Problem I with ρ = σ = 0 \rho =\sigma =0 and K = 3 {\mathcal{K}}=3 .$

Figure 1

Plots of the exact and numerical results ( Z and Z K ) of Problem I with ρ = σ = 0 and K = 3 .

$Figure 2 Plots of the exact and numerical results ( Q {\mathcal{Q}} and Q K {{\mathcal{Q}}}_{{\mathcal{K}}} )of Problem I with ρ = σ = 0 \rho =\sigma =0 and K = 3 {\mathcal{K}}=3 .$

Figure 2

Plots of the exact and numerical results ( Q and Q K )of Problem I with ρ = σ = 0 and K = 3 .

$Figure 3 Plots of the AE ( E Z {E}_{{\mathcal{Z}}} ) of Problem I with ρ = σ = 0 \rho =\sigma =0 and K = 3 {\mathcal{K}}=3 .$

Figure 3

Plots of the AE ( E Z ) of Problem I with ρ = σ = 0 and K = 3 .

4.2 Problem II

The nonlinear singular second-order coupled FD model of LE type is written as follows:

(24) Z ″ ( 2 x − 1 ) + 3 x Z ′ ( 3 x ) + Q ( x ) Z ( x + 1 ) = ℱ ( x ) Q ″ ( 2 x − 1 ) + 2 x Q ′ ( 3 x ) + Z ( x ) Q ( x + 1 ) = ℋ ( x ) , 0 ≤ x ≤ 1 , Z ( 1 ) = 1 + cos ( 1 ) , Z ′ ( 0 ) = 0 , Q ( 1 ) = 1 − cos ( 1 ) , Q ′ ( 0 ) = 0 ,

where ℱ ( x ) and ℋ ( x ) are selected for the exact solutions as follows: Z ( x ) = 1 + cos ( x ) , Q ( x ) = 1 − cos ( x ) .

Table 2 highlights the accurate obtained results for the ℳ E Z and ℳ E Q using the spectral collocation method. Moreover, the logarithmic graphs of ℳ E Z and ℳ E Q are plotted using the current scheme for different values of ρ , σ , and ( K = 2 , 4 , … , 18 ) in Figures 4 and 5. Take ρ = σ = − 1 2 , the obtained form of the numerical solution becomes as follows:

Z 18 ( x ) = 2 + 2.64225 × 10 − 17 x − 0.5 x 2 + 2.93062 × 10 − 8 x 3 + 0.0416666 x 4 + 2.36602 × 10 − 9 x 5 − 0.00138888 x 6 − 5.80661 × 10 − 9 x 7 + 0.000024802 x 8 + 7.2645 × 10 − 10 x 9 − 2.75948 × 10 − 7 x 10 + 7.96816 × 10 − 11 x 11 + 2.09201 × 10 − 9 x 12 − 9.37722 × 10 − 12 x 13 − 7.83276 × 10 − 12 x 14 − 8.39506 × 10 − 13 x 15 + 1.72737 × 10 − 13 x 16 − 1.12777 × 10 − 14 x 17 + 3.31034 × 10 − 16 x 18 ,

Q 18 ( x ) = 1.21754 × 10 − 9 − 1.89035 × 10 − 17 x + 0.5 x 2 + 1.22541 × 10 − 8 x 3 − 0.0416667 x 4 − 1.16041 × 10 − 8 x 5 + 0.00138889 x 6 + 2.43626 × 10 − 9 x 7 − 0.0000248026 x 8 + 1.25299 × 10 − 11 x 9 + 2.75684 × 10 − 7 x 10 − 4.29394 × 10 − 11 x 11 − 2.08151 × 10 − 9 x 12 + 1.24306 × 10 − 12 x 13 + 1.05259 × 10 − 11 x 14 + 2.72614 × 10 − 13 x 15 − 9.49649 × 10 − 14 x 16 + 4.89297 × 10 − 15 x 17 − 9.21167 × 10 − 17 x 18 .

Table 2

ℳ ℰ Z and ℳ ℰ Q of Problem II

K	ρ = 0 , σ = 1 2		ρ = σ = 1 2		ρ = σ = 0
K	ℳ ℰ Z	ℳ ℰ Q	ℳ ℰ Z	ℳ ℰ Q	ℳ ℰ Z	ℳ ℰ Q
2	1.34 × 10 − 1	1.21 × 10 − 1	1.04 × 10 − 1	9.47 × 10 − 2	1.04 × 10 − 1	9.47 × 10 − 2
6	3.09 × 10 − 4	9.56 × 10 − 4	3.43 × 10 − 4	9.73 × 10 − 4	4.61 × 10 − 4	1.14 × 10 − 3
10	7.16 × 10 − 6	3.04 × 10 − 5	7.28 × 10 − 6	2.91 × 10 − 5	7.88 × 10 − 6	2.97 × 10 − 5
14	6.57 × 10 − 8	2.94 × 10 − 7	5.35 × 10 − 8	2.82 × 10 − 7	4.72 × 10 − 8	2.86 × 10 − 7
18	3.93 × 10 − 9	1.28 × 10 − 8	6.46 × 10 − 9	1.00 × 10 − 8	1.16 × 10 − 8	1.22 × 10 − 9

$Figure 4 Plots of the AE ( E Q {E}_{{\mathcal{Q}}} ) of Problem I with ρ = σ = 0 \rho =\sigma =0 and K = 3 {\mathcal{K}}=3 .$

Figure 4

Plots of the AE ( E Q ) of Problem I with ρ = σ = 0 and K = 3 .

$Figure 5 ℳ E Z { {\mathcal M} }_{{E}_{{\mathcal{Z}}}} convergence of Problem II.$

Figure 5

ℳ E Z convergence of Problem II.

4.3 Problem III

Consider the following nonlinear singular second kind of coupled FD LE system model (Figure 6):

(25) Z ″ ( 2 x − 1 ) + 3 x Z ′ ( 3 x ) + Q ( x ) Z ( x + 1 ) = ℱ ( x ) Q ″ ( 2 x − 1 ) + 2 x Q ′ ( 3 x ) + Z ( x ) Q ( x + 1 ) = ℋ ( x ) , 0 ≤ x ≤ 1 , Z ( 1 ) = 1 + cos ( 1 ) , Z ′ ( 0 ) = 0 , Q ( 1 ) = 1 − cos ( 1 ) , Q ′ ( 0 ) = 0 ,

where ℱ ( x ) and ℋ ( x ) are selected for the exact solutions, i.e., Z ( x ) = x + e − x , Q ( x ) = x − e − x . Table 3 provides accurate results for ℳ E Z and ℳ E Q using the spectral collection method. Moreover, the sketches in Figures 7 and 8 show the logarithmic graphs of ℳ E Z and ℳ E Q , which are obtained from the present scheme for different values of ρ , σ and ( K = 2 , 4 , … , 18 ) .

$Figure 6 ℳ E Q { {\mathcal M} }_{{E}_{{\mathcal{Q}}}} convergence of Problem II.$

Figure 6

ℳ E Q convergence of Problem II.

Table 3

ℳ ℰ Z and ℳ ℰ Q of Problem III

K	ρ = σ = 0		ρ = 0 , σ = 1 2		ρ = − 1 2 , σ = 0
K	ℳ ℰ Z	ℳ ℰ Q	ℳ ℰ Z	ℳ ℰ Q	ℳ ℰ Z	ℳ ℰ Q
2	8.36 × 10 − 2	3.18 × 10 − 1	1.06 × 10 − 1	4.28 × 10 − 1	1.12 × 10 − 1	4.63 × 10 − 1
6	6.52 × 10 − 4	4.17 × 10 − 3	5.44 × 10 − 4	3.78 × 10 − 3	5.94 × 10 − 4	4.15 × 10 − 3
10	8.98 × 10 − 6	1.10 × 10 − 4	8.74 × 10 − 6	1.15 × 10 − 4	8.96 × 10 − 6	1.17 × 10 − 4
14	3.96 × 10 − 8	1.06 × 10 − 6	3.83 × 10 − 8	1.107 × 10 − 6	3.92 × 10 − 8	1.12 × 10 − 6
18	2.23 × 10 − 9	2.59 × 10 − 9	6.10 × 10 − 9	8.50 × 10 − 9	4.55 × 10 − 9	7.78 × 10 − 9

$Figure 7 ℳ E Z { {\mathcal M} }_{{E}_{{\mathcal{Z}}}} convergence of Problem III.$

Figure 7

ℳ E Z convergence of Problem III.

$Figure 8 ℳ E Q { {\mathcal M} }_{{E}_{{\mathcal{Q}}}} convergence of Problem III.$

Figure 8

ℳ E Q convergence of Problem III.

Taking ρ = σ = 0 , the numerical solutions of Problem III are given as follows:

Z 18 ( x ) = 1 + 2.74967 × 10 − 17 x + 0.5 x 2 − 0.16765 x 3 + 0.0523669 x 4 − 0.00433457 x 5 + 0.00238683 x 6 − 0.000198414 x 7 + 0.0000248017 x 8 − 2.75562 × 10 − 7 x 9 + 2.75502 × 10 − 7 x 10 − 2.50341 × 10 − 8 x 11 + 3.08738 × 10 − 9 x 12 − 1.62179 × 10 − 10 x 13 + 1.21612 × 10 − 11 x 14 − 9.28464 × 10 − 13 x 15 + 7.09953 × 10 − 14 x 16 − 4.45469 × 10 − 15 x 17 + 1.51078 × 10 − 16 x 18 ,

Q 18 ( x ) = − 1 − 1.58323 × 10 − 17 x − 0.5 x 2 − 0.168767 x 3 − 0.0415439 x 4 − 0.00833331 x 5 − 0.00267886 x 6 − 0.000198417 x 7 − 0.0000247997 x 8 − 2.75582 × 10 − 6 x 9 − 2.75762 × 10 − 7 x 10 − 2.49717 × 10 − 8 x 11 − 2.10074 × 10 − 9 x 12 − 1.62453 × 10 − 10 x 13 − 9.7797 × 10 − 12 x 14 − 1.24893 × 10 − 12 x 15 + 2.79433 × 10 − 14 x 16 − 8.70301 × 10 − 15 x 17 − 7.23672 × 10 − 17 x 18 .

5 Conclusion

To model, the nonlinear LE system of FD equations and numerical presentations is not easy to handle. However, the solutions of the model are numerically presented by taking three different variants and compared with the true results, which depict the competency of the designed form of the system. The numerical spectral collocation method is the best choice to handle such complicated, singular, coupled nonlinear FD form models, whereas the traditional/conventional schemes do not work. Consequentially, the adopted numerical approach is an effective and suitable form to solve such systems. The spectral collocation method is a fast track of convergent approach, which can be implemented effectively with many types of linear/nonlinear, fractional/integer, singular/non-singular, and FD models. The present investigations show that the spectral collocation method is an effective and suitable scheme for solving the nonlinear LE second-order system of FD equations.

In the future, the designed method is an alternate promising solver to be exploited to examine the computational models of fluid dynamics, wire coating model, thin-film flow, squeezing flow systems, Jeffery Hamel type of systems, stretching flow problems, calendaring models, food processing systems, and related research areas [50,51,52,53,54]

Funding: The first author is partially supported by Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-097198-B-I00 and Fundación Séneca de la Región de Murcia grant number 20783/PI/18.

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Received: 2020-03-08

Revised: 2020-07-28

Accepted: 2020-09-17

Published Online: 2020-11-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model

Abstract

1 Introduction

2 Shifted Jacobi polynomials

3 Methodology of shifted Jacobi collocation method

4 Numerical results and comparisons

4.1 Problem I

4.2 Problem II

4.3 Problem III

5 Conclusion

References

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