A Numerical Algorithm for the Solutions of ABC Singular Lane–Emden Type Models Arising in Astrophysics Using Reproducing Kernel Discretization Method

: This paper deals with the numerical solutions and convergence analysis for general singular Lane–Emden type models of fractional order, with appropriate constraint initial conditions. A modiﬁed reproducing kernel discretization technique is used for dealing with the fractional Atangana–Baleanu–Caputo operator. In this tendency, novel operational algorithms are built and discussed for covering such singular models in spite of the operator optimality used. Several numerical applications using the well-known fractional Lane–Emden type models are examined, to expound the feasibility and suitability of the approach. From a numerical viewpoint, the obtained results indicate that the method is intelligent and has several features stability for dealing with many fractional models emerging in physics and mathematics, using the new presented derivative.

We are standing for the following: l ∈ J [0, 1]; δ ∈ (1, 2]; α, β, κ 1 , κ 2 ∈ R with κ 1 0 κ 2 ; ψ ∈ C(J → R); q 1 , q 2 ∈ C(J → R) while q 1 (0) = 0 = q 2 (0); Φ ∈ C(J × R → R); T ∈ C(J − {0} → R). We are recording ABC 0 ∂ δ l ψ(l) to sign the ABC fractional derivative of ψ in l over J of order δ with ABC 0 in which l = 0 is a base point acquaint at l ∈ J − B(J ) and ψ ∈ S 2 (J − B(J )), whereas S 2 is the Sobolev functions' space of order 2 on the domain J except the boundary B(J ) of J erected as The FLETM is categorized as a singular differential problem and supplied as an instrument in the formulation of the phenomena that emerge with various applications across mathematical physics and astrophysics. It characterizes the equilibrium thickness allocation in the self-gravitating sphere of polytrophic isothermal gas and, at the origin, contained singularity nodes. The FLETM has weight in the domain of modeling the clusters of galaxies, stellar structure, and radiative cooling. Interested reader can go through [32][33][34][35][36], to identify more details, properties, results, and applications on such singular models.

Concrete Structure of the RKDM
This part is dedicated to describing some adaptive necessary rules and preliminaries for the RKDM, especially those concerning the kernel functions and independency. We take AC(J ) to denote the set of absolutely continuous functions on J and we take L 2 (J ) to denote the set of square-integrable functions on J.
Assume that H is a reproducing kernel Hilbert space. From the Riesz representation theorem, it follows that for every k ∈ J, there exists only one H l (k) ∈ H, such that for every F ∈ H, we have ∀k ∈ J : F(k), H l (k) H = F(l).
Definition 1. [23] Let Π(J ) be a Hilbert space with inner product and functional structures, as is given below: Definition 2. [23] Let ∆(J ) be a Hilbert space with inner product and functional structures, as is given below: Theorem 1. [23] The space Π(J ) is a complete reproducing kernel with rule Theorem 2. [23] The space ∆(J ) is a complete reproducing kernel with rule When applied the RKDM, one must firstly split the convex compact set J into regular sections encoded with l i . This assumes that the acquired set {l i } ∞ i=1 will be dense in J. We attempt to cover J, as well as the numerical procedure ought to end in finite phases.
To examine the independency, suppose {θ i } m i=1 are not all zero, such that m i=1 θ i F l i (k) = 0. Take h s (k) ∈ Π(J ), such that h s (k i ) = δ i,k , ∀i = 1, 2, . . . , m, then is linearly independent for all m ≥ 1 and, thus, is linearly independent too.
From (5) an orthonormal functions system, then Theorem 3. The subsequent are achieved:
The n-term numerical solution of Equation (15) fulfills: Proof. Assume that ε ij are orthogonalization coefficients for the orthonormal functions systems . Then For the numerical computations, we truncate the series in (32) using the n-term numerical solution of ψ(l).
The attached steps focusing on the computational steps require using an appropriate software package for solving (15) using RKDM, and in order to evaluate the numerical approximation ψ n of ψ in Π(J ).

Algorithm 2.
Steps of RKDM for numerical approximations model of FLETM in ABC derivative: Step I: Fix l, k in J and do Phases 1 and 2: Phase 1: Set l i = 1 n i in the index i = 0, 1, . . . , n.
Output: the orthogonal function system Λ i (l).

Convergence Analysis
In this part, the convergence of numerical solution and error behavior are presented. Using convergent series representation, the following two theorems explain that FLETM described in Equation (15) is conditionally formulated and consistent.
To achieve our goal, we assume ψ n−1 Π is bounded whenever n → ∞ and {l i } ∞ i=1 is dense on J. Then the error E n = ψ − ψ n 2 Π is decreasing for sufficiently large n, since we have The convergence of ∞ i=1 ψ(l), Λ i (l) Π Λ i (l) yields E n → 0 whenever n → ∞ as long as ψ(l) and ψ n (l) are extracted from (32) and (33).

Model Experiments and Computational Results
In this important portion; in order to solve FLETM in (1) and (2) numerically using the RKDM, three models are presented in certain specific form. In the examples, we demonstrate the performance and efficiency of the proposed approach in term of tables and figures, with some scientific explanations' comments.

Certain Examples
In the subsequent FLETM, the readers should note that ψ(0) and ∂ l ψ(0) are known and may not be homogeneous. The forcing term T(l) can be obtain by substituting ψ(l) through the given model.

Results and Discussions
Take into consideration Algorithms 1 and 2, following the RKDM, using l i = i n , i = 0, 1, . . . , n = 50 the numerical validations for different values of grid points l i ∈ J will be exhibited. For this purpose, Tables 1-3 tabulates the evolution of the absolute errors Ab as for Examples 1, 2, and 3, simultaneously.    From the tables, we observe that the RKDM numerical outcomes are unanimous with analytic solutions during in the area of interest. Additional iterations will lead to more refined solutions along the memory and heritage characteristics of δ. The ABC fractional derivative orders have powerful belongings on the model shapes, which head for lead to remarkable behaviors in the incident of a considerable departure from the value of δ = 2.
The 3D surfaces plot of the RKDM numerical solutions for Examples 1, 2, and 3 are drawn in Figure 1a-c simultaneously, for different values of grid points l i ∈ J when δ ∈ (1,2]. It appears that all figures almost look identical in their behaviors, and in good agreement with each other, particularly when comparing the case of δ = 2. Moreover, the RKDM numerical solutions are very close at the CICs.

Conclusion and Outline
The attractive RKDM has been successfully employed to construct and predict the numerical/analytic solutions for FLETM under the ABC fractional sense. Convergence and consistency were discussed, which turns out that the proposed scheme has decreasing absolute error in the Π space. Three FLETM models have been given to test applicability and straightforwardness of the presented approach. The gained numerical data reveal that the numerical solutions are conformable with each other at the selected parameters and nods. Finally, one can see that the RKDM is a methodical and convenient scheme to address various fractional differential/integral problems across applied sciences and engineering area.
In the near future, we intend to conduct more research as a continuation of this work. One of these research studies is related to the applications of the RKDM to solve numerically the Lane-Emden type models that contain functions with singularities or weak regularity, subject to CICs or constraint boundary conditions.

Conclusions and Outline
The attractive RKDM has been successfully employed to construct and predict the numerical/analytic solutions for FLETM under the ABC fractional sense. Convergence and consistency were discussed, which turns out that the proposed scheme has decreasing absolute error in the Π(J ) space. Three FLETM models have been given to test applicability and straightforwardness of the presented approach. The gained numerical data reveal that the numerical solutions are conformable with each other at the selected parameters and nods. Finally, one can see that the RKDM is a methodical and convenient scheme to address various fractional differential/integral problems across applied sciences and engineering area.
In the near future, we intend to conduct more research as a continuation of this work. One of these research studies is related to the applications of the RKDM to solve numerically the Lane-Emden type models that contain functions with singularities or weak regularity, subject to CICs or constraint boundary conditions.