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

Abstract

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.

1. Prolegomena and Presentation

Fractional calculus is dealing with investigations and applications of derivatives and integrations of arbitrary order that provide an attractive mechanism for explaining the memory and hereditary effort of complex systems [1,2,3,4,5]. Fractional calculus theory dates back to Leibniz in the sixteenth century, and after that, many forms of fractional operators have been introduced in the classical theory. Among them are the Riemann–Liouville, Caputo–Liouville, Atangana–Baleanu–Caputo, and Riesz operator approaches [6,7,8,9,10,11,12,13,14,15,16,17,18]. Fractional calculus attracted focus in current scientific research, due to its nonlocal nature and its ability to handle the effects of external forces of phenomena that cannot be modeled in a traditional way. Recently, a new definition of fractional derivative has proposed, the ABC fractional derivative [19,20,21,22,23,24,25,26,27,28,29,30,31].

At all events, it is not easy to find accurate numerical solutions to FLETM due to the complexity that occurs inside the Mittag–Leffler function in the fractional ABC derivatives. Thereafter, software computer programming is ordinarily used to obtain numerical solutions in acceptable accuracy. Over this treatise, we face to utilize the RKDM to gain approximate numerical outcomes of FLETM, utilizing the ABC fractional sense. More specifically, we consider the subsequent form [32,33].

$${}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)+\frac{{\kappa }_1}{q_1\left(l\right)}{\partial }_l\psi \left(l\right)+\frac{{\kappa }_2}{q_2\left(l\right)}\psi \left(l\right)+\mathsf{\Phi }\left(l,\psi \left(l\right)\right)=\mathrm{T}\left(l\right),$$

(1)

subject to the CICs

$$\psi \left(0\right)=\alpha \mathrm{and} {\partial }_l\psi \left(0\right)=\beta .$$

(2)

We are standing for the following: $l\in \mathcal{J}≔\left[0,1\right]$; $\delta \in (1,2]$; $\alpha ,\beta ,{\kappa }_1,{\kappa }_2\in ℝ$ with ${\kappa }_1\ne 0\ne {\kappa }_2$; $\psi \in C\left(\mathcal{J}\to ℝ\right)$; $q_1,q_2\in C\left(\mathcal{J}\to ℝ\right)$ while $q_1\left(0\right)=0=q_2\left(0\right)$; $\mathsf{\Phi }\in C\left(\mathcal{J}\times ℝ\to ℝ\right)$; $\mathrm{T}\in C\left(\mathcal{J}-\left\{0\right\}\to ℝ\right)$. We are recording ${}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)$ to sign the ABC fractional derivative of $\psi$ in $l$ over $\mathcal{J}$ of order $\delta$ with

$${}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)=\left(1-\delta \right)\left(1-\delta +\delta {\mathsf{\Gamma }}^{-1}\left(\delta \right)\right){\int }_0^l{\partial }^2\psi \left(k\right){\displaystyle \sum }_{n=0}^{\infty }\frac{{\left(-1\right)}^n}{\mathsf{\Gamma }\left(n\delta +1\right)}{\left(\frac{\delta }{2-\delta }\right)}^n{\left(l-k\right)}^{n\delta }dk,$$

(3)

in which $l=0$ is a base point acquaint at $l\in \mathcal{J}-\mathfrak{B}\left(\mathcal{J}\right)$ and $\psi \in {\mathcal{S}}^2\left(\mathcal{J}-\mathfrak{B}\left(\mathcal{J}\right)\right)$, whereas ${\mathcal{S}}^2$ is the Sobolev functions’ space of order 2 on the domain $\mathcal{J}$ except the boundary $\mathfrak{B}\left(\mathcal{J}\right)$ of $\mathcal{J}$ erected as

$${\mathcal{S}}^2\left(\mathcal{J}-\mathfrak{B}\left(\mathcal{J}\right)\right)=\left\{\psi \in L^2\left(\mathcal{J}-\mathfrak{B}\left(\mathcal{J}\right)\right):{\partial }_l\psi \left(l\right),{\partial }_l^2\psi \left(l\right)\in L^2\left(\mathcal{J}-\mathfrak{B}\left(\mathcal{J}\right)\right)\right\}.$$

(4)

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.

The standard RKDM main field topics are in the modelling and simulation of sundry-dimensional issues in applied computational physical, applied mathematics, and engineering [37,38,39], it has been used in creating numerical and approximate solutions for integral and differential models in the shape of infinite convergent series with floor extent of calculations, without any limited assumptions. This approach adjusted has been utilized as a solver technique to deal with complexes’ nonlinear and discontinues shapes of integral/differential problems arising in various applications area range from engineering to physics as utilized in [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59].

2. 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 $AC\left(\mathcal{J}\right)$ to denote the set of absolutely continuous functions on $\mathcal{J}$ and we take $L^2\left(\mathcal{J}\right)$ to denote the set of square-integrable functions on $\mathcal{J}$.

Assume that $ℍ$ is a reproducing kernel Hilbert space. From the Riesz representation theorem, it follows that for every $k\in \mathcal{J}$, there exists only one $H_l\left(k\right)\in ℍ$, such that for every $F\in ℍ$, we have

$$\forall k\in \mathcal{J}: ⟨F\left(k\right),H_l{\left(k\right)⟩}_{ℍ}=F\left(l\right).$$

(5)

Definition 1. 

[23] Let$\mathsf{\Pi }\left(\mathcal{J}\right)$be a Hilbert space with inner product and functional structures, as is given below:

$$\{\begin{array}{l}\mathsf{\Pi }\left(\mathcal{J}\right)=\{\psi \left(l\right):\psi \left(l\right),{\partial }_l\psi \left(l\right),{\partial }_l^2\psi \left(l\right)\in AC\left(\mathcal{J}\right); {\partial }_l^3\psi \left(l\right)\in L^2\left(\mathcal{J}\right);\psi \left(0\right)={\partial }_l\psi \left(0\right)=0\},\hfill \\ ⟨{\psi }_1\left(l\right),{\psi }_2{\left(l\right)⟩}_{\mathsf{\Pi }}={\partial }_l^2{\psi }_1\left(0\right){\partial }_l^2{\psi }_2\left(0\right)+{\int }_{\mathcal{J}}{\partial }_l^3{\psi }_1\left(l\right){\partial }_l^3{\psi }_2\left(l\right)dl,\hfill \\ {\Vert \psi \Vert }_{\mathsf{\Pi }}=\sqrt{\psi \left(l\right),\psi {\left(l\right)}_{\mathsf{\Pi }}}.\hfill \end{array}$$

(6)

Definition 2. 

[23] Let$\Delta \left(\mathcal{J}\right)$be a Hilbert space with inner product and functional structures, as is given below:

$$\{\begin{array}{l}\mathsf{\Delta }\left(\mathcal{J}\right)=\left\{\psi \left(l\right):\psi \left(l\right)\in AC\left(\mathcal{J}\right); {\partial }_l\psi \left(l\right)\in L^2\left(\mathcal{J}\right)\right\},\hfill \\ ⟨{\psi }_1\left(l\right),{\psi }_2{\left(l\right)⟩}_{\mathsf{\Delta }}={\int }_{\mathcal{J}}{\psi }_1\left(l\right){\psi }_2\left(l\right)dl+{\partial }_l{\psi }_1\left(l\right){\partial }_l{\psi }_2\left(l\right)dl,\hfill \\ {\Vert \psi \Vert }_{\mathsf{\Delta }}=\sqrt{\psi \left(l\right),\psi {\left(l\right)}_{\mathsf{\Delta }}}.\hfill \end{array}$$

(7)

Theorem 1. 

[23] The space$\mathsf{\Pi }\left(\mathcal{J}\right)$is a complete reproducing kernel with rule

$$F_l\left(k\right)=\frac{1}{120}\{\begin{array}{l}{l}^2\left(-5k^{2}l+k^3+10l^2\left(3+k\right)\right),k\le l,\hfill \\ k^2\left(-5l^{2}k+l^3+10k^2\left(3+l\right)\right),k>l.\hfill \end{array}$$

(8)

Theorem 2. 

[23] The space$\Delta \left(\mathcal{J}\right)$is a complete reproducing kernel with rule

$$G_l\left(k\right)=\frac{sinh\left(1\right)}{2}\{\begin{array}{l}cosh\left(l+k-1\right)+cosh\left(l-k-1\right),k\le l,\hfill \\ cosh\left(l+k-1\right)+cosh\left(k-l-1\right),k>l.\hfill \end{array}$$

(9)

When applied the RKDM, one must firstly split the convex compact set $\mathcal{J}$ into regular sections encoded with $l_i$. This assumes that the acquired set ${\left\{l_i\right\}}_{i=1}^{\infty }$ will be dense in $\mathcal{J}$. We attempt to cover $\mathcal{J}$, as well as the numerical procedure ought to end in finite phases.

To examine the independency, suppose ${\left\{{\theta }_i\right\}}_{i=1}^m$ are not all zero, such that $\sum _{i=1}^m{\theta }_i{F}_{l_i}\left(k\right)=0$. Take $h_s\left(k\right)\in \mathsf{\Pi }\left(\mathcal{J}\right)$, such that $h_s\left(k_i\right)={\delta }_{i,k}$, $\forall i=1,2,\dots ,m$, then

$$\begin{array}{cc}0=\hfill & ⟨h_s\left(k\right),{\displaystyle \sum _{i=1}^m}{\theta }_i{F}_{l_i}{\left(k\right)⟩}_{\mathsf{\Pi }}\hfill \\ & ={\displaystyle \sum _{i=1}^m}{\theta }_i{h}_{\mathrm{s}}\left(k\right),F_{l_i}{\left(k\right)}_{\mathsf{\Pi }}\hfill \\ & ={\displaystyle \sum _{i=1}^m}{\theta }_i{h}_s\left(k_i\right)\hfill \\ & ={\theta }_i,\hfill \end{array}$$

(10)

for $i=1,2,\dots ,m$. This shows that ${\left\{F_{l_i}\left(k\right)\right\}}_{i=1}^m$ is linearly independent for all $m\ge 1$ and, thus, ${\left\{F_{l_i}\left(k\right)\right\}}_{i=1}^{\infty }$ is linearly independent in $\mathsf{\Pi }\left(\mathcal{J}\right)$. Similarly, one can find that ${\left\{G_{l_i}\left(k\right)\right\}}_{i=1}^{\infty }$ is linearly independent too.

3. Solutions Shape of FLETM

Multiplying (1) by $q_1\left(l\right)q_2\left(l\right)$, we get

$$q\left(l\right){}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)+{\kappa }_1{q}_2\left(l\right){\partial }_l\psi \left(l\right)+{\kappa }_2{q}_1\left(l\right)\psi \left(l\right)+ \overline{\mathsf{\Phi }}\left(l,\psi \left(l\right)\right)= \overline{\mathrm{T}}\left(l\right),$$

(11)

in which $q\left(l\right)=q_1\left(l\right)q_2\left(l\right)$, $\overline{\mathsf{\Phi }}\left(l,\psi \left(l\right)\right)=q_1\left(l\right)q_2\left(l\right)\mathsf{\Phi }\left(l,\psi \left(l\right)\right)$, and $\overline{\mathrm{T}}\left(l\right)=q_1\left(l\right)q_2\left(l\right)\mathrm{T}\left(l\right)$.

The replacement $\psi \left(l\right):\to \psi \left(l\right)-\left(\beta l+\alpha \right)$ converts FLETM given by (11) and (2) into homogenous one. We are denoting the new equation by

$$q\left(l\right){}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)+{\kappa }_1{q}_2\left(l\right){\partial }_l\psi \left(l\right)+{\kappa }_2{q}_1\left(l\right)\psi \left(l\right)+ \overline{\mathsf{\Phi }}\left(l,\psi \left(l\right)-\left(\beta l+\alpha \right)\right)=\overline{\overline{\mathrm{T}}}\left(l\right),$$

(12)

subject to the CICs

$$\psi \left(0\right)=0 \mathrm{and} {\partial }_l\psi \left(0\right)=0.$$

(13)

The function $\overline{\overline{\mathrm{T}}}\left(l\right)$ appears from substituting $\psi \left(l\right)-\left(\beta l+\alpha \right)$ instead of $\psi \left(l\right)$ in (12). In fact, after simplification and transform the extra mathematical terms into the right hand side of (12), one can write $\overline{\overline{\mathrm{T}}}\left(l\right)=q\left(l\right){}_0{}^{ABC}\partial {}_l^{\delta }\left(\beta l+\alpha \right)+{\kappa }_1{q}_2\left(l\right){\partial }_l\left(\beta l+\alpha \right)+{\kappa }_2{q}_1\left(l\right)\left(\beta l+\alpha \right)+\overline{\mathrm{T}}\left(l\right)$.

Define the linear operator $\mathcal{A}:\mathsf{\Pi }\left(\mathcal{J}\right)\to \mathsf{\Delta }\left(\mathcal{J}\right)$ and its map $\mathcal{A}\left[\psi \right]\left(l\right)$ as

$$\mathcal{A}\left[\psi \right]\left(l\right)=q\left(l\right){}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)+{\kappa }_1{q}_2\left(l\right){\partial }_l\psi \left(l\right)+{\kappa }_2{q}_1\left(l\right)\psi \left(l\right).$$

(14)

Putting $\zeta \left(l,\psi \left(l\right),{\partial }_l\psi \left(l\right)\right):=\overline{\overline{\mathrm{T}}}\left(l\right)- \overline{\mathsf{\Phi }}\left(l,\psi \left(l\right)-\left(\beta l+\alpha \right)\right)$, then (14) and (13) are converted into the form

$$\{\begin{array}{l}\mathcal{A}\left[\psi \right]\left(l\right)=\zeta \left(l,\psi \left(l\right),{\partial }_l\psi \left(l\right)\right),\hfill \\ \psi \left(0\right)=0 and {\partial }_l\psi \left(0\right)=0.\hfill \end{array}$$

(15)

We arrange the system of orthogonal functions taking ${\mathfrak{S}}_i\left(l\right)=G_{l_i}\left(l\right)$, ${\Lambda }_i\left(l\right)={\mathcal{A}}^{*}\left[{\mathfrak{S}}_i\right]\left(l\right)$, $i=1,2,3,\dots$, such that ${\left\{l_i\right\}}_{i=1}^{\infty }$ is dense on $\mathcal{J}$. The Gram–Schmidt orthogonalization process is used to generate the system of orthonormal functions ${\left\{{ \overline{\Lambda }}_i\left(l\right)\right\}}_{i=1}^{\infty }$ on $\mathsf{\Pi }\left(\mathcal{J}\right)$, where

$${ \overline{\Lambda }}_i\left(l\right)={\displaystyle \sum }_{j=1}^i{\epsilon }_{ij}{\Lambda }_j\left(l\right),$$

(16)

with the orthogonalization coefficients ${\epsilon }_{ij}$ with the indexes $i=2,3,\dots$, and $j=1,2,\dots ,i-1$ are computed in the subsequent algorithm.

Algorithm 1. Steps of the orthonormal Gram–Schmidt process:

Step 1: For $i=2,3,\dots$ and $j=1,2,\dots ,i-1$, do the following:      $$\begin{array}{c}{\epsilon }_{11}=\frac{1}{\Vert {\Lambda }_1\Vert {}_{\mathsf{\Pi }}},\\ {\epsilon }_{ii}=\frac{1}{\sqrt{\Vert {\Lambda }_i\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^{i-1}⟨{\Lambda }_i\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}},i\ne 1,\\ {\epsilon }_{ij}=-\frac{1}{\sqrt{\Vert {\Lambda }_i\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^{i-1}⟨{\Lambda }_i\left(t\right),{ \overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}{\displaystyle \sum _{p=j}^{i-1}}⟨{\Lambda }_i\left(t\right),{ \overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{pj},i>j.\end{array}$$ (17)     Output: The orthogonalization coefficients ${\epsilon }_{ij}$.     Step 2: For $i=1,2,3,\dots$ set $${\overline{\Lambda }}_i\left(l\right)={\displaystyle \sum }_{j=1}^i{\epsilon }_{ij}{\Lambda }_j\left(l\right).$$ (18)     Output: System of orthonormal functions ${\left\{{ \overline{\Lambda }}_i\left(l\right)\right\}}_{i=1}^{\infty }$.

Remark 1. 

In the third formula of (17), the calculations of ${\epsilon }_{ij}$,$i>j$are obtaining recursively as follows:

• If$i=2$and$j=1$, then

  $${\epsilon }_{21}=-\frac{1}{\sqrt{\Vert {\Lambda }_2\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^1⟨{\Lambda }_1\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}{\displaystyle \sum }_{p=1}^1⟨{\Lambda }_2\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p1},$$

  (19)

  where

  $$\begin{array}{c}{\displaystyle \sum _{p=1}^1}⟨{\Lambda }_2\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p1}=⟨{\Lambda }_2\left(t\right),{\overline{\Lambda }}_1{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{11}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨{\Lambda }_2\left(t\right),{\overline{\Lambda }}_1{\left(t\right)⟩}_{\mathsf{\Pi }}\frac{1}{\Vert {\Lambda }_1\Vert {}_{\mathsf{\Pi }}}.\end{array}$$

  (20)
• If$i=3$and$j=1,2$, then

  $${\epsilon }_{31}=-\frac{1}{\sqrt{\Vert {\Lambda }_3\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^2⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}{\displaystyle \sum _{p=1}^2}⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p1},$$

  (21)

  where

  $$\begin{array}{c}{\displaystyle \sum _{p=1}^2}⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p1}=⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_1{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{11}+⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{21}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_1{\left(t\right)⟩}_{\mathsf{\Pi }}\frac{1}{\Vert {\Lambda }_1\Vert {}_{\mathsf{\Pi }}}+⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{21}\end{array}$$

  (22)

  in which${\epsilon }_{21}$comes from (19). Similarly,

  $${\epsilon }_{32}=-\frac{1}{\sqrt{\Vert {\Lambda }_3\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^2⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}{\displaystyle \sum _{p=2}^2}⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p2},$$

  (23)

  where

  $$\begin{array}{c}{\displaystyle \sum _{p=2}^2}⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p2}=⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{22}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}\frac{1}{\sqrt{\Vert {\Lambda }_2\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^1⟨{\Lambda }_2\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}.\hfill \end{array}$$

  (24)
• If$i=4$and$j=1,2$, 3 then

  $${\epsilon }_{41}=-\frac{1}{\sqrt{\Vert {\Lambda }_4\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^3⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}{\displaystyle \sum _{p=1}^3}⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p1},$$

  (25)

  where

  $$\begin{array}{c}{\displaystyle \sum _{p=1}^3}⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p1}=⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_1{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{11}+⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{21}+⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_3{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{31}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_1{\left(t\right)⟩}_{\mathsf{\Pi }}\frac{1}{\Vert {\Lambda }_1\Vert {}_{\mathsf{\Pi }}}+⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{21}+⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_3{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{31},\end{array}$$

  (26)

  in which${\epsilon }_{21}$comes from (19) and${\epsilon }_{31}$comes from (21). Similarly,

  $${\epsilon }_{42}=-\frac{1}{\sqrt{\Vert {\Lambda }_4\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^3⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}{\displaystyle \sum _{p=2}^3}⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p2},$$

  (27)

  where

  $$\begin{array}{c}{\displaystyle \sum _{p=2}^3}⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p2}=⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{22}+⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_3{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{32}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_2{\left(t\right)⟩}_{\mathsf{\Pi }}\frac{1}{\sqrt{\Vert {\Lambda }_i\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^{i-1}⟨{\Lambda }_i\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}+⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_3{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{32},\end{array}$$

  (28)

  in which${\epsilon }_{32}$comes from (23). Finally,

  $${\epsilon }_{43}=-\frac{1}{\sqrt{\Vert {\Lambda }_4\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^3⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}{\displaystyle \sum _{p=3}^3}⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p3},$$

  (29)

  where

  $$\begin{array}{c}{\displaystyle \sum _{p=3}^3}⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{p3}=⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_3{\left(t\right)⟩}_{\mathsf{\Pi }}{\epsilon }_{33}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨{\Lambda }_4\left(t\right),{\overline{\Lambda }}_3{\left(t\right)⟩}_{\mathsf{\Pi }}\frac{1}{\sqrt{\Vert {\Lambda }_3\Vert {}_{\mathsf{\Pi }}^2-{\sum }_{p=1}^2⟨{\Lambda }_3\left(t\right),{\overline{\Lambda }}_p{\left(t\right)⟩}_{\mathsf{\Pi }}^2}}.\end{array}$$

  (30)

Analogue for the remaining indexes$i=4, 5,\dots$and$j=1,2,\dots ,i-1$.

Lemma 1. 

The system${\left\{{\Lambda }_i\left(l\right)\right\}}_{i=1}^{\infty }$is complete on $\mathsf{\Pi }\left(\mathcal{J}\right).$

Proof. 

${\Lambda }_i\left(l\right)={\mathcal{A}}^{*}\left[{\mathfrak{S}}_i\right]\left(l\right)$ assures that ${\Lambda }_i\left(l\right)\in \mathsf{\Pi }\left(\mathcal{J}\right)$. For each $\psi \left(l\right)\in \mathsf{\Pi }\left(\mathcal{J}\right)$, if $⟨\psi \left(l\right),{\Lambda }_i{\left(l\right)⟩}_{\mathsf{\Pi }}=0$, $i=1,2,\dots$, then

$$\begin{array}{cc}⟨\psi \left(l\right),{\Lambda }_i{\left(l\right)⟩}_{\mathsf{\Pi }}& =⟨\psi \left(l\right),{\mathcal{A}}^{*}\left[{\mathfrak{S}}_i\right]{\left(l\right)⟩}_{\mathsf{\Pi }}\\ & =⟨\mathcal{A}\left[\psi \right]\left(l\right),{\mathfrak{S}}_i{\left(l\right)⟩}_{\mathsf{\Delta }}\\ & =\mathcal{A}\left[\psi \right]\left(l_i\right)\\ & =0.\end{array}$$

(31)

From (5) we get $\mathcal{A}\left[\psi \right]\left(l_i\right)=⟨\mathcal{A}\left[\psi \right]\left(l\right),{\mathfrak{S}}_i{\left(l\right)⟩}_{\mathsf{\Pi }}=0$. By the density of ${\left\{l_i\right\}}_{i=1}^{\infty }$ in $\mathcal{J}$, we have $\mathcal{A}\left[\psi \right]\left(l\right)=0$. The existence of ${\mathcal{A}}^{-1}$ yields $\psi \left(l\right)=0$. Subsequently, ${\left\{{\Lambda }_i\left(l\right)\right\}}_{i=1}^{\infty }$ is complete on $\mathsf{\Pi }\left(\mathcal{J}\right)$. □

Definition 3. 

[60] If$\psi$is a continuous function and${\left\{{\overline{\Lambda }}_i\left(l\right)\right\}}_{i=1}^{\infty }$an orthonormal functions system, then$⟨\psi \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathsf{\Pi }}$, $i=1,2,\dots$are called Fourier functions of$\psi$with respect to the system${\left\{{\overline{\Lambda }}_i\left(l\right)\right\}}_{i=1}^{\infty }$and$\psi \left(l\right)={\sum }_{i=1}^{\infty }⟨\psi \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathsf{\Pi }}{\overline{\Lambda }}_i\left(l\right)$is called its Fourier expansion.

Theorem 3. 

The subsequent are achieved:

• Whenever$n\to \infty$the analytic solution of (15) fulfills:

  $$\psi \left(l\right)={\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^i}{\epsilon }_{ij}\zeta \left(l_j,\psi \left(l_j\right),{\partial }_l\psi \left(l_j\right)\right){\overline{\Lambda }}_i\left(l\right).$$

  (32)
• The$n$-term numerical solution of Equation (15) fulfills:

  $${\psi }^n\left(l\right)={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^i}{\epsilon }_{ij}\zeta \left(l_j,\psi \left(l_j\right),{\partial }_l\psi \left(l_j\right)\right){\overline{\Lambda }}_i\left(l\right).$$

  (33)

Proof. 

Assume that ${\epsilon }_{ij}$ are orthogonalization coefficients for the orthonormal functions systems ${\left\{{ \overline{\Lambda }}_i\left(l\right)\right\}}_{i=1}^{\infty }$. Then

$$\begin{array}{c}\psi \left(l\right)={\displaystyle \sum _{i=1}^{\infty }}⟨\psi \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathsf{\Pi }}{\overline{\Lambda }}_i\left(l\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\infty }}⟨\psi \left(l\right),{\displaystyle \sum _{j=1}^i}{\epsilon }_{ij}{\Lambda }_j{\left(l\right)⟩}_{\mathsf{\Pi }}{\overline{\Lambda }}_i\left(l\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^i}{\epsilon }_{ij}⟨\psi \left(l\right),{\mathcal{A}}^{*}\left[{\mathfrak{S}}_j\right]{\left(l\right)⟩}_{\mathsf{\Pi }}{\overline{\Lambda }}_i\left(l\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^i}{\epsilon }_{ij}⟨\mathcal{A}\left[\psi \right]\left(l\right),{\mathfrak{S}}_j{\left(l\right)⟩}_{\mathsf{\Delta }}{ \overline{\Lambda }}_i\left(l\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^i}{\epsilon }_{ij}⟨\zeta \left(l,\psi \left(l\right),{\partial }_l\psi \left(l\right)\right),{\mathfrak{S}}_j{\left(l\right)⟩}_{\mathsf{\Delta }}{ \overline{\Lambda }}_i\left(l\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{j=1}^i}{\epsilon }_{ij}\zeta \left(l_j,\psi \left(l_j\right),{\partial }_l\psi \left(l_j\right)\right){\overline{\Lambda }}_i\left(l\right).\end{array}$$

(34)

For the numerical computations, we truncate the series in (32) using the $n$-term numerical solution of $\psi \left(l\right)$. □

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 ${\psi }^n$ of $\psi$ in $\mathsf{\Pi }\left(\mathcal{J}\right)$.

Algorithm 2. Steps of RKDM for numerical approximations model of FLETM in ABC derivative:

Step I: Fix $l,k$ in $\mathcal{J}$ and do Phases 1 and 2:      Phase 1: Set $l_i=\frac{1}{n}i$ in the index $i=0,1,\dots ,n$.      Phase 2: Set ${\Lambda }_i\left(l\right)={\mathcal{A}}^{*}\left[{\mathfrak{S}}_i\right]\left(l\right)$ in the index $i=1,2,\dots ,n$.   Output: the orthogonal function system ${\Lambda }_i\left(l\right)$.   Step II: For the indices $i=1,2,\dots$ and $j=1,2,\dots ,i-1$ do Algorithm 1.   Output: the orthogonalization coefficients ${\epsilon }_{ij}$.   Step III: Set ${ \overline{\Lambda }}_i\left(l\right)={\sum }_{j=1}^i{\epsilon }_{ij}{\Lambda }_j\left(l\right)$ in the indices $i=1,2,\dots ,n$.   Output: the orthonormal function system ${ \overline{\Lambda }}_i\left(l\right)$.   Step IV: Set ${\psi }^0\left(l_1\right)=0$ and with the indices $i=1,2,\dots ,n$ do Phases 1, 2, and 3:      Phase 1: Set ${\psi }^i\left(l_i\right)={\psi }^{i-1}\left(l_i\right)$.      Phase 2: Set ${ℱ}_i={\sum }_{j=1}^i{\epsilon }_{ij}\zeta \left(l_j,\psi \left(l_j\right),{\partial }_l\psi \left(l_j\right)\right)$.      Phase 3: Set ${\psi }^n\left(l\right)={\sum }_{j=1}^i{ℱ}_j{\overline{\Lambda }}_j\left(l\right)$.   Output: The $n$-term numerical approximation ${\psi }^n\left(l\right)$ of $\psi \left(l\right)$.

4. 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 $\Vert {\psi }^{n-1\Vert }{}_{\mathsf{\Pi }}$ is bounded whenever $n\to \infty$ and ${\left\{l_i\right\}}_{i=1}^{\infty }$ is dense on $\mathcal{J}$. Then the error ${ℰ}^n=\Vert \psi -{\psi }^n\Vert {}_{\mathsf{\Pi }}^2$ is decreasing for sufficiently large $n$, since we have

$$\begin{array}{c}{ℰ}^n-{ℰ}^{n-1}=\Vert {\displaystyle \sum _{i=n+1}^{\infty }}⟨\psi \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathcal{A}}{ \overline{\Lambda }}_i{\left(l\right)\Vert }_{\mathsf{\Pi }}^2-\Vert {\displaystyle \sum _{i=n}^{\infty }}⟨\mu \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathcal{A}}{ \overline{\Lambda }}_i{\left(l\right)\Vert }_{\mathsf{\Pi }}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{i=n+1}^{\infty }}⟨\psi \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathsf{\Pi }}^2-{\displaystyle \sum _{i=n}^{\infty }}⟨\psi \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathsf{\Pi }}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}<0.\end{array}$$

(35)

The convergence of ${\sum }_{i=1}^{\infty }⟨\psi \left(l\right),{\overline{\Lambda }}_i{\left(l\right)⟩}_{\mathsf{\Pi }}{\overline{\Lambda }}_i\left(l\right)$ yields ${ℰ}^n\to 0$ whenever $n\to \infty$ as long as $\psi \left(l\right)$ and ${\psi }^n\left(l\right)$ are extracted from (32) and (33).

Lemma 2. 

For$\psi \in \mathsf{\Pi }\left(\mathcal{J}\right)$, it holds$\left|\psi \left(l\right)\right|\le 3.5\Vert \psi {\Vert }_{\mathsf{\Pi }}$,$\left|{\partial }_l\psi \left(l\right)\right|\le 3\Vert \psi {\Vert }_{\mathsf{\Pi }}$, and$\left|{\partial }_l^2\psi \left(l\right)\right|\le 2\Vert \psi {\Vert }_{\mathsf{\Pi }}$.

Proof. 

The proof is straightforward from ${\partial }_l^2\psi \left(l\right)-{\partial }_l^2\psi \left(0\right)={\int }_0^l{\partial }_p^2\psi \left(p\right)dp$, Holder’s inequality, and (6). □

If $\Vert {\psi }^{n-1}-{\psi \Vert }_{\mathsf{\Pi }}\to 0$ and $l_n\to k$ whenever $n\to \infty$ then $\zeta \left(l_n,{\psi }^{n-1}\left(l_n\right),{\partial }_l{\psi }^{n-1}\left(l_n\right)\right)\to \zeta \left(k,\psi \left(k\right),{\partial }_l\psi \left(k\right)\right)$. This can be seen directly from Lemma 2 and the fact that $\zeta \left(l,\psi \left(l\right),{\partial }_l\psi \left(l\right)\right)\in C\left(\mathcal{J}\times ℝ\times ℝ,ℝ\right)$. We denote ${ℱ}_i={\sum }_{j=1}^i{\epsilon }_{ij}\zeta \left(l_j,\psi \left(l_j\right),{\partial }_l\psi \left(l_j\right)\right)$. This allows us to rewrite ${\psi }^n\left(l\right)$ as

$${\psi }^n\left(l\right)={\displaystyle \sum _{i=1}^n}{ℱ}_i{\overline{\Lambda }}_i\left(l\right).$$

(36)

Theorem 4. 

From (36), it holds that${\psi }^n\left(l\right)\to \psi \left(l\right)$whenever$n\to \infty$.

Proof. 

Clearly, ${\psi }^{n+1}\left(l\right)={\psi }^n\left(l\right)+{ℱ}_{n+1}{\overline{\Lambda }}_{n+1}\left(l\right)$. From the orthogonality of ${\left\{{\overline{\Lambda }}_i\left(l\right)\right\}}_{i=1}^{\infty }$, we get

$$\begin{array}{c}\Vert {\psi }^{n+1}\Vert {}_{\mathsf{\Pi }}^2=\Vert {\psi }^n\Vert {}_{\mathsf{\Pi }}^2+{ℱ}_{n+1}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\Vert {\psi }^{n-1}\Vert {}_{\mathsf{\Pi }}^2+{ℱ}_n^2+{ℱ}_{n+1}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\dots \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\Vert {\psi }^0\Vert {}_{\mathsf{\Pi }}^2+{\displaystyle \sum _{i=1}^{n+1}}{ℱ}_i^2.\end{array}$$

(37)

This implies $\Vert {\psi }^{n+1}\Vert {}_{\mathsf{\Pi }}\ge \Vert {\psi }^n\Vert {}_{\mathsf{\Pi }}$ and there exists $\gamma$ in $ℝ$ such that ${\sum }_{i=1}^{\infty }{ℱ}_i^2=\gamma$, which means that ${\left\{{ℱ}_i^2\right\}}_{i=1}^{\infty }\in l^2$. In order to have

$${\psi }^m\left(l\right)-{\psi }^{m-1}\left(l\right)\perp {\psi }^{m-1}\left(l\right)-{\psi }^{m-2}\left(l\right)\perp \dots \perp {\psi }^{n+1}\left(l\right)-{\psi }^n\left(l\right),$$

(38)

it is sufficient to have for $m>n$ that

$$\begin{array}{c}\Vert {\psi }^m-{\psi }^n\Vert {}_{\mathsf{\Pi }}^2=\Vert {\psi }^m-{\psi }^{m-1}+{\psi }^{m-1}-\dots +{\psi }^{n+1}-{\psi }^n\Vert {}_{\mathsf{\Pi }}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\Vert {\psi }^m-{\psi }^{m-1}\Vert {}_{\mathsf{\Pi }}^2+\Vert {\psi }^{m-1}-{\psi }^{m-2}\Vert {}_{\mathsf{\Pi }}^2+\dots +\Vert {\psi }^{n+1}-{\psi }^n\Vert {}_{\mathsf{\Pi }}^2,\end{array}$$

(39)

whereas, $\Vert {\psi }^m-{\psi }^{m-1}\Vert {}_{\mathsf{\Pi }}^2={ℱ}_m^2$. As $n,m\to \infty$ one has $\Vert {\psi }^m-{\psi }^n\Vert {}_{\mathsf{\Pi }}^2={\sum }_{l=n+1}^m{ℱ}_i^2\to 0$. By the completeness, $\exists {\psi }^n\left(l\right)\in \mathsf{\Pi }\left(\mathcal{J}\right)$ such that ${\psi }^n\left(l\right)\to \psi \left(l\right)$ as $n\to \infty$. □

Theorem 5. 

One has$\psi \left(l\right)={\sum }_{i=1}^{\infty }{ℱ}_i{\overline{\Lambda }}_i\left(l\right)$whenever$n\to \infty$in (36).

Proof. 

Taking the $\underset{n\to \infty }{\lim }$ on both sides of (36), one get $\psi \left(l\right)={\sum }_{i=1}^{\infty }{ℱ}_i{\overline{\Lambda }}_i\left(l\right)$. Thus

$$\begin{array}{cc}\mathcal{A}\left[\psi \right]\left(l_k\right)& ={\displaystyle \sum _{i=1}^{\infty }}{ℱ}_i⟨\mathcal{A}\left[{\overline{\Lambda }}_i\right]\left(l\right),{\mathfrak{S}}_k{\left(l\right)⟩}_{\mathsf{\Delta }}\\ & ={\displaystyle \sum _{i=1}^{\infty }}{ℱ}_i⟨{ \overline{\Lambda }}_i\left(l\right),{\mathcal{A}}^{*}\left[{\mathfrak{S}}_k\right]{\left(l\right)⟩}_{\mathsf{\Pi }}\\ & ={\displaystyle \sum _{i=1}^{\infty }}{ℱ}_i⟨{\overline{\Lambda }}_i\left(l\right),{\Lambda }_k{\left(l\right)⟩}_{\mathsf{\Pi }},\end{array}$$

(40)

and

$$\begin{array}{cc}{\displaystyle \sum _{k^{\prime }=1}^l}{\epsilon }_{k{k}^{\prime }}\mathcal{A}\left[\psi \right]\left(l_{k^{\prime }}\right)& ={\displaystyle \sum _{i=1}^{\infty }}{ℱ}_i⟨{\overline{\Lambda }}_i\left(l\right),{\displaystyle \sum _{k^{\prime }=1}^l}{\epsilon }_{k{k}^{\prime }}{\Lambda }_{k^{\prime }}{\left(l\right)⟩}_{\mathsf{\Pi }}\\ & ={\displaystyle \sum _{i=1}^{\infty }}{ℱ}_i⟨{\overline{\Lambda }}_i\left(t\right),{\overline{\Lambda }}_{k^{\prime }}{\left(l\right)⟩}_{\mathsf{\Pi }}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={ℱ}_l.\end{array}$$

(41)

If $l=1$, then $\mathcal{A}\left[\psi \right]\left(l_1\right)=\zeta \left(l_1,{\psi }^0\left(l_1\right),{\partial }_l{\psi }^0\left(l_1\right)\right)$ and if $l=2$, then $\mathcal{A}\left[\psi \right]\left(l_2\right)=\zeta \left(l_2,{\psi }^1\left(l_2\right),{\partial }_l{\psi }^1\left(l_2\right)\right)$. Generally, we have $\mathcal{A}\left[\psi \right]\left(l_n\right)=\zeta \left(l_n,{\psi }^{n-1}\left(l_n\right),{\partial }_l{\psi }^{n-1}\left(l_n\right)\right)$. By the density condition, $\forall k\in \mathcal{J}$; $\exists {\left\{l_{n_q}\right\}}_{q=1}^{\infty }$ such that $l_{n_q}\to k$ whenever $q\to \infty$ or $\mathcal{A}\left[\psi \right]\left(l_{n_q}\right)=\zeta \left(l_{n_q},{\psi }^{n_q-1}\left(l_{n_q}\right),{\partial }_l{\psi }^{n_q-1}\left(l_{n_q}\right)\right)$. Letting $j\to \infty$ one can get $\mathcal{A}\left[\psi \right]\left(k\right)=\zeta \left(k,\psi \left(k\right),{\partial }_l\psi \left(k\right)\right)$. Since ${ \overline{\Lambda }}_i\left(l\right)\in \mathsf{\Pi }\left(\mathcal{J}\right)$, then $\psi \left(l\right)$ satisfies (15). □

5. 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.

5.1. Certain Examples

In the subsequent FLETM, the readers should note that $\psi \left(0\right)$ and ${\partial }_l\psi \left(0\right)$ are known and may not be homogeneous. The forcing term $\mathrm{T}\left(l\right)$ can be obtain by substituting $\psi \left(l\right)$ through the given model.

Example 1. 

Consider the following FLETM in ABC sense:

$${}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)+\frac{1}{l}{\partial }_l\psi \left(l\right)+\frac{1}{l}\psi \left(l\right)+ln\left(\psi \left(l\right)\right)=\mathrm{T}\left(l\right),$$

(42)

subject to the CICs

$$\psi \left(0\right)=1 \mathrm{and} {\partial }_l\psi \left(0\right)=\delta ,$$

(43)

where the analytic solution is given by

$$\psi \left(l\right)=e^{\delta l}.$$

(44)

Example 2. 

Consider the following FLETM in ABC sense:

$${}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)+\frac{1}{sin\left(l\right)}{\partial }_l\psi \left(l\right)-\psi \left(l\right)+sin\left(\psi \left(l\right)\right)=\mathrm{T}\left(l\right),$$

(45)

subject to the CICs

$$\psi \left(0\right)=0 \mathrm{and} {\partial }_l\psi \left(0\right)=0,$$

(46)

where the analytic solution is given by

$$\psi \left(l\right)=l^{2\delta }-l^3.$$

(47)

Example 3. 

Consider the following FLETM in ABC sense:

$${}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)+\frac{1}{l^2}{\partial }_l\psi \left(l\right)-\frac{1}{l^4}\psi \left(l\right)+\sqrt[3]{\psi \left(l\right)}=\mathrm{T}\left(l\right),$$

(48)

subject to the CICs

$$\psi \left(0\right)=0 \mathrm{and} {\partial }_l\psi \left(0\right)=\delta ,$$

(49)

where the analytic solution is given by

$$\psi \left(l\right)=\delta l-l^2.$$

(50)

Recall that $l\in \left[0,1\right]$; $\delta \in (1,2]$; $\psi \in C\left(\mathcal{J}\to ℝ\right)$; and $\mathrm{T}\in C\left(\mathcal{J}-\left\{0\right\}\to ℝ\right)$, while ${}_0{}^{ABC}\partial {}_l^{\delta }\psi \left(l\right)$ denotes the ABC fractional derivative of $\psi$ in $l$ over $\mathcal{J}$ of order $\delta$.

5.2. Results and Discussions

Take into consideration Algorithms 1 and 2, following the RKDM, using $l_i=\frac{i}{n}$, $i=0,1,\dots ,n=50$ the numerical validations for different values of grid points $l_i\in \mathcal{J}$ will be exhibited. For this purpose, Table 1, Table 2 and Table 3 tabulates the evolution of the absolute errors $Ab$ as

$$Ab\left[{\psi }^{50}\right]\left(l_i\right)=\left|\psi \left(l_i\right)-{\psi }^{50}\left(l_i\right)\right|,$$

(51)

for Examples 1, 2, and 3, simultaneously.

Table 1. Numerical results of Example 1 using RKDM.

	$\mathit{\delta }=2$	$\mathit{\delta }=1.75$	$\mathit{\delta }=1.5$	$\mathit{\delta }=1.25$
${\mathit{l}}_{\mathit{i}}$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$
$0$	$0$	$0$	$0$	$0$
$0.1$	$5.58923\times {10}^{-7}$	$2.39589\times {10}^{-6}$	$9.76028\times {10}^{-6}$	$1.26481\times {10}^{-5}$
$0.2$	$7.29487\times {10}^{-7}$	$8.43183\times {10}^{-6}$	$5.53946\times {10}^{-5}$	$4.51370\times {10}^{-4}$
$0.3$	$5.10731\times {10}^{-7}$	$9.31378\times {10}^{-6}$	$6.44253\times {10}^{-5}$	$8.45264\times {10}^{-4}$
$0.4$	$4.86899\times {10}^{-7}$	$4.02242\times {10}^{-6}$	$5.96694\times {10}^{-5}$	$6.39510\times {10}^{-4}$
$0.5$	$3.73586\times {10}^{-7}$	$6.44048\times {10}^{-6}$	$9.76767\times {10}^{-5}$	$8.49810\times {10}^{-4}$
$0.6$	$5.96425\times {10}^{-7}$	$9.83946\times {10}^{-6}$	$3.13543\times {10}^{-5}$	$7.18295\times {10}^{-4}$
$0.7$	$8.30602\times {10}^{-7}$	$8.11052\times {10}^{-6}$	$1.78901\times {10}^{-5}$	$2.28708\times {10}^{-4}$
$0.8$	$3.09230\times {10}^{-7}$	$3.03588\times {10}^{-6}$	$8.12647\times {10}^{-5}$	$5.51276\times {10}^{-4}$
$0.9$	$7.82026\times {10}^{-6}$	$2.81165\times {10}^{-6}$	$5.67583\times {10}^{-5}$	$6.57153\times {10}^{-4}$
$1.0$	$6.27080\times {10}^{-6}$	$3.20502\times {10}^{-5}$	$2.14565\times {10}^{-4}$	$3.76831\times {10}^{-4}$

Table 2. Numerical results of Example 2 using RKDM.

	$\mathit{\delta }=2$	$\mathit{\delta }=1.75$	$\mathit{\delta }=1.5$	$\mathit{\delta }=1.25$
${\mathit{l}}_{\mathit{i}}$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$
$0$	$0$	$0$	$0$	$0$
$0.1$	$1.66034\times {10}^{-7}$	$3.79764\times {10}^{-6}$	$8.68106\times {10}^{-5}$	$7.78427\times {10}^{-4}$
$0.2$	$2.58021\times {10}^{-7}$	6.04130$\times {10}^{-6}$	$6.66813\times {10}^{-5}$	$4.83384\times {10}^{-4}$
$0.3$	$2.47746\times {10}^{-7}$	$2.43097\times {10}^{-6}$	$5.42359\times {10}^{-5}$	$4.13100\times {10}^{-4}$
$0.4$	$6.83007\times {10}^{-8}$	$7.36473\times {10}^{-6}$	$1.28745\times {10}^{-5}$	$9.26837\times {10}^{-4}$
$0.5$	$1.54628\times {10}^{-7}$	$7.49228\times {10}^{-6}$	$7.24921\times {10}^{-5}$	$2.34712\times {10}^{-4}$
$0.6$	$4.61756\times {10}^{-7}$	$9.86819\times {10}^{-6}$	$3.86754\times {10}^{-5}$	$7.72225\times {10}^{-4}$
$0.7$	$8.34165\times {10}^{-8}$	$6.41801\times {10}^{-6}$	$2.32842\times {10}^{-5}$	$2.55304\times {10}^{-4}$
$0.8$	$9.75666\times {10}^{-8}$	$6.05864\times {10}^{-6}$	$6.41085\times {10}^{-5}$	$5.48677\times {10}^{-4}$
$0.9$	$1.96125\times {10}^{-7}$	$1.54146\times {10}^{-6}$	$5.49951\times {10}^{-5}$	$6.03551\times {10}^{-4}$
$1.0$	$2.45293\times {10}^{-7}$	$9.86166\times {10}^{-6}$	$2.78153\times {10}^{-5}$	$9.96146\times {10}^{-4}$

Table 3. Numerical results of Example 3 using RKDM.

	$\mathit{\delta }=2$	$\mathit{\delta }=1.75$	$\mathit{\delta }=1.5$	$\mathit{\delta }=1.25$
${\mathit{l}}_{\mathit{i}}$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$	$\mathit{A}\mathit{b}\left[{\mathit{\psi }}^{50}\right]\left({\mathit{l}}_{\mathit{i}}\right)$
$0$	$0$	$0$	$0$	$0$
$0.1$	$8.72538\times {10}^{-7}$	$8.35309\times {10}^{-6}$	$4.24210\times {10}^{-5}$	$9.37424\times {10}^{-4}$
$0.2$	$7.62576\times {10}^{-7}$	$5.90241\times {10}^{-6}$	$1.89126\times {10}^{-5}$	$4.86929\times {10}^{-4}$
$0.3$	$4.31937\times {10}^{-7}$	$7.87242\times {10}^{-6}$	$9.13479\times {10}^{-5}$	$2.67124\times {10}^{-4}$
$0.4$	$4.71124\times {10}^{-7}$	$3.19969\times {10}^{-6}$	$7.50933\times {10}^{-5}$	$7.64597\times {10}^{-4}$
$0.5$	$3.51107\times {10}^{-6}$	$6.03111\times {10}^{-6}$	$4.86751\times {10}^{-5}$	$7.01821\times {10}^{-4}$
$0.6$	$8.69821\times {10}^{-6}$	$6.45966\times {10}^{-5}$	$2.98386\times {10}^{-5}$	$8.42708\times {10}^{-4}$
$0.7$	$2.50806\times {10}^{-6}$	$2.80152\times {10}^{-5}$	$7.89215\times {10}^{-5}$	$4.27468\times {10}^{-4}$
$0.8$	$1.25974\times {10}^{-6}$	$3.66312\times {10}^{-5}$	$4.03998\times {10}^{-5}$	$8.68758\times {10}^{-4}$
$0.9$	$2.14535\times {10}^{-6}$	$5.01585\times {10}^{-5}$	$1.76836\times {10}^{-4}$	$8.38715\times {10}^{-4}$
$1.0$	$2.93206\times {10}^{-6}$	$1.57354\times {10}^{-5}$	$2.81862\times {10}^{-4}$	$9.03151\times {10}^{-4}$

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 $\delta$. 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 $\delta =2$.

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 $l_i\in \mathcal{J}$ when $\delta \in (1,2]$. It appears that all figures almost look identical in their behaviors, and in good agreement with each other, particularly when comparing the case of $\delta =2$. Moreover, the RKDM numerical solutions are very close at the CICs.

Figure 1. The $3D$ computational values of the RKDM when $l_i\in \mathcal{J}$ and $\delta \in (1,2]$: (a) Example 1; (b) Example 2; and (c) Example 3.

6. 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 $\mathsf{\Pi }\left(\mathcal{J}\right)$ 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.