The utilization of fractional order calculus finds extensive application in various disciplines and, thus, significantly influences the swift progress of scientific investigations. The three most common definitions of fractional order derivatives are: Grünwald-Letnikov (G-L), Caputo, and Riemann-Liouville (R-L) [11] .

Definition 1. [12] Define the α-order Riemann-Liouville derivative of the function $f\left(t\right)$ as follows:

${}_a^{RL}{D}_t^{\alpha }f\left(t\right)=\frac{{\text{d}}^n}{\text{d}{t}^n}{}_{a}D{}_t^{-\left(n-\alpha \right)}f\left(t\right)=\frac{1}{\text{Γ}\left(n-\alpha \right)}\frac{{\text{d}}^n}{\text{d}{t}^n}{\displaystyle {\int }_a^t{\left(t-\tau \right)}^{n-\alpha -1}f\left(\tau \right)\text{d}\tau },$(1)

where $n-1<\alpha <n$, $n\in Z^{+}$, and $\text{Γ}\left(z\right)={\displaystyle {\int }_0^{\infty }{t}^{z-1}{\text{e}}^{-t}\text{d}t}$ is a gamma function.

While the R-L definition has a very high status in theoretical analysis, the Caputo derivative with initial value conditions is more applicable in practical applications.

Definition 2. [12] Define the α-order Caputo derivative of the function $f\left(t\right)$ as follows:

${}_a^C{D}_t^{\alpha }f\left(t\right)={}_{0}D{}_t^{-\left(n-\alpha \right)}\frac{{\text{d}}^n}{\text{d}{t}^n}f\left(t\right)=\frac{1}{\text{Γ}\left(n-\alpha \right)}{\displaystyle {\int }_a^t{\left(t-\tau \right)}^{n-\alpha -1}{f}^{\left(n\right)}\left(\tau \right)\text{d}\tau },$(2)

where $n-1<\alpha <n$, $n\in Z^{+}$.

Definition 3. [12] Define the α-order Grünwald-Letnikov derivative of the function $f\left(t\right)$ as follows:

$\begin{array}{c}{}_a^{GL}{D}_t^{\alpha }f\left(t\right)=\underset{\begin{array}{c}h\to 0\\ \varrho h=t\end{array}}{\lim }{h}^{-\alpha }{\displaystyle \underset{r=0}{\overset{\varrho }{\sum }}{\left(-1\right)}^r}\left(\begin{array}{l}\alpha \hfill \\ r\hfill \end{array}\right)f\left(t-rh\right)\\ ={\displaystyle \underset{k=0}{\overset{n}{\sum }}\frac{f^{\left(k\right)}\left(\alpha \right){\left(t-\alpha \right)}^{k-\alpha }}{\Gamma \left(k+1-\alpha \right)}}+\frac{1}{\Gamma \left(\varrho +1-\alpha \right)}\cdot {\displaystyle {\int }_a^t{\left(t-v\right)}^{\varrho -\alpha }}{f}^{\left(\varrho +1\right)}\left(v\right)\text{d}\nu ,\end{array}$(3)

the function $f\left(t\right)$ possesses continuous derivatives up to order n within the interval $\left[a,t\right]$.

The R-L derivative is equivalent to the G-L derivative based on the definition of the fractional order derivative. However, it is important to note that the Caputo derivative differs from the R-L derivative and can be represented in the following manner:

${}_a^{RL}{D}_t^{\alpha }f\left(t\right)={}_0{}^{C}D{}_t^{\alpha }f\left(t\right)+\underset{k=0}{\overset{n-1}{{\displaystyle \sum }}}{r}_k^{\alpha }\left(t\right)f^{\left(k\right)}\left(a\right),$(4)

here, $n-1<\alpha <n$, $n\in Z^{+}$, and $f^{\left(k\right)}\left(a\right)$ with $k=0,1,\cdots ,n-1$ represent the initial conditions, and $r_k^{\alpha }=\frac{t^{k-\alpha }}{\text{Γ}\left(k+1-\alpha \right)}$.

In this article, our preference lies in employing the Caputo operator and examining and discussing the scenario for the case of $n=1$, where $\alpha \in \left(0,1\right)$. The aforementioned Equation (4) can be written as follows:

${}_a^{RL}{D}_t^{\alpha }f\left(t\right)={}_0{}^{C}D{}_t^{\alpha }f\left(t\right)+r_0^{\alpha }f\left(a\right).$(5)

Here, we consider the case where $r_0^{\alpha }=\frac{t^{-\alpha }}{\text{Γ}\left(1-\alpha \right)}$.

Multiple simulations using the SIR model were conducted to study the spread of COVID-19 in various countries. These simulations found that the predicted number of confirmed cases can be further improved for the most part [10] . This is mainly due to the fact that during the course of the outbreak, people who tested positive were quarantined and some asymptomatic infected people did not know whether they were carrying the virus or not, resulting in a shortage of active cases. Thus, we have introduced a novel parameter p that signifies the individuals who have undergone the test and tested positive, as well as those who were infected and quarantined, and thus an improved SIR model is extracted in this paper [13] . The expressed by the differential equation as

$\{\begin{array}{l}{S}^{\prime }\left(t\right)=A-\frac{\beta S\left(t\right)I\left(t\right)}{N\left(t\right)}-\delta S\left(t\right),\\ I^{\prime }\left(t\right)=\frac{\beta S\left(t\right)I\left(t\right)}{N\left(t\right)}-pI\left(t\right)-\gamma I\left(t\right)-\left(\delta +d\right)I\left(t\right),\\ T{P}^{\prime }\left(t\right)=pI\left(t\right)-\gamma TP\left(t\right)-\left(\delta +d\right)TP\left(t\right),\\ {R^{\prime }}_1\left(t\right)=\gamma I\left(t\right)-\delta R_1\left(t\right),\\ {R^{\prime }}_2\left(t\right)=\gamma TP\left(t\right)-\delta R_2\left(t\right),\end{array}$(6)

where

$N\left(0\right)=S\left(0\right)+I\left(0\right)+TP\left(0\right)+R_1\left(0\right)+R_2\left(0\right).$

In the system (6), $TP\left(t\right)$ represents infected individuals who test positive and are thus quarantined, $R_1\left(t\right)$ represents the number of individuals who have been removed (recovered + dead) from the group I, and $R_2\left(t\right)$ represents the number of individuals removed (recovered + dead) from TP. In recent times, the use of fractional differential equations has become increasingly widespread. Many researchers have discovered that models incorporating fractional derivatives can produce more accurate measurement data than traditional classical models, demonstrating a higher level of consistency [14] . Therefore, a fractional order COVID-19 system of the same order is described as follows. The meaning of the parameters is shown in Table 1 .

$\{\begin{array}{l}{\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }S\left(t\right)=A-\frac{\beta S\left(t\right)I\left(t\right)}{N\left(t\right)}-\delta S\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }I\left(t\right)=\frac{\beta S\left(t\right)I\left(t\right)}{N\left(t\right)}-pI\left(t\right)-\gamma I\left(t\right)-\left(\delta +d\right)I\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }TP\left(t\right)=pI\left(t\right)-\gamma TP\left(t\right)-\left(\delta +d\right)TP\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }{R}_1\left(t\right)=\gamma I\left(t\right)-\delta R_1\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }{R}_2\left(t\right)=\gamma TP\left(t\right)-\delta R_2\left(t\right),\end{array}$(7)

where $N\left(0\right)=S\left(0\right)+I\left(0\right)+TP\left(0\right)+R_1\left(0\right)+R_2\left(0\right)$, and $\alpha \in \left(0,1\right)$.

To keep the system concise, we define $R_i=R_1+R_2$. Consequently, we obtain

$\{\begin{array}{l}{\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }S\left(t\right)=A-\frac{\beta S\left(t\right)I\left(t\right)}{N\left(t\right)}-\delta S\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }I\left(t\right)=\frac{\beta S\left(t\right)I\left(t\right)}{N}-pI\left(t\right)-\gamma I\left(t\right)-\left(\delta +d\right)I\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }TP\left(t\right)=pI\left(t\right)-\gamma TP\left(t\right)-\left(\delta +d\right)TP\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }{R}_i\left(t\right)=\gamma I\left(t\right)+\gamma TP\left(t\right)-\delta R_i\left(t\right),\end{array}$(8)

where $N\left(0\right)=S\left(0\right)+I\left(0\right)+TP\left(0\right)+R_i\left(0\right)$, and $\alpha \in \left(0,1\right)$.

It can be observed that the Equations (7) and (8) exhibit dissimilar units on the left and right sides. More precisely, their respective units are expressed as (days)^−α and (days)⁻¹. By dividing the left-hand side of the equations in systems (7) and (8) by ${\lambda }_{\alpha }$, the dimensions become (days)^α−1, thereby enabling the attainment of identical units on both sides of the system equations, namely (days)⁻¹. Generally, in a fractional order system, it becomes feasible to acquire ${\lambda }_{\alpha }=1$. The purpose of this paper is to utilize the MH-NMSS-PSO approach in order to identify an $\alpha$ appropriate collection of fractional orders as well as parameters. Subsequent subsections will establish the essential prerequisites for ensuring the boundedness of the system and determining the feasible domain.

Denote ${ℝ}_{+}^4:=\left\{\vartheta \in {ℝ}^4:\vartheta \ge 0\right\}$ and $\vartheta \left(t\right)={\left(S\left(t\right),I\left(t\right),TP\left(t\right),R\left(t\right)\right)}^{\text{T}}$.

To establish the primary outcome, it is necessary to demonstrate that the positivity and invariance of ${ℝ}_{+}^4$ are imperative, which is very helpful for us to prove the boundedness of the solution.

Theorem 3.1. System (8) has a unique solution, which remains within the set of positive real numbers ( ${ℝ}_{+}^4$).

Proof. The uniqueness of the existence of a solution for system (8) is obtained from theorem 3.1 [15] . Subsequently, we will demonstrate that ${ℝ}_{+}^4$ maintains positivity throughout, since

$\begin{array}{l}{{}^C{D}_{0,t}^{\alpha }S\left(t\right)|}_{S=0}=A\ge 0,\\ {{}^C{D}_{0,t}^{\alpha }I\left(t\right)|}_{I=0}=0\ge 0,\\ {{}^C{D}_{0,t}^{\alpha }TP\left(t\right)|}_{TP=0}=pI\ge 0,\\ {{}^C{D}_{0,t}^{\alpha }R\left(t\right)|}_{R=0}=\gamma I+\gamma Tp\ge 0.\end{array}$(9)

By using the equation above (9) and theorem 1 [16] , we get that ${ℝ}_{+}^4$ is positively invariant.

$\underset{t\to \infty }{\lim \sup }N\left(t\right)\le \frac{A}{\delta }.$(14)

Based on the derivation and proof presented above, we can conclude that a feasible domain and bounded solutions of the system can be determined by the set $\mathcal{D}$.

Given that the initial three equations in the system (8) do not rely on the last variable R, we can assess the system’s equilibrium and stability using the subsequent methodology.

$\{\begin{array}{l}{}^{C}D{}_{0,t}^{\alpha }S\left(t\right)=A-\frac{\beta S\left(t\right)I\left(t\right)}{N\left(t\right)}-\delta S\left(t\right),\\ {}^{C}D{}_{0,t}^{\alpha }I\left(t\right)=\frac{\beta S\left(t\right)I\left(t\right)}{N\left(t\right)}-pI\left(t\right)-\gamma I\left(t\right)-\left(\delta +d\right)I\left(t\right),\\ {}^{C}D{}_{0,t}^{\alpha }TP\left(t\right)=pI\left(t\right)-\gamma TP\left(t\right)-\left(\delta +d\right)TP\left(t\right),\\ {}^{C}D{}_{0,t}^{\alpha }N\left(t\right)=A-\delta N\left(t\right)-d\left(I\left(t\right)+TP\left(t\right)\right).\end{array}$(15)

Now, to find the stable point of the system (15), let

$\{\begin{array}{l}{}^{C}D{}_{0,t}^{\alpha }S\left(t\right)=0,\\ {}^{C}D{}_{0,t}^{\alpha }I\left(t\right)=0,\\ {}^{C}D{}_{0,t}^{\alpha }TP\left(t\right)=0,\\ {}^{C}D{}_{0,t}^{\alpha }N\left(t\right)=0.\end{array}$(16)

Next, the system’s equilibrium points can be represented as $E_0=\left(\frac{A}{\delta },0,0,\frac{A}{\delta }\right)$ and $E_1=\left(\bar{S},\bar{I},\overline{TP},\bar{N}\right)$, where

$\begin{array}{l}\bar{S}=\frac{-A\left(\gamma +\delta \right)}{\delta \left(d-\beta +\frac{p\beta }{p+\gamma +\delta +d}\right)},\\ \bar{I}=\frac{A}{p+\gamma +\delta +d}+\frac{A\left(\gamma +\delta \right)}{d\left(p+\gamma +\delta +d\right)-\beta \left(\gamma +\delta +d\right)},\\ \overline{TP}=\frac{p}{\gamma +\delta +d}\left[\frac{A}{p+\gamma +\delta +d}+\frac{A\left(\gamma +\delta \right)}{d\left(p+\gamma +\delta +d\right)-\beta \left(\gamma +\delta +d\right)}\right],\\ \bar{N}=\frac{-A\beta \left(\gamma +\delta \right)}{\delta \left[d\left(p+\gamma +\delta +d\right)-\beta \left(\gamma +\delta +d\right)\right]}.\end{array}$(17)

Jacobian matrix is given by

$J=\left(\begin{array}{cccc}-\frac{\beta I}{N}-\delta & -\frac{\beta S}{N}& 0& \frac{\beta SI}{N^2}\\ \frac{\beta I}{N}& \frac{\beta S}{N}-\left(p+\gamma +\delta +d\right)& 0& -\frac{\beta SI}{N^2}\\ 0& p& -\left(\gamma +\delta +d\right)& 0\\ 0& -d& -d& -\delta \end{array}\right).$(18)

Theorem 3.3. If $\frac{\beta }{p+\gamma +\delta +d}<1$, the disease-free equilibrium $E_0=\left(\frac{A}{\delta },0,0,\frac{A}{\delta }\right)$ of the system is locally asymptotically stable; otherwise, if $\frac{\beta }{p+\gamma +\delta +d}>1$ it is unstable.

Proof. The Jacobian matrix is obtained by following these steps at $E_0=\left(\frac{A}{\delta },0,0,\frac{A}{\delta }\right)$

$J\left(E_0\right)=\left(\begin{array}{cccc}-\delta & -\beta & 0& 0\\ 0& \beta -\left(p+\gamma +\delta +d\right)& 0& 0\\ 0& p& -\left(\gamma +\delta +d\right)& 0\\ 0& -d& -d& -\delta \end{array}\right).$(19)

By solving the characteristic equation of the Jacobian matrix $J\left(E_0\right)$, the eigenvalues of $J\left(E_0\right)$ can be determined. This leads to the equation.

${\left(\lambda +\delta \right)}^2\left(\lambda +\gamma +\delta +d\right)\left[\lambda -\left(\beta -p-\gamma -\delta -d\right)\right]=0.$(20)

Therefore, the eigenvalues of the Jacobian matrix $J\left(E_0\right)$ are

$\begin{array}{l}{\lambda }_1=-\delta ,\\ {\lambda }_2=-\delta ,\\ {\lambda }_3=\beta -\left(p+\gamma +\delta +d\right),\\ {\lambda }_4=-\left(\gamma +\delta +d\right).\end{array}$(21)

Based on the analysis presented above, we can observe that if $\frac{\beta }{p+\gamma +\delta +d}<1$,

all the eigenvalues will have negative real parts. Consequently, the disease-free equilibrium point $E_0$ will be locally asymptotically stable. However, if

$\frac{\beta }{p+\gamma +\delta +d}>1$, the disease-free equilibrium point $E_0$ will be unstable. In this context, the ratio $\frac{\beta }{p+\gamma +\delta +d}$ is referred to as the basic reproduction number,

represented by $R_0$. From a biological perspective, $R_0$ can be interpreted as follows: if $R_0$ is less than one, the infection will eventually disappear; whereas if it is greater than one, the infection will persist. Therefore, we can now proceed to discuss the asymptotic stability of the endemic equilibrium in the given system.

Now, let $M=p+\gamma +\delta +d$. The obtained Jacobian matrix is denoted as $J\left(E_1\right)$,

$J\left(E_1\right)=\left(\begin{array}{cccc}\frac{\delta \left[dM-\beta \left(M-p\right)\right]}{M\left(\gamma +\delta \right)}& -M& 0& \frac{\delta \left[dM-\beta \left(M-p\right)\right]}{-\beta \left(\gamma +\delta \right)}-\frac{M\delta }{\beta }\\ \frac{\delta \left[dM-\beta \left(M-p\right)\right]}{-M\left(\gamma +\delta \right)}-\delta & 0& 0& \frac{\delta \left[dM-\beta \left(M-p\right)\right]}{\beta \left(\gamma +\delta \right)}+\frac{M\delta }{\beta }\\ 0& p& p-M& 0\\ 0& -d& -d& -\delta \end{array}\right).$(22)

Let $Q=\frac{dM-\beta \left(M-p\right)}{M\left(\gamma +\delta \right)}$. Then, the characteristic equation for the Jacobian matrix $J\left(E_1\right)$ is to be simplified,

$T\left(\lambda \right)=\left(\delta +\lambda \right)X\left(\lambda \right)\text{and}X\left(\lambda \right)={\lambda }^3+D_1{\lambda }^2+D_2\lambda +D_3,$(23)

where

$\begin{array}{l}{D}_1=M-p-\delta Q,\\ D_2=\delta Q\left(p-M\right)+\delta \left(Q+M\right)\frac{d-\beta }{\beta },\\ D_3=\delta \left(Q+M\right)\left[\left(p-M\right)+\frac{dM}{\beta }\right].\end{array}$(24)

Regarding the polynomial $X\left(\lambda \right)$, we can have the following definition:

$D\left(X\right)=18D_1{D}_2{D}_3+D_1^2{D}_2^2-4D_3{D}_1^3-4D_2^3-27D_3^3.$(25)

Using Proposition 1 given in the article [18] , if we observe that the stable point $E_1$ is asymptotically steady, then $D\left(X\right)>0$, $D_1>0$, $D_3>0$, and $D_1{D}_2>D_3$.

By combining the equations for $TP\left(t\right)$ and $R_2\left(t\right)$ outlined in system (8), the equation for $C\left(t\right)$, representing the total number of confirmed cases at time t, can be derived. Now classifying everyone as susceptible, and therefore with a negligible recruitment rate A for the susceptible population, the exact system is as follows:

$\{\begin{array}{l}{\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }S\left(t\right)=-\frac{\beta S\left(t\right)I\left(t\right)}{N}-\delta S\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }I\left(t\right)=\frac{\beta S\left(t\right)I\left(t\right)}{N}-pI\left(t\right)-\gamma I\left(t\right)-\left(\delta +d\right)I\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }C\left(t\right)=pI\left(t\right),\\ {\lambda }_{\alpha }{}_0{}^{C}D{}_t^{\alpha }{R}_1\left(t\right)=\gamma I\left(t\right)-\delta R_1\left(t\right),\end{array}$(26)

where $N\left(0\right)=S\left(0\right)+I\left(0\right)+C\left(0\right)+R_1\left(0\right)$, and $\alpha \in \left(0,1\right)$.

A number of numerical methods are currently used to find fractional order systems, including the prediction correction method, the Millin transformation method, etc. In this research paper, we employ the GMMP method to compute the numerical solution of the system (26). The system (26) can be expressed as: $\lambda \odot {}_a{}^{C}D{}_t^{\alpha }z\left(t\right)=f\left(t,z\left(t\right)\right)$, here we can observe that $z\left(t\right)={\left(S\left(t\right),I\left(t\right),C\left(t\right),R_1\left(t\right)\right)}^{\text{T}}$. Assuming the adoption of a uniform grid over the interval $\left[a,t\right]$, as follows: $a=t_0<t_1<t_2<\cdots <t_N=t$, and $\text{Δ}t=t_{i+1}-t_i=h$. Also assume that $z\left(t\right)$ is sequential in each limited interval $\left(a,t\right)$.

First, we rely on the classical finite-difference notation [19] to proceed with our analysis and $\alpha \in \left(0,1\right)$.

$\frac{1}{h^{\alpha }}{\text{Δ}}_h^{\alpha }z\left(t\right)=\frac{1}{h^{\alpha }}\left(z\left(x_n\right)-\underset{k=0}{\overset{n}{{\displaystyle \sum }}}{c}_k^{\alpha }z\left(x_{n-k}\right)\right).$(27)

In the Equation (27), $c_k^{\alpha }={\left(-1\right)}^k\left(\begin{array}{l}\alpha \hfill \\ k\hfill \end{array}\right)$ represents the most commonly used binomial coefficients.

Both R-L and G-L fractional order derivatives can be expressed in the following format according to Equation (27).

${}_a{}^{RL}D{}_t^{\alpha }z\left(t\right)={}_a{}^{GL}D{}_t^{\alpha }z\left(t\right)=\underset{h\to 0}{\lim }\frac{1}{h^{\alpha }}{\Delta }_h^{\alpha }z\left(t\right)\approx \frac{1}{h^{\alpha }}{\Delta }_h^{\alpha }z\left(t\right).$(28)

The error term $r_0^{\alpha }$ in Equation (5) can also be represented using a uniform grid, leading to the following expression for the Caputo fractional order derivative [20]

${}_a^C{D}_t^{\alpha }z\left(t\right)\approx \frac{1}{h^{\alpha }}\left(z\left(x_n\right)-\underset{k=0}{\overset{n}{{\displaystyle \sum }}}{c}_k^{\alpha }z\left(x_{n-k}\right)\right)-r_n^{\alpha }z\left(a\right),$(29)

where $z\left(a\right)$ is the initial condition, and $r_n^{\alpha }=r_0^{\alpha }\left(t_n\right)={\omega }_{0,-1}^{\alpha }{(n)}^{-\alpha }$. The function $\omega$ is given by ${\omega }_{\mu ,\nu }^{\alpha }=\frac{\text{Γ}\left(\mu \alpha +1\right)}{\text{Γ}\left(v\alpha +1\right)}$, $\mu ,\nu \in {\mathbb{N}}_0\cup \left\{-1\right\}$. And $r_n^{\alpha }z\left(a\right)$ tends to zero when $n\to \infty$ [21] .

Table 2 below shows the fundamental procedure of GMMP (Gorenflo-Mainardi-Moretti-Paradisi Programme).

In a general sense, the fractional order nonlinear equation can be expressed in terms of the Caputo operator as follows:

$\begin{array}{l}\lambda \odot {}_a{}^{C}D{}_t^{\alpha }z\left(t\right)=f\left(t,z\left(t\right)\right),0\le t\le T,\\ z^{\left(k\right)}\left(a\right)=z_0^{\left(k\right)},k=0,1,\cdots ,n-1.\end{array}$(30)

Finally, combining (29) and (30), The equation can be obtained through deduction as follows:

$z\left(x_n\right)=h^{\alpha }\oslash \lambda \odot f\left(t_n,z\left(x_n\right)\right)+\underset{k=0}{\overset{n}{{\displaystyle \sum }}}{c}_k^{\alpha }z\left(x_{n-k}\right)+h^{\alpha }{r}_n^{\alpha }{z}_0.$(31)

The reasoning presented above leads to the ultimate formulation (31) of the fractional order nonlinear equation, which takes the form of an implicit difference scheme. One can interpret (31) as an equation concerning the unfamiliar variable $z\left(x_n\right)$, which appears on both ends of the equation. Hence, Equation (31) can be utilized to numerically solve any initial value problem associated with a Caputo fractional order differential equation. To effectively tackle such problems, it is necessary to introduce the relevant equations and employ Matlab software (R2020a) for their resolution. Since the Equation (31) possesses an unfamiliar variable $z\left(x_n\right)$ on both ends, the Newton’s algorithm is selected to obtain the value of $z\left(x_n\right)$ through Equation (31).