Fractal–Fractional Analysis of a Water Pollution Model Using Fractional Derivatives

Abstract

Water pollution is a significant threat for human health, particularly in developed countries. This study advances the mathematical understanding of WP transmission dynamics by developing a fractional–fractal derivative framework with non-singular kernels and the Mittage–Leffler function, which successfully preserves the non-local behavior of pollutants. The fractional–fractal derivatives in sense of the Atangana–Baleanu–Caputo formulation inherently captures the non-local and memory-dependent behavior of pollutant diffusion, addressing limitations of classical differential operators. A novel parameter, $\gamma$, is introduced to represent the recovery rate of water systems through treatment processes, explicitly modeling the bridge between natural purification mechanisms and engineered remediation efforts. Furthermore, this study establishes stability analysis, and the existence and uniqueness of the solution are established through fixed-point theory to ensure the mathematical stability of the system. Moreover, a numerical scheme based on the Newton polynomial is formulated, by obtaining significant simulations of pollution dynamics under various conditions. Graphical results show the effect of important parameters on pollutant evolution, providing useful information about the behavior of the system.

1. Introduction

Water pollution (WP) introduces vital challenges in the development of the world’s nations, due to its drastic impacts on the economy, human health, and the environment. Although water is an essential part of living organisms, the rapid growth in industrialization has badly affected freshwater contamination, and WP has thus become the most critical environmental issue in the current age. In particular, contaminated water serves as a carrier of waterborne diseases, which include cholera, diarrhea, typhoid, and hepatitis [1].

Mathematical modeling (MM) plays an important role in understanding and predicting the dynamics of complex systems across disciplines such as environmental pollution [2], epidemiology [3], fluid dynamics [4], and control theory [5]. MM acts as an interface between real-world phenomena and their mathematical representation, which allows us to analyze and investigate problems empirically. In 2024, Xu et al. [6] modeled a plankton–oxygen dynamical model and explored bifurcation along with its stability. Dorshan and Elisov [7] found chaotic attractors, while modeling a multi-rotor attitude system. Further application of MM can be observed in cubic equivalent systems [8,9,10], especially with the analysis of complex systems.

Apart from the above applications, MM has a great role in water power [11]. In 2023, Laskar et al. [12] studied the quality of spring water near Tuirial in a city in India. In the north-eastern state of India, two models, i.e., AHP-TOPSIS and AHP-VIKOR, were considered for the suitability of spring water [13]. Ren et al. [14] collected 168 water samples from seven different locations from the Laixi to the Yangtze Rivers and applied several statistical tools to check the quality of sub-watersheds. The control analysis of water in hilly areas, affecting the surface water, due to anthropogenic factors was studied by Wang et al. [15]. Recently, Cheng et al. [16] discussed the role of leaching in landfall and, indirectly, its impact on water pollution, by determining a strategy for leach reduction and pollution control. A new fractional model of WP based on the Levenberg–Marqurdt back-propagation method for control analysis was modeled for predictive water quality in various contexts [17].

Dynamical systems are further categorized into integer and fractional orders and are an important feature of Fractional Calculus (FC), which is the development of actual dynamical systems in terms of fractional differential equations [18,19]. There exist many studies about the usage of fractional derivatives in dynamical systems. In 2009, El-Sayed [20] considered a biological population system and transformed it into a fractional-order one to find its exact solution. The numerical modeling of a biological model along with a fractional derivative can be found in the work of Rihan [21]. A pine wilt disease-based model [22] with the Caputo–Febrizio derivative was studied to confirm the uniqueness of the solution and its stability. A system based on pantograph fractional-order differential equations [23] was studied to confirm the existence and uniqueness of the solution. In 2020, Ali et al. [24] performed mathematical analysis of dynamical systems along their boundary conditions. Marwan et al. [25] worked on the existence of a solution in a fractional-order attitude chaotic system. In 2024, a tumor–immune interaction dynamical model was investigated with the aid of a fractional derivative [26]. Recently, a non-local operator was considered in a fractional-order coronavirus disease 2019 model [27], which was modeled not only to study its existence and uniqueness but also to discuss its stability.

The new class of fractional operators has a variety of applications in chemical [28], financial [29], and biological systems [30] to develop new systems of differential equations (DEs). In accordance with applications of fractional differential equations (FDEs), a new method for fractal–fractional derivatives (FFDs) was introduced in FC [31] to enhance the analysis of fractional derivatives using different kernels and applications [32,33,34].

There have been several dynamical models designed so far including quadrotor unmanned aerial vehicles [35,36,37], cubic switching [38,39], energy supply and demand [40], switch–twist manifolds [41], and many more beyond these. Similarly, in addition to the above-mentioned dynamical systems, WP is an important issue for all living organisms, consisting of four dimensional systems of ordinary differential equations (ODEs) [42] representing the mathematical form of the systems as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\mathbb{W}\left(\tau \right)}{dt}}& =\Lambda -{\alpha }_1\mathbb{W}\mathbb{S}-{\alpha }_2\mathbb{W}\mathbb{I}+\rho {\alpha }_2\mathbb{I}-\mu \mathbb{W},\hfill \\ \hfill {\displaystyle \frac{d\mathbb{S}\left(\tau \right)}{dt}}& ={\alpha }_1\mathbb{W}\mathbb{S}+\delta \mathbb{I}-({\theta }_1+\mu )\mathbb{S},\hfill \\ \hfill {\displaystyle \frac{d\mathbb{I}\left(\tau \right)}{dt}}& ={\alpha }_2\mathbb{W}\mathbb{I}-\rho {\alpha }_2\mathbb{I}-(\delta +{\theta }_2+\mu )\mathbb{I},\hfill \\ \hfill {\displaystyle \frac{d\mathbb{T}\left(\tau \right)}{dt}}& ={\theta }_1\mathbb{S}+{\theta }_2\mathbb{I}-\mu \mathbb{T},\hfill \end{array}$$

(1)

with the initial conditions $\mathbb{W}\left(0\right)={\mathbb{W}}_0$; $\mathbb{S}\left(0\right)={\mathbb{S}}_0$; $\mathbb{I}\left(0\right)={\mathbb{I}}_0$; and $\mathbb{T}\left(0\right)={\mathbb{T}}_0$. The state variables $\mathbb{W}$, $\mathbb{S}$, $\mathbb{I}$, and $\mathbb{T}$ in System (1) represent the number of polluted water sources (WSs), the WSs susceptible to pollution, the WSs infected due to WP, and the WSs recovered due to treatment for the insoluble class, respectively. Regarding the involved parameters in our considered dynamical system, $\Lambda$ is the rate of WP, ${\alpha }_1$ is the rate of soluble CW, ${\alpha }_2$ is the rate of insoluble contaminated water (CW), $\rho$ represents the rate at which insoluble water pollutants (WPs) turn into WP, $\mu$ shows the vanishing rate of WPs, $\delta$ is for the conversion rate of insoluble WPs into solute, ${\theta }_1$ and ${\theta }_2$ are the preserved rate of soluble and insoluble WPs, respectively, and $\gamma$ is used for the re-susceptibility of filtered water.

Regardless of the progress made in these areas, there is still demand for further research on the intersection of FFDs and existence theory. This research work aims to explore this intersection and develop a numerical technique for solving nonlinear dynamical models via the results of existence theory and the Newton polynomial scheme.

This research secures significant importance by addressing the serious issue of WP through advanced MM and analysis. A novel parameter, $\gamma$, is introduced to represent the recovery rate of water pollution, which signifies the effectiveness of water treatment processes that not only reduce pollutant concentrations but also restore water quality. A higher value of $\gamma$ indicates more efficiency in the control of water pollution. Furthermore, the intervention of $\gamma$ explicitly bridges human intervention strategies with natural self-cleaning mechanisms, employing a comprehensive measure of water quality dynamics restoration. Further, utilizing FFDs in sense of the Atangana–Baleanu–Caputo (ABC) formulation [43,44,45], the model captures the memory effects and non-local dynamics inherent in pollution systems, providing a more accurate representation of real-world scenarios compared to traditional integer-order models. This study establishes the existence and stability of the solution using fixed-point theory (FPT), ensuring the mathematical robustness of the model.

Furthermore, the development of a Newton polynomial (NP) numerical scheme offers an efficient and accurate method for solving the system, enabling the simulation of pollution dynamics under various conditions. This research not only advances the field of environmental modeling but also provides practical insights for policymakers and stakeholders to design effective strategies for mitigating water pollution and promoting sustainable water resource management.

2. Preliminaries

In this section, we present some important lemmas and definitions about our research, in the context of fractal–fractional operators, defined as follows:

Definition 1

([45]). The Gamma function is defined as

$$\Gamma \left(\mathtt{p}\right)={\int }_0^{\infty }{\eta }^{\mathtt{p}-1}{e}^{\eta }d\eta ,whereRe\left(\mathtt{p}\right)>0.$$

(2)

where $Re\left(p\right)$ means the real part of p.

Definition 2

([46]). The authors define Euler’s first kind of integral as

$$\mathbb{B}({\mathtt{p}}_1,{\mathtt{p}}_2)={\int }_0^1{\eta }^{{\mathtt{p}}_1-1}{(1-\eta )}^{{\mathtt{p}}_2-1}d\eta ,$$

(3)

for ${\mathtt{p}}_1$, ${\mathtt{p}}_2$∈$\mathbb{C}$, such that $Re\left({\mathtt{p}}_1\right)$, $Re\left({\mathtt{p}}_2\right)>0$.

Definition 3

([45]). The relationship of the Gamma function and the Beta function is given by

$$\mathbb{B}({\mathtt{p}}_1,{\mathtt{p}}_2)={\displaystyle \frac{\Gamma \left({\mathtt{p}}_1\right)\Gamma \left({\mathtt{p}}_2\right)}{\Gamma ({\mathtt{p}}_1+{\mathtt{p}}_2)}}.$$

(4)

Definition 4

([31]). The FFD in the sense of the ABC framework for the function $\mathbb{H}\left(\tau \right)$, incorporating a generalized Mittag–Leffler function as a nonsingular kernel, is defined as

$${}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}D_{0,\tau }^{{\mathtt{p}}_1,{\mathtt{p}}_2}={\displaystyle \frac{M\left({\mathtt{p}}_1\right)}{1-{\mathtt{p}}_1}}{\displaystyle \frac{d}{d{\tau }^{{\mathtt{p}}_2}}}{\int }_0^{\tau }{\mathbb{E}}_{{\mathtt{p}}_1}{P}^{\ast }\mathbb{H}\left(s\right)ds,$$

(5)

where $P^{\ast }=-{\displaystyle \frac{{\mathtt{p}}_1}{1-{\mathtt{p}}_1}}{(\tau -s)}^{{\mathtt{p}}_1}$, ${\mathtt{p}}_1>0,{\mathtt{p}}_2\le 1\in \mathbb{N}$, where ${\displaystyle \frac{d}{d{\tau }^{{\mathtt{p}}_2}}}$ represents the fractal dimension of the measure ${\mathtt{p}}_2$ and $\mathbb{E}$ describes a Mittage–Leffler function and $M\left({\mathtt{p}}_1\right)=1-{\mathtt{p}}_1+{\displaystyle \frac{{\mathtt{p}}_1}{\Gamma \left({\mathtt{p}}_1\right)}}.$

Definition 5

([47]). The fractal derivative of the function $M\left(\tau \right)$, with the fractal measure ${\tau }^{\mathtt{p}}$ (where q is a fractal scaling parameter), is defined as

$${\displaystyle \frac{dM\left(\tau \right)}{d{\tau }^{\mathtt{p}}}}=\underset{{\tau }_1^{\mathtt{p}}\to {\tau }^{\mathtt{p}}}{lim}{\displaystyle \frac{M{\left(\tau \right)}_1-M\left(\tau \right)}{{\tau }_1^{\mathtt{p}}-{\tau }^{\mathtt{p}}}}\mathrm{for}\tau >0.$$

(6)

Definition 6

([31]). The fractal–fractional integral (FFI) in sense of the ABC derivative for the function $\mathbb{H}\left(\tau \right)$, incorporating a generalized Mittag–Leffler function as a nonsingular kernel, is defined as

$$\begin{array}{cc}\hfill {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}I_{0,\tau }^{{\mathtt{p}}_1,{\mathtt{p}}_2}& ={\displaystyle \frac{{\mathtt{p}}_1{\mathtt{p}}_2}{M\left({\mathtt{p}}_1\right)\Gamma \left({\mathtt{p}}_1\right)}}{\int }_0^{\tau }{(\tau -s)}^{{\mathtt{p}}_1}{s}^{{\mathtt{p}}_2-1}\mathcal{H}\left(s\right)ds\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{p}}_2(1-{\mathtt{p}}_1){\tau }^{{\mathtt{p}}_2-1}\mathcal{H}\left(\tau \right)}{M\left({\mathtt{p}}_1\right)}},\hfill \end{array}$$

(7)

satisfying that, ${\mathtt{p}}_1>0,{\mathtt{p}}_2\le 1\in \mathbb{N}$ also combine with, $M\left({\mathtt{p}}_1\right)=1-{\mathtt{p}}_1+{\displaystyle \frac{{\mathtt{p}}_1}{\Gamma \left({\mathtt{p}}_1\right)}}.$

Remark 1.

The FFD of the function $\mathbb{H}\left(\tau \right)$, in the ABC sense, which incorporates a generalized Mittag–Leffler function as a nonsingular kernel, is defined as

$$\begin{array}{c}\hfill {\displaystyle \frac{d\mathbb{H}\left(\tau \right)}{d{\tau }^{\mathtt{p}}}}={\displaystyle \frac{d\mathbb{H}}{d\tau }}{\displaystyle \frac{d\tau }{d{\tau }^{\mathtt{p}}}}={\displaystyle \frac{{\tau }^{1-\mathtt{p}}}{\mathtt{p}}}{\displaystyle \frac{d\mathbb{H}}{d\tau }}.\end{array}$$

Lemma 1.

Let the function $\mathfrak{q}\left(b\right)\in \mathcal{C}\left(\mathbb{I}\right)\cap \mathcal{C}\left(\mathbb{I}\right)$; then, the solution of

$$\begin{array}{cc}\hfill {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\mathfrak{q}\left(b\right)& =\mathtt{w}\left(\mathtt{b}\right)forall{\mathtt{r}}_1,{\mathtt{r}}_2\in (0,1),\hfill \\ \hfill \mathfrak{q}\left(0\right)& ={\mathfrak{q}}_0,\hfill \end{array}$$

(8)

is given by

$$\begin{array}{cc}\hfill \mathfrak{q}\left(b\right)={\mathfrak{q}}_0& +{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{M\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^b{(b-s)}^{{\mathtt{r}}_1}{s}^{{\mathtt{r}}_2-1}\mathtt{w}\left(\mathtt{s}\right)ds+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1)b^{{\mathtt{r}}_2-1}\mathtt{w}\left(\mathtt{b}\right)}{M\left({\mathtt{r}}_1\right)}}.\hfill \end{array}$$

(9)

Definition 7.

The function $\mathbb{A}$ is Lipschitz continuous (LC) on R with the Lipschitz constant $\mathtt{p}$ if there exists a constant, $0\le \mathtt{p}$, such that the inequality holds for all pairs:

$$\parallel {\mathbb{A}}_1\left({\mathcal{H}}_1\left(\tau \right)\right)-{\mathbb{A}}_1\left({\mathcal{H}}_2\left(\tau \right)\right)\parallel \le \mathtt{p}\parallel {\mathcal{H}}_1-{\mathcal{H}}_2\parallel$$

(10)

for all ${\mathcal{H}}_1,{\mathcal{H}}_2\in R$.

Theorem 1.

Consider a closed convex subset, $\mathtt{z}$, within the Banach space $\mathbb{B}$. Let $\mathbb{H}:\mathbb{B}\to \mathbb{B}$, where $\mathbb{B}$ is a completely continuous operator in this space; then, $\mathbb{H}$ is guaranteed to possess at least one fixed point.

Assumption 1.

The following norm properties are satisfied for any two functions.

1.

$\parallel M_1\left(\tau \right)·M_2\left(\tau \right)\parallel = \parallel M_1\left(\tau \right)\parallel ·\parallel M_2\left(\tau \right)\parallel .$

2.

$\parallel M_1\left(\tau \right)-M_2\left(\tau \right)\parallel = \parallel M_2\left(\tau \right)-M_1\left(\tau \right)\parallel .$

3. Generalized FFWP Model

In this research work, the authors generalize the structure of the ordinary WP model (1), in the sense of the fractal–fractional differential operator, and introduce the new parameter $\gamma$, representing the re-susceptibility of filter water. The generalized WP model in the $\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}$ sense is

$$\begin{array}{cc}\hfill & {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\mathbb{W}\left(\tau \right)=\Lambda -{\alpha }_1\mathbb{W}\mathbb{S}-{\alpha }_2\mathbb{W}\mathbb{I}+\rho {\alpha }_2\mathbb{I}-\mu \mathbb{W},\hfill \\ \hfill & {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\mathbb{S}\left(\tau \right)={\alpha }_1\mathbb{W}\mathbb{S}+\delta \mathbb{I}-({\theta }_1+\mu )\mathbb{S}+\gamma \mathbb{T},\hfill \\ \hfill & {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\mathbb{I}\left(\tau \right)={\alpha }_2\mathbb{W}\mathbb{I}-\rho {\alpha }_2\mathbb{I}-(\delta +{\theta }_2+\mu )\mathbb{I},\hfill \\ \hfill & {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\mathbb{T}\left(\tau \right)={\theta }_1\mathbb{S}+{\theta }_2\mathbb{I}-\mu \mathbb{T}-\gamma \mathbb{T}.\hfill \end{array}$$

(11)

Subject to the conditions given as $\mathbb{W}\left(0\right)={\mathbb{W}}_0$, $\mathbb{T}\left(0\right)={\mathbb{T}}_0$, $\mathbb{S}\left(0\right)={\mathbb{S}}_0$, and $\mathbb{I}\left(0\right)={\mathbb{I}}_0$, System (11) can be rewritten in the form of (12) with the kernels defined in (13).

$$\begin{array}{cc}\hfill {}^{\mathcal{ABC}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1}\mathbb{W}\left(\tau \right)& ={\mathtt{r}}_2{t}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1(\mathbb{W},\tau ),\hfill \\ \hfill {}^{\mathcal{ABC}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1}\mathbb{S}\left(\tau \right)& ={\mathtt{r}}_2{t}^{{\mathtt{r}}_2-1}{\mathcal{F}}_2(\mathbb{S},\tau ),\hfill \\ \hfill {}^{\mathcal{ABC}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1}\mathbb{I}\left(\tau \right)& ={\mathtt{r}}_2{t}^{{\mathtt{r}}_2-1}{\mathcal{F}}_3(\mathbb{I},\tau ),\hfill \\ \hfill {}^{\mathcal{ABC}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1}\mathsf{D}\left(\tau \right)& ={\mathtt{r}}_2{t}^{{\mathtt{r}}_2-1}{\mathcal{F}}_4(\mathbb{T},\tau ),\hfill \end{array}$$

(12)

where

$$\begin{array}{cc}\hfill {\mathcal{F}}_1(\mathbb{W},\tau )& =\Lambda -{\alpha }_1\mathbb{W}\mathbb{S}-{\alpha }_2\mathbb{W}\mathbb{I}+\rho {\alpha }_2\mathbb{I}-\mu \mathbb{W},\hfill \\ \hfill {\mathcal{F}}_2(\mathbb{S},\tau )& ={\alpha }_1\mathbb{W}\mathbb{S}+\delta \mathbb{I}-({\theta }_1+\mu )\mathbb{S}+\gamma \mathbb{T},\hfill \\ \hfill {\mathcal{F}}_3(\mathbb{I},\tau )& ={\alpha }_2\mathbb{W}\mathbb{I}-\rho {\alpha }_2\mathbb{I}-(\delta +{\theta }_2+\mu )\mathbb{I},\hfill \\ \hfill {\mathcal{F}}_4(\mathbb{T},\tau )& ={\theta }_1\mathbb{S}+{\theta }_2\mathbb{I}-\mu \mathbb{T}-\gamma \mathbb{T},\hfill \end{array}$$

(13)

defines the kernels for the generalized model (11).

4. Existence of the Solutions

In this section, the focus is on the existence and uniqueness of the solution (EUS) of the proposed model (11), with particular emphasis on the boundedness and non-negativity.

4.1. Existence

To investigate existence of the system’s solution (11), the Picard–Lindolof method is adopted in this subsection.

Theorem 2.

The solutions of the model in (11) are uniformly bounded and non-negative.

Proof. 

System (14) can be computed with ease using Equation (13)

$$\left\{\begin{array}{c}{}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathbb{W}\left(\tau \right)|}_{W=0}=\Lambda +\rho {\alpha }_2\mathbb{I}\ge 0,\hfill \\ {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\mathbb{S}\left(\tau \right){|}_{S=0}=\delta \mathbb{I}-({\theta }_1+\gamma \mathbb{T}\ge 0,\hfill \\ {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathbb{I}\left(\tau \right)|}_{I=0}=0,\hfill \\ {}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{D}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathbb{T}\left(\tau \right)|}_{T=0}={\theta }_1\mathbb{S}\ge 0.\hfill \end{array}\right.$$

(14)

Then, the solutions remain confined to the hyperplane if $((\mathbb{W}\left(0\right),\mathbb{S}\left(0\right),\mathbb{T}\left(0\right),\mathbb{I}\left(0\right))\in {\mathbb{R}}_{+}^4)$. Since trajectories cannot escape into negative regions, ${\mathbb{R}}_{+}^4$ is a positively invariant set; solutions starting within it remain non-negative for all future time, preserving biological or physical feasibility. □

Theorem 3.

The kernels ${\mathcal{F}}_1$, ${\mathcal{F}}_2$, ${\mathcal{F}}_3$, and ${\mathcal{F}}_4$ satisfy the contraction and LC aspect.

Proof. 

To make the kernels LC, let us take the difference of ${\mathbb{W}}_1$ and ${\mathbb{W}}_2$; we obtain

$$\begin{array}{cc}\hfill \parallel {\mathcal{F}}_1({\mathbb{W}}_1,\tau )-{\mathcal{F}}_1({\mathbb{W}}_2,\tau )\parallel & \\ & =\parallel \Lambda -{\alpha }_1{\mathbb{W}}_1\mathbb{S}-{\alpha }_2{\mathbb{W}}_1\mathbb{I}+\rho {\alpha }_2\mathbb{I}-\mu {\mathbb{W}}_1\hfill \\ & -\left(\Lambda -{\alpha }_1{\mathbb{W}}_2\mathbb{S}-{\alpha }_2{\mathbb{W}}_2\mathbb{I}+\rho {\alpha }_2\mathbb{I}-\mu {\mathbb{W}}_2\right)\parallel ,\hfill \\ \hfill & \le ({\alpha }_1\parallel \mathbb{S}\left(\tau \right)\parallel +\mu +{\alpha }_2\parallel \mathbb{I}\parallel )∥{\mathbb{W}}_1\left(\tau \right)-{\mathbb{W}}_2\left(\tau \right)∥,\hfill \\ \hfill & \le {ħ}_1∥{\mathbb{W}}_1\left(\tau \right)-{\mathbb{W}}_2\left(\tau \right)∥.\hfill \end{array}$$

By deploying Assumption 1 and employing ${ћ}_1={\alpha }_1{ϵ}_1+\mu +{\alpha }_2{ϵ}_2<1$, the kernel ${\mathcal{F}}_1$ thus satisfies the Lipschitz condition. Moreover,

$$\begin{array}{cc}\hfill \parallel {\mathcal{F}}_2({\mathbb{S}}_1,\tau )-{\mathcal{F}}_2({\mathbb{S}}_2,\tau )\parallel & =∥{\alpha }_1\mathbb{W}({\mathbb{S}}_1-{\mathbb{S}}_2)-({\theta }_1+\mu )({\mathbb{S}}_1-{\mathbb{S}}_2)∥,\hfill \\ \hfill & \le {\alpha }_1\parallel \mathbb{W}\parallel \parallel {\mathbb{S}}_1-{\mathbb{S}}_2\parallel +({\theta }_1+\mu )\parallel {\mathbb{S}}_1-{\mathbb{S}}_2\parallel ,\hfill \\ \hfill & =\left({\alpha }_1\parallel \mathbb{W}\parallel +{\theta }_1+\mu \right)\parallel {\mathbb{S}}_1-{\mathbb{S}}_2\parallel ,\hfill \\ \hfill & ={ħ}_2\parallel {\mathbb{S}}_1-{\mathbb{S}}_2\parallel ,\hfill \end{array}$$

using Assumption 1 and taking ${ħ}_2={\alpha }_1ϵ-{\theta }_1-\mu <1$. Thus, the kernel ${\mathcal{F}}_2$ satisfies the Lipschitz condition. Similarly,

$$\begin{array}{cc}\hfill \parallel {\mathcal{F}}_3({\mathbb{I}}_1,\mathrm{t})-{\mathcal{F}}_3({\mathbb{I}}_2,\mathrm{t})\parallel & =∥{\alpha }_2\mathbb{W}{\mathbb{I}}_1-\rho {\alpha }_2{\mathbb{I}}_1-(\delta +{\theta }_2+\mu ){\mathbb{I}}_1\hfill \\ \hfill & -\left[{\alpha }_2\mathbb{W}{\mathbb{I}}_2-\rho {\alpha }_2{\mathbb{I}}_2-(\delta +{\theta }_2+\mu ){\mathbb{I}}_2\right]∥,\hfill \\ \hfill & \le ({\alpha }_2\mathbb{W}-\rho {\alpha }_2-\delta -{\theta }_2-\mu )∥\left({\mathbb{I}}_2-{\mathbb{I}}_1\right)∥,\hfill \\ \hfill & \le {ħ}_3∥\left({\mathbb{I}}_1\left(t\right)-{\mathbb{I}}_2\left(t\right)\right)∥,\hfill \end{array}$$

where ${ħ}_3={\alpha }_2ϵ-\rho {\alpha }_2-(\delta +{\theta }_2+\mu )<1$. Thus, the kernel ${\mathcal{F}}_3$ satisfies the Lipschitz condition. Similarly,

$$\begin{array}{cc}\hfill \parallel {\mathcal{F}}_4({\mathbb{T}}_1,\tau )-{\mathcal{F}}_4({\mathbb{T}}_2,\tau )\parallel & =∥{\theta }_1\mathbb{S}+{\theta }_2\mathbb{I}-\mu {\mathbb{T}}_1-\gamma {\mathbb{T}}_1\hfill \\ \hfill & -\left[{\theta }_1\mathbb{S}+{\theta }_2\mathbb{I}-\mu {\mathbb{T}}_2-\gamma {\mathbb{T}}_2\right]∥,\hfill \\ \hfill & \le (\mu +\gamma )∥\left({\mathbb{T}}_2\left(t\right)-{\mathbb{T}}_1\left(t\right)\right)∥,\hfill \\ \hfill & \le {ħ}_4∥\left({\mathbb{T}}_2\left(t\right)-{\mathbb{T}}_1\left(t\right)\right)∥,\hfill \end{array}$$

for ${ħ}_4=\mu +\gamma <1$; therefore, ${\mathcal{F}}_4$ is LC. As a result, ${\mathcal{F}}_1\left(\mathbb{W}\left(t\right)\right)$, ${\mathcal{F}}_2\left(\mathbb{S}\left(t\right)\right)$, ${\mathcal{F}}_3\left(\mathbb{I}\left(t\right)\right)$, and ${\mathcal{F}}_4\left(\mathbb{T}\left(t\right)\right)$ satisfy the LC condition. □

4.2. Uniqueness

Theorem 4.

The solution of the generalized WP model (11) is unique, if the following inequality,

$$\left({\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1)}{N\left({\mathtt{r}}_1\right)}}+{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2\Gamma \left({\mathtt{r}}_2\right)}{N\left({\mathtt{r}}_1\right)\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}{\chi }_i\le 1\right),i\in {\mathbb{N}}_i^4,$$

(15)

is true, for first four non-negative integers.

Proof. 

From the definition of the fractal–fractional integral in the ABC sense, we obtain

$$\begin{array}{cc}\hfill & \mathbb{W}\left(\tau \right)=\mathbb{W}\left(0\right)+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)+{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)d\mathtt{s},\hfill \\ \hfill & \mathbb{S}\left(\tau \right)=\mathbb{S}\left(0\right)+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_2\left(\mathbb{S}\left(\tau \right)\right)+{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_2\left(\mathbb{S}\left(\tau \right)\right)d\mathtt{s},\hfill \\ \hfill & \mathbb{I}\left(\tau \right)=\mathbb{I}\left(0\right)+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_3\left(\mathbb{I}\left(\tau \right)\right)+{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_3\left(\mathbb{I}\left(\tau \right)\right)d\mathtt{s},\hfill \\ \hfill & \mathbb{T}\left(\tau \right)=\mathbb{T}\left(0\right)+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_4\left(\mathbb{T}\left(\tau \right)\right)+{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_4\left(\mathbb{T}\left(\tau \right)\right)d\mathtt{s}.\hfill \end{array}$$

(16)

Picard’s operators $A_1$, $A_2$, $A_3$ and $A_4$ are generated using the FFI method. Therefore, Equation (16) becomes

$$\begin{array}{cc}\hfill & \mathbb{W}\left(\tau \right)=\mathbb{W}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)\right\},\hfill \\ & \mathbb{S}\left(\tau \right)=\mathbb{S}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_2\left(\mathbb{S}\left(\tau \right)\right)\right\},\hfill \\ \hfill & \mathbb{I}\left(\tau \right)=\mathbb{I}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_3\left(\mathbb{I}\left(\tau \right)\right)\right\},\hfill \\ \hfill & \mathbb{T}\left(\tau \right)=\mathbb{T}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_4\left(\mathbb{T}\left(\tau \right)\right)\right\}.\hfill \end{array}$$

(17)

If $A_1$, $A_2$, $A_3$, and $A_4$ are Picard’s operators, then we can obtain

$$\begin{array}{cc}\hfill & A_1\mathbb{W}\left(\tau \right)=\mathbb{W}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)\right\},\hfill \\ & A_2\mathbb{S}\left(\tau \right)=\mathbb{S}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_2\left(\mathbb{S}\left(\tau \right)\right)\right\},\hfill \\ \hfill & A_3\mathbb{I}\left(\tau \right)=\mathbb{I}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_3\left(\mathbb{I}\left(\tau \right)\right)\right\},\hfill \\ \hfill & A_4\mathbb{T}\left(\tau \right)=\mathbb{T}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_4\left(\mathbb{T}\left(\tau \right)\right)\right\}.\hfill \end{array}$$

(18)

Hence, the boundedness can be formulated by employing the supremum norm to Picard’s operators as follows:

$$\begin{array}{cc}\hfill ∥A_1\mathbb{W}\left(\tau \right)∥& =∥\mathbb{W}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le ∥\mathbb{W}\left(0\right)∥+∥{}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{I}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le \mathbb{W}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}∥{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)∥,\hfill \\ \hfill ∥A_2\mathbb{S}\left(\tau \right)∥& =∥\mathbb{S}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_2\left(\mathbb{S}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le ∥\mathbb{S}\left(0\right)∥+∥{}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{I}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_2\left(\mathbb{S}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le \mathbb{S}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}∥{\mathcal{F}}_2\left(\mathbb{S}\left(\tau \right)\right)∥,\hfill \\ \hfill ∥A_3\mathbb{I}\left(\tau \right)∥& =∥\mathbb{I}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_3\left(\mathbb{I}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le ∥\mathbb{I}\left(0\right)∥+∥{}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{I}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_3\left(\mathbb{I}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le \mathbb{I}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}∥{\mathcal{F}}_3\left(\mathbb{I}\left(\tau \right)\right)∥,\hfill \\ \hfill ∥A_4\mathbb{T}\left(\tau \right)∥& =∥\mathbb{T}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_4\left(\mathbb{T}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le ∥\mathbb{T}\left(0\right)∥+∥{}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{I}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left\{{\mathcal{F}}_4\left(\mathbb{T}\left(\tau \right)\right)\right\}∥,\hfill \\ \hfill & \le \mathbb{T}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}∥{\mathcal{F}}_4\left(\mathbb{T}\left(\tau \right)\right)∥.\hfill \end{array}$$

(19)

In view of Theorem 3, $\mathcal{F}(\mathbb{W},\mathbb{S},\mathbb{I},\mathbb{T})$ is LC, which means that $\mathcal{F}\left(\mathbb{W}\right(\tau ),\mathbb{S}(\tau ),\mathbb{I}(\tau ),\mathbb{T}(\tau ),\tau )$ is bounded. Thus, there exist the constants ${\wp }_1$, ${\wp }_2$, ${\wp }_3$, and ${\wp }_4$, such that

$$\parallel {\mathcal{F}}_1(\mathbb{W},\tau )\parallel \le {\wp }_1,\parallel {\mathcal{F}}_2(\mathbb{S},\tau )\parallel \le {\wp }_2,$$

$$\parallel {\mathcal{F}}_3(\mathbb{I},\tau )\parallel \le {\wp }_3\mathrm{and}\parallel {\mathcal{F}}_4(\mathbb{T},\tau )\parallel \le {\wp }_4.$$

Thus,

$$\begin{array}{cc}\hfill ∥A_1\mathbb{W}\left(\tau \right)∥& \le ∥\mathbb{W}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_1\left(\mathbb{W}\left(\tau \right)\right)∥,\hfill \\ & \le \mathbb{W}\left(0\right){+}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left({\wp }_1\right),\hfill \\ \hfill & \le \mathbb{W}\left(0\right)+{\wp }_1^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \end{array}$$

(20)

by employing $\tau =1$ and letting there exist ${\mathtt{s}}_1\in R$.

$$\begin{array}{cc}\hfill \parallel A_1\mathbb{W}\left(\tau \right)\parallel & \le \mathbb{W}\left(0\right)+{\wp }_1\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_1^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right].\hfill \end{array}$$

(21)

In Equation (21), we determined that $A_1$ is a bounded operator. Similarly, we can also determine the boundedness of the operators $A_2$, $A_3$, and $A_4$, i.e.,

$$\begin{array}{cc}\hfill \parallel A_2\mathbb{S}\left(\tau \right)\parallel & \le \mathbb{S}\left(0\right)+{\wp }_1\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_2^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right],\hfill \end{array}$$

Let ∃${\mathtt{s}}_2\in \Re$ where $0\le \tau \le {\mathtt{s}}_2$.

$$\begin{array}{cc}\hfill \parallel A_3\mathbb{I}\left(\tau \right)\parallel & \le \mathbb{I}\left(0\right)+{\wp }_1\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_3^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right].\hfill \end{array}$$

Considering ${\mathtt{s}}_3$, ${\mathtt{s}}_4$$\in \Re$ for $\tau \in [0,{\mathtt{s}}_3]$ and $\tau \in [0,{\mathtt{s}}_4]$, respectively to obtain

$$\begin{array}{cc}\hfill \parallel A_4\mathbb{T}\left(\tau \right)\parallel & \le \mathbb{I}\left(0\right)+{\wp }_1\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_4^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right].\hfill \end{array}$$

The operators $A_1$, $A_2$, $A_3$, and $A_4$ are Picard bounded and, consequently are contraction mappings if the following conditions holds:$A_1$, $A_2$, $A_3$, and $A_4$ are contraction mappings if the following conditions holds:

$$\begin{array}{cc}\hfill \parallel A_1{W}_1-A_1{W}_2\parallel =& {\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_1\left({\mathbb{W}}_1\left(\tau \right)\right){-}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_1\left({\mathbb{W}}_2\left(\tau \right)\right)\parallel ,\hfill \\ \hfill & \le \parallel {\mathcal{F}}_1\left({\mathbb{W}}_1\left(\tau \right)\right)-{\mathcal{F}}_1\left({\mathbb{W}}_2\left(\tau \right)\right){\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ \hfill & \le ({\alpha }_1{ϵ}_1+{\alpha }_2{ϵ}_2+\mu ){\parallel {\mathbb{W}}_1-{\mathbb{W}}_2\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ \hfill & \le ({\alpha }_1{ϵ}_1+{\alpha }_2{ϵ}_2+\mu )\parallel {\mathbb{W}}_1\left(\tau \right)-{\mathbb{W}}_2\left(\tau \right)\parallel ,\hfill \\ & \le \left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathbb{X}}_1^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right].\hfill \end{array}$$

Moreover, if $K_1<{\displaystyle \frac{1}{({\alpha }_1{ϵ}_1+{\alpha }_2{ϵ}_2+\mu )}}$ such that $K_1=\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathbb{X}}_1^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right]$, then the operator $A_1$ satisfies the contraction principle. Also,

$$\begin{array}{cc}\hfill \parallel A_2{S}_1-A_2{S}_2\parallel & {=\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_2\left({\mathbb{S}}_1\left(\tau \right)\right){-}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_2\left({\mathbb{S}}_2\left(\tau \right)\right)\parallel ,\hfill \\ \hfill & \le \parallel {\mathcal{F}}_1\left({\mathbb{S}}_1\left(\tau \right)\right)-{\mathcal{F}}_2\left({\mathbb{S}}_2\left(\tau \right)\right){\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ & \le ({\alpha }_1{ϵ}_1-{\theta }_1-\mu ){\parallel {\mathbb{S}}_1\left(\tau \right)-{\mathbb{S}}_2\left(\tau \right)\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ \hfill & \le ({\alpha }_1{ϵ}_1-{\theta }_1-\mu )\parallel {\mathbb{S}}_1\left(\tau \right)-{\mathbb{S}}_2\left(\tau \right)\parallel ,\hfill \\ \hfill & \le \left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_2^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right].\hfill \end{array}$$

Suppose $K_2=\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_2^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right]$, then $0<K_2<{\displaystyle \frac{1}{({\alpha }_1{ϵ}_1-{\theta }_1-\mu )}}$ satisfies the contraction principle for $A_2$. Moreover, from the third equation in System (11), we have

$$\begin{array}{cc}\hfill \parallel A_3{I}_1-A_3{I}_2\parallel & {=\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_3({\mathbb{I}}_1,\left(\tau \right)){-}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_3({\mathbb{I}}_2,\left(\tau \right))\parallel ,\hfill \\ \hfill & \le \parallel {\mathcal{F}}_3\left({\mathbb{I}}_1\left(\tau \right)\right)-{\mathcal{F}}_3\left({\mathbb{I}}_2\left(\tau \right)\right){\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ & \le ({\alpha }_2ϵ+\rho {\alpha }_2+\delta +{\theta }_2+\mu )\parallel {∥{\mathbb{I}}_1-{\mathbb{I}}_2∥}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right).\hfill \end{array}$$

Now, let $K_3{=}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right)=\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathbb{X}}_3^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right]$, then $0<K_3<{\displaystyle \frac{1}{\left({\alpha }_2ϵ+\rho {\alpha }_2+\delta +{\theta }_2+\mu \right)}}$ satisfies the contraction principle for $A_3$. Similarly, for the last equation in System (11),

$$\begin{array}{cc}\hfill \parallel A_4{T}_1-A_4{T}_2\parallel & {=\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_4({\mathbb{T}}_1,\left(\tau \right)){-}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_4({\mathbb{T}}_2,\left(\tau \right))\parallel ,\hfill \\ \hfill & \le \parallel {\mathcal{F}}_4\left({\mathbb{T}}_1\left(\tau \right)\right)-{\mathcal{F}}_T\left({\mathbb{T}}_2\left(\tau \right)\right){\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ & \le (\mu +\gamma \parallel {\mathbb{T}}_1\left(\tau \right)-{\mathbb{T}}_2\left(\tau \right){\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ \hfill & \le (\mu +\gamma )∥{\mathbb{T}}_1\left(\tau \right)-{\mathbb{T}}_2\left(\tau \right)∥,\hfill \\ & \le \left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathbb{X}}_4^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right].\hfill \end{array}$$

Further, suppose that $K_4=\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathbb{X}}_4^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right]$, then the operator $A_4$ satisfies the contraction principle for $0<K_4<{\displaystyle \frac{1}{(\mu +\gamma )}}$. Thus, the operators $A_1$, $A_1$, $A_3$, and $A_4$ satisfy the contraction principle. Finally, we establish the uniqueness of the solution by assuming that both ${\mathbb{W}}_1\left(\tau \right)$ and ${\mathbb{W}}_2\left(\tau \right)$ satisfy the initial equation in System (11).

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbb{W}}_1\left(\tau \right)-{\mathbb{W}}_0\left(\tau \right){=}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_1({\mathbb{W}}_1,\left(\tau \right)),\\ \hfill {\mathbb{W}}_2\left(\tau \right)-{\mathbb{W}}_0\left(\tau \right){=}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_1({\mathbb{W}}_2,\left(\tau \right)).\end{array}\end{array}$$

(22)

The difference of ${\mathbb{W}}_1$ and ${\mathbb{W}}_2$ in Equation (22) gives

$$\begin{array}{cc}\hfill ∥{\mathbb{W}}_1-{\mathbb{W}}_2∥& =∥{}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{I}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_1({\mathbb{W}}_1,\left(\tau \right)){-}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_1({\mathbb{W}}_2,\left(\tau \right))∥,\hfill \\ \hfill \parallel {\mathbb{W}}_1-{\mathbb{W}}_2\parallel & \le {∥{\mathcal{F}}_1({\mathbb{W}}_1,\tau )-{\mathcal{F}}_1({\mathbb{W}}_2,\left(\tau \right))∥}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right),\hfill \\ & \le ({\alpha }_1{ϵ}_1+{\alpha }_2{ϵ}_2+\mu ){\parallel {\mathbb{W}}_1-{\mathbb{W}}_2\parallel }^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right).\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill {(1-({\alpha }_1{ϵ}_1+{\alpha }_2{ϵ}_2+\mu ))}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right)∥{\mathbb{W}}_1\left(\tau \right)-{\mathbb{W}}_2\left(\tau \right)∥& \le 0.\hfill \end{array}$$

Thus, the term

$$(1-{\alpha }_1{ϵ}_1-{\alpha }_2{ϵ}_2-\mu )\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_1^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right]\ne 0$$

implies that ${\mathbb{W}}_1\left(\tau \right)={\mathbb{W}}_2\left(\tau \right).$ Further, let ${\mathbb{S}}_1$ and ${\mathbb{S}}_2$ denote the two solutions of the second equation in the proposed model (11), such that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbb{S}}_1-{\mathbb{S}}_0{=}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_2({\mathbb{S}}_1,\left(\tau \right)),\\ \hfill {\mathbb{S}}_2-{\mathbb{S}}_0{=}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_2({\mathbb{S}}_2,\left(\tau \right)).\end{array}\end{array}$$

The difference of these two equations gives

$$\begin{array}{cc}\hfill \parallel {\mathbb{S}}_1-{\mathbb{S}}_2\parallel & =∥{}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}\mathbb{I}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_2({\mathbb{S}}_1,\tau ){-}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}{\mathcal{F}}_2({\mathbb{S}}_2,\tau )∥,\hfill \\ \hfill & \le {∥{\mathcal{F}}_2({\mathbb{S}}_1,\tau )-{\mathcal{F}}_2({\mathbb{S}}_2,\tau )∥}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}\left(1\right).\hfill \end{array}$$

(23)

Shifting the right side of Equation (23), we obtain that ${\left(1-({\alpha }_1{ϵ}_1-{\theta }_1-\mu )\right)}^{\mathcal{F}\mathcal{F}\mathcal{A}\mathcal{B}}{\mathbb{I}}_{0,\tau }^{{\mathtt{r}}_1,{\mathtt{r}}_2}∥{\mathbb{S}}_1\left(\tau \right)-{\mathbb{S}}_2\left(\tau \right)∥$ is negative. The term

$$1-({\alpha }_1{ϵ}_1-{\theta }_1-\mu )\left[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\mathtt{s}}_2^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\right]\ne 0$$

and ${\mathbb{S}}_1\left(\tau \right)={\mathbb{S}}_2\left(\tau \right)$. Similarly, one can compute the conditions for $\mathbb{I}\left(\tau \right)$ and $\mathbb{T}\left(\tau \right)$ to obtain ${\mathbb{I}}_1\left(\tau \right)={\mathbb{I}}_2\left(\tau \right)$ and ${\mathbb{T}}_1\left(\tau \right)={\mathbb{T}}_2\left(\tau \right)$, respectively. Consequently, the operators $A_1$, $A_2$, $A_3$, and $A_4$ are well defined and adhere to contraction mapping principles. This ensures the uniqueness of the solution for the generalized WP model (11). □

4.3. Stability Analysis

This subsection analyzes the stability properties of the proposed model 11, specifically examining the criteria for Hyers–Ulam stability.

Definition 8.

The generalized WP model (11) is defined as Hyers–Ullam stable within the FFI framework if for every index $\mathbb{I}\in {\mathbb{N}}_i^4$, the positive constants ${\beta }_i>0$ satisfy all subsequent conditions:

$$\begin{array}{cc}\hfill & \left|\mathbb{W}\left(\tau \right)-{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1(\mathbb{W}\left(\tau \right),\tau )d\mathtt{s}-{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1(\mathbb{W}\left(\tau \right),\tau )\right|\le {\beta }_1,\hfill \\ \hfill & \left|\mathbb{S}\left(\tau \right)-{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_2(\mathbb{S}\left(\tau \right),\tau )d\mathtt{s}-{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_2(\mathbb{S}\left(\tau \right),\tau )\right|\le {\beta }_2,\hfill \\ \hfill & \left|\mathbb{I}\left(\tau \right)-{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_3(\mathbb{I}\left(\tau \right),\tau )d\mathtt{s}-{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_3(\mathbb{I}\left(\tau \right),\tau )\right|\le {\beta }_3,\hfill \\ \hfill & \left|\mathbb{T}\left(\tau \right)-{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_4(\mathbb{T}\left(\tau \right),\tau )d\mathtt{s}-{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_4(\mathbb{T}\left(\tau \right),\tau )\right|\le {\beta }_4.\hfill \end{array}$$

(24)

For the generalized model (11), there exists an approximate solution (${\mathbb{W}}^{\ast }\left(\tau \right)$, ${\mathbb{S}}^{\ast }\left(\tau \right)$, ${\mathbb{I}}^{\ast }\left(\tau \right)$, ${\mathbb{T}}^{\ast }\left(\tau \right)$) so that the following equations are satisfied:

$$\begin{array}{cc}\hfill & {\mathbb{W}}^{\ast }\left(\tau \right)={\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1({\mathbb{W}}^{\ast }\left(\tau \right),\tau )d\mathtt{s}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1({\mathbb{W}}^{\ast }\left(\tau \right),\tau ),\hfill \\ \hfill & {\mathbb{S}}^{\ast }\left(\tau \right)={\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1({\mathbb{S}}^{\ast }\left(\tau \right),\tau )d\mathtt{s}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1({\mathbb{S}}^{\ast }\left(\tau \right),\tau ),\hfill \\ \hfill & {\mathbb{I}}^{\ast }\left(\tau \right)={\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1({\mathbb{I}}^{\ast }\left(\tau \right),\tau )d\mathtt{s}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1({\mathbb{I}}^{\ast }\left(\tau \right),\tau ),\hfill \\ \hfill & {\mathbb{T}}^{\ast }\left(\tau \right)={\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1({\mathbb{T}}^{\ast }\left(\tau \right),\tau )d\mathtt{s}+{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1({\mathbb{T}}^{\ast }\left(\tau \right),\tau ).\hfill \end{array}$$

(25)

Thus, Equation (11) is said to be Hyers–Ullam stable if

$$\begin{array}{cc}\hfill & |\mathbb{W}-{\mathbb{W}}^{\ast }|\le {\mathtt{p}}_1{\beta }_1,|\mathbb{S}-{\mathbb{S}}^{\ast }|\le {\mathtt{p}}_2{\beta }_2,\hfill \\ \hfill & |\mathbb{I}-{\mathbb{I}}^{\ast }|\le {\mathtt{p}}_3{\beta }_3,|\mathbb{T}-{\mathbb{T}}^{\ast }|\le {\mathtt{p}}_4{\beta }_4.\hfill \end{array}$$

(26)

Theorem 5.

If System (26) holds, then System (11) is said to be Hyers–Ullam stable.

Proof. 

Consider the first equation of System (26); we have

$$\begin{array}{cc}\hfill |\mathbb{W}-{\mathbb{W}}^{\ast }|& =|{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1(\mathbb{W}\left(\tau \right),\tau )d\mathtt{s}\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1(\mathbb{W}\left(\tau \right),\tau )\hfill \\ \hfill & -[{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}{\mathcal{F}}_1({\mathbb{W}}^{\ast }\left(\tau \right),\tau )d\mathtt{s},\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathcal{F}}_1({\mathbb{W}}^{\ast }\left(\tau \right),\tau )\left]\right|,\hfill \\ \hfill & =|{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1){\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\left[{\mathcal{F}}_1(\mathbb{W}\left(\tau \right),\tau )-{\mathcal{F}}_1({\mathbb{W}}^{\ast }\left(\tau \right),\tau )\right]\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{\tau }{(\tau -\psi )}^{{\mathtt{r}}_1-1}{\psi }^{{\mathtt{r}}_2-1}\hfill \\ & \times \left[{\mathcal{F}}_1(\mathbb{W}\left(\tau \right),\tau )-{\mathcal{F}}_1({\mathbb{W}}^{\ast }\left(\tau \right),\tau )\right]d\psi |,\hfill \\ \hfill & \le {\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1)}{N\left({\mathtt{r}}_1\right)}}{\psi }_1\parallel \mathbb{W}\left(\tau \right)-{\mathbb{W}}^{\ast }\left(\tau \right)\parallel ,\hfill \\ \hfill & \le {\psi }_1\left[{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1)}{N\left({\mathtt{r}}_1\right)}}+{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\right].\hfill \end{array}$$

This gives $|\mathbb{W}-{\mathbb{W}}^{\ast }| \le$ $\parallel \mathbb{W}\left(\tau \right)-{\mathbb{W}}^{\ast }\left(\tau \right)\parallel$. Moreover, suppose that ${\psi }_1={\beta }_1$ and

$$\left[{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1)}{N\left({\mathtt{r}}_1\right)}}+{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}.{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\right]\parallel \mathbb{W}\left(\tau \right)-{\mathbb{W}}^{\ast }\left(\tau \right)\parallel ={\mathtt{p}}_1,$$

then, we obtain $|\mathbb{W}\left(\tau \right)-{\mathbb{W}}^{\ast }\left(\tau \right)|\le {\mathtt{p}}_1{\beta }_1.$ Similarly, for other state variables, the following inequalities hold: $|\mathbb{S}-{\mathbb{S}}^{\ast }|\le {\mathtt{p}}_2{\beta }_2,|\mathbb{I}-{\mathbb{I}}^{\ast }|\le {\mathtt{p}}_3{\beta }_3,|\mathbb{T}-{\mathbb{T}}^{\ast }|\le {\mathtt{p}}_4{\beta }_4$ for ${\psi }_2={\beta }_2$, ${\psi }_3={\beta }_3$, ${\psi }_4={\beta }_4$ and

$${\mathtt{p}}_2={\mathtt{p}}_3={\mathtt{p}}_4=\left[{\displaystyle \frac{{\mathtt{r}}_2(1-{\mathtt{r}}_1)}{N\left({\mathtt{r}}_1\right)}}+{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)}}{\displaystyle \frac{\Gamma \left({\mathtt{r}}_2\right)}{\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\right].$$

Hence, $\mathbb{W}$, $\mathbb{S}$, $\mathbb{I}$, and $\mathbb{T}$ satisfy Equation (26) thus, the generalized model (11) is Hyers–Ullam stable. □

5. Numerical Scheme Based on Proposed Model

The Newton polynomial numerical scheme for the considered FFDEs is ${}_{0,\tau }{}^{FFAB}{\mathtt{r}}_1,{\mathtt{r}}_2\mathtt{z}\left(\tau \right)=\mathcal{F}(\tau ,\mathtt{z})$. Applying FFI on both sides, we obtain

$$\begin{array}{cc}\hfill & \mathtt{z}\left(\tau \right)=\mathtt{z}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2{\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\mathcal{F}(\tau ,\mathtt{z}\left(\tau \right))+{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{{\tau }_{n+1}}{(\tau -\mathtt{s})}^{{\mathtt{r}}_1-1}{\mathtt{s}}^{{\mathtt{r}}_2-1}\mathcal{F}(s,\mathtt{z}\left(s\right))d\mathtt{s},\hfill \end{array}$$

(27)

We replace “$\tau$” with “${\mathtt{t}}_{\mathtt{n}+\mathtt{1}}$” in Equation (27) to obtain

$$\begin{array}{cc}\hfill & \mathtt{z}\left({\mathrm{t}}_{\mathrm{n}+1}\right)=\mathtt{z}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2{\tau }_{n+1}^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\mathcal{F}\left(\tau ,\mathtt{z}\left(\tau \right)\right)+\hfill \\ \hfill & {\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}{\int }_0^{{\mathrm{t}}_{\mathrm{n}+1}}{({\mathrm{t}}_{\mathrm{n}+1}-\mathtt{x})}^{{\mathtt{r}}_1-1}{\mathtt{x}}^{{\mathtt{r}}_2-1}\mathcal{F}(\mathtt{x},\mathtt{z}\left(\mathtt{x}\right))d\mathtt{x}.\hfill \end{array}$$

(28)

We apply a two-step Lagrange polynomial (LP) to obtain

$$\begin{array}{cc}\hfill \mathcal{F}(t,\mathtt{z}(\tau \left)\right)& ={\displaystyle \frac{\mathcal{F}\left({\tau }_{\mathtt{x}},\mathtt{z}\left({\tau }_{\mathtt{x}-1}\right)\right)(\mathtt{x}-{\tau }_{\mathtt{x}-1})}{\mathfrak{h}}}\hfill \\ \hfill & -{\displaystyle \frac{\mathcal{F}\left({\tau }_{\mathtt{x}-1},\mathtt{z}\left({\tau }_{\mathtt{x}-1}\right)\right)(\mathtt{x}-{\tau }_{\mathtt{x}})}{\mathfrak{h}}},\hfill \end{array}$$

where $\mathfrak{h}={\tau }_{\mathtt{x}}-{\tau }_{\mathtt{x}-1}$ denotes the step size (the interval between two consecutive points in a set). Applying LP methods to Equation (28) results in

$$\begin{array}{cc}\hfill & \mathtt{z}\left({\mathrm{t}}_{\mathrm{n}+1}\right)=\mathtt{z}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2{\tau }_{n+1}}{N\left({\mathtt{r}}_1\right)}}\mathcal{F}\left(\tau ,\mathtt{z}\left(\tau \right)\right)+{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2}{N\left({\mathtt{r}}_1\right)\Gamma \left({\mathtt{r}}_1\right)}}\times \hfill \\ \hfill & \sum _{\mathtt{x}=1}^n\left({\displaystyle \frac{\mathcal{F}({\tau }_{\mathtt{x}},\mathtt{z}\left({\tau }_{\mathtt{x}}\right))}{\mathfrak{h}}}{\int }_0^{t_{n+1}}{({\tau }_{n+1}-\mathtt{x})}^{{\mathtt{r}}_1-1}{\mathtt{x}}^{{\mathtt{r}}_2-1}(\mathtt{x}-t_{\mathtt{x}-1})d\mathtt{x}\right)\hfill \\ & -{\displaystyle \frac{\mathcal{F}(t_{\mathtt{x}-1},\mathtt{z}\left(t_{\mathtt{x}-1}\right))}{\mathfrak{h}}}{\int }_0^{{\mathrm{t}}_{\mathrm{n}+1}}{({\mathrm{t}}_{\mathrm{n}+1}-\mathtt{x})}^{{\mathtt{r}}_1-1}{\mathtt{x}}^{{\mathtt{r}}_2-1}(\mathtt{x}-{\tau }_{\mathtt{x}})d\mathtt{x}.\hfill \end{array}$$

Solving the integrals in the aforementioned equation, we obtain

$$\begin{array}{cc}\hfill & \mathtt{z}\left({\mathrm{t}}_{\mathrm{n}+1}\right)=\mathtt{z}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2^2{\tau }_{n+1}{\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}\mathrm{U}\left({\tau }_n,\mathtt{z}\left({\tau }_n\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2^2{\tau }^{{\mathtt{r}}_2-1}\Gamma \left({\mathtt{r}}_2\right)}{\mathfrak{h}N\left({\mathtt{r}}_1\right)\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\times \sum _{\mathtt{x}=1}^n\left(\mathrm{U}({\tau }_{\mathtt{x}},\mathtt{z}\left({\tau }_{\mathtt{x}}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}-1}}{{\tau }_{n+1}}}\right)\right)\hfill \\ & -\mathrm{U}(t_{\mathtt{x}-1},\mathtt{z}\left(t_{\mathtt{x}-1}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}}}{{\tau }_{n+1}}}\right),\hfill \end{array}$$

where ${\Psi }_1$=${\displaystyle \frac{{\left(t_{n+1}\right)}^{{\mathtt{r}}_1+{\mathtt{r}}_2}{\mathtt{r}}_2}{({\mathtt{r}}_1+{\mathtt{r}}_2)}}$. Further, employing the NP scheme in the model in (11) gives

$$\begin{array}{cc}\hfill & \mathbb{W}\left({\mathrm{t}}_{\mathrm{n}+1}\right)=\mathbb{W}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2^2{\tau }_{n+1}{\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathrm{U}}_1\left({\tau }_n,\mathbb{W}\left({\tau }_n\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2^2{\tau }^{{\mathtt{r}}_2-1}\Gamma \left({\mathtt{r}}_2\right)}{\mathfrak{h}N\left({\mathtt{r}}_1\right)\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\times \sum _{\mathtt{x}=1}^n\left({\mathrm{U}}_1({\tau }_{\mathtt{x}},\mathbb{W}\left({\tau }_{\mathtt{x}}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}-1}}{{\tau }_{n+1}}}\right)\right)\hfill \\ & -{\mathrm{U}}_1(t_{\mathtt{x}-1},\mathbb{W}\left(t_{\mathtt{x}-1}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}}}{{\tau }_{n+1}}}\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \mathbb{S}\left({\mathrm{t}}_{\mathrm{n}+1}\right)=\mathbb{S}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2^2{\tau }_{n+1}{\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathrm{U}}_2\left({\tau }_n,\mathbb{S}\left({\tau }_n\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2^2{\tau }^{{\mathtt{r}}_2-1}\Gamma \left({\mathtt{r}}_2\right)}{\mathfrak{h}N\left({\mathtt{r}}_1\right)\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\times \sum _{\mathtt{x}=1}^n\left({\mathrm{U}}_2({\tau }_{\mathtt{x}},\mathbb{S}\left({\tau }_{\mathtt{x}}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}-1}}{{\tau }_{n+1}}}\right)\right)\hfill \\ & -{\mathrm{U}}_2(t_{\mathtt{x}-1},\mathbb{S}\left(t_{\mathtt{x}-1}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}}}{{\tau }_{n+1}}}\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \mathbb{I}\left({\mathrm{t}}_{\mathrm{n}+1}\right)=\mathbb{I}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2^2{\tau }_{n+1}{\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathrm{U}}_3\left({\tau }_n,\mathbb{I}\left({\tau }_n\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2^2{\tau }^{{\mathtt{r}}_2-1}\Gamma \left({\mathtt{r}}_2\right)}{\mathfrak{h}N\left({\mathtt{r}}_1\right)\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\times \sum _{\mathtt{x}=1}^n\left({\mathrm{U}}_3({\tau }_{\mathtt{x}},\mathbb{I}\left({\tau }_{\mathtt{x}}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}-1}}{{\tau }_{n+1}}}\right)\right)\hfill \\ & -{\mathrm{U}}_3(t_{\mathtt{x}-1},\mathbb{I}\left(t_{\mathtt{x}-1}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}}}{{\tau }_{n+1}}}\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \mathbb{T}\left({\mathrm{t}}_{\mathrm{n}+1}\right)=\mathbb{T}\left(0\right)+{\displaystyle \frac{(1-{\mathtt{r}}_1){\mathtt{r}}_2^2{\tau }_{n+1}{\tau }^{{\mathtt{r}}_2-1}}{N\left({\mathtt{r}}_1\right)}}{\mathrm{U}}_4\left({\tau }_n,\mathbb{T}\left({\tau }_n\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{{\mathtt{r}}_1{\mathtt{r}}_2^2{\tau }^{{\mathtt{r}}_2-1}\Gamma \left({\mathtt{r}}_2\right)}{\mathfrak{h}N\left({\mathtt{r}}_1\right)\Gamma ({\mathtt{r}}_1+{\mathtt{r}}_2)}}\times \sum _{\mathtt{x}=1}^n\left({\mathrm{U}}_4({\tau }_{\mathtt{x}},\mathbb{T}\left({\tau }_{\mathtt{x}}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}-1}}{{\tau }_{n+1}}}\right)\right)\hfill \\ & -{\mathrm{U}}_4(t_{\mathtt{x}-1},\mathbb{T}\left(t_{\mathtt{x}-1}\right))\left({\Psi }_1-{\displaystyle \frac{{\tau }_{\mathtt{x}}}{{\tau }_{n+1}}}\right).\hfill \end{array}$$

(32)

In all above cases, ${\Psi }_1=$${\displaystyle \frac{{\left(t_{n+1}\right)}^{{\mathtt{r}}_1+{\mathtt{r}}_2}{\mathtt{r}}_2}{({\mathtt{r}}_1+{\mathtt{r}}_2)}}$, with ${\mathrm{U}}_1,{\mathrm{U}}_2,{\mathrm{U}}_3$, and ${\mathrm{U}}_4$ representing the corresponding kernels for each compartment.

6. Graphical Representations and Discussion

This section of this research work focuses on the graphical representation of the proposed model (11), using the assumed values of the compartment and parameters with the step size $h=0.5$. MATLAB 2019b, a mathematical software tool, was employed to generate and analyze these graphs in detail. The following figures illustrate the analysis of different compartments for various values of fractal–fractional orders, i.e., $r_1$ and $r_2$.

From the graphical analysis of the combined Figure 1a, it is observed that the number of polluted water sources decreased for different values of $r_1$ and $r_2$. As a result, we can come to the conclusion that the fractional order $r_1$ and the fractal dimension $r_2$ had a significant impact on reducing the number of polluted water sources, $\mathbb{W}\left(\tau \right)$. In Figure 1b, we observe that the number of susceptible water sources decreased for different values of $r_1$ and $r_2$. Consequently, the fractional order $r_1$ and the fractal dimension $r_2$ had a significant impact on reducing the number of susceptible water sources, $\mathbb{S}\left(\tau \right)$. In Figure 1c, it is observed that the number of infected water sources decreased for various values of $r_1$ and $r_2$, which concludes that the fractional order $r_1$ and the fractal dimension $r_2$ reduced the number of infected water sources, $\mathbb{I}\left(\tau \right)$. Similar was the case shown in Figure 1d, where the number of recovered water sources increased with the increase in $r_1$ and $r_2$, and the fractional order $r_1$ along with the fractal dimension $r_2$ reduced the number of polluted water sources, $\mathbb{I}\left(\tau \right)$.

In Figure 2, we observe that the polluted, susceptible, and infected water sources decreased, while recovered water sources increased for various values of $r_1$ and $r_2$, based on the inspection of graphical analysis; as the fractional order $r_1$ and the fractal dimension $r_2$ increased, the polluted, susceptible, and infected water sources decreased and recovered water increased. Thus, the fractal–fractional operator had a vital influence on the proposed model.

Comparative Study

To check the effectiveness of the proposed numerical scheme with ordered classical numerical methods, such as the Runge–Kutta fourth-order (RK-4) method and modified Euler method (EM), by converting the considered model to a conventional integral-order model, e.g., $r_1=r_2=1$. We performed the numerical simulations for the compartment $\mathbb{W}\left(\tau \right)$ of the proposed model with the step size $h=0.5$ using MATLAB-15 with Central Processing Unit (CPU) time comparison.

Thus, we compared the aforementioned numerical schemes for CPU time to see which method is cheaper regarding the consumption of time presented in

Table 1. CPU time comparison for extended simulation intervals using NP, RK-4, and EM.

$\mathit{N}=\frac{\mathit{t}}{\mathit{h}}$	Time Range t	RK-4 CPU Time (s)	EM CPU Time (s)	NP (AB) CPU Time (s)
100	50	0.129	0.114	0.092
200	100	0.251	0.228	0.188
3000	150	0.372	0.340	0.284
400	200	0.503	0.451	0.369
500	250	0.634	0.567	0.457

. From Table 1, it is clear that our proposed method is more reliable than both the RK-4 method and Euler's methods. Furthermore, the NP scheme remains applicable and advantageous for fractional-order settings $(r_1,r_2<1)$, where classical methods like RK-4 are not valid. This flexibility makes the NP approach particularly suited for modeling real-world processes with memory and non-locality, such as water pollution dynamics.

7. Conclusions

In this current study, a novel mathematical model for water pollution was examined using FFDs in the ABC sense, which effectively captured the memory and non-local dynamics of pollutant behavior. The inclusion of the parameter $\gamma$ emphasized the critical role of improving efforts in relieving pollution. Moreover, the existence, uniqueness, and stability of solutions using fixed-point theory were mathematically analyzed. An NP numerical scheme was proposed to solve the system efficiently, by providing accurate simulations of pollution dynamics under changing conditions.

This research contributes to the field of environmental modeling by integrating advanced mathematical tools with practical applications, providing a framework for understanding and addressing WP. The analysis highlighted the importance of fractional-order modeling in capturing complex environmental processes and the results offer a platform for upcoming studies. Furthermore, the model serve as a valuable tool for policymakers and stakeholders to evaluate and design effective strategies for sustainable water resource management.