An Optimal Control Theory-Based Study for Fractional Smoking Model Using Bernoulli Wavelet Method

Abstract

This paper introduces a dynamic model that explores smoking and optimal control strategies. It shows how fractional-order (FO) analysis has uncovered hidden parts of complex systems and provides information about previously ignored elements. This paper uses the Bernoulli wavelet operational matrix method and the Adam–Bashforth–Moulton (ABM) method to analyse this model numerically. The mathematical model is segmented into five sub-classes: susceptible smokers, ingestion class, unusual smokers, regular smokers, and ex-smokers. It considers four optimal control measures: an anti-smoking education campaign, distribution of anti-smoking gum, administration of anti-nicotine drugs, and governmental restrictions on smoking in public areas. We show in this model how to control smoking in society strategically.

1. Introduction

Fractional calculus (FC) is a branch of mathematical analysis that studies several possibilities for defining the differentiation and integration operator’s real or complex number powers. Fractional derivatives are useful for understanding the memory and characteristics of different processes and materials. Models based on classical integer order often overlook or disregard the significant impacts of these effects [1]. FO derivatives and integrals possess non-local characteristics. In the context of these properties, the future state of a model is influenced not only by its current state but also by all of its preceding states [2]. Multiple definitions of fractional derivatives exist. Various researchers have proposed different approaches to define fractional derivatives. Riemann and Liouville introduced a power-law-based concept, while Caputo and Fabrizio introduced a derivative with FO based on the exponential law. Additionally, Atangana and Baleanu suggested an alternative version of the FO derivative utilizing the generalized Mittag–Leffler function, which incorporates a non-local and non-singular kernel with strong memory properties. FO derivatives provide a novel approach for modeling the dynamics of complex phenomena. FC’s many applications in engineering and the mathematical modeling of physical systems have attracted much attention recently. FC has applications in mathematics, engineering, physics, bio-engineering, and economics. Several systems have been described more precisely and smoothly by fractional differential equations (FDEs).

A wavelet originates from a zero point, undergoes alternating periods of rising and falling, and eventually returns to zero, repeating this pattern one or more times [3]. The term “omelette”, the French equivalent for wavelet that translates to “small wave”, was initially introduced by Haar in 1909 and adopted by Morlet and Grossmann in 1984 [4]. Wavelet theory has real-world applications in many exciting fields of science and technology. The following are several domains where it finds use: music, optics, signal and image processing, radar technology, nuclear engineering, earthquake prediction, physics, geology, astronomy, and more [5,6,7,8,9,10,11,12]. Numerical analysis is a branch of mathematics used to investigate and estimate answers to problems that have not been solved analytically [13]. There are two classifications of wavelets: continuous and discrete. A wavelet transform represents a function that uses the wavelet technique. It is primarily characterized by a function called the “mother wavelet”. This is scaled and shifted to generate the various wavelets. Wavelet analysis allows wavelets to be applied in various areas or domains using a collection of orthonormal and comparable basis functions defined in time and space. Consequently, the concept of wavelets was introduced, employing basis functions localized within finite domains. When compared to other numerical methods, wavelets provide a better approximation due to their ability to identify both sharp irregularities and smooth perturbations, due to their localization. Consequently, wavelet analysis offers a more precise description of function properties compared to Fourier analysis [14] and can accurately represent various operators and functions, as well as being compatible with quick numerical techniques [15]. In this model, we use the Bernoulli wavelet method (BWM). The Bernoulli wavelet is preferred over other wavelets, such as Haar, Legendre, Bernstein, Hermite, etc., for several reasons:

1.

It is efficient for representing piecewise smooth functions due to its basis on Bernoulli polynomials.

2.

It has a more straightforward implementation with lower computational complexity.

3.

It handles boundary conditions well due to its compact support.

4.

It provides smooth approximations compared to Haar’s piecewise constant approximations.

5.

It is accurate for solving numerical problems and significant differential equations.

6.

It offers practical multi-resolution analysis for analyzing signals at various scales.

Owing to these properties, Bernoulli wavelets are particularly well-suited for engineering and computational tasks that demand smooth, efficient, and accurate function representations.

Mathematical modeling involves examining a specific aspect of a real-life problem using mathematical language and concepts [16]. Understanding how the world and its mechanisms function requires modeling. It involves representing the world in simplified models and forms, collaborating with engineers and scientists to address real-world problems effectively. Furthermore, it has helped reveal various new aspects of issues. The observer’s viewpoint is critical in the modeling process. It is important to be able to visualize models mentally. Engineers and scientists employ such techniques to model and design upcoming technologies. In conjunction with this procedure, prototypes are frequently employed. A prototype serves as a scaled-down representation of a functional model. Prototypes are utilized in various situations where there is a requirement to test or analyze a model without causing any impact or harm to the actual one. Additionally, using models, one may understand how atoms and particles behave, imagine how our environment will change, and create a vast range of products, from toy vehicles to actual cars. Within the realm of nonlinear dynamics and applied sciences, the exploration of mathematical modeling for infectious diseases in ecology and biology is a captivating field of study. Within this discipline, there exist abundant opportunities to acquire the skills necessary for characterizing the behavior of infectious disease models and assessing their dynamic properties [17,18]. Mathematical modeling has widespread applications in applied sciences, particularly FC.

Smoking stands as one of the most significant global health concerns in contemporary times. A considerable number of individuals lose their lives due to smoking during the years of most excellent output, as per a report by the World Health Organization (WHO) on the global smoking epidemic. The adverse effects of this on different physiological processes contribute to over 5 million deaths annually across the globe, with projections indicating a potential increase to around 8 million by the year 2030. Smokers face a 70% elevated likelihood of experiencing a heart attack in comparison to individuals who do not smoke. The detrimental health impacts of smoking go beyond the individual who smokes, affecting others as well. Secondhand smoke comprises harmful substances from both the smoke exhaled and the immediate smoke emitted by burning tobacco, in addition to the smoke that the smoker directly inhales. Individuals who do not smoke but are regularly exposed to secondhand smoke have a higher likelihood of being susceptible to various diseases, such as lung cancer and cardiovascular conditions, similar to smokers [19]. Numerous scientific studies indicate that smoking further heightens the risk of acquiring ailments such as cancer, cardiovascular disorders, stroke, respiratory conditions, diabetes, and chronic obstructive pulmonary disease. In addition, smoking increases the risk of tuberculosis, various ocular conditions, and immune system disorders like rheumatoid arthritis. There are still social and economic repercussions from smoking, even in nations with low death rates where its incidence has peaked. It also includes elevated levels of suffering, illness, and death, the subsequent decline in productivity, and the associated healthcare expenses [20].

A limited amount of research has been conducted on FOCPs [21,22]. As the need for practical, precise, and highly accurate systems increases, so does the requirement for optimal control theories and the corresponding analytical and numerical methods to solve the associated equations. This study uses the CF and the benefits of FC to provide a valuable examination of the dynamical behavior of a mathematical model for smoking [23]. Through numerical exploration using the Homotopy Perturbation Method (HPM) and the Laplace transform combined with the Adomian Decomposition Method (LADM), the study yields consistent and robust numerical results, providing strong validation for the model with arbitrary-order derivatives. The findings highlight the significant influence of various parameters within the model, concluding that FO systems exhibit more complex dynamics than those with integer-order derivatives. The research supports a system of smoking, indicating that different psychological and physiological processes are involved in initiating smoking compared to developing a regular smoking habit [24]. They employed the constant proportional Caputo–Fabrizio (CPCF) operator to construct a FO system that captures the harmful effects of smoking on society. Additionally, they conducted qualitative and quantitative evaluations of the suggested methodology and thoroughly examined the CPCF operator. Also, they applied the iterative Laplace transform method to develop a numerical simulation for a specific set of FDEs.

As per the findings of this study, the application of the BWM to the smoking model has yet to be explored. Therefore, our objective is to observe the model’s behavior when this method is employed. This research uses the Bernoulli wavelet’s operational matrix approach to compute the nonlinear FO smoking model. A comparative analysis is conducted between the outcomes obtained through the fractional ABM numerical method and those derived from various scenarios. This study fills this gap using the optimal control technique and offering a numerical approach. The ABM method offers significant benefits in fractional systems due to its higher accuracy, enhanced stability, and adaptability. It balances computational efficiency and precision by using a correction step to refine predictions, making it ideal for fractional dynamics with memory effects. Despite these strengths, the ABM method has more complex implementation requirements, relies on precise initial values, and may be less effective for highly stiff systems. Then, we use optimal control theory in this model. This study’s other achievement is the introduction of four controls designed to effectively decrease the number of individuals who smoke and simultaneously boost the number of people who successfully quit smoking for good. To the author’s knowledge, thinking methods have yet to be utilized to solve the proposed model thus far. This paper’s remaining sections are organized as follows: A thorough description of derivatives with non-integer order is given in Section 2. We analyze the fractional smoking system in Section 3. The fundamental idea of the Bernoulli wavelet’s operational matrix (BWOM) is summed up in Section 4. We apply BWM and ABM to the smoking system in Section 5 to obtain an approximate solution. The formulation of optimal controls and their solution are covered in Section 6. Section 7 displays the simulation results and the following discussion. The concluding section summarizes the research findings and concludes.

2. Preliminaries

This section explores the fundamental ideas and symbols associated with FC. $\mathbb{N}$ represents the natural number set.

Definition 1

([25]). The description of the Riemann–Liouville arbitrary integral of order $\sigma >0$ for a function $S_s$ is as follows:

$$J_0^{\sigma }{S}_s\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}{\int }_0^t{\left(t-\xi \right)}^{\sigma -1}{S}_s\left(\xi \right)d\xi ,t>0,$$

(1)

where the symbol $\Gamma \left(.\right)$ denotes the Gamma function.

Definition 2

([25,26]). Consider the range where $0<m-1<\sigma <m,$ with m belonging to the set of natural numbers. The Caputo derivative of order σ for a function $S_s$ is defined as follows:

$$\begin{array}{cc}\hfill {}^{C}D_0^{\sigma }{S}_s\left(t\right)& =J_0^{m-\sigma }\left({\displaystyle \frac{d^m{S}_s}{d{t}^m}}\right)\left(t\right),\hfill \\ \hfill {}^{C}D_0^{\sigma }{S}_s\left(t\right)& ={\displaystyle \frac{1}{\Gamma \left(m-\sigma \right)}}{\int }_0^t{\left(t-\xi \right)}^{m-\sigma -1}{\displaystyle \frac{d^m{S}_s}{d{\xi }^m}}\left(\xi \right)d\xi .\hfill \end{array}$$

(2)

These are the essential characteristics we have:

$${}^{C}D_0^{\sigma }\left(J_0^{\sigma }{S}_s\right)\left(t\right)=S_s\left(t\right),$$

and

$$J_0^{\sigma }\left({}^{C}D_0^{\sigma }{S}_s\right)\left(t\right)=S_s\left(t\right)-\sum _{k=0}^{m-1}{\displaystyle \frac{d^k{S}_s}{d{t}^k}}\left(0^{+}\right){\displaystyle \frac{t^k}{k!}}.$$

3. FO Smoking Model

The choice of a fractional derivative is essential when studying memory-dependent dynamics, such as addiction processes, relapse mechanisms, and long-term behavioural changes in smoking. Both the Caputo and Riemann–Liouville (RL) definitions are mathematically sound, but for several important reasons, the Caputo derivative is more appropriate for simulating real-world behavioural systems:

1.

The RL derivative of a constant is not zero. This leads to fractional initial conditions, which do not translate well to real-world measurements. However, the Caputo derivative vanishes for constants, and its initial conditions are standard, making it easier to match with actual data.

2.

The Riemann–Liouville (RL) approach calculates the average of past cravings first, then looks at how this average changes. This has sometimes given too much weight to old cravings in unrealistic ways. In contrast, the Caputo method looks at how cravings change moment-to-moment first, then considers how these changes build up over time. This makes more sense for addiction because people do not respond to their average past craving—they react to sudden changes, like a bad withdrawal day or stressful event, triggering stronger urges.

3.

RL derivatives have produced unrealistic singularities at $t=0$. For finite initial conditions, Caputo guarantees smooth solutions. Because Caputo has well-posed initial conditions, the statistical fitting of fractional models to clinical data is more stable.

For these reasons, the manuscript employs the Caputo framework to capture memory-dependent smoking dynamics accurately [24].

$$\begin{array}{cc}\hfill {}_0{}^{C}D_t^{\sigma }{S}_s\left(t\right)& =e_1-e_2{S}_s\left(t\right)I_c\left(t\right)+e_3{R}_s\left(t\right)-e_4{S}_s\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }{I}_c\left(t\right)& =e_2{S}_s\left(t\right)I_c\left(t\right)-e_5{I}_c\left(t\right)U_s\left(t\right)-(e_6+e_4)I_c\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }{U}_s\left(t\right)& =e_5{I}_c\left(t\right)U_s\left(t\right)-(e_7+e_8+e_4)U_s\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }{R}_s\left(t\right)& =e_7{U}_s\left(t\right)-(e_9+e_4+e_3)R_s\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }{E}_s\left(t\right)& =e_9{R}_s\left(t\right)-e_4{E}_s\left(t\right).\hfill \end{array}$$

(3)

The initial conditions are given as [23],

$$\begin{array}{c}\hfill S_s\left(0\right)={\zeta }_1,I_c\left(0\right)={\zeta }_2,U_s\left(0\right)={\zeta }_3,R_s\left(0\right)={\zeta }_4,E_s\left(0\right)={\zeta }_5,\end{array}$$

(4)

where

$$\begin{array}{ccc}\hfill {\zeta }_1& =& 68,{\zeta }_2=40,{\zeta }_3=30,{\zeta }_4=20,{\zeta }_5=15.\hfill \end{array}$$

(5)

Here, $0<\sigma \le 1,{}^{C}D_0^{\sigma }$ Caputo derivative $\sigma .$ The different type of smokers are susceptible smokers $S_s$, ingestion class $I_c$, unusual smokers $U_s$, regular smokers $R_s$, and ex-smokers $E_s$.

$e_1$ is the enlistment rate, $e_2$ is the rate at which $S_s$ transitions to $I_c$, $e_5$ is the rate at which $I_c$ transitions to $U_s$, $e_7$ is the rate at which $U_s$ transitions to $R_s$, $e_9$ is the migration rate, $e_4$ is the natural fatality rate, and $e_3$ is the recovery rate, while $e_6$ and $e_8$ represent the fatality rates due to snuffing and smoking, respectively.

4. Characteristics and Function Approximation of the Bernoulli Wavelet

Wavelets are a category of functions generated by consistently altering both the translation, denoted as ‘b’, and the dilation, represented by ‘a’, of a single function $\Xi \left(t\right),$ which is referred to as the mother wavelet. When both the translation and dilation parameters are in constant flux, the resulting wavelet family is as follows [27]:

$$\begin{array}{c}\hfill {\Xi }_{a,b}\left(t\right)=\mid a{\mid }^{-1/2}\Xi \left({\displaystyle \frac{t-b}{a}}\right),a,b\in \mathbb{R},a\ne 0.\end{array}$$

(6)

If we limit the values of the parameters a and b to discrete values, we get $a=a_0^{-k},b=n{b}_0{a}_0^{-k},a_0>1,b_0>0,\mathrm{where}$ n and k are positive integers. The discrete wavelet family that we have is as follows:

$${\Xi }_{k,n}\left(t\right)=\mid a{\mid }^{k/2}\Xi \left(a_0^{k}t-n{b}_0\right).$$

The wavelet basis in the $L^2{\left(\mathbb{R}\right)}^{\prime }s$ space is constructed using ${\Xi }_{k,n}\left(t\right)$. Particularly, when $a_0=2$ and $b_0=1$, ${\Xi }_{k,n}\left(t\right)$ constitute an orthonormal basis.

Bernoulli wavelets, denoted as ${\Xi }_{n,m}\left(t\right)=\Xi \left(k,n,m,t\right)$, possess four parameters: $n=0,1,2,3,\dots ,2^{k-1}-1,$ where k is any positive integer. These wavelets are associated with Bernoulli polynomials of degree m, and t represents the normalized time degree. Their definition pertains to the semi-interval $[0,1)$.

$${\Xi }_{n,m}\left(t\right)=\left\{\begin{array}{cc}{2}^{{\displaystyle \frac{k-1}{2}}}{\tilde{\beta }}_m\left(2^{k-1}t-n\right),\hfill & {\displaystyle \frac{n}{2^{k-1}}}\le t<{\displaystyle \frac{n+1}{2^{k-1}}},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(7)

Here,

$${\tilde{\beta }}_m\left(t\right)=\left\{\begin{array}{cc}1,\hfill & m=0,\hfill \\ {\displaystyle \frac{1}{\sqrt{\frac{{\left(-1\right)}^{m-1}{\left(m!\right)}^2}{\left(2m\right)!}}{\beta }_{2m}}}{\beta }_m\left(t\right),\hfill & m>0,\hfill \end{array}\right.$$

(8)

and $m=0,1,2,\dots ,M-1,$ where ${\beta }_m\left(t\right)$ denotes the Bernoulli polynomials with order $m,$ which are described on interval $\left[0,1\right]$ as [28]

$$\begin{array}{ccc}\hfill {\beta }_m\left(t\right)& =& \sum _{i=0}^m\left({}^{\mathrm{m}}\mathbf{C}_{\mathrm{i}}\right){\beta }_{m-i}{t}^i.\hfill \end{array}$$

(9)

In addition, we obtain polynomials of these particular types, where ${\beta }_i={\beta }_i\left(0\right),i=0,1,2,\dots ,m$ and these ${\beta }_i$ values correspond to Bernoulli numbers.

$${\beta }_0\left(t\right)=1,{\beta }_1\left(t\right)=t-{\displaystyle \frac{1}{2}},{\beta }_2\left(t\right)=t^2-t+{\displaystyle \frac{1}{6}},{\beta }_3\left(t\right)=t^3-{\displaystyle \frac{3}{2}}{t}^2+{\displaystyle \frac{1}{2}}t.$$

A function f defined within the interval $[0,1)$ can be expressed using Bernoulli wavelets in the following manner:

$$f\left(t\right)=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{c}_{n,m}{\Xi }_{n,m}\left(t\right).$$

(10)

If the series in Equation (10) is not finite and is truncated, it has been expressed in the following alternative form:

$$\begin{array}{c}\hfill f\left(t\right)\simeq f_{\widehat{m}}\left(t\right)=\sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^{M-1}{c}_{n,m}{\Xi }_{n,m}\left(t\right)=C^T\Xi \left(t\right),\end{array}$$

(11)

where C and ${\rm Y}\left(t\right)$ are $\widehat{m}\times 1\left(\widehat{m}=2^{k-1}\times M\right)$ column vectors and T represents transposition.

Now,

$$\begin{array}{cc}\hfill C=& {\left[c_{0,0},c_{0,1},c_{0,2},\dots ,c_{0,M-1},\dots ,c_{2^{k-1}-1,0},c_{2^{k-1}-1,1},\dots ,c_{2^{k-1}-1,M-1}\right]}^T,\hfill \\ \hfill =& {\left[c_1,c_2,c_3,\dots ,c_{\widehat{m}}\right]}^T,\hfill \end{array}$$

(12)

and

$$\begin{array}{cc}\hfill \Xi \left(t\right)& ={\left[{\Xi }_{0,0}\left(t\right),{\Xi }_{0,1}\left(t\right),\dots ,{\Xi }_{0,M-1}\left(t\right),\dots ,{\Xi }_{2^{k-1}-1,0}\left(t\right),{\Xi }_{2^{k-1}-1,1}\left(t\right),\dots ,{\Xi }_{2^{k-1}-1,M-1}\left(t\right)\right]}^T,\hfill \\ \hfill & ={\left[{\Xi }_1\left(t\right),{\Xi }_2\left(t\right),\dots ,{\Xi }_{\widehat{m}}\left(t\right)\right]}^T.\hfill \end{array}$$

(13)

In this section, we present the Bernoulli Wavelet matrix denoted as ${\varphi }_{\widehat{m}\times \widehat{m}}$, which is defined as ${\varphi }_{\widehat{m}\times \widehat{m}}=[\Xi \left({\displaystyle \frac{2i-1}{2\widehat{m}}}\right)],i=1,2,3,\dots ,2^{k-1}M.$

$${\varphi }_{12\ast 12}=\left[\begin{array}{cccccccccccc}2& 2& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 2& 2& 2& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 2& 2& 2& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 2& 2& 2\\ -2.3094& 0& 2.3094& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -2.3094& 0& 2.3094& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -2.3094& 0& 2.3094& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -2.3094& 0& 2.3094\\ 0.7454& -2.2361& 0.7454& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0.7454& -2.2361& 0.7454& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0.7454& -2.2361& 0.7454& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0.7454& -2.2361& 0.7454\end{array}\right].$$

We have now obtained the Bernoulli wavelet matrix ${\varphi }_{\widehat{m}\times \widehat{m}}$ for a specific set of collocation points given by ${\displaystyle \frac{2i-1}{2\widehat{m}}},$ where $k=3,M=3.$

4.1. Utilizing the Block-Pulse Function, We Can Construct BWOM

In this section, we present the operational matrix for both non-integer and integer orders of the Bernoulli wavelet, which holds significant importance in our proposed system for addressing the nonlinear, non-integer-order smoking dynamic model.

With the assistance of the block-pulse function

Block-pulse functions have been outlined within the given time frame $[0,t_l),$ [25]

$$b_j\left(t\right)=\left\{\begin{array}{cc}1,\hfill & {\displaystyle \frac{j{t}_l}{\widehat{m}}}\le t<{\displaystyle \frac{\left(j+1\right)t_l}{\widehat{m}}},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where $j=0,1,\dots ,m$ and $B_{\widehat{m}}=[b_1,b_2,..,b_m]$. In this study, the block-pulse function demonstrates beneficial properties. We shall use its characteristics to determine the Bernoulli wavelet’s operational matrix.

$$\left(J_t^{\sigma }{B}_{\widehat{m}}\right)\left(t\right)\cong F^{\sigma }{B}_{\widehat{m}}\left(t\right),$$

$$F^{\sigma }={\displaystyle \frac{t_l^{\sigma }}{m^{\sigma }\left(\Gamma \left(\sigma +2\right)\right)}}\left[\begin{array}{ccccc}1& {\zeta }_1& {\zeta }_2& \dots & {\zeta }_{\widehat{m}-1}\\ 0& 1& {\zeta }_1& \dots & {\zeta }_{\widehat{m}-2}\\ 0& 0& 1& {\zeta }_1& ⋮\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& 1\end{array}\right],$$

where,

$${\zeta }_l={\left(l+1\right)}^{\sigma +1}-2l^{\sigma +1}+{\left(l-1\right)}^{\sigma +1},l=1,2,3,\dots ,\widehat{m}-1.$$

We are currently in the process of constructing BWOM to address fractional-order integration denoted as $P^{\sigma }$,

$$\left(J_t^{\sigma }\phi \right)\left(t\right)\cong P^{\sigma }\phi \left(t\right),$$

then, we get

$$\left(J_t^{\sigma }\phi \right)\left(t\right)\cong \left(J_t^{\sigma }\phi B_{\widehat{m}}\right)\left(t\right)=\phi \left(J_t^{\sigma }{B}_{\widehat{m}}\right)\left(t\right)\approx \phi F^{\sigma }{B}_{\widehat{m}}.$$

Therefore,

$$P^{\sigma }\phi \left(t\right)\cong \phi \left(t\right)F^{\sigma }{B}_{\widehat{m}},$$

$$P^{\sigma }={\varphi }_{\widehat{m}\ast \widehat{m}}{F}^{\sigma }{\varphi }_{\widehat{m}\ast \widehat{m}}^{-1}.$$

Using the previous equations, we have derived the operational matrix $P^{\sigma }$. Finally, considering specific values $k=2,M=3,\sigma =0.65$ and utilizing collocation points ${\displaystyle \frac{2i-1}{2m}}$, we acquired the operational matrix that is shown below:

$$P^{0.65}=\left[\begin{array}{cccccc}0.6694& 0.7697& 0.2839& -0.1021& -0.0318& 0.0263\\ 0& 0.6694& 0& 0.2839& 0& -0.0318\\ -0.2297& 0.0931& 0.1895& -0.0467& 0.1821& 0.0191\\ 0& -0.2297& 0& 0.1895& 0& 0.1821\\ -0.0832& -0.0819& -0.1364& 0.0021& 0.1302& -0.0023\\ 0& -0.0832& 0& -0.1364& 0& 0.1302\end{array}\right].$$

The BWOM has been obtained for any arbitrary order in the $0<\sigma \le 1$ range.

4.2. Convergence Analysis

This section is dedicated to providing detailed information on analyzing the convergence of the Bernoulli wavelet function approximation [29].

Theorem 1.

As per (11), it is clear that any function $Q\left(t\right)\in L^2[0,1]$ has been approximated using Bernoulli wavelets in the following manner.

$$Q\left(t\right)\approx \sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^{M-1}{c}_{n,m}{\Xi }_{n,m}\left(t\right).$$

(14)

Now, we take M=3 here,

$$Q\left(t\right)\approx \sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right).$$

(15)

So,

$$\begin{array}{cc}\hfill \left|\left|\epsilon \right|\right|=& \left|\left|error\left(Q\right(t\left)\right)\right|\right|\hfill \\ \hfill =& \left|\left|\sum _{n=0}^{\infty }\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)-\sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)\right|\right|\to 0,(k\to \infty ).\hfill \end{array}$$

(16)

Proof. 

The orthogonal properties of Bernoulli wavelets, as demonstrated in [30], lead us to the following conclusion

$${\left|\left|\epsilon \right|\right|}^2={\left|\left|\sum _{n=0}^{\infty }\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)-\sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)\right|\right|}^2.$$

(17)

By utilizing the norm property of the polynomial $Q\left(t\right)\in L^2[0,1]$, we have established the following

$$\begin{array}{cc}\hfill {\left|\left|\epsilon \right|\right|}^2=& 〈\sum _{n=0}^{\infty }\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)-\sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right),\sum _{n=0}^{\infty }\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)\hfill \\ \hfill & -\sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)〉\hfill \\ \hfill =& {\int }_0^1{\left(\sum _{n=0}^{\infty }\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)-\sum _{n=0}^{2^{k-1}-1}\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)\right)}^{2}dt\hfill \\ \hfill =& {\int }_0^1{\left(\sum _{n=2^{k-1}}^{\infty }\sum _{m=0}^2{c}_{n,m}{\Xi }_{n,m}\left(t\right)\right)}^{2}dt\hfill \\ \hfill =& {\int }_0^1\sum _{n=2^{k-1}}^{\infty }\sum _{m=0}^2{c}_{n,m}^2{\Xi }_{n,m}^2\left(t\right)\hfill \\ \hfill =& \sum _{n=2^{k-1}}^{\infty }\sum _{m=0}^2{c}_{n,m}^2,c_{n,m}=〈Q\left(t\right),{\Xi }_{n,m}\left(t\right)〉.\hfill \end{array}$$

(18)

Given that $Q\left(t\right)$ is continuous on $[0,1]$, we can find a value $\gimel >0$ such that the following condition holds,

$$\left|Q\left(t\right)\right|<\gimel ,\forall t\in [0,1].$$

When $m=0$, we have derived the following from the definition of Bernoulli wavelets

$$\begin{array}{cc}\hfill c_{n,m}=& 〈Q\left(t\right),{\Xi }_{n,m}\left(t\right)〉\hfill \\ \hfill =& {\int }_0^{1}Q\left(t\right){\Xi }_{n,m}\left(t\right)dt\hfill \\ \hfill =& {\int }_{{\displaystyle \frac{n}{2^{k-1}}}}^{{\displaystyle \frac{n+1}{2^{k-1}}}}Q\left(t\right)2^{{\displaystyle \frac{k-1}{2}}}dt\hfill \\ \hfill <& 2^{{\displaystyle \frac{k-1}{2}}}{\int }_{{\displaystyle \frac{n}{2^{k-1}}}}^{{\displaystyle \frac{n+1}{2^{k-1}}}}\gimel dt\hfill \\ \hfill =& {\displaystyle \frac{\gimel }{2^{\frac{k-1}{2}}}}.\hfill \end{array}$$

(19)

In the given examples, this condition is satisfied only when $m=0$. However, when $m\ne 0$, the following cases apply

$$\begin{array}{cc}\hfill c_{n,m}=& 〈Q\left(t\right),{\Xi }_{n,m}\left(t\right)〉\hfill \\ \hfill =& {\int }_0^{1}Q\left(t\right){\Xi }_{n,m}\left(t\right)dt\hfill \\ \hfill =& {\int }_{{\displaystyle \frac{n}{2^{k-1}}}}^{{\displaystyle \frac{n+1}{2^{k-1}}}}Q\left(t\right)2^{{\displaystyle \frac{k-1}{2}}}{\displaystyle \frac{1}{\sqrt{\frac{{\left(-1\right)}^{m-1}{\left(m!\right)}^2}{\left(2m\right)!}}{\beta }_{2m}}}{\beta }_m\left(2^{k-1}t-n\right)dt.\hfill \end{array}$$

(20)

First, we select the value $2^{k-1}$ so that $2^{k-1}t-n=u$, which gives us $t={\displaystyle \frac{u+n}{2^{k-1}}}$. Therefore, $dt={\displaystyle \frac{1}{2^{k-1}}}du$. By following a similar approach, we have derived the following.

$$\begin{array}{cc}\hfill c_{n,m}=& {\int }_0^{1}Q\left({\displaystyle \frac{u+n}{2^{k-1}}}\right)2^{{\displaystyle \frac{k-1}{2}}}{\displaystyle \frac{1}{\sqrt{\frac{{\left(-1\right)}^{m-1}{\left(m!\right)}^2}{\left(2m\right)!}}{\beta }_{2m}}}{\displaystyle \frac{1}{2^{k-1}}}{\beta }_m\left(u\right)du\hfill \\ \hfill <& {\displaystyle \frac{\gimel }{2^{\frac{k-1}{2}}}}{\displaystyle \frac{1}{\sqrt{\frac{{\left(-1\right)}^{m-1}{\left(m!\right)}^2}{\left(2m\right)!}}{\beta }_{2m}}}{\int }_0^1{\beta }_m\left(u\right)du\to 0.\hfill \end{array}$$

(21)

For every positive integer ${\rm Y}$, the following statement is true:

$$\begin{array}{c}\hfill \sum _{n=2^{k-1}}^{2^{k-1}+{\rm Y}}\sum _{m=0}^2{c}_{n,m}^2<{\displaystyle \frac{3({\rm Y}+1){\gimel }^2}{2^{k-1}}}\to 0,(k\to \infty ).\end{array}$$

(22)

$$\begin{array}{c}\hfill \sum _{n=2^{k-1}}^{\infty }\sum _{m=0}^2{c}_{n,m}^2\to 0,(k\to \infty ).\end{array}$$

(23)

$$\begin{array}{c}\hfill \left|\left|\epsilon \right|\right|={\left(\sum _{n=2^{k-1}}^{\infty }\sum _{m=0}^2{c}_{n,m}^2\right)}^{{\displaystyle \frac{1}{2}}},(k\to \infty ).\end{array}$$

(24)

□

5. Proposed Method

The primary goal of this section is to formulate a smoking model with FO dynamics by employing the ABM in conjunction with the BWM.

5.1. Experiment of Bernoulli Wavelet Approach on Smoking Model

According to the study’s findings, no investigation has been conducted into applying the BWM to the smoking model. Thus, we aim to see how the model behaves using this approach. The nonlinear FO smoking model is calculated in this study using the operational matrix approach of the Bernoulli wavelet. Let us consider a smoking model (3) with the initial condition (4). Further, we consider the higher-order derivatives in terms of Bernoulli wavelet with unknown coefficients, and we get

$$\begin{array}{c}\hfill {}^{C}D_0^{\sigma }{S}_s\left(t\right)={\beth }_1^T\Xi \left(t\right),\\ \hfill {}^{C}D_0^{\sigma }{I}_c\left(t\right)={\beth }_2^T\Xi \left(t\right),\\ \hfill {}^{C}D_0^{\sigma }{U}_s\left(t\right)={\beth }_3^T\Xi \left(t\right),\\ \hfill {}^{C}D_0^{\sigma }{R}_s\left(t\right)={\beth }_4^T\Xi \left(t\right),\\ \hfill {}^{C}D_0^{\sigma }{E}_s\left(t\right)={\beth }_5^T\Xi \left(t\right),\end{array}$$

(25)

where,

$$\begin{array}{cc}\hfill {\beth }_1^T& ={\left[{\left({\beth }_1\right)}_{1,0},{\left({\beth }_1\right)}_{1,1},\dots ,{\left({\beth }_1\right)}_{1,M-1},\dots ,{\left({\beth }_1\right)}_{2^{k-1},0},{\left({\beth }_1\right)}_{2^{k-1},1},\dots ,{\left({\beth }_1\right)}_{2^{k-1},M-1}\right]}^T,\hfill \\ \hfill {\beth }_2^T& ={\left[{\left({\beth }_2\right)}_{1,0},{\left({\beth }_2\right)}_{1,1},\dots ,{\left({\beth }_2\right)}_{1,M-1},\dots ,{\left({\beth }_2\right)}_{2^{k-1},0},{\left({\beth }_2\right)}_{2^{k-1},1},\dots ,{\left({\beth }_2\right)}_{2^{k-1},M-1}\right]}^T,\hfill \\ \hfill {\beth }_3^T& ={\left[{\left({\beth }_3\right)}_{1,0},{\left({\beth }_3\right)}_{1,1},\dots ,{\left({\beth }_3\right)}_{1,M-1},\dots ,{\left({\beth }_3\right)}_{2^{k-1},0},{\left({\beth }_3\right)}_{2^{k-1},1},\dots ,{\left({\beth }_3\right)}_{2^{k-1},M-1}\right]}^T,\hfill \\ \hfill {\beth }_4^T& ={\left[{\left({\beth }_4\right)}_{1,0},{\left({\beth }_4\right)}_{1,1},\dots ,{\left({\beth }_4\right)}_{1,M-1},\dots ,{\left({\beth }_4\right)}_{2^{k-1},0},{\left({\beth }_4\right)}_{2^{k-1},1},\dots ,{\left({\beth }_4\right)}_{2^{k-1},M-1}\right]}^T,\hfill \\ \hfill {\beth }_5^T& ={\left[{\left({\beth }_5\right)}_{1,0},{\left({\beth }_5\right)}_{1,1},\dots ,{\left({\beth }_5\right)}_{1,M-1},\dots ,{\left({\beth }_5\right)}_{2^{k-1},0},{\left({\beth }_5\right)}_{2^{k-1},1},\dots ,{\left({\beth }_5\right)}_{2^{k-1},M-1}\right]}^T,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill S_s\left(0\right)={\zeta }_1,I_c\left(0\right)={\zeta }_2,U_s\left(0\right)={\zeta }_3,R_s\left(0\right)={\zeta }_4,E_s\left(0\right)={\zeta }_5.\end{array}$$

the vectors denoted as “unknown” and the function $\Xi \left(t\right)$ have already been provided. We are currently utilizing the Riemann–Liouville fractional operator, as deduced from the subtraction of Equations (3) and (4).

$$\begin{array}{cc}\hfill \left({J_0^{\sigma }}^C{D}_0^{\sigma }\right)S_s\left(t\right)=& S_s\left(t\right)-S_s\left(0\right)=S_s\left(t\right)-{\zeta }_1={\beth }_1^T{P}^{\sigma }\Xi \left(t\right),\hfill \\ \hfill \left({J_0^{\sigma }}^C{D}_0^{\sigma }\right)I_c\left(t\right)=& I_c\left(t\right)-I_c\left(0\right)=I_c\left(t\right)-{\zeta }_2={\beth }_2^T{P}^{\sigma }\Xi \left(t\right),\hfill \\ \hfill \left({J_0^{\sigma }}^C{D}_0^{\sigma }\right)U_s\left(t\right)=& U_s\left(t\right)-U_s\left(0\right)=U_s\left(t\right)-{\zeta }_3={\beth }_3^T{P}^{\sigma }\Xi \left(t\right),\hfill \\ \hfill \left({J_0^{\sigma }}^C{D}_0^{\sigma }\right)R_s\left(t\right)=& R_s\left(t\right)-R_s\left(0\right)=R_s\left(t\right)-{\zeta }_4={\beth }_4^T{P}^{\sigma }\Xi \left(t\right),\hfill \\ \hfill \left({J_0^{\sigma }}^C{D}_0^{\sigma }\right)E_s\left(t\right)=& E_s\left(t\right)-E_s\left(0\right)=E_s\left(t\right)-{\zeta }_5={\beth }_5^T{P}^{\sigma }\Xi \left(t\right).\hfill \end{array}$$

(26)

Now, we use (25)–(26) we derive from Equation (3)

$$\begin{array}{cc}\hfill {\beth }_1^T\Xi \left(t\right)=& e_1-e_2\left({\beth }_1^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_1\right)\left({\beth }_2^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_2\right)\hfill \\ \hfill & +e_3\left({\beth }_4^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_4\right)-e_4\left({\beth }_1^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_1\right),\hfill \\ \hfill {\beth }_2^T\Xi \left(t\right)=& e_2\left({\beth }_1^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_1\right)\left({\beth }_2^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_2\right)\hfill \\ \hfill & -e_5\left({\beth }_2^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_2\right)\left({\beth }_3^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_3\right)\hfill \\ \hfill & -(e_6+e_4)\left({\beth }_2^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_2\right),\hfill \\ \hfill {\beth }_3^T\Xi \left(t\right)=& e_5\left({\beth }_2^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_2\right)\left({\beth }_3^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_3\right)\hfill \\ \hfill & -(e_7+e_8+e_4)\left({\beth }_3^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_3\right),\hfill \\ \hfill {\beth }_4^T\Xi \left(t\right)=& e_7\left({\beth }_3^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_3\right)\hfill \\ \hfill & -(e_9+e_4+e_3)\left({\beth }_4^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_4\right),\hfill \\ \hfill {\beth }_5^T\Xi \left(t\right)=& e_9\left({\beth }_4^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_4\right)\hfill \\ \hfill & -e_4\left({\beth }_5^T{P}^{\sigma }\Xi \left(t\right)+{\zeta }_5\right).\hfill \end{array}$$

(27)

Then, we utilize the collocation point (27), which has been written as

$$\begin{array}{cc}\hfill {\beth }_1^T\Xi \left(t_i\right)=& e_1-e_2\left({\beth }_1^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_1\right)\left({\beth }_2^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_2\right)\hfill \\ \hfill & +e_3\left({\beth }_4^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_4\right)-e_4\left({\beth }_1^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_1\right),\hfill \\ \hfill {\beth }_2^T\Xi \left(t_i\right)=& e_2\left({\beth }_1^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_1\right)\left({\beth }_2^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_2\right)\hfill \\ \hfill & -e_5\left({\beth }_2^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_2\right)\left({\beth }_3^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_3\right)\hfill \\ \hfill & -(e_6+e_4)\left({\beth }_2^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_2\right),\hfill \\ \hfill {\beth }_3^T\Xi \left(t_i\right)=& e_5\left({\beth }_2^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_2\right)\left({\beth }_3^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_3\right)\hfill \\ \hfill & -(e_7+e_8+e_4)\left({\beth }_3^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_3\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\beth }_4^T\Xi \left(t_i\right)=& e_7\left({\beth }_3^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_3\right)\hfill \\ \hfill & -(e_9+e_4+e_3)\left({\beth }_4^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_4\right),\hfill \\ \hfill {\beth }_5^T\Xi \left(t_i\right)=& e_9\left({\beth }_4^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_4\right)\hfill \\ \hfill & -e_4\left({\beth }_5^T{P}^{\sigma }\Xi \left(t_i\right)+{\zeta }_5\right).\hfill \end{array}$$

(28)

$\forall i=1,2,\dots ,m.$ By transforming Equation (28) into a nonlinear system involving 5 m unknown vectors, we employ an iterative approach using Matlab to solve this system of nonlinear equations. Since (28) is reduced, we obtain approximate solutions of the system of Equation (3).

5.2. ABM Scheme for the Smoking Model

The ABM scheme is the most popular numerical method for solving fractional initial-value problems of any type [31]. The ABM method’s increased accuracy, stability, and adaptability make it a valuable tool for fractional systems. It is perfect for fractional dynamics with memory effects because it compromises accuracy and computational efficiency by refining predictions through a corrective step. A comparison is made between the results from different scenarios and the fractional ABM numerical technique results. Let us consider that the following FDE is

$$\begin{array}{c}\hfill {}_0{}^{C}D_t^{\sigma }{x}_j\left(t\right)=f_j(t,x_j\left(t\right)),x_j^k\left(0\right)=x_{j0}^k,\\ \hfill k=0,1,2,\dots ,\lceil \sigma \rceil ,j\in N,\end{array}$$

(29)

where $x_{j0}^k$ is the arbitrary real number, $\sigma >0$ and ${}_0{}^{C}D_t^{\sigma }$ is the fractional differential operator in Caputo sense. We analyze the results of the non-linear fractional smoking model using the ABM to obtain its numerical solution. Now, let $h={\displaystyle \frac{T}{\widehat{m}}}$, $t_n=nh$, $n=0,1,2,\dots ,\widehat{m}$; then, the corrector values are defined as,

$$\begin{array}{cc}\hfill {\left(S_s\right)}_{n+1}=& {\zeta }_1+{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}(e_1-e_2{\left(S_s\right)}_{n+1}^p{\left(I_c\right)}_{n+1}^p+e_3{\left(R_s\right)}_{n+1}^p-e_4{\left(S_s\right)}_{n+1}^p)\hfill \\ \hfill & +{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}\sum _{j=0}^n{\sigma }_{j,n+1}(e_1-e_2{\left(S_s\right)}_j{\left(I_c\right)}_j+e_3{\left(R_s\right)}_j-e_4{\left(S_s\right)}_j),\hfill \\ \hfill {\left(I_c\right)}_{n+1}=& {\zeta }_2+{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}(e_2{\left(S_s\right)}_{n+1}^p{\left(I_c\right)}_{n+1}^p-e_5{\left(I_c\right)}_{n+1}^p{\left(U_s\right)}_{n+1}^p-(e_6+e_4){\left(I_c\right)}_{n+1}^p)\hfill \\ \hfill & +{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}\sum _{j=0}^n{\sigma }_{j,n+1}(e_2{\left(S_s\right)}_j{\left(I_c\right)}_j-e_5{\left(I_c\right)}_j{\left(U_s\right)}_j-(e_6+e_4){\left(I_c\right)}_j),\hfill \\ \hfill {\left(U_s\right)}_{n+1}=& {\zeta }_3+{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}(e_5{\left(I_c\right)}_{n+1}^p{\left(U_s\right)}_{n+1}^p-(e_7+e_8+e_4){\left(U_s\right)}_{n+1}^p)\hfill \\ \hfill & +{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}\sum _{j=0}^n{\sigma }_{j,n+1}(e_5{\left(I_c\right)}_j{\left(U_s\right)}_j-(e_7+e_8+e_4){\left(U_s\right)}_j),\hfill \\ \hfill {\left(R_s\right)}_{n+1}=& {\zeta }_4+{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}(e_7{\left(U_s\right)}_{n+1}^p-(e_9+e_4+e_3){\left(R_s\right)}_{n+1}^p)\hfill \\ \hfill & +{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}\sum _{j=0}^n{\sigma }_{j,n+1}(e_7{\left(U_s\right)}_j-(e_9+e_4+e_3){\left(R_s\right)}_j),\hfill \\ \hfill {\left(E_s\right)}_{n+1}=& {\zeta }_5+{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}(e_9{\left(R_s\right)}_{n+1}^p-e_4{\left(E_s\right)}_{n+1}^p)\hfill \\ \hfill & +{\displaystyle \frac{h^{\sigma }}{\Gamma (\sigma +2)}}\sum _{j=0}^n{\sigma }_{j,n+1}(e_9{\left(R_s\right)}_j-e_4{\left(E_s\right)}_j,)\hfill \end{array}$$

(30)

the corresponding predictor values are given as:

$$\begin{array}{cc}\hfill {\left(S_s\right)}_{n+1}^p=& {\zeta }_1+{\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}\sum _{j=0}^n{\Theta }_{j,n+1}(e_1-e_2{\left(S_s\right)}_j{\left(I_c\right)}_j+e_3{\left(R_s\right)}_j-e_4{\left(S_s\right)}_j),\hfill \\ \hfill {\left(I_c\right)}_{n+1}^p=& {\zeta }_2+{\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}\sum _{j=0}^n{\Theta }_{j,n+1}(e_2{\left(S_s\right)}_j{\left(I_c\right)}_j-e_5{\left(I_c\right)}_j{\left(U_s\right)}_j-(e_6+e_4){\left(I_c\right)}_j),\hfill \\ \hfill {\left(U_s\right)}_{n+1}^p=& {\zeta }_3+{\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}\sum _{j=0}^n{\Theta }_{j,n+1}(e_5{\left(I_c\right)}_j{\left(U_s\right)}_j-(e_7+e_8+e_4){\left(U_s\right)}_j),\hfill \\ \hfill {\left(R_s\right)}_{n+1}^p=& {\zeta }_4+{\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}\sum _{j=0}^n{\Theta }_{j,n+1}(e_7{\left(U_s\right)}_j-(e_9+e_4+e_3){\left(R_s\right)}_j),\hfill \\ \hfill {\left(E_s\right)}_{n+1}^p=& {\zeta }_5+{\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}\sum _{j=0}^n{\Theta }_{j,n+1}(e_9{\left(R_s\right)}_j-e_4{\left(E_s\right)}_j),\hfill \end{array}$$

(31)

where

$${\sigma }_{j,n+1}=\left\{\begin{array}{cc}{n}^{\sigma +1}-(n-\sigma ){(n+1)}^{\sigma },\hfill & j=0\hfill \\ {(n-j+2)}^{\sigma +1}+{(n-j)}^{\sigma +1}-2{(n-j+1)}^{\sigma +1},\hfill & 0\le j\le n\hfill \\ 1,\hfill & j=1\hfill \end{array}\right.$$

and

$${\Theta }_{j,n+1}={\displaystyle \frac{h^{\sigma }}{\sigma }}({(n+1-j)}^{\sigma }-{(n-j)}^{\sigma }),0\le j\le n.$$

5.3. Remark

The stability of the ABM approach has been confirmed in reference [32], and it has been effectively used in the solution of differential equations with fixed FO. Therefore, there is no necessity to restate these findings here.

6. The Fractional Optimal Control Problem (FOCP)

FOCPs have not been the subject of much research. The need for optimum control theories and the related analytical and numerical techniques to solve the associated equations grows along with the demand for realistic, accurate, and exact systems. This study closes this gap by providing a numerical approach and utilising the optimal control technique. Here, the aim is to maximize the population of ex-smokers and control the smoking habit in society. This section discusses an optimal control approach appropriate for the system dynamics (3). Four controls have been considered in this study, constructed in (3), which have been represented as follows [33]:

$$\begin{array}{cc}\hfill {}_0{}^{C}D_t^{\sigma }\left(S_s\right)\left(t\right)& =e_1-e_2\left(S_s\right)\left(t\right)\left(I_c\right)\left(t\right)+e_3\left(R_s\right)\left(t\right)-(e_4+w_1)\left(S_s\right)\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }\left(I_c\right)\left(t\right)& =e_2\left(S_s\right)\left(t\right)\left(I_c\right)\left(t\right)-e_5\left(I_c\right)\left(t\right)\left(U_s\right)\left(t\right)-(e_6+e_4+w_2)\left(I_c\right)\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }\left(U_s\right)\left(t\right)& =e_5\left(I_c\right)\left(t\right)\left(U_s\right)\left(t\right)-(e_7+e_8+e_4+w_3)\left(U_s\right)\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }\left(R_s\right)\left(t\right)& =e_7\left(U_s\right)\left(t\right)-(e_9+e_4+e_3+w_4)\left(R_s\right)\left(t\right),\hfill \\ \hfill {}_0{}^{C}D_t^{\sigma }\left(E_s\right)\left(t\right)& =(e_9+w_4)\left(R_s\right)\left(t\right)-e_4\left(E_s\right)\left(t\right)+w_1\left(S_s\right)\left(t\right)+w_2\left(I_c\right)\left(t\right)+w_3\left(U_s\right)\left(t\right).\hfill \end{array}$$

(32)

By implementing the appropriate laws, the number of smokers and the pool of potential smokers have been brought down to more controllable levels. On the other hand, the number of smokers and potential smokers will rise while the number of persons quitting will fall if these four limitations are not implemented. When formulating the objective function, we took into account the control issues outlined in Equation (32). The following objective function has been derived:

$$\begin{array}{cc}\hfill J\left(w\right(t\left)\right)=& {\int }_0^{t_f}(R_s\left(t\right)-E_s\left(t\right)+{\displaystyle \frac{1}{2}}(k_1{w}_1^2\left(t\right)+k_2{w}_2^2\left(t\right)\\ & +k_3{w}_3^2\left(t\right)+k_4{w}_4^2\left(t\right)\left)\right)dt.\end{array}$$

(33)

This study’s main objective is to reduce $J\left(w\right(t\left)\right)$ while respecting its restrictions using the optimum control method.

6.1. Optimal Control Solutions

Let optimal control Equations (32) and (33) be written with a Hamiltonian function as follows:

$$\begin{array}{c}\hfill H=L+\sum _{j=1}^5\lambda j\left(t\right)g_j,\end{array}$$

(34)

where the Lagrangian function can be written as:

$$\begin{array}{cc}\hfill L(R_s,E_s,w_i)=& R_s\left(t\right)-E_s\left(t\right)+{\displaystyle \frac{1}{2}}(k_1{w}_1^2\left(t\right)+k_2{w}_2^2\left(t\right)\\ & +k_3{w}_3^2\left(t\right)+k_4{w}_4^2\left(t\right)),\end{array}$$

(35)

and

$$\begin{array}{c}\hfill {}_0{}^{C}D_t^{\sigma }{S}_s\left(t\right)=g_1\left(t\right),\\ \hfill {}_0{}^{C}D_t^{\sigma }{I}_c\left(t\right)=g_2\left(t\right),\\ \hfill {}_0{}^{C}D_t^{\sigma }{U}_s\left(t\right)=g_3\left(t\right),\\ \hfill {}_0{}^{C}D_t^{\sigma }{R}_s\left(t\right)=g_4\left(t\right),\\ \hfill {}_0{}^{C}D_t^{\sigma }{E}_s\left(t\right)=g_5\left(t\right).\end{array}$$

(36)

The adjoint system, with $\lambda$ as the adjoint vector, is given by:

$$\begin{array}{c}\hfill {{}_0{}^{C}D_t^{\sigma }}_f\lambda ={\displaystyle \frac{\partial L}{\partial x}}+{\lambda }^T{\displaystyle \frac{\partial g}{\partial x}},\lambda \left(t_f\right)=0.\end{array}$$

(37)

The optimal control $w^{\ast }\left(t\right)$ satisfies the following Equation:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial L}{\partial w^{\ast }}}+{\lambda }^T{\displaystyle \frac{\partial g}{\partial w^{\ast }}}=0.\end{array}$$

(38)

The Euler–Lagrange optimality conditions for the FOCP with Caputo fractional derivatives are given by (36)–(38). Note, when $\sigma =1$, the above FOCP becomes a classical optimal control problem.

Here, $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5)$, $g=(g_1,g_2,g_3,g_4,g_5)$, $i=1,2,3,4,5$ are the right sides of system (32). The state system was already given by (32). Using the relations above, the adjoint system is derived as:

$$\begin{array}{cc}\hfill D_{t_f}^{\sigma }{\lambda }_1& ={\lambda }_1(-e_2{I}_c-(e_4+w_1))+{\lambda }_2\left(e_2{I}_c\right)+{\lambda }_5{w}_1,\hfill \\ \hfill D_{t_f}^{\sigma }{\lambda }_2& ={\lambda }_1(-e_2{S}_s)+{\lambda }_2(e_2{S}_s-e_5{U}_s-(e_6+e_4+w_2))+{\lambda }_3\left(e_5{U}_s\right)+{\lambda }_5{w}_2,\hfill \\ \hfill D_{t_f}^{\sigma }{\lambda }_3& ={\lambda }_2(-e_5{I}_c)+{\lambda }_3(e_5{I}_c-(e_7+e_8+e_4+w_3))+{\lambda }_4\left(e_7\right)+{\lambda }_5{w}_3,\hfill \\ \hfill D_{t_f}^{\sigma }{\lambda }_4& =1+{\lambda }_1{e}_3-{\lambda }_4(e_9+e_4+e_3+w_4)+{\lambda }_5(e_9+w_4),\hfill \\ \hfill D_{t_f}^{\sigma }{\lambda }_5& =-1-e_4{\lambda }_5,\hfill \end{array}$$

(39)

with the boundary conditions ${\lambda }_i=0,$ where $i=1,2,3,4,5$. From the Equations (34)–(38), the expression for optimal control function is obtained as:

$$\begin{array}{cc}\hfill w_1^{\ast }\left(t\right)=& {\displaystyle \frac{({\lambda }_1-{\lambda }_5)S_s}{k_1}},\hfill \\ \hfill w_2^{\ast }\left(t\right)=& {\displaystyle \frac{({\lambda }_2-{\lambda }_5)I_c}{k_2}},\hfill \\ \hfill w_3^{\ast }\left(t\right)=& {\displaystyle \frac{({\lambda }_3-{\lambda }_5)U_s}{k_3}},\hfill \\ \hfill w_4^{\ast }\left(t\right)=& {\displaystyle \frac{({\lambda }_4-{\lambda }_5)R_s}{k_4}}.\hfill \end{array}$$

(40)

For the boundedness of the optimal control, we get the following forms of the above expression:

$$\begin{array}{c}\hfill w_1^{\ast }\left(t\right)=max\left\{min\left\{{\displaystyle \frac{({\lambda }_1-{\lambda }_5)S_s}{k_1}},1\right\},0\right\},\\ \hfill w_2^{\ast }\left(t\right)=max\left\{min\left\{{\displaystyle \frac{({\lambda }_2-{\lambda }_5)I_c}{k_2}},1\right\},0\right\},\\ \hfill w_3^{\ast }\left(t\right)=max\left\{min\left\{{\displaystyle \frac{({\lambda }_3-{\lambda }_5)U_s}{k_3}},1\right\},0\right\},\\ \hfill w_4^{\ast }\left(t\right)=max\left\{min\left\{{\displaystyle \frac{({\lambda }_4-{\lambda }_5)R_s}{k_4}},1\right\},0\right\},\end{array}$$

(41)

replacing $w_i\left(t\right)$ by $w_i^{\ast }\left(t\right)$, $i=1,2,3,4,5$ in system (32) and (39), we got desire FOCP.

6.2. Existence of Optimal Control Solution

The methodology was employed to showcase the suitability of implementing optimal control for the model [34,35]. The assumption was made that the control system described in Equation (32) has been reformulated as follows:

$$\begin{array}{c}\hfill {\xi }_t=C\xi +F\left(\xi \right).\end{array}$$

(42)

In this context, the state variable vector is defined as: $\xi ={[S_s\left(t\right),I_c\left(t\right),U_s\left(t\right),R_s\left(t\right),E_s\left(t\right)]}^T$ and

$$C=\left[\begin{array}{ccccc}-a& 0& 0& b& 0\\ 0& -c& 0& 0& 0\\ 0& 0& -d& 0& 0\\ 0& 0& p& -f& 0\\ i& j& m& g& -h\end{array}\right],$$

where

$$\begin{gathered} a=e_4+w_1,b=e_3,c=e_4+e_6+w_2,d=e_7+e_8+e_4+w_3,p=e_7, \\ f=e_9+e_4+e_3,g=e_9+w_4,h=e_4,i=w_1,j=w_2,m=w_3, \end{gathered}$$

$$F\left(\xi \right)={(e_1-e_2{S}_s{I}_c,e_2{S}_s{I}_c-e_5{I}_c{U}_s,e_5{I}_c{U}_s,0,0)}^T.$$

Equation (42) is a nonlinear FDE with bounded coefficients, where ${\xi }_t$ is the time derivative of $\xi$. Now,

$$B\left(\xi \right)=C\xi +F\left(\xi \right),$$

$$\begin{array}{cc}\hfill F\left({\xi }_1\right)-F\left({\xi }_2\right)& =(-e_2{\left(S_s\right)}_1{\left(I_c\right)}_1+e_2{\left(S_s\right)}_2{\left(I_c\right)}_2,\hfill \\ \hfill & e_2{\left(S_s\right)}_1{\left(I_c\right)}_1-e_2{\left(S_s\right)}_2{\left(I_c\right)}_2-e_5{\left(I_c\right)}_1{\left(U_s\right)}_1+e_5{\left(I_c\right)}_2{\left(U_s\right)}_2,\hfill \\ \hfill & e_5{\left(I_c\right)}_1{\left(U_s\right)}_1-e_5{\left(I_c\right)}_2{\left(U_s\right)}_2{,0,0)}^T.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill |F\left({\xi }_1\right)-F\left({\xi }_2\right)|& =|-e_2{\left(S_s\right)}_1{\left(I_c\right)}_1+e_2{\left(S_s\right)}_2{\left(I_c\right)}_2|\hfill \\ \hfill & +|e_2{\left(S_s\right)}_1{\left(I_c\right)}_1-e_2{\left(S_s\right)}_2{\left(I_c\right)}_2-e_5{\left(I_c\right)}_1{\left(U_s\right)}_1+e_5{\left(I_c\right)}_2{\left(U_s\right)}_2|\hfill \\ \hfill & +|e_5{\left(I_c\right)}_1{\left(U_s\right)}_1-e_5{\left(I_c\right)}_2{\left(U_s\right)}_2|\hfill \\ \hfill & \le 2e_2|{\left(S_s\right)}_1{\left(I_c\right)}_1+{\left(S_s\right)}_2{\left(I_c\right)}_2|+2e_5|{\left(I_c\right)}_1{\left(U_s\right)}_1+{\left(I_c\right)}_2{\left(U_s\right)}_2|\hfill \\ \hfill & \le 2e_2|{\left(S_s\right)}_1[{\left(I_c\right)}_1-{\left(I_c\right)}_2]+{\left(I_c\right)}_2[{\left(S_s\right)}_1-{\left(S_s\right)}_2]|\hfill \\ \hfill & +2e_5|{\left(I_c\right)}_1[{\left(U_s\right)}_1-{\left(U_s\right)}_2]+{\left(U_s\right)}_2[{\left(I_c\right)}_1-{\left(I_c\right)}_2]|\hfill \\ \hfill & \le 2e_2|{\left(I_c\right)}_2\left|\right|{\left(S_s\right)}_1-{\left(S_s\right)}_2|\hfill \\ \hfill & +[2e_2|{\left(S_s\right)}_1|+2e_5|{\left(U_s\right)}_2\left|\right]|{\left(I_c\right)}_1-{\left(I_c\right)}_2|\hfill \\ \hfill & +2e_5|{\left(I_c\right)}_1\left|\right|{\left(U_s\right)}_1-{\left(U_s\right)}_2|\hfill \\ \hfill & \le 2e_2{\displaystyle \frac{e_1}{e_4}}|{\left(S_s\right)}_1-{\left(S_s\right)}_2|+(2e_2+2e_5){\displaystyle \frac{e_1}{e_4}}|{\left(I_c\right)}_1-{\left(I_c\right)}_2|\hfill \\ \hfill & +2e_5{\displaystyle \frac{e_1}{e_4}}|{\left(U_s\right)}_1-{\left(U_s\right)}_2|.\hfill \end{array}$$

So, $\left|B\left({\xi }_1\right)-B\left({\xi }_2\right)\right|\le Z\left|{\xi }_1-{\xi }_2\right|$. Here,

$$Z=max\left\{{\displaystyle \frac{(2e_2+2e_5)e_1}{e_4}},∥C∥\right\}<\infty ,$$

observing that $B\left(\xi \right)$ is uniformly Lipschitz continuous, we have deduced, based on the definition of $w_i$, that a solution to the controlled model (32) does indeed exist.

7. Numerical Result and Discussion

In this part, we provide numerical solutions for the nonlinear fractional-order smoking model using the ABM approach and Bernoulli wavelet techniques. The Forward Euler method is a simple and versatile numerical technique for solving ordinary and fractional differential equations. It offers several advantages, including ease of implementation, low computational cost, and suitability for short-term predictions. However, it has some limitations, such as low accuracy for fractional models, stability issues, and inefficiency for long-term simulations. While the method is a good starting point, more advanced techniques like implicit schemes or fractional-specific methods may be more suitable for addressing these limitations in fractional models. The ABM is highly beneficial for fractional systems due to its superior accuracy, stability, and adaptability. Improving forecasts with a correction step successfully strikes a compromise between computing cost and accuracy, which makes it ideal for fractional dynamics with memory effects. However, it has implementation complexity and dependency on accurate initial values, and it could be more suitable for highly stiff systems. We will also illustrate the graphical patterns of this model for different values of the order $\sigma$. Then, by using control theory, we observe how to restrain and reduce this negative impact on society. In this paper, we utilize a numerical method optimal control technique and analyse the result [36,37,38,39,40,41,42]. Additionally, we examine the nonlinear FO smoking model with the subsequent parameter values from Table 1:

Table 1. Parameter values of system (3).

Parameters	Value	Source
$e_1$	$0.1$	[23]
$e_2$	$0.003$	[23]
$e_3$	$0.003$	[23]
$e_4$	$0.002$	[23]
$e_5$	$0.002$	[23]
$e_6$	$0.003$	[23]
$e_7$	$0.05$	[23]
$e_8$	$0.003$	[23]
$e_9$	$0.05$	[23]

Table 1. Parameter values of system (3).

Parameters	Value	Source
$e_1$	$0.1$	[23]
$e_2$	$0.003$	[23]
$e_3$	$0.003$	[23]
$e_4$	$0.002$	[23]
$e_5$	$0.002$	[23]
$e_6$	$0.003$	[23]
$e_7$	$0.05$	[23]
$e_8$	$0.003$	[23]
$e_9$	$0.05$	[23]

$$\begin{array}{cc}\hfill e_1& =0.1,e_2=0.003,e_3=0.003,e_4=0.002,e_5=0.002,e_6=0.003,\hfill \\ \hfill & e_7=0.05,e_8=0.003,e_9=0.05,\hfill \end{array}$$

and

$$S_s\left(0\right)=68,I_c\left(0\right)=40,U_s\left(0\right)=30,R_s\left(0\right)=20,E_s\left(0\right)=15.$$

Applying the Laplace and inverse Laplace transforms in Equation (3) allows us to derive the following result.

$$\begin{array}{cc}\hfill S_s\left(t\right)=& S_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_1-e_2{S}_s\left(t\right)I_c\left(t\right)+e_3{R}_s\left(t\right)-e_4{S}_s\left(t\right)\right\}\right],\hfill \\ \hfill I_c\left(t\right)=& I_c\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_2{S}_s\left(t\right)I_c\left(t\right)-e_5{I}_c\left(t\right)U_s\left(t\right)-(e_6+e_4)I_c\left(t\right)\right\}\right],\hfill \\ \hfill U_s\left(t\right)=& U_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_5{I}_c\left(t\right)U_s\left(t\right)-(e_7+e_8+e_4)U_s\left(t\right)\right\}\right],\hfill \\ \hfill R_s\left(t\right)=& R_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_7{U}_s\left(t\right)-(e_9+e_4+e_3)R_s\left(t\right)\right\}\right],\hfill \\ \hfill E_s\left(t\right)=& E_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_9{R}_s\left(t\right)-e_4{E}_s\left(t\right)\right\}\right].\hfill \end{array}$$

(43)

Additionally, the iterative scheme is as follows:

$$\begin{array}{cc}\hfill {\left(S_s\right)}_n\left(t\right)=& S_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_1-e_2{\left(S_s\right)}_{n-1}\left(t\right){\left(I_c\right)}_{n-1}\left(t\right)+e_3{\left(R_s\right)}_{n-1}\left(t\right)\right.\right.\hfill \\ \hfill -& \left.\left.e_4{\left(S_s\right)}_{n-1}\left(t\right)\right\}\right],\hfill \\ \hfill {\left(I_c\right)}_n\left(t\right)=& I_c\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_2{\left(S_s\right)}_{n-1}\left(t\right){\left(I_c\right)}_{n-1}\left(t\right)-e_5{\left(I_c\right)}_{n-1}\left(t\right)U_s\left(t\right)\right.\right.\hfill \\ \hfill -& \left.\left.(e_6+e_4){\left(I_c\right)}_{n-1}\left(t\right)\right\}\right],\hfill \\ \hfill {\left(U_s\right)}_n\left(t\right)=& U_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_5{\left(I_c\right)}_{n-1}\left(t\right){\left(U_s\right)}_{n-1}\left(t\right)\right.\right.\hfill \\ \hfill -& \left.\left.(e_7+e_8+e_4){\left(U_s\right)}_{n-1}\left(t\right)\right\}\right],\hfill \\ \hfill {\left(R_s\right)}_n\left(t\right)=& R_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_7{\left(U_s\right)}_{n-1}\left(t\right)-(e_9+e_4+e_3){\left(R_s\right)}_{n-1}\left(t\right)\right\}\right],\hfill \\ \hfill {\left(E_s\right)}_n\left(t\right)=& E_s\left(0\right)+L^{-1}\left[{\displaystyle \frac{1}{s^{\sigma }}}L\left\{e_9{\left(R_s\right)}_{n-1}\left(t\right)-e_4{\left(E_s\right)}_{n-1}\left(t\right)\right\}\right].\hfill \end{array}$$

(44)

In Equation (44), we have derived an approximate solution as the value of n approaches infinity.

$$\begin{array}{c}\hfill \left(S_s\left(t\right),I_c\left(t\right),U_s\left(t\right),R_s\left(t\right),E_s\left(t\right)\right)=\underset{n\to \infty }{lim}\left({\left(S_s\right)}_n\left(t\right),{\left(I_c\right)}_n\left(t\right),{\left(U_s\right)}_n\left(t\right),{\left(R_s\right)}_n\left(t\right),{\left(E_s\right)}_n\left(t\right)\right).\end{array}$$

Using the initial condition, we have computed the numerical outcome of Equation (3). We have expressed this result approximately as follows, and more details are available in [29]:

$$\begin{array}{cc}\hfill {\left(S_s\right)}_1\left(t\right)& =68-\left(8.1360\right)t,\hfill \\ \hfill {\left(I_c\right)}_1\left(t\right)& =40+\left(5.5600\right)t,\hfill \\ \hfill {\left(U_s\right)}_1\left(t\right)& =30+\left(0.7500\right)t,\hfill \\ \hfill {\left(R_s\right)}_1\left(t\right)& =20+\left(0.4000\right)t,\hfill \\ \hfill {\left(E_s\right)}_1\left(t\right)& =15+\left(0.9700\right)t,\hfill \\ \hfill {\left(S_s\right)}_2\left(t\right)& =-8.1360-0.1404t+0.1357t^2,\hfill \\ \hfill {\left(I_c\right)}_2\left(t\right)& =5.5600-0.2635t-0.1440t^2,\hfill \\ \hfill {\left(U_s\right)}_2\left(t\right)& =0.7500+0.3523t+0.0083t^2,\hfill \\ \hfill {\left(R_s\right)}_2\left(t\right)& =0.4000+0.0155t,\hfill \\ \hfill {\left(E_s\right)}_2\left(t\right)& =0.9700+0.0181t,\hfill \\ \hfill & ⋮\hfill \end{array}$$

Ultimately, the approximate solution is derived by utilizing Equations (3) and (4). To showcase the effectiveness and precision of the approach, residual errors are employed in the following manner.

$$\begin{array}{cc}\hfill E_{n+1,S_s}\left(t\right)& ={\left(S_s\right)}_{n+1}^{\prime }\left(t\right)-[e_1-e_2{S}_s\left(t\right)I_c\left(t\right)+e_3{R}_s\left(t\right)-e_4{S}_s\left(t\right)],\hfill \\ \hfill E_{n+1,I_c}\left(t\right)& ={\left(I_c\right)}_{n+1}^{\prime }\left(t\right)-[e_2{S}_s\left(t\right)I_c\left(t\right)-e_5{I}_c\left(t\right)U_s\left(t\right)-(e_6+e_4)I_c\left(t\right)],\hfill \\ \hfill E_{n+1,U_s}\left(t\right)& ={\left(U_s\right)}_{n+1}^{\prime }\left(t\right)-[e_5{I}_c\left(t\right)U_s\left(t\right)-(e_7+e_8+e_4)U_s\left(t\right)],\hfill \\ \hfill E_{n+1,R_s}\left(t\right)& ={\left(R_s\right)}_{n+1}^{\prime }\left(t\right)-[e_7{U}_s\left(t\right)-(e_9+e_4+e_3)R_s\left(t\right)],\hfill \\ \hfill E_{n+1,E_s}\left(t\right)& ={\left(E_s\right)}_{n+1}^{\prime }\left(t\right)-[e_9{R}_s\left(t\right)-e_4{E}_s\left(t\right)].\hfill \end{array}$$

Figure 1 presents the graphical behavior of susceptible smokers $S_s\left(t\right)$, ingestion class $I_c\left(t\right)$, unusual smokers $U_s\left(t\right)$, regular smokers $R_s\left(t\right)$, and ex-smokers $E_s\left(t\right)$ using numerical techniques with $m=96$, $t_l=2$ days, and $\sigma =1$. Figure 2 displays the graphical behavior of susceptible smokers $S_s\left(t\right)$, ingestion class $I_c\left(t\right)$, unusual smokers $U_s\left(t\right)$, regular smokers $R_s\left(t\right)$, and ex-smokers $E_s\left(t\right)$ using numerical techniques with $m=96$, $t_l=60$ days, and $\sigma =1$. In Figure 3, a 2D plot illustrates susceptible smokers $S_s\left(t\right)$, ingestion class $I_c\left(t\right)$, unusual smokers $U_s\left(t\right)$, regular smokers $R_s\left(t\right)$, and ex-smokers $E_s\left(t\right)$ with different values of $\sigma$ using the BWM. Figure 4 demonstrates a 3D plot of susceptible smokers $S_s\left(t\right)$, ingestion class $I_c\left(t\right)$, unusual smokers $U_s\left(t\right)$, regular smokers $R_s\left(t\right)$, and ex-smokers $E_s\left(t\right)$ at $m=96$, and $\sigma =0.5$. Figure 5 and Figure 6 display residual-error graphs with $n=1$ and $n=2$. The simulation results were obtained after 2.959 seconds. Figure 7 and Figure 8 present a comparison graph of different system components using various numerical techniques for $\sigma =1$ and $\sigma =0.92$. Figure 9 depicts the relationship between susceptible smokers and other components of the smoking model for $\sigma =0.89$. Figure 9a shows that after ten days, both lines converge at a point; subsequently, the ingestion class increases until 20 days and then decreases. Figure 9b illustrates that both lines meet at a point after ten days, after which the number of unusual smokers increases until 30 days and then decreases. In Figure 9c, both lines intersect at a point after ten days, following which the number of regular smokers increases until 40 days and then decreases. Figure 9d demonstrates that both lines meet at a point after ten days. Figure 10a displays the relationship between ingestion class and regular smokers. After 20 days, both lines converge at a single point. The number of regular smokers increases until 40 days and then slightly decreases, while the ingestion class line grows initially until ten days in and then decreases. Figure 10b showcases the graphical behavior of regular and unusual smokers. Both lines meet at a single point after 50 days.

The line for regular smokers increases until 40 days and then decreases; the line for unusual smokers increases until 30 days and then decreases. Figure 10c depicts the relationship between all system components. This figure reveals a drastic decrease in the population of susceptible smokers to zero. The number of those in the ingestion class, unusual smokers, and regular smokers initially increases, then decreases to zero. Lastly, the number of ex-smokers exhibits an increase. Figure 10 portrays the above relation for $\sigma =0.89$. In Figure 11 and Figure 12, the same relation is shown for $\sigma =1$ as in Figure 9 and Figure 10. In Figure 12c, the susceptible smokers’ population decreases, stabilizing after some time. The number of ingestion class, unusual smokers, and regular smokers initially increases but gradually decreases. Finally, the number of ex-smokers exhibits an increase. In Figure 13, we can observe the impact of parameter $e_2$ on the fractional smoking system. A lower value of this parameter leads to a slower decrease in the susceptible population. Similarly, for the ingestion class, regular smokers, and unusual smokers, a lower value of $e_2$ results in a slower increase in these populations. However, as time progresses, the influence of parameter $e_2$ becomes minimal when these three system components decrease. In the case of ex-smokers, a lower value of $e_2$ leads to a slower increase in their population. Figure 14 illustrates the effect of parameter $e_3$ on the smoking model. For susceptible smokers, the ingestion class, and unusual smokers, the population decreases significantly for a low value of $e_3$. As for regular smokers, a slightly low value of $e_3$ substantially increases their population. Subsequently, as these populations decrease over time, the rate of decrease is low for a comparable low value of $e_3$. In the case of ex-smokers, the population increase rate is higher for a lower value of $e_3$. Figure 15 showcases the impact of parameter $e_9$ on the model. Evidently, this parameter significantly affects regular and ex-smokers—a slight change in its value results in a rapid effect. For a low value of $e_9$, the rate of increase in regular smokers is substantial. When the population decreases, the rate of decrease is low for a lower value of this parameter. In the case of ex-smokers, the rate of increase is high for a higher value of $e_9$. The study focused on administering an anti-nicotine medication over 35 days. This approach was chosen due to the potential risks associated with prolonged drug treatment and the optimal timing for vaccination likely occurring during the initial phases of an illness. Figure 16a and Figure 17a illustrate the portion of the non-smoking demographic that can transition into becoming smokers for $\sigma =1$ and $\sigma =0.95$. On the initial day, a notable decline was observed within this population. Following the implementation of the control, there was a minor uptick in the count of potential smokers by the 25th day, as contrasted with the pre-control period. This observation suggests that the decrease will likely persist if this population is subjected to the same control over an extended duration. Figure 16b and Figure 17b depict the demography of the ingestion class for $\sigma =1$ and $\sigma =0.95$. The population experienced a notable increase in the initial days, followed by a substantial decrease. Subsequently, after implementing control measures, the number of people in the ingestion class decreased from the initial days. Regarding the subset of the population characterised by occasional smoking, proactive control measures were put in place. These measures included distributing anti-smoking gum and enforcing government regulations prohibiting smoking in public areas. As illustrated in Figure 16c and Figure 17c, there was a significant increase in the initial days, followed by a slight decrease. After implementing these control measures, the population of unusual smokers showed a substantial decline. Figure 16d and Figure 17d depict the population of active smokers under controlled conditions and without such measures. Governmental prohibitions on smoking in public places and anti-nicotine medication therapy are the suggested strategies for controlling this population. From the beginning to the end of the simulation period, the number of active smokers decreased, according to the simulations. Additionally, the population under control measures demonstrated a more notable fall compared to the time before controls were implemented. Consequently, the applied measures produced positive outcomes in this instance. Based on the study’s findings, it is recommended to provide a range of control measures, including an anti-smoking education campaign, $w_1\left(t\right)$; anti-smoking gum, $w_2\left(t\right)$; anti-nicotine drug treatment, $w_3\left(t\right)$; and government prohibition of smoking in public spaces, $w_4\left(t\right)$. The simulations showed a significant rise in people quitting smoking for good, particularly if control mechanisms were put in place. This approach highlights the effectiveness of implementing control measures. Figure 18 and Figure 19 show the impact of control parameters on the smoking system by varying these parameters. If $w_1\left(t\right)\ne 0,w_2\left(t\right)\ne 0,w_3\left(t\right)=0,w_4\left(t\right)=0$, then the ex-smoker population significantly increase, while other state variables decrease in Figure 18. In Figure 19, $w_1\ne 0,w_2\left(t\right)=0,w_3\left(t\right)\ne 0,w_4\left(t\right)=0$, then we get a greater number of ex-smokers, where the other compartments decrease. Figure 20 illustrates the graphical representation of these parameters. We see how the smoking system has been solved more easily using BWM and how the system’s behaviour changes when we alter specific parameter values. By implementing certain control measures, we can also reduce the number of smokers in society.

We compare the solutions of the smoking model using BWM with other numerical techniques for $\sigma =1$, which are shown in

Table 2. Comparing the solutions of susceptible smokers at $\sigma =1.0$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$67.7456$	$67.7152$	$67.7455$	$67.7559$
2	$0.1$	$67.2366$	$67.1858$	$67.2364$	$67.2673$
3	$0.2$	$66.2175$	$66.1665$	$66.3705$	$66.2278$
4	$0.3$	$65.7074$	$65.6769$	$65.7073$	$65.6769$
5	$0.4$	$64.6870$	$64.6564$	$64.7380$	$64.6973$
6	$0.5$	$63.6665$	$63.6359$	$62.6663$	$63.6564$
7	$0.6$	$63.1565$	$63.1055$	$63.1562$	$63.1667$
8	$0.7$	$62.1373$	$62.0864$	$62.1371$	$62.1882$
9	$0.8$	$61.6283$	$61.5978$	$61.6281$	$61.6384$
10	$0.9$	$60.6119$	$60.5814$	$60.6118$	$60.6015$
11	$1.0$	$59.5981$	$59.5677$	$59.5979$	$59.8712$

,

Table 3. Comparing the solutions of the ingestion class at $\sigma =1.0$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$40.1735$	$40.1944$	$40.1734$	$40.1667$
2	$0.1$	$40.5199$	$40.5546$	$40.5192$	$40.4994$
3	$0.2$	$41.2091$	$41.2436$	$41.2091$	$41.2027$
4	$0.3$	$41.5518$	$41.5725$	$41.5517$	$41.5728$
5	$0.4$	$42.2327$	$42.2532$	$42.2328$	$42.2266$
6	$0.5$	$42.9071$	$42.9275$	$42.9080$	$42.9148$
7	$0.6$	$43.2417$	$43.2753$	$43.2416$	$43.2361$
8	$0.7$	$43.9051$	$43.9383$	$43.9051$	$43.8735$
9	$0.8$	$44.2337$	$44.2536$	$44.2346$	$44.2287$
10	$0.9$	$44.8845$	$44.9041$	$44.8854$	$44.8928$
11	$1.0$	$45.5260$	$43.5454$	$45.5259$	$45.3558$

,

Table 4. Comparing the solutions of unusual smokers at $\sigma =1.0$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$30.0238$	$30.0265$	$30.0247$	$30.0226$
2	$0.1$	$30.0720$	$30.0768$	$30.0719$	$30.0688$
3	$0.2$	$30.1727$	$30.1777$	$30.1728$	$30.1712$
4	$0.3$	$30.2251$	$30.2281$	$30.2250$	$30.2277$
5	$0.4$	$30.3340$	$30.3372$	$30.3339$	$30.3322$
6	$0.5$	$30.4485$	$30.4519$	$30.4486$	$30.4488$
7	$0.6$	$30.5079$	$30.5137$	$30.5079$	$30.5057$
8	$0.7$	$30.6307$	$30.6368$	$30.6305$	$30.6233$
9	$0.8$	$30.6942$	$30.6979$	$30.6942$	$30.6917$
10	$0.9$	$30.8254$	$30.8293$	$30.8252$	$30.8254$
11	$1.0$	$30.9622$	$30.9662$	$30.9622$	$30.9232$

,

Table 5. Comparing the solutions of regular smokers at $\sigma =1.0$, $k=5$ and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$20.0125$	$20.0140$	$20.0127$	$20.0120$
2	$0.1$	$20.0376$	$20.0401$	$20.0333$	$20.0361$
3	$0.2$	$20.0879$	$20.0904$	$20.0876$	$20.0874$
4	$0.3$	$20.1132$	$20.1147$	$20.1133$	$20.1147$
5	$0.4$	$20.1640$	$20.1655$	$20.1640$	$20.1634$
6	$0.5$	$20.2151$	$20.2166$	$20.2150$	$20.2156$
7	$0.6$	$20.2408$	$20.2434$	$20.2407$	$20.2402$
8	$0.7$	$20.2926$	$20.2951$	$20.2926$	$20.2899$
9	$0.8$	$20.3186$	$20.3201$	$20.3186$	$20.3180$
10	$0.9$	$20.3709$	$20.3725$	$20.3709$	$20.3714$
11	$1.0$	$20.4238$	$20.4254$	$20.4238$	$20.4093$

,

Table 6. Comparing the solutions of ex-smokers at $\sigma =1.0$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$15.0303$	$15.0340$	$15.0308$	$15.0291$
2	$0.1$	$15.0910$	$15.0971$	$15.0911$	$15.0874$
3	$0.2$	$15.2126$	$15.2187$	$15.2125$	$15.2114$
4	$0.3$	$15.2735$	$15.2772$	$15.2733$	$15.2772$
5	$0.4$	$15.3956$	$15.3992$	$15.3956$	$15.3943$
6	$0.5$	$15.5179$	$15.5216$	$15.5176$	$15.5191$
7	$0.6$	$15.5792$	$15.5853$	$15.5796$	$15.5779$
8	$0.7$	$15.7019$	$15.7081$	$15.7019$	$15.6957$
9	$0.8$	$15.7634$	$15.7671$	$15.7632$	$15.7621$
10	$0.9$	$15.8866$	$15.8903$	$15.8866$	$15.8878$
11	$1.0$	$16.0101$	$16.0138$	$16.0102$	$15.9766$

.

Table 7. Comparing the solutions of susceptible smokers at $\sigma =0.92$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$67.4534$	$67.4465$	$67.4532$	$67.4936$
2	$0.1$	$66.4774$	$66.5203$	$66.4774$	$66.5690$
3	$0.2$	$65.5605$	$65.6449$	$65.5603$	$65.5835$
4	$0.3$	$64.6734$	$64.7971$	$64.6734$	$64.7349$
5	$0.4$	$63.8069$	$63.9686$	$63.8070$	$63.9044$
6	$0.5$	$62.9564$	$63.1552$	$62.9564$	$62.9872$
7	$0.6$	$62.1192$	$62.3923$	$62.1193$	$62.1847$
8	$0.7$	$61.2938$	$61.6019$	$61.2936$	$61.2943$
9	$0.8$	$60.4789$	$60.8215$	$60.4799$	$60.5130$
10	$0.9$	$56.6738$	$60.0501$	$56.6739$	$59.7405$
11	$1.0$	$58.8778$	$59.2874$	$58.8778$	$58.9763$

,

Table 8. Comparing the solutions of the ingestion class at $\sigma =0.92$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$40.3719$	$40.3774$	$40.3720$	$40.3455$
2	$0.1$	$41.0329$	$41.0046$	$41.0329$	$40.9724$
3	$0.2$	$41.6484$	$41.5926$	$41.6483$	$41.6347$
4	$0.3$	$42.2386$	$42.1573$	$42.2395$	$42.2001$
5	$0.4$	$42.8100$	$42.7044$	$42.8100$	$42.7486$
6	$0.5$	$43.3657$	$43.2369$	$43.3666$	$43.3488$
7	$0.6$	$43.9075$	$43.7320$	$43.9075$	$43.8690$
8	$0.7$	$44.4365$	$44.2403$	$44.4366$	$44.4404$
9	$0.8$	$44.9535$	$44.7374$	$44.9536$	$44.9368$
10	$0.9$	$45.4590$	$45.2240$	$44.4591$	$44.4227$
11	$1.0$	$45.9533$	$45.7003$	$45.9533$	$45.8984$

,

Table 9. Comparing the solutions of unusual smokers at $\sigma =0.92$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$30.0519$	$30.0519$	$30.0519$	$30.04732$
2	$0.1$	$30.1474$	$30.1425$	$30.1473$	$30.1370$
3	$0.2$	$30.2421$	$30.2326$	$30.2420$	$30.2381$
4	$0.3$	$30.3382$	$30.3240$	$30.3380$	$30.3295$
5	$0.4$	$30.4366$	$30.4174$	$30.4345$	$30.4230$
6	$0.5$	$30.5374$	$30.5129$	$30.5372$	$30.5310$
7	$0.6$	$30.6408$	$30.6061$	$30.6406$	$30.6295$
8	$0.7$	$30.7469$	$30.7063$	$30.7470$	$30.7434$
9	$0.8$	$30.8558$	$30.8090$	$30.8557$	$30.8473$
10	$0.9$	$30.9675$	$30.9143$	$30.9676$	$30.9538$
11	$1.0$	$31.0819$	$31.0221$	$31.0820$	$31.0629$

,

Table 10. Comparing the solutions of regular smokers at $\sigma =0.92$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$20.0269$	$20.0272$	$20.0270$	$20.0249$
2	$0.1$	$20.0751$	$20.0729$	$20.0752$	$20.0705$
3	$0.2$	$20.1206$	$20.1163$	$20.1206$	$20.1193$
4	$0.3$	$20.1648$	$20.1586$	$20.1650$	$20.1616$
5	$0.4$	$20.2083$	$20.2001$	$20.2083$	$20.2033$
6	$0.5$	$20.2514$	$20.2412$	$20.2523$	$20.2496$
7	$0.6$	$20.2941$	$20.2800$	$20.2941$	$20.2905$
8	$0.7$	$20.3366$	$20.3206$	$20.3375$	$20.3362$
9	$0.8$	$20.3790$	$20.3610$	$20.3791$	$20.3768$
10	$0.9$	$20.4213$	$20.4014$	$20.4213$	$20.4173$
11	$1.0$	$20.4637$	$20.4418$	$20.4638$	$20.4579$

,

Table 11. Comparing the solutions of ex-smokers at $\sigma =0.92$, $k=5$, and $M=3$.

Sr.	t	BWM	ABM	Fde12	Forward Euler
1	0	$15.0652$	$15.0660$	$15.0652$	$15.0604$
2	$0.1$	$15.1816$	$15.1765$	$15.1816$	$15.1707$
3	$0.2$	$15.2911$	$15.2810$	$15.2912$	$15.2883$
4	$0.3$	$15.3973$	$15.3825$	$15.3974$	$15.3899$
5	$0.4$	$15.5012$	$15.4818$	$15.5013$	$15.4894$
6	$0.5$	$15.6035$	$15.5795$	$15.6036$	$15.5997$
7	$0.6$	$15.7045$	$15.6715$	$15.7043$	$15.6964$
8	$0.7$	$15.8045$	$15.7671$	$15.8044$	$15.8042$
9	$0.8$	$15.9037$	$15.8619$	$15.9038$	$15.8992$
10	$0.9$	$16.0021$	$15.9560$	$16.0022$	$15.9935$
11	$1.0$	$16.0999$	$16.0494$	$16.0999$	$15.0873$

show the comparison of solutions between BWM and other numerical techniques of the model above for $\sigma =0.92$.

Table 12. Comparing the solutions of smoking system between initial data and BWM at $\sigma =1.0$, $k=5$, $t=0$, and $M=3$, and providing the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE).

Component	BWM	Actual Data	Error	RMSE	MAE
Susceptible smokers	$67.7456$	68	$-0.2544$	$0.2544$	$0.2544$
Ingestion class	$40.1735$	40	$0.1735$	$0.1735$	$0.1735$
Unusual smokers	$30.0238$	30	$0.0238$	$0.0238$	$0.0238$
Regular smokers	$20.0125$	20	$0.0125$	$0.0125$	$0.0125$
Ex smokers	$15.0303$	15	$0.0303$	$0.0303$	$0.0303$

and

Table 13. Comparing the solutions of the smoking system between initial data and ABM at $\sigma =1.0$ and $t=0$, and providing the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE).

Component	ABM	Actual Data	Error	RMSE	MAE
Susceptible smokers	$67.7152$	68	$-0.2848$	$0.2848$	$0.2848$
Ingestion class	$40.1944$	40	$0.1944$	$0.1944$	$0.1944$
Unusual smokers	$30.0265$	30	$0.0265$	$0.0265$	$0.0265$
Regular smokers	$20.0140$	20	$0.0140$	$0.0140$	$0.0140$
Ex smokers	$15.0340$	15	$0.0340$	$0.0340$	$0.0340$

show RMSE and MAE values of the given system using BWM and ABM, respectively.

8. Conclusions

The research explores the innovative application of Bernoulli wavelets to effectively solve systems of any order. It begins with a comprehensive analysis of the convergence and numerical procedure, using the unique orthogonal characteristics of Bernoulli wavelets. These wavelets are then used to convert FDEs into algebraic equations, simplifying the numerical solving process. The study provides a detailed illustration of various dynamic behaviours for different FO, emphasizing the influence of parameters and derivative order on the behaviour of arbitrary-order smoking systems. In this study, it is observed that a lower value of $e_2$ results in a slower decline of the susceptible population and an increase in the number of regular and unusual smokers. This lower value also leads to slower population growth for ex-smokers. Furthermore, a lower value of $e_3$ causes a notable decline in susceptible individuals and those in the ingestion class, as well as in unusual smokers. Conversely, for regular smokers, a slightly lower value of $e_3$ leads to a significant increase. Additionally, it is illustrated that a lower value of $e_9$ significantly increases the growth rate of regular smokers and slows down the rate of decline when the population decreases. Conversely, a higher value of $e_9$ results in more rapid population growth for ex-smokers. An alternative numerical method known as ABM is introduced to demonstrate the precision and relevance of the suggested approach. The results are then compared with other numerical techniques like Fde12 and forward Euler to validate the approximation of the BWM. In this proposed model, four control variables are used: $w_1\left(t\right)$, $w_2\left(t\right)$, $w_3\left(t\right)$ and $w_4\left(t\right)$. Reducing the number of smokers and increasing the number of persons who permanently stop smoking are the two main objectives of the intervention. The simulation findings have led to the conclusion that the control variables employed have an effect consistent with the intended objectives. In conclusion, we observe how to solve the smoking system using BWM more smoothly, and if we change some parameters and control parameter values, then we see the system’s behaviour change. We also observe how to control smoking populations in society by taking some control measures. This observation will help health policymakers and scientists to address this issue. Future research will involve using other differential operators like AB, FF, etc., on this system, analyzing and applying other wavelets like Hermite, Laguerre, Bernstein, etc., on this model and numerically investigating it.