Intelligent Automation & Soft Computing DOI:10.32604/iasc.2022.017801 | |

Article |

Optimal Control and Spectral Collocation Method for Solving Smoking Models

1Department of Mathematics, College of Science, Taif University, Taif, 21944, Saudi Arabia

2Department of Mathematics, College of Science, Bisha University, Bisha, P.O. Box 61922, Saudi Arabia

*Corresponding Author: Amr M. S. Mahdy. Email: amattaya@tu.edu.sa

Received: 11 February 2021; Accepted: 10 June 2021

Abstract: In this manuscript, we solve the ordinary model of nonlinear smoking mathematically by using the second kind of shifted Chebyshev polynomials. The stability of the equilibrium point is calculated. The schematic of the model illustrates our proposition. We discuss the optimal control of this model, and formularize the optimal control smoking work through the necessary optimality cases. A numerical technique for the simulation of the control problem is adopted. Moreover, a numerical method is presented, and its stability analysis discussed. Numerical simulation then demonstrates our idea. Optimal control for the model is further discussed by clarifying the optimal control through drawing before and after control. Fractional request differential equations (FDEs) are usually used to display frameworks that have memory and exist in a few thermoelasticity models and organic standards. FDEs show the realistic biphasic decline of infection of diseases but at a slower rate. FDEs are more suitable than integer order ones in modeling complex systems, such as biological systems.

Keywords: Shifted second kind chebyshev; smoking model; stability; hamiltonian; lagrange multipliers; optimal control

Infectious diseases have a tremendous influence on human life. Every year billions of people suffer from or die due to various infectious diseases. Mathematical modeling is of considerable importance in epidemiology because it may provide understanding of the underlying mechanisms that influence the spread of a disease and thus may be used to offer control strategies. Many scientists explored the detection of illnesses and pests [1–72], wherein the idea that mathematical modeling helps in discovering diseases’ spread [17]. In 1766, Islamic scholars presented for the first time what was considered the beginning of modern epidemiology discovery to be since followed. Model smoking is one of the most famous and important model commonly used by researchers [8–14], [18–21], [22–36]. According to the World Health Organization [14], the global tobacco pandemic killed several million people. The cost of health care is increasing and its budgets are decreasing as developing economies are shrinking [17]. Smoking negatively affects overall health as well [1–4]. According to [1–3], more than five million individuals are killed every year from smoking tobacco, as such it is the main cause of preventable deaths. Serious illnesses caused by it include asthma, lung cancer, heart disease, and mouth ulcers. In [8–9], we discussed the smoking dynamics of the integer {of the integer what}. Moreover, [13] analyzed the above using the Homotopy analysis method. Four kinds of polynomials, Chebyshev [35], [57–59]: Chebyshev's model includes four kinds of polynomials. The outcomes of each and their applications can be found in several books. Moreover, many researchers use several of these polynomials in a single model [33,35]. However, the shifted Chebyshev polynomial of the second kind

In the present manuscript, we discuss the analytical solutions for alpha = 1, for the following order of smoking models [4,5,56]:

The initial conditions are as follows:

In Fig. 1, a schematic proposition graph is a tool that is applied to display the interrelation between model states and enables the application of graph-theoretic tools to discover novel features of the model.

Tab. 1 describes the parameters of the model.

The organization of the remainder of this paper is as follows. The stability of equilibrium points is studied in Section 2. Optimal control for the smoking model is discussed in Section 3. In Section 4, the second kind of Chebyshev polynomials and their properties are given. In Section 5, the numerical implementation is given. Finally, a conclusion is presented in Section 6.

2 Stability of Equilibrium Points

For the stability behavior of this model at E0=(1, 0, 0, 0, 0), we use transformations [36]:

U = 1-P, P = 1-U, L=L, S=S, Q=Q, R=R,

U*= −P*, L*=L*, S*=S*, Q*=Q*, R*=R*.

It follows that

The system (2.1) has

P1=(0, 0, 0, 0, 0), P2=(U*, L*, S*, Q*, R*), P3=(U**, L**, S**, Q**, R**),

P4=(U***, L***, S***, Q***, R***) and P5=(U****, L****, S****, Q****, R****),

where, U*=1-P*, U**=1-P**, U***=1-P*** and U****=1-P****.

Through Taylor approximation, the linearized form of the model is

where X=U-0, Y=L-0, Z=S-0, V=Q-0, W=R-0, and

The stability of (2.1) is the same as the linearized stability (2.3). The stability of (2.3) depends on eigenvalues B. The point P1 is asymptotically locally stable if eigenvalues are real negative parts; conversely, P1 is unstable if at least one eigenvalue of B is a nonnegative real part. For details on stability see [56,63].

3 Optimal Control for Smoking Model

Consider the state presented (1.2), in R5, with control functions admissible [28–32]:

where Tf is the final time, and uV(.) and uR(.) are functions controls.

The objective function is defined as

where A, B, and C represent the number of occasional smokers, the rate of contact between smokers and quitters, and temporarily who regain support to smoking, and rate of smoking quitting.

We minimize the objective function as follows [28–32]:

which is subjected to the constraint

The following initial conditions are satisfied:

OCP is defined, and we consider the following modified objective (cost) function:

where the Hamiltonian and control smoking objective functions are defined as

From (3.5) and (3.7), the conditions, necessary and sufficient for OPC are

where λk, k = 1, 2, 3, 4, 5 are Lagrange multipliers. Eqs. (3.9)–(3.10) clarify the conditions of the Hamiltonian for the OPC.

We construct a theorem similar to that presented in [28–32], [44–47].

Theorem 1.

If uV and uR are optimal controls with states corresponding to

Co-state equation

With condition transversality:

Optimality conditions

Therefore, the function controls

For more on optimal controls for solving models, see [29–32,44,47].

4 Properties of the Second Kind of Chebyshev Polynomials

4.1 Second Kind of Chebyshev Polynomials

Chebyshev polynomials Zn(y) of the second kind are rectangular polynomials of stage n in x presented on the interval [−1, 1] [34,35,57–59]:

Polynomials have rectangular with rating to the products indoor

where

Zn(y) can be generated by recurring relations

Zn(y) = 2y Zn−1(y) − Zn−2(y), n = 2, 3, …, n, with Z0(y) = 1. Z1(y) = 2y.

The analytical form Zn(y) of stage n is given by [34]:

Here,

4.2 Second Kind of Shifted Chebyshev Polynomials

On the interval y ∈ [0, 1],

The following inner product is orthogonal on the interval [0, 1]:

with weight function

The following formula represents the analytical

The solution of this model can be written as

Let g(y) be a square integral in [0, 1], and then the second kind of shifted Chebyshev polynomials can be represented as follows:

The coefficients aj, j = 0, 1, …, are expressed as

or

We use only the (r + 1) terms. Then,

Using the practice to construct an integral collocation style then

By integrating (4.8), we can obtain the following:

From (4.8) and (4.12), we then have

We presently register Eqs. (4.9)–(4.13) at (r + 1) points yp, p = 0, 1, …, r as follows:

where

5 Style Integral Collocation for Resolving the Nonlinear Smoking Model

In this section, we present the model's implementation through the following steps:

(i)We first approximate the function using Eqs. (4.8)–(4.14) with m = 5 , as follows:

where

Then, the nonlinear smoking model (1.1) is transformed to

Now, we collocate Eq. (5.2) at (r + 1 = 6) points ty, y = 0 − 5; as follows:

The roots of shifted Chebyshev polynomial

(ii) By putting the initial Eqs. (1.2) into (5.1), we can obtain five equations.

Eqs. (5.1) and (5.3) obtained in step (ii) represent nonlinear algebraic system equations.

(iii) We use Newton's iteration to resolve the system and solve for the unknowns.

Figs. 2–4 show the nonlinear smoking model's behavior before control.

Figs. 5–8, show the show the nonlinear smoking model's behavior after control.

In this manuscript, a mathematical nonlinear smoking model is studied. The optimal control of this smoking model is discussed. The stability of the equilibrium point is calculated, and a schematic of the proposed model is presented. Moreover, the integral collocation style is used to obtain the approximate solutions of the model. ICM using the shifted Chebyshev polynomials of the second kind is a new technique for solving these problems. Under the application with the necessary optimality conditions, we have studied the problem control with numerical techniques for the simulation. Moreover, a numerical method and its stability is discussed.

Acknowledgement: The authors are grateful for Taif University. Taif University researchers support project number (TURSP-2020/160), Taif University, Taif, Saudi Arabia.

Funding Statement: This manuscript has been funded through “Taif University's Researchers Supporting Project Number (TURSP-2020/160), Taif University, Taif, Saudi Arabia”.

Conflicts of Interest: The authors declare that there are no conflicts of interest.

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