Article
Kyungpook Mathematical Journal 2020; 60(4): 753765
Published online December 31, 2020
Copyright © Kyungpook Mathematical Journal.
The Multistep Adomian Decomposition Method for Approximating a Fractional Smoking Habit Model
Mohammad Zuriqat^{*}, Asad Freihat
Department of Mathematics, Al alBayt University, P. O. Box 130095 Mafraq, Jordan
email : moh_zur@hotmail.com
Department of Applied Science, Ajloun College, AlBalqa Applied University, Ajloun 26816, Jordan
email : asadfreihat@yahoo.com
Received: June 10, 2015; Accepted: June 21, 2019
Abstract
Smoking is one of the main causes of health problems and continues to be one of the world’s most significant health challenges. In this paper, we use the multistep Adomian decomposition method (MSADM) to obtain approximate analytical solutions for a mathematical fractional model of the evolution of the smoking habit. The proposed MSADM scheme is only a simple modification of the Adomian decomposition method (ADM), in which ADM is treated algorithmically with a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding problems. A comparative study between the new algorithm and the classical RungeKutta method is presented in the case of integerorder derivatives. The solutions obtained are also presented graphically. The results reveal that the method is effective and convenient for solving linear and nonlinear differential equations of fractional order.
Keywords: fractional differential equations, multistep Adomian decomposition method, smoking model, numerical analysis, mathematical fractional model
1. Introduction
Fractional order ordinary differential equations, as generalizations of classical integer order ordinary differential equations, are increasingly used to model problems in fluid flow, mechanics, viscoelasticity, biology, physics, engineering and other applications [17, 9]. Fractional differential equations are the result of mathematical modeling of complex processes and phenomena. The most fundamental characteristic of these models is their nonlocal characteristic which does not exist in the differential operators of integer order. This property means that the next aspect of a model relates not only upon its present state but also upon all of its history. As a consequence, there is an already very large, and still growing, number of scientific and engineering problems involving fractional derivatives. The fractional calculus (that is, derivatives and integrals of any real or complex order) will perhaps be the calculus of the twentyfirst century [3, 12, 22, 24]. The solutions of fractional differential equations are quite involved. In general, there exists no method that yields exact solutions to fractional differential equations. Only approximate solutions can be derived. Several methods have been used to solve fractional differential equations, such as Laplace transform method [18, 20], Fourier transform method [14], homotopy perturbation method [19, 25], homotopy analysis method [2, 28, 27], Adomian decomposition method [1, 5, 13, 21] and differential transform method [6, 7, 8]. In this paper, we investigate the applicability and effectiveness of ADM when treated as an algorithm with a sequence of intervals (i.e. time step) for finding accurate approximate solutions to a timefractional epidemic model for the habit of smoking in a community in Spain. This modified method is named as the multistep Adomian decomposition method. It can be found that the corresponding numerical solutions obtained by using ADM are valid only for a short time. While the ones obtained by using MSADM are valid and more accurate over a longer time. The agree closely with the RK45 numerical solutions in the case of integerorder systems.
Smoking is the leading cause of preventable death, and is estimated to kill more than 5 million people worldwide each year. This number is expected to grow. Smoking or tobacco is a known or probable cause of cancers of the oral cavity, larynx, lung, oesophagus, bladder, pancreas, renal pelvis, stomach, and cervix. Smoking is also a cause of heart disease, strokes, peripheral vascular diseases, chronic obstructive lung diseases and other respiratory diseases, and lowbirth weight babies [15].
There have been several attempts [16, 26], since the 2000s, to mathematically model the effort to give up smoking. In 2000, CastilloGarsow et al. [4] proposed a simple mathematical model for giving up smoking. The fact that epidemic models consist of a system of nonlinear differential equations underlines the importance of having reliable methods for solving them. This type of model can be integrated using any standard numerical method. However, it is known that these algorithms are subject to problems such as numerical instabilities, oscillations or false equilibrium states. This means that the numerical solution may not correspond to the real solution of the original system of differential equations. This is the reason why we are interested in obtaining a continuous solution in the form of an analytical approximation to the real solution. The epidemic model for smoking is a system of nonlinear differential equations without closed solution. The interest of this fractional model is that it has been able to describe the general evolution of the spread of smoking. It was constructed using real data for the initial values and for the parameters of the system. Constant population is assumed by taking birth and death rates equal and different from zero. The structure of this paper is as follows. In Section 2, we present some necessary definitions and notation related to fractional calculus. In Section 3, we construct the general form of MSADM for a system of fractional differential equations. In Section 4, we describe the MSADM of system of a timefractional epidemic model for smoking in a community and numerical simulations are presented graphically. Finally, we make our conclusions in Section 5
2. Preliminaries
In this section, we introduce the linear operators of fractional integration and fractional differentiation in the framework of the RiemannLiouville and Caputo fractional calculus.
A function
Hence, we have the following properties [9, 17]
3. Multistep Adomian Decomposition Method
The ADM is used to provide approximate solutions for nonlinear problems in the form of convergent series with easily computable components. It has been shown that the approximated solution obtained by this method are not always valid for large
subject to the initial conditions
where
where
for all
We employ the Adomian decomposition method to solve the systems of equations (3.4) [13]. For all
and
where
We set
and
In order to determine the Adomian polynomials, we introduce a parameter λ, so (3.7) becomes
Let
where
In view of (3.11), (3.12) we get
Hence (3.9), (3.13) lead to the following recurrence relations:
We can approximate the solution
For the convergence of the above method, if system (3.1) admits a unique solution, then the method will produce the unique solution and If the system (3.1) does not admit a unique solution, the decomposition method will give a solution among many (possible) other solutions [1]. The solutions of system (3.1) in each subinterval
and the solution of system (3.1) in the interval
where
4. Numerical Results
This model was presented in [10, 11, 23] to describe and predict the evolution of the habit of smoking in Spain and to quantify the impact of Spanish smokefree policies. Smoking has traditionally been modeled by ordinary differential equations. The following system of ordinary differential equations models the dynamics between the different subpopulations considered.
The subpopulations included in the model are:
for any instant of time. The objective of the present paper is to use the MSADM to obtain the approximate solution of the following timefractional epidemic model for smoking:
subject to the initial conditions
In order to perform the resumption methods, we set the values of the parameters as in [4]:
In order to solve the system (4.3), (4.4), we define the nonlinear terms by
where
and
So in this case we have to satisfy the initial condition at each of the subintervals. Accordingly, the initial values will be changed for each subinterval, i.e.
Where
The solutions of system (4.3), (4.4) in each subinterval
and the solution in the interval
System (4.3) with transformed initial conditions were solved analytically using the MSADM and numerically using the classical Runge Kutta method in the case of integerorder derivative. To demonstrate the effectiveness of the proposed algorithm as an approximate tool for solving the nonlinear system of fractional differential equations (4.3) with initial conditions (4.4) for larger t, we use a small time step. We assume that the optimal campaign continues for 200 days, so apply the proposed algorithm on the interval [0,200]. We choose to divide the interval [0, 200] into subintervals with time step

Figure 1. Comparing the result of the MSADM; dotted line, the RK4; solid line, the numerical solution of the system with
${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}={\alpha}_{4}=1$

Figure 2. Phase plot of
n(t),s(t),c(t) ande(t) , with
${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}={\alpha}_{4}=0.95$

Figure 3. Phase plot of
n(t),s(t),c(t) ande(t) , with
${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}={\alpha}_{4}=0.001$
4. Numerical Results
The analytical approximations to the solutions of the model for the habit of smoking habit are reliable and confirm the power and ability of the MSADM as an easy device for computing the solution of nonlinear problems. In this paper, a fractional differential model of smoking is studied and its approximate solution is presented using a MSADM. Comparisons of the results obtained by using the MSADM with that obtained by the classical RungeKutta method in the integer case reveal that the approximate solutions obtained by ADM are only valid for a small time, while the ones obtained by MSADM are highly accurate and valid for a long time. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability. Finally, the recent appearance of nonlinear fractional differential equations as models in fields such as science and engineering motivates investigating various solutions methods for such equations. We hope that this work is a step in this direction.
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