Analytical analysis of fractional-order sequential hybrid system with numerical application

We investigate a general sequential hybrid class of fractional differential equations in the Caputo and Atangana–Baleanu fractional senses of derivatives. We consider the existence and uniqueness of solutions and the Hyers–Ulam (H-U) stability for a general class. We use the Banach and Leray–Schauder alternative theorems for the existence criteria. With the help of nonnegative Green’s functions, the fractional-order class is turned into m-equivalent integral forms. As an application of our problem, a fractional-order smoking model in terms of the Atangana–Baleanu derivative is presented as a particular case.


Introduction
Mathematical modeling of dynamical systems and their numerical simulations are widely studied in science and engineering. One of the useful and mostly studied approaches for generalizing the classical models uses the fractional-order operators. The fractional-order operators have a long history from local to nonlocal and from singular to nonsingular kernels. These aspects were recently highlighted in some useful articles. For details, we refer the readers to [1][2][3][4].
Fractional-order operators have recently been researched in engineering and science for modeling system dynamics. In the literature, singular and nonsingular kernels are recently well studied. It is difficult to say which one is the greatest right now, but academics always examine several operators for new applications and features. For details, we refer the researchers to [5][6][7].
The numerical techniques play an important role in the study of dynamical models. For the fractional-order operators, recently, some numerical techniques were developed and applied. For example, the readers can see [8][9][10][11][12][13][14]. Using various methodologies, a novel class of mathematical modelings based on hybrid fractional differential equations with hybrid or nonhybrid boundary value conditions has attracted the interest of numerous academics. Nonhomogeneous physical phenomena that occur in their form can be modeled and described using fractional hybrid differential equations. Hybrid differential equations are significant because they incorporate a variety of dynamical systems as special instances. The derivative of an unknown function hybrid with nonlinearity is included in this family of differential equations. In addition, hybrid differential equations can be found in several subjects of mathematics and physics, such as the deflection of a curved beam with constant or varying cross-section, a three-layer beam, electromagnetic waves, or gravity-driven flows, and so on. For details, we refer the readers to [15][16][17][18][19][20][21][22][23][24][25].
The general classes of the fractional-order differential equations (FDEs) were considered by experts. This area is still open for sequential fractional differential equations, hybrid FDEs, mixed fractional functional equations, and many more. Dhage [26][27][28] initiated hybrid FDEs and divided them into two subclasses called the linear and quadratic differential equations. More relevant studies on the hybrid FDEs can be found in [29][30][31][32][33][34][35][36]. In this paper, we present a system of hybrid sequential FDEs with two different fractional operators, the Caputo and Atangana-Baleanu operators. Our presumed system of hybrid sequential FDEs with initial and boundary conditions is . . , m) satisfy the Carathéodory conditions and are continuous functions. The c D α i are in the Caputo sense, whereas ABC D i , i = 1, 2, . . . , m, are in the ABC-sense of fractional derivatives. This sort of general sequential hybrid problems have not been studied in the literature. To know whether such problems can have solutions and applications, we consider the existence, uniqueness, stability, and applications in the dynamical systems. Dhage [26][27][28] and the references therein give more information to the readers on the theory and applications of the problem. We use the fixed point approach for the theoretical analysis and Euler discretization technique in application aspects. For details of the nonlinear models and their simulations, we refer the readers to [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54].

Basic definitions of fractional calculus
The concept of a nonsingular kernel was given by Caputo and Fabrizio [55] by replacing the singular kernel by exponential function. This work was then studied for some essential properties in [56]. Later on, Atangana and Baleanu [57] modified the concept of a nonsingular kernel with the replacement of the exponential kernel by the Mittag-Lefflerkernel. They called the new fractional-order differential operator the Atangana-Baleanu fractional derivative, which was recently well studied by many authors. We refer the readers to some related work on these operators and its applications in [55][56][57] and the references therein.

Definition 1.1 ([57])
The ABC-fractional differential operator on ψ ∈ H 1 (a, b), b > a, for where B( 1 ) is a normalizer function satisfying B(0) = B(1) = 1. If the function does not belong to H 1 (a, b), b > a, then the fractional-order derivative of order 1 ∈ [0, 1] has the form where H 1 (a, b) is the set of functions with continuous first derivatives.
The ABC and ABR are related to each other by the following relation.
In our results, we will need to the following result.

Existence of solution
In this section, we obtain the existence of a solution for the suggested hybrid FDE (1) with the help of fixed point technique.
In this paper, we consider the Banach space The G i (t, s) and H i (t, s), i = 1, 2, . . . , m, are given in (9) and (10), respectively. By (15) the fixed points of the operators T i give the solutions of the hybrid system (1). By (9) and (10) the functions G i (s, t) and H i (s, t) are clearly positive operators for both t ≤ s and t ≥ s, for t, s ∈ (0, 1] and i = 1, 2, . . . , m. and for η i < 1, i = 1, 2, . . . , m. Then (1) has a unique solution.
Similarly, for v i ∈ S η i and t ∈ [0, k], we have Furthermore, for t ≥ s, by (9) we have and for t ≤ s, we have Now, consider the Green's functions H i (t, s) given by (10). For the case t ≥ s, we have and for t ≤ s, we have With the help of (15), for t ≥ s, we have and for t ≤ s, we have This implies T i S η i ⊂ S η i . Further, assuming that u l , u j ∈ C([0, k], R e ) and k ≥ 1, for t ≥ s ∈ [0, k], we get and for t ≤ s ∈ [0, k], a calculation for i = 1, 2, . . . , m leads to If η i < 1, where η i are given by (18), then T i are contraction operators, and by Banach's fixed point theorem the sequential hybrid fractional-order system (1) has a unique solution, represented by the fixed points of T i .

Theorem 2.3 Under the assumptions of Lemma 2.2, the sequential hybrid system of FDEs
(1) has a solution.
Proof In Lemma 2.2, we proved that T i are bounded. Furthermore, let t 1 , t 2 ∈ [0, k] with t 2 > t 1 and k ≤ 1. In the case t ≥ s, we have This implies that T i u(t 2 ) → T i u(t 1 ) as t 2 → t 1 . This implies |T i u i (t 2 ) -T i u i (t 1 )| → 0 as t 2 → t 1 . Hence T i are equicontinuous operators for t ≥ s. The case t ≤ s is similar, and thus we omit it. Next, for any u ∈ {u ∈ C([0, k], R e ) : u = T i (u), for ∈ [0, 1]}, we have for i = 1, 2, . . . , m. With the help of (31), (32), and (33) we have for i = 1, 2, . . . , m. Hence the requirement of the Leray-Schauder alternative theorem is ensured, and therefore the system of sequential hybrid FDEs (1) has a solution.

H-U-stability
Here we consider the H-U stability of system (15). The following definition plays a vital role in the stability.

Definition 3.1
The fractional-order integral system (15) is H-U-stable if for some ζ i > 0, there exist i > 0, for each solution u i with there areū i (t) of the operators system (15) with such that for all i = 1, 2, . . . , m.

Numerical scheme
Let us consider where P(0) = P 0 . Applying the AB fractional integral, we get Replacing (t) by t n+1 , we have and H t, By applying the Lagrange polynomial we further have Now solving the integrals, we get Replacing the value of H(t, P(t)) by the functions, we obtain the following numerical scheme: -G 1 (t i-1 , P i-1 ) (n + 1i) 1 +1 -(n + 1i + 1)(ni) 1 , In Fig. 1, we present a graphical representation of the simulation of the potential smokers P(t) in (43) for the fractional orders 1.0, 0.99, 0.98, 0.97. They increase up to 100 days and then decrease to a certain value. Comparing the results of fractional orders with integer orders, we see that the fractional-order results get closer to the classical results on values of the fractional orders closer to 1. Figures 2 and 3 show the computational results for the light L and smokers S.
In Fig. 4, a comparative analysis is given for the numerical simulations of the Q class of the smoking model (43). The final Fig. 5 shows a join solution of the model for order 1.

Conclusions
We considered a general class of fractional-order differential equations (FDEs). This area is still open for consideration of the sequential fractional differential equations, hybrid FDEs, mixed fractional functional equations, and many more. The hybrid FDEs consist of two subclasses called the linear and quadratic differential equations. In this paper, we aimed to present a system of hybrid sequential FDEs with two different fractional operators, the Caputo and Atangana-Baleanu operators. We studied the existence and uniqueness of a solution. We have observed that some essential conditions are required for the existence of a solution of the fractional-order hybrid problem (1). There are two basic reasons for the importance. We have studied in the literature that the sequential hybrid class of FDS have not been considered for the presumed class for the existence of solution and stability analysis. Among them, one is a combination of the fractional derivative operators (the Caputo fractional differential operator and the Atangana-Baleanu fractional operator), whereas the second importance of the problem is the coupling of n FDEs with complex boundary conditions. This can further motivate the readers to the combination of other operators and make more research problems for the initial and boundary conditions. The problem is converted into its equivalent integral form using Green's functions. Then the H-U stability is illustrated. Mathematical modeling of dynamical systems and their numerical simula- tions are considered as an application of the work. This aspect of the paper consists of a fractional-order smoking model studied for the numerical analysis. The numerical results are illustrated by some graphics based on our numerical scheme for model (43).