SOLUTION OF AMBARTSUMIAN DELAY DIFFERENTIAL EQUATION IN THE q-CALCULUS

The Ambartsumian equation in view of the q-calculus is investigated in this paper. This equation is of practical interest in the theory of surface brightness in the Milky Way. Two approaches are applied to obtain the closed form solution. The first approach implements a direct series assumption while the second approach is based on the Adomian decomposition method. The two approaches lead to a unique power series of arbitrary powers. Furthermore, the convergence of the obtained series is theoretically proven. In addition, we showed that the present solution reduces to the results in the relevant literature when the quantum calculus parameter tends to 1.


Introduction
In regular calculus, we usually use the limits to calculate the derivatives of any given real functions. While the quantum calculus (q-calculus) provides the derivatives without implementing limits. Euler obtained the basic formulae in q-calculus in the eighteenth century. However, the notion of the definite q-derivative and q-integral were introduced by Jackson [1] to the first time. In the present time, there is a great interest in the applications of the q-calculus in various fields such as mathematics, number theory, and combinatorics [2]. Besides, Ernst [3,4] pointed out that the majority of scientists who use q-calculus are physicists. In addition, Baxter [5] introduced the exact solutions of several models in Statistical Mechanics. Also, Bettaibi and Mezlini [6] obtained the solutions of some q-heat/q-wave equations. In the literature [7][8][9][10][11][12], several interesting results for the q-calculus have been discussed by several authors. This paper considers a q-calculus model of the Ambartsumian delay equation (ADE) in the form: where ρ > 1 and λ is a constant. The system (1-2) is a generalized form of the standard ADE which describes the surface brightness in the Milky Way [13]. When q → 1, the system (1-2) was investigated by the authors [14][15]. However, the fractional model was solved in [16] using the homotopy transform analysis method by means of the Caputo's definition. Here, we consider the q-derivative to deal with the present model. In the literature, there are several analytical methods to deal with the system (1-2) such as the Adomian decomposition method (ADM) [17][18], the homotopy perturbation method (HPM) [19][20], and the homotopy analysis method (HAM) [16].
In this paper, two different approaches are suggested to analytically solve the system (1-2). The first approach is a regular power series approach while the second is based on the ADM. Such approaches are preferred here, especially, in proving the convergence of the resulting series solution. The paper is organized as follows. The main aspects of the q-calculus are presented in Section 2. In addition, a basic lemma for the integrals arise from the ADM is to be proved in Section 2. Sections 3 discusses the application of the q-calculus to solving the current model. Section 4 is devoted to the application of the ADM. In addition it will be shown that the present solution reduces to that one in the literature as q → 1. Section 5 includes an analysis of convergence. Finally, section 6 outlines the conclusions.

Preliminaries
Let q ∈ R and n ∈ N, then [n] q is defined as (first chapter in [21]) and as q → 1, we have The q-factorial [n] q ! of a positive integer n is given by and as q → 1, we have if f is differentiable at t, and we have at t = 0 that According to (2.5) we have The definite Jackson q-integral is defined by and hence, In order to apply the ADM on the system (1.1-1.2), we need to introduce and prove the following lemma.

Direct series solution
In order to solve Eq. (1.1), we assume the solution in the series form: and therefore [n] q a n t n−1 , [n] q a n t n−1 , [n + 1] q a n+1 t n . [n + 1] q a n+1 t n = − ∞ n=0 a n t n + 1 ρ ∞ n=0 a n t ρ n , = − ∞ n=0 a n t n + ∞ n=0 1 ρ n+1 a n t n , [n + 1] q a n+1 − 1 ρ n+1 − 1 a n t n = 0, which requires that (3.5) [n + 1] q a n+1 − 1 ρ n+1 − 1 a n = 0. Therefore (3.6) a n+1 = ρ −(n+1) − 1 [n + 1] q a n , n ≥ 0 From (3.6), we have This n-term coefficient can expressed in terms of the q-factorial [n] q ! as Thus y(t) = a 0 + ∞ n=1 a n t n , Applying the initial condition (1.2) on (3.9), yields a 0 = λ. Hence, the closed-form solution of the system (1.1-1.2) is finally given by which is the solution of ADE in the q-calculus. Moreover, the solution (3.10) as q → 1 reduces to which is the same closed form solution obtained by the authors [14] for the standard model of Ambartsumian equation.

Application of the ADM
Integrating Eq. (1.1) and based on Eq. (2.10), we have Following the ADM [17][18], we assume that and hence, where Lemma 1 is implemented to calculate the involved integral. Similarly, at k = 2, we have Proceeding as above, we obtain From Eq. (2.3), we can rewrite Eq. (4.6) as Thus which is the same closed form series solution that was obtained in the previous section. The series (3.11) which is equivalent to (4.8) will be proved for convergence in the next section.

Analysis of convergence
In order to proving the convergence of (3.11), we assume that Accordingly, we obtain the theorem below.

Theorem 1:
The radius of convergence of the series (3.11) is

Proof:
Assume that µ is the radius of convergence. Therefore, we have from (5.1) and the ratio test that .
Inserting (5.3) into (5.2), we obtain which completes the proof, where the following property: was implemented in deducing (5.5). It is noticed from (5.6) that as q → 1 then 1 µ → 0. Hence, the series (3.11) has an infinite radius of convergence at such special case which is in full agreement with the results obtained by [14] for the standard ADE.

Conclusion
In this paper, the quantum calculus was applied to generalize the Ambartsumian equation. The resulting q-differential equation was analytically solved via the power series approach and the ADM. The convergence of the obtained power series was theoretically proven. The implemented approaches led to the same closed form series solution. In addition, we showed that the present solution reduces to the results in the literature when the quantum calculus parameter tends to 1. Finally, the present work can be further extended to explore several physical models in view of the q-calculus.
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Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.