A study of fractional order Ambartsumian equation involving exponential decay kernel

Abstract: Recently, non-singular fractional operators have a significant role in the modeling of realworld problems. Specifically, the Caputo-Fabrizio operators are used to study better dynamics of memory processes. In this paper, under the non-singular fractional operator with exponential decay kernel, we analyze the Ambartsumian equation qualitatively and computationally. We deduce the result of the existence of at least one solution to the proposed equation through Krasnoselskii’s fixed point theorem. Also, we utilize the Banach fixed point theorem to derive the result concerned with unique solution. We use the concept of functional analysis to show that the proposed equation is Ulam-Hyers and Ulam-Hyers-Rassias stable. We use an efficient analytical approach to compute a semi-analytical solution to the proposed problem. The convergence of the series solution to an exact solution is proved through non-linear analysis. Lastly, we present the solution for different fractional orders.


Introduction and motivation
Ambartsumian derived the standard Ambartsumian equation (SAE) [1]. The absorption of light by interstellar matter has been defined in this equation. In the theory of surface brightness in the Milky Way, the Ambartsumian delay equation is used. We consider the fractional Ambartsumian equation (FAE) in this paper as [2]: In the applied analysis, we investigate two types of solutions the analytical and the numerical, for which different analytical and computational techniques are used respectively. To obtain the analytical and the numerical solutions for FDEs is of great interest among the researchers. Therefore, the researchers introduce different methods to solve FDEs. Padey et al., used Homotopy analysis Sumudu transform method to solve the third order dispersive PDE under fractional operator [30]. Collocation method was used to solve nonsingular fractional order differential equations by Dumitru with his co-authors [31]. In these methods, LADM is the best one for solving nonlinear FDEs. The Laplace transform method and the Adomian decomposition method are combined to form LADM. Like Runge-Kutta and collocation methods LAMD doesn't require any predefined size and discretization of data which require extra memory and time-consuming process. These methods are expensive. Moreover, the homotopy perturbation methods needed the auxiliary parameters which control both the technique and the solution. While in LADM, neither discretization of date nor auxiliary parameters are required [32]. It produces the same solution generated by the other methods. Therefore, we considered LADM, an ideal for the solution of the proposed equation. The comparison between ADM and LADM is available in [33]. Some of the applications of LADM are also available in [34,35]. These motivated us to study the Ambartsumian equation by the aforementioned fractional derivative. The Ambartsumian equation under the Caputo-Fabrizio fractional derivative is given by In the current article, we study the qualitative and quantitative aspects of the Eq (1.2). We use fixed point results for qualitative analysis of the considered equation. For quantitative approach, we utilize an efficient and accurate analytical method (LADM). We prove the convergence of the suggested method via nonlinear analysis.

Preliminaries
Let For the sake of simplicity denote the exponential kernel as Definition 2.2.
[7] The Caputo-Fabrizio fractional integral is defined as is given by 3. Existence and stability theory

Existence theory
In this section, we use fixed point theory approach to confirm the existence of solution of the proposed equation under the nonsingular fractional derivative. Consider the proposed model as To derive the existence and uniqueness results, we define a Banach space B = C [P, R]; where P = [0, T] and 0 ≤ t ≤ T < ∞. We define a norm for Banach space as follows Let us define an operator T : We impose growth and Lipschitz condition on Q as • Under the continuity of Q, for K Q > 0, we define Q : Theorem 3.1. Assume that the conditions (3.5) and (3.6) hold. Then there is at least one solution of the Eq (3.2), if be a convex and closed set. Now, we define two operators as such that GA(t) + HA(t) = T A(t). First, we prove that G is a contractive mapping, for this let A, A ∈ B, one has By the hypothesis ( It follows that the operator H is bounded. Next, to show the equi-continuity of H consider t 1 < t 2 , one has when t 1 → t 2 , then |HA(t 1 ) − HA(t 2 )| → 0. Also the continuity and boundeness of H implies that Thus H is completely continuous by the "Arzelá-Ascoli theorem". Thus, by "Krasnoselskii's fixed point theorem [40]", our proposed equation has at least one solution.
Now, we show that our proposed equation possess at most one solution. To achieve this goal, we will use Banach contraction theorem.
By hypothesis (3.9), T is contraction. Thus by "Banach contraction theorem", the proposed equation has unique solution.

Stability theory
The stability of mathematical models is an important aspect of the DEs. To study the different kinds of stability, one can find the most interesting kind is Ulam-Hyers stability introduced by Ulam [36], further generalized by Rassias [37]. The more general form of this stability is known as Ulam-Hyers-Rassias stability. This stability is studied by many authors in the last few years [38,39]. Therefore, in this article, we also use UH stability for the mentioned problem.

Ulam-Hyers stability
there exists a unique solution A(t) ∈ B of the proposed equation with U q > 0 such that Remark 3.4. We perturb our proposed equation by taking a small perturbation ∆(t) ∈ C [P, R] which depends on A and satisfies the following The corresponding perturbed equation is given by (3.12) Now, we prove an important Lemma, which will be used for further analysis.
Lemma 3.5. The following result holds for the perturbed Eq (3.12) .
Proof. Applying the fractional integral to the perturbed Eq (3.12), we get

Now using Eq (3.4), we have
This complete the required result.
Theorem 3.6. Under the above Lemma 3.5, the solution of the proposed equation is Ulam-Hyers stable and also generalized-Ulam-Hyers stable if U q < 1.

This implies that
Hence, the proposed problem is Ulam-Hyers stable. Consequently, it is generalized Ulam-Hyers stable.
there exists a unique solution A(t) ∈ B of the proposed equation with U q > 0 such that , then the proposed equation is generalized Ulam-Hyers-Rassias stable.
Remark 3.8. We perturb our proposed equation by taking a small perturbation ∆(t) ∈ C [P, R] which depends on A and satisfies the following . Lemma 3.9. The following result holds for the perturbed Eq (3.12)

16)
Proof. The proof is similar to the above Lemma 3.5.
Theorem 3.10. The solution of the proposed problem is Ulam-Hyers-Rassias stable if U q < 1.

This implies that
Thus the proposed equation is Ulam-Hyers-Rassias stable. Consequently, it is generalized Ulam-Hyers-Rassias stable.

Solution of the proposed equation and simulations
In this section, we will deduce a solution of (1.2) by an efficient analytical technique called Laplace transform. Via Laplace transform, we will find the concerned equation for semi-analytical solution. Now applying the Laplace transform on (1.2), we get To get approximate solution, we assume the solution in series form as which gives The recursion formula (3.18) gives the following result , (3.20) and so on. Now applying inverse Laplace transform to (3.17) and (3.19), we get So the first two term of the infinite series solution is given by Next we prove by fixed point theory that the obtained series solution is a convergent series, i.e., it uniformly converges to the exact solution.
Consider G α (A) = Ā ∈ B : A −Ȳ < α and assume that A ∈ B, then A j ∈ G α (A) and lim j→∞ A j = A.
Proof. To prove the required result, we use the concept of mathematical induction, for j = 1, we have which is true for j = 1. Assume that the result is true for j − 1, then this shows that A j ∈ G α (A). Now, we prove the second part. Since A j − A ≤ Φ j A 0 − A and Φ ∈ (0, 1), therefore, Φ j → 0 as j → ∞. Consequently, A j − A → 0 as j → ∞. Thus, lim j→∞ A j = A.

Conclusions
In this paper, we have investigated the Ambartsumian equation under a non-singular fractional operator called the Caputo-Fabrizio operator. We have deduced the existence and uniqueness results by Krasnoselskii's fixed point theorem and Banach fixed point theorem. We have used the notion of functional analysis to show that the proposed equation is Ulam-Hyers and Ulam-Hyers-Rassias stable. We have used an efficient analytical method to find a novel series solution of the proposed equation. We have proved that the series solution is convergent to the exact solution of the equation. Lastly, we have simulated the obtained results for different non-integer orders belongs to (0,1] in Figures 1-6. We have shown through graphs that when the fractional order tends to unity, then solution curves at noninteger orders tend to solution curves at integer order. Thus, we conclude that nonsingular fractional operators are appropriate for the study of the dynamics of a model at fractional orders. In the future, we will study the concerned equation under more generalized non-integers derivatives.