Abstract
The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.
Highlights
The mathematical model describing the absorption of light by the interstellar matter was derived more than two decades earlier by Ambartsumian [1]
The classical Ambartsumian equation was physically derived by Ambartsumian [1], the derivation of the fractional form of this equation needs to introduce some materials about the physics of interstellar matter
The convergence of the Homotopy Perturbation Method (HPM) has been investigated in detail by Ayati and Biazar [11] for ordinary differential equations (ODEs) and by Touchent et al [13] and Sene and Fall [14] for fractional partial differential equations (FPDEs)
Summary
The mathematical model describing the absorption of light by the interstellar matter was derived more than two decades earlier by Ambartsumian [1]. Equation (1) reduces to the classical model with ordinary derivative as α → 1. The present work is focused on deriving accurate approximations of the fractional model (1–2) when compared with other approaches in the relevant literature. Patade and Bhalekar [2] obtained the solution of the ordinary model (as α → 1) as a power. Patade [9] obtained a closed-form power series solution for the present FAE using an iterative method. We will show in this paper that the solution of the system (1–2) can be obtained in terms of the Mittag-Leffler functions which converge in the whole domain of the current model. The HPM is proposed to solve the current fractional model. The advantages of the HPM over the closed-form solution [9] will be discussed
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