Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method

Abstract

The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to establish a closed-form solution. The present closed-form solution is reported for the first time for the Ambartsumian equation. Numerically, the calculations have demonstrated a rapid rate of convergence of the obtained approximate solutions, which are displayed in several graphs. It has also been shown that only a few terms of the new approximate solution were sufficient to achieve extremely accurate numerical results. Furthermore, comparisons of the present results with the existing methods in the literature were introduced.

Solution of Ambartsumian Delay Differential Equation with Conformable Derivative

“…which agrees with the third-order approximate solution in [5] for the model in Equations (1)- (3). Hence, the present results are applicable for 0 < α ≤ 1 and more general than those in relevant literature [3][4][5][6] when α → 1. The applicability and validity of the obtained results are illustrated in the next section, where the convergence of the approximate solutions in Equation 28is valid in the whole domain t ≥ 0 for all values of q > 1.…”

“…Moreover, the impacts of the initial condition, the delay parameter, and the arbitrary order of the conformable derivative on the residual error are discussed in detail. It is also declared that the residual achieved by using only ten terms is very small for moderate values of the delay parameter while such residual approaches zero as the the delay parameter increases.The obtained results are applicable for 0 < α ≤ 1 and more general than those in relevant literature [3][4][5][6] in which α → 1. The obtained residual errors in Figures 7-9 are less than 3 × 10 −6 , which confirms the accuracy of the approximate solution.…”

“…The existence and uniqueness of the standard model (Equations (2) and (3)) was analyzed by Kato and McLeod [2]. In the literature [3][4][5][6], the standard model Equations (2) and (3)) was solved via several analytical methods. Patade and Bhalekar [3] applied the Daftardar-Gejji and Jafari Method to obtain a power series solution and provided a theoretical analysis for convergence.…”

“…Patade and Bhalekar [3] applied the Daftardar-Gejji and Jafari Method to obtain a power series solution and provided a theoretical analysis for convergence. Later, Bakodah and Ebaid [4] obtained the exact solution of the standard model (Equations (2) and (3)), i.e., as α → 1, by means of Laplace transform. In addition, Ebaid et al [5] implemented the Adomian decomposition method (ADM) to derive an accurate approximate solution.…”

A Possible Generalized Model of the Chlorine Concentration Decay in Pipes: Exact Solution

“…Such LT method has been successfully applied on several models such as diffusions [5], heat transfer of nanofluids suspended with carbon-nanotubes [6], singular boundary value problems (SBVPs) related to fluid flow of carbon-nanotubes [7,8], and the MHD Marangoni convection over a flat plate [9]. Furthermore, the LT was successfully applied to solve the Ambartsumian delay equation [10]. Moreover, one can find in Refs.…”