Solution of Ambartsumian Delay Differential Equation with Conformable Derivative

Khaled, S. M.; El‐Zahar, Essam R.; Ebaid, Abdelhalim

doi:10.3390/math7050425

Cited by 19 publications

(10 citation statements)

References 23 publications

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“…which agrees with the 3-term approximate solution in the literature [4,6] and also agrees with Ref. [7] (see Equation (30)) for ordinary derivative (α → 1).…”

Section: As α →supporting

confidence: 92%

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Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

Alharbi

Petrovskii

2020

Mathematics

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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.

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“…which agrees with the 3-term approximate solution in the literature [4,6] and also agrees with Ref. [7] (see Equation (30)) for ordinary derivative (α → 1).…”

Section: As α →supporting

confidence: 92%

“…Patade and Bhalekar [2] obtained the solution of the ordinary model (as α → 1) as a power series. Additional results were also reported by Khaled et al [7] for the Ambartsumian equation using the conformable derivative. However, the FAE has been investigated via the homotopy transform analysis method (HTAM) in Ref.…”

Section: Introductionsupporting

confidence: 74%

Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

Alharbi

Petrovskii

2020

Mathematics

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“…Additional application of the PDDE arises in astronomy at specific values of α, β, and γ. For declaration, the PDDE model becomes the Ambartsumian model [9][10][11][12][13] when α = −1 and β = γ = 1/q (q > 1). The Ambartsumian model studies the surface brightness in the Milky Way.…”

Section: Introductionmentioning

confidence: 99%

On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach

Alshomrani,

Ebaid,

Aldosari

et al. 2024

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The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters.

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“…The standard Pantograph model has been analyzed by several authors utilizing various techniques [2][3][4][5][6][7][8]. A special case of the Pantograph model is known as Ambartsumian equation which is of practical applications in Astronomy [9][10][11][12][13][14][15][16][17][18][19]. However, an interest is recently given to another special case of such model when γ = −1, given by [20] φ (t) = αφ(t) + βφ (−t) ,…”

Section: Introductionmentioning

confidence: 99%

Exact and Numerical Treatment of a Special Kind of the Pantograph Model via Laplace Technique

Alharbi¹

2022

Int. J. Anal. Appl.

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The Pantograph is a device of practical application in electric trains, by which the current is collected. The mathematical problem of this device is generally given by the delay differential equation φ’(t) = αy(t) + βφ(γt), where α and β are real constants and γ is a proportional delay parameter. In the literature, a special attention has been given to the particular case γ = −1. The objective of this paper is to extend the application of the Laplace transform (LT) combined with the Adomian decomposition method (ADM) to analyze the above model at such particular case of γ. The solution will be determined in exact form which agrees with the corresponding results in the literature. Various properties of the obtained exact solution are discussed in detail. Moreover, it will be declared that for sufficiently small values of α compared to β there exists an accurate approximate solution. The accuracy of the approximate solution is numerically validated. In addition, some numerical results are conducted for the behavior of the present solution at selected values of α and β.

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Solution of Ambartsumian Delay Differential Equation with Conformable Derivative

Cited by 19 publications

References 23 publications

Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method

On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach

Exact and Numerical Treatment of a Special Kind of the Pantograph Model via Laplace Technique

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