On the Exact Solution of the Functional Differential Equation

Ebaid, Abdelhalim; Al-Jeaid, Hind K.

doi:10.17654/0974324322003

Cited by 10 publications

(11 citation statements)

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“…which has been also mentioned in Ref. [20]. However, after taking a deep insight at the periodicity (27) for sufficiently small values of α compared to β one can detect that P ≈ 2π β .…”

Section: 2supporting

confidence: 67%

“…Some of these properties were addressed in Ref. [20] but represented here just for adding some materials and observations as follows.…”

Section: Properties Of Solutionmentioning

confidence: 99%

“…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%

“…In Ref. [20], the standard series method (SSM) has been employed to solve the model given by Eqs. (1)(2).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: 2supporting

confidence: 67%

“…Some of these properties were addressed in Ref. [20] but represented here just for adding some materials and observations as follows.…”

Section: Properties Of Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Ref. [20], the standard series method (SSM) has been employed to solve the model given by Eqs. (1)(2).…”

Section: Introductionmentioning

confidence: 99%

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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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“…Following Ebaid and Al-Jeaid [13], one can obtain the exact solution of Eqs. (5.2) in the periodic form:…”

Section: Exact Solution Of Inhomogeneous Differential-difference Equa...mentioning

confidence: 99%

Analytical study to solving the inhomogeneous pantograph delay equation: the exact solution

El-Zahar,

Ebaid,

Seddek

2024

J. Math. Computer Sci.

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This paper solves the pantograph delay equation which includes inhomogeneous term. The inhomogeneous term is in the form of a class of exponential functions. An efficient transformation is introduced to reduce the inhomogeneous pantograph delay differential equation (IPDDE) to the homogeneous pantograph delay differential equation (HPDDE), which is a homogeneous model. It is found that the solution of the IPDDE depends mainly on the solution of HPDDE. It is well-known in the literature that the analytical solution of the HPDDE is already obtained in a closed series form. Such ready solution of the HPDDE is invested in this paper and accordingly, the solution of the IPDDE under consideration is established. Also, several exact solutions of the present model are determined at specific conditions and values of the parameters. The solutions of some examples in the literature are obtained in exact forms as special cases of the current results. Moreover, the properties of the obtained solutions are theoretically and graphically addressed.

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Developed analytical approach for a special kind of differential-difference equation: Exact solution

Seddek,

Ebaid,

El-Zahar

2024

J. Math. Computer Sci.

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This paper investigates a differential-difference equation with a variable coefficient of exponential order in the formφ (t) = αφ(t) + βe σt φ (−t). In literature, periodic solution has been obtained at the special case σ = 0. In this paper, an effective approach is developed to determine the exact solution in terms of exponential and trigonometric functions. In addition, the exact solution is expressed in terms of exponential and hyperbolic functions under specific conditions of the involved parameters. Exact solutions of several special cases are derived and found in full agreement with the corresponding results in the relevant literature. Some theoretical results are presented and proved which can be generalized to include other complex models. The behavior of the obtained shows periodicity in the absence of σ while the damped oscillations are shown graphically when σ is assigned to negative values.

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On the Exact Solution of the Functional Differential Equation

Cited by 10 publications

References 0 publications

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

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

Analytical study to solving the inhomogeneous pantograph delay equation: the exact solution

Developed analytical approach for a special kind of differential-difference equation: Exact solution

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