The Mittag–Leffler Functions for a Class of First-Order Fractional Initial Value Problems: Dual Solution via Riemann–Liouville Fractional Derivative

Ebaid, Abdelhalim; Al-Jeaid, Hind K.

doi:10.3390/fractalfract6020085

Cited by 14 publications

(10 citation statements)

References 27 publications

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“…Besides, several authors ([3]- [8]) implemented different analytical and numerical methods to treating the system (1)- (5). The LT method was widely applied to solve various physical models [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. However, it requires massive calculations as seen in Refs.…”

Section: Introductionmentioning

confidence: 99%

A Simple Approach for Explicit Solution of The Neutron Diffusion Kinetic System

Al-Jeaid¹

2023

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This paper introduces a new approach to directly solve a system of two coupled partial differential equations (PDEs) subjected to physical conditions describing the diffusion kinetic problem with one delayed neutron precursor concentration in Cartesian geometry. In literature, many difficulties arise when dealing with the current model using various numerical/analytical approaches. Normally, mathematicians search for simple but effective methods to solve their physical models. This work aims to introduce a new approach to directly solve the model under investigation. The present approach suggests to transform the given PDEs to a system of linear ordinary differential equations (ODEs). The solution of this system of ODEs is obtained by a simple analytical procedure. In addition, the solution of the original system of PDEs is determined in explicit form. The main advantage of the current approach is that it avoided the use of any natural transformations such as the Laplace transform in the literature. It also gives the solution in a direct manner; hence, the massive computational work of other numerical/analytical approaches is avoided. Hence, the proposed method is effective and simpler than those previously published in the literature. Moreover, the proposed approach can be further extended and applied to solve other kinds of diffusion kinetic problems.

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Section: Introductionmentioning

confidence: 99%

A Simple Approach for Explicit Solution of The Neutron Diffusion Kinetic System

Al-Jeaid¹

2023

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“…Moreover, Aljohani et al [21] obtained the exact solution of the chlorine transport model in fractional form in terms of the Mittag-Leffler function. Furthermore, the application of the RLFD on a class of engineering oscillatory problems was addressed by Ebaid and Al-Jeaid [22] for a class of first-order fractional initial value problems in which the dual solution was obtained. In addition, Seddek et al [23] applied the RLFD to solve non-homogeneous fractional differential system containing periodic terms.…”

Section: Introductionmentioning

confidence: 99%

Application of Riemann–Liouville Derivatives on Second-Order Fractional Differential Equations: The Exact Solution

Albidah

2023

Fractal Fract

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This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fractional differential equations under appropriate initial conditions. It is shown that the zeroth lower bound implies implicit solutions in terms of the Mittag–Leffler functions while explicit solutions are derived when negative infinity is taken as a lower bound. Such explicit solutions are obtained for the current two classes in terms of trigonometric and hyperbolic functions. Some theoretical results are introduced to facilitate the solutions procedures. Moreover, the characteristics of the obtained solutions are discussed and interpreted.

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“…Ebaid et al [16] developed a new approach to solve a class of first-order fractional initial value problems based on Riemann-Liouville fractional derivative. Failla et al [17] proposed a numerical method to investigate the dynamical response of the fractional-order system with random excitation.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Behavior of Fractional-Order Delayed Feedback Control on the Mathieu Equation by Incremental Harmonic Balance Method

Wen

Shen

Niu

et al. 2022

Shock and Vibration

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In this study, the dynamical analysis of the Mathieu equation with multifrequency excitation under fractional-order delayed feedback control is investigated by the incremental harmonic balance method (IHBM). IHBM is applied to the fractional-order delayed feedback control system, and the general formulas of the first-order approximate periodic solution for the Mathieu equation are derived. Caputo’s definition is adopted to process the fractional-order delayed feedback term. The general formulas of this system are suitable for not only the weakly but also the strongly nonlinear fractional-order system. Through the analysis of the general formulas of this system, it shows that fractional-order delayed feedback control has two functions, which are velocity delayed feedback control and displacement delayed feedback control. Next, the numerical simulation of the system is carried out. The comparison between the approximate analytical solution and the numerical iterative result is made, and the accuracy of the approximate analytical result by IHBM is proved to be high. At last, the effects of the time delay, feedback coefficient, and fractional order are investigated, respectively. It is generally known that time delay is common and inevitable in the control system. But the fractional order can be used to adjust the influence caused by time delay in fractional-order delayed feedback control. Those new system characteristics will provide theoretical guidance to the design and the control of this kind system.

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The Mittag–Leffler Functions for a Class of First-Order Fractional Initial Value Problems: Dual Solution via Riemann–Liouville Fractional Derivative

Cited by 14 publications

References 27 publications

A Simple Approach for Explicit Solution of The Neutron Diffusion Kinetic System

A Simple Approach for Explicit Solution of The Neutron Diffusion Kinetic System

Application of Riemann–Liouville Derivatives on Second-Order Fractional Differential Equations: The Exact Solution

Dynamical Behavior of Fractional-Order Delayed Feedback Control on the Mathieu Equation by Incremental Harmonic Balance Method

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