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1

A New Class of Fractional Cumulative Residual Entropy - Some Theoretical Results

Benmahmoud Slimane

Journal of Telecommunications and Information Technology

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2023

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nr 1

25--29

EN

In this paper, by differentiating the entropy’s generating function (i.e., h(t) = R SX̄F tX (x)dx) using a Caputo fractional-order derivative, we derive a generalized non-logarithmic fractional cumulative residual entropy (FCRE). When the order of differentiation α → 1, the ordinary Rao CRE is recovered, which corresponds to the results from first-order ordinary differentiation. Some properties and examples of the proposed FCRE are also presented.

2

The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach

Maayah Banan, Moussaoui Asma, Bushnaq Samia, Arqub Omar Abu

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

963--977

EN

COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, researchers have been concentrating their efforts on understanding the complex dynamics of this widespread illness. Mathematics plays a big role in understanding the mechanism of the spread of this disease by modeling it and trying to find approximate solutions. In this study, we implement a new technique for an approximation of the analytic series solution called the multistep Laplace optimized decomposition method for solving fractional nonlinear systems of ordinary differential equations. The proposed method is a combination of the multistep method, the Laplace transform, and the optimized decomposition method. To show the ability and effectiveness of this method, we chose the COVID-19 model to apply the proposed technique to it. To develop the model, the Caputo-type fractional-order derivative is employed. The suggested algorithm efficacy is assessed using the fourth-order Runge-Kutta method, and when compared to it, the results show that the proposed approach has a high level of accuracy. Several representative graphs are displayed and analyzed in two dimensions to show the growth and decay in the model concerning the fractional parameter α values. The central processing unit computational time cost in finding graphical results is utilized and tabulated. From a numerical viewpoint, the archived simulations and results justify that the proposed iterative algorithm is a straightforward and appropriate tool with computational efficiency for several coronavirus disease differential model solutions.

3

Distributed optimal control problems driven by space-time fractional parabolic equations

Mehandiratta Vaibhav, Mehra Mani, Leugering Günter

Control and Cybernetics

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2022

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Vol. 51, No. 2

191--226

EN

We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial fractional derivatives of Sturm-Liouville type. We first prove existence and uniqueness of solutions of STFPEs on an open bounded interval and study their regularity. Then we show existence and uniqueness of solutions to a quadratic distributed optimal control problem. We derive an adjoint problem using the right-Caputo derivative in time and provide optimality conditions for the control problem. Moreover, we propose a finite difference scheme to find the approximate solution of the considered optimal control problem. In the proposed scheme, the well-known L1 method has been used to approximate the time-fractional Caputo derivative, while the spatial derivative is approximated using the Grünwald-Letnikov formula. Finally, we demonstrate the accuracy and the performance of the proposed difference scheme via examples.

4

Extremal solutions to a coupled system of nonlinear fractional differential equations with ψ–Caputo fractional derivatives

Derbazi Choukri, Baitiche Zidane, Benchohra Mouffak, Graef John R.

Journal of Mathematics and Applications

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2021

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Vol. 44

19--34

EN

Using the well-known monotone iterative technique together with the method of upper and lower solutions, the authors investigate the existence of extremal solutions to a class of coupled systems of nonlinear fractional differential equations involving the ψ–Caputo derivative with initial conditions. As applications of this work, two illustrative examples are presented.

5

Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method

Owolabi Kolade M.

Journal of Applied Analysis

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2021

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Vol. 27, nr 2

269--282

EN

In this work, synchronization of fractional dynamics of chaotic system is presented. The suggested dynamics is governed by a system of fractional differential equations, where the fractional derivative operator is modeled by the novel Caputo operator. The nature of fractional dynamical system is non-local which often rules out a closed-form solution. As a result, an efficient numerical method based on shifted Chebychev spectral collocation method is proposed. The error and convergence analysis of this scheme is also given. Numerical results are given for different values of fractional order and other parameters when applied to solve chaotic system, to address any points or queries that may occur naturally.

6

L1-solutions of the boundary value problem for implicit fractional order differentia equations

Telli Benoumran, Souid Mohammed Said

Journal of Applied Analysis

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2021

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Vol. 27, nr 1

121--128

EN

The aim of this paper is to present new results on the existence of solutions for a class of the boundary value problem for fractional order implicit differential equations involving the Caputo fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

7

Existence of solutions of BVPs for fractional Langevin equations involving Caputo fractional derivatives

Kiyamehr Zohre, Baghani Hamid

Journal of Applied Analysis

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2021

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Vol. 27, nr 1

47--55

EN

This article investigates a nonlinear fractional Caputo-Langevin equation Dβ(Dα + λ)x(t) = f(t, x(t)), 0 < t < 1, 0 < α ≤ 1, 1 < β ≤ 2, subject to the multi-point boundary conditions x(0) = 0, D2αx(1) + λDαx(1) = 0, x(1) =η∫0 x(τ) dτ for some 0 < η < 1, where Dα is the Caputo fractional derivative of order α, f : [0, 1] × ℝ → ℝ is a given continuous function, and λ is a real number. Some new existence and uniqueness results are obtained by applying an interesting fixed point theorem.

8

Mittag-Leffler stability for a Timoshenko problem

Tatar Nasser-eddine

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 2

219--232

EN

A Timoshenko system of a fractional order between zero and one is investigated here. Using a fractional version of resolvents, we establish an existence and uniqueness theorem in an appropriate space. Moreover, it is proved that lower order fractional terms (in the rotation component) are capable of stabilizing the system in a Mittag-Leffler fashion. Therefore, they deserve to be called damping terms. This is shown through the introduction of some new functionals and some fractional inequalities, and the establishment of some properties, involving fractional derivatives. In the case of different wave speeds of propagation we obtain convergence to zero.

9

On theoretical and practical aspects of Duhamel’s integral

Różański Michał, Sikora Beata, Smuda Adrian, Wituła Roman

Archives of Control Sciences

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2021

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Vol. 31, No. 4

815--847

EN

The paper is a new approach to the Duhamel integral. It contains an overview of formulas and applications of Duhamel’s integral as well as a number of new results on the Duhamel integral and principle. Basic definitions are recalled and formulas for Duhamel’s integral are derived via Laplace transformation and Leibniz integral rule. Applications of Duhamel’s integral for solving certain types of differential and integral equations are presented. Moreover, an interpretation of Duhamel’s formula in the theory of operator semigroups is given. Some applications of Duhamel’s formula in control systems analysis are discussed. The work is also devoted to the usage of Duhamel’s integral for differential equations with fractional order derivative.

10

A new numerical technique for solving fractional Bratu’s initial value problems in the Caputo and Caputo-Fabrizio sense

Khalouta Ali, Kadem Abdelouahab

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

43--56

EN

The purpose of this paper is to propose a new numerical technique called the natural decomposition method (NDM) for solving fractional Bratu’s initial value problems (FBIVP) in the Caputo and Caputo-Fabrizio sense. The NDM is a combined form of the natural transform method and the Adomian decomposition method. The numerical example is provided in order to validate the efficiency and reliability of the proposed method. The obtained results reveal that the proposed method is a very efficient and simple tool for solving fractional differential equations.

11

A new modification of the reduced differential transform method for nonlinear fractional partial differential equations

Khalouta Ali, Kadem Abdelouahab

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 3

45--58

EN

The objective of this study is to present a new modification of the reduced differential transform method (MRDTM) to find an approximate analytical solution of a certain class of nonlinear fractional partial differential equations in particular, nonlinear time-fractional wave-like equations with variable coefficients. This method is a combination of two different methods: the Shehu transform method and the reduced differential transform method. The advantage of the MRDTM is to find the solution without discretization, linearization or restrictive assumptions. Three different examples are presented to demonstrate the applicability and effectiveness of the MRDTM. The numerical results show that the proposed modification is very effective and simple for solving nonlinear fractional partial differential equations.

12

On LQ optimization problem subject to fractional order irregular singular systems

Muhafzan, Nazra Admi, Yulianti Lyra, Zulakmal, Revina Refi

Archives of Control Sciences

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2020

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Vol. 30, No. 4

745--756

EN

In this paper we discuss the linear quadratic (LQ) optimization problem subject to fractional order irregular singular systems. The aim of this paper is to find the control-state pairs satisfying the dynamic constraint of the form a fractional order irregular singular systems such that the LQ objective functional is minimized. The method of solving is to convert such LQ optimization into the standard fractional LQ optimization problem. Under some particularly conditions we find the solution of the problem under consideration.

13

The Generalized Differential Transform Method for solution of a free vibration linear differential equation with fractional derivative damping

Das Deepanjan

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 2

19--29

EN

In the present paper, the Generalized Differential Transform Method (GDTM) is used for obtaining the approximate analytic solutions of a free vibration linear differential equation of a single-degree-of-freedom (SDOF) system with fractional derivative damping. The fractional derivatives are described in the Caputo sense.

14

Application of LDG scheme to solve semi-differential equations

Izadi Mohammad

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 4

27--39

EN

In the current work, we investigate a technique based on discontinuous Galerkin method for the numerical approximation of semi-differential equations with Caputo’s fractional derivative. In this approach, using the natural upwind fluxes enables us to solve the model problem element by element locally in each subintervals and there is no need to solve a full global matrix. Numerical experiments are given to verify the efficiency and accuracy of the proposed method. Numerical solutions are compared with the exact solutions as well as the numerical solutions obtained by other available well-established computational procedures. The results show that the LDG method is more accurate for solving this class of differential equation with relatively low degrees of polynomials and number of elements.

15

Existence and stability of impulsive coupled system of fractional integrodifferential equations

Zada Akbar, Waheed Hira, Alzabut Jehad, Wang Xiaoming

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

296--335

EN

In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

16

Nonlinear fractional differential equations with non-instantaneous impulses in Banach spaces

Benchohra M., Slimane M.

Journal of Mathematics and Applications

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2018

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Vol. 41

39--51

EN

This paper is devoted to study the existence of solutions for a class of initial value problems for non-instantaneous impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Monch's fixed point theorem and the technique of measures of noncompactness.

17

Existence and convergence results for Caputo fractional Volterra integro-differential equations

Hamoud A. A., Bani Issa M. Sh., Ghadle K. P., Abdulghani M.

Journal of Mathematics and Applications

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2018

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Vol. 41

109--121

EN

In this article, homotopy analysis method is successfully applied to find the approximate solution of Caputo fractional Volterra integro-differential equation. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximate. Moreover, we proved the existence and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

18

The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption

Povstenko Y., Klekot J.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 2

101--112

EN

The time-fractional heat conduction equation with heat absorption proportional to temperature is considered in the case of central symmetry. The fundamental solutions to the Cauchy problem and to the source problem are obtained using the integral transform technique. The numerical results are presented graphically.

19

An analytical solution to the problem of time-fractional heat conduction in a composite sphere

Kukla S., Siedlecka U.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2017

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Vol. 65, nr 2

179--186

EN

An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative.

20

On fractional random differential equations with delay

Vu H., Phung N. N., Phuong N.

Opuscula Mathematica

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2016

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Vol. 36, no. 4

541--556

EN

In this paper, we consider the existence and uniqueness of solutions of the fractional random differential equations with delay. Moreover, some kind of boundedness of the solution is proven. Finally, the applicability of the theoretical results is illustrated with some real world examples.