Publications Scientifiques

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    An efficient approach for solving differential equations in the frame of a new fractional derivative operator
    (MDPI, 2023) Attia, Nourhane; Akgül, Ali; Seba, Djamila; Nour, Abdelkader; la Sen, Manuel De; Bayram, Mustafa
    Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann–Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for short) for solving certain fractional differential equations involving this new derivative. This is the first time that the application of the RKHSM is employed for solving some differential equations with the new operator. We illustrate the convergence analysis of the applicability and reliability of the suggested approaches. The results confirm that the RKHSM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed method
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    A novel method for fractal-fractional differential equations
    (Elsevier, 2022) Attia, Nourhane; Akgül, Ali; Seba, Djamila; Nour, Abdelkader; Asad, Jihad
    We consider the reproducing kernel Hilbert space method to construct numerical solutions for some basic fractional ordinary differential equations (FODEs) under fractal fractional derivative with the generalized Mittag–Leffler (M-L) kernel. Deriving the analytic and numerical solutions of this new class of differential equations are modern trends. To apply this method, we use reproducing kernel theory and two important Hilbert spaces. We provide three problems to illustrate our main results including the profiles of different representative approximate solutions. The computational results are compared with the exact solutions. The results obtained clearly show the effect of the fractal fractional derivative with the M-L kernel in the obtained outcomes. Meanwhile, the compatibility between the approximate and exact solutions confirms the applicability and superior performance of the method
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    Controllability of nonlocal Hilfer fractional delay dynamic inclusions with non-instantaneous impulses and non-dense domain
    (Springer, 2022) Boudjerida, Assia; Seba, Djamila
    The controllability of a class of nondensely defined fractional dynamic delay inclusions containing Hilfer fractional derivative, nonlocal conditions, and non-instantaneous impulses in abstract spaces is investigated without compactness assumption. The existence of an integral solution and the controllability for the given problem are established relying on a condensing fixed point theorem of multivalued maps. In support, an example is given to clarify the obtained theoretical outcomes
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    Local bifurcation analysis of one parameter in the greitzer’s model with a general compressor characteristic
    (Springer, 2022) Naima Meskine, Naima; Kessal, Mohand; Seba, Djamila
    Based on the Greitzer’s reduced model, an analytical study on the instabilities phenomena of the operating point is presented using some basic properties of the nonlinear dynamic system. Moreover, a proposal of a general compressor characteristic curve, that suits the stationary system, is given. The Routh–Hurwitz theorem is applied to determine the stability conditions on the model parameters. An analysis along with a discussion is presented when the compression system goes to the Hopf bifurcation point during surge. For the Hopf bifurcation case, an approximate expression, for the periodic cycle of the system’s solution from the equilibrium point, is obtained and the direction is determined using Lyapunov’s stability theory. A numerical simulation is executed to illustrate the theoretical results
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    Neighbourhood star selection properties in bitopological spaces
    (University of Nis, 2021) Lakehal, Rachid; Kočinac, Ljubiˇsa D.R.; Seba, Djamila
    In this paper we introduce and study some new types of star-selection principles ((i, j)-NSM, (i, j)-NSR and (i, j)-NSH) in bitopologivcal spaces. Various properties of these selection properties are established and their relations with known selection principles are discussed. Several examples are given
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    Numerical solutions to the time-fractional swift–hohenberg equation using reproducing kernel hilbert space method
    (Springer, 2021) Attia, Nourhane; Akgül, Ali; Seba, Djamila; Nour, Abdelkader
    In this work, a numerical approach based on the reproducing kernel theory is presented for solving the fractional Swift–Hohenberg equation (FS-HE) under the Caputo time-fractional derivative. Such equation is an effective model to describe a variety of phenomena in physics. The analytic and approximate solutions of FS-HE in the absence and presence of dispersive terms have been described by applying the reproducing kernel Hilbert space method (RKHSM). The benefit of the proposed method is its ability to get the approximate solution of the FS-HE easily and quickly. The current approach utilizes reproducing kernel theory, some valuable Hilbert spaces, and a normal basis. The theoretical applicability of the RKHSM is demonstrated by providing the convergence analysis. By testing some examples, we demonstrated the potentiality, validity, and effectiveness of the RKHSM. The computational results are compared with other available ones. These results indicate the superiority and accuracy of the proposed method in solving complex problems arising in widespread fields of technology and science
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    Numerical solution of the fractional relaxation-oscillation equation by using reproducing kernel hilbert space method
    (Springer, 2021) Attia, Nourhane; Akgül, Ali; Seba, Djamila; Nour, Abdelkader
    In this article, the reproducing kernel Hilbert space is proposed and analyzed for the relaxation-oscillation equation of fractional order (FROE). The relaxation oscillation is a type of oscillator based on the way that the physical system’s returns to its equilibrium after being disturbed. We make use of the Caputo fractional derivative. The approximate solution can be obtained by taking n-terms of the analytical solution that is in term of series formula. The numerical experiments are used to prove the convergence of the approximate solution to the analytical solution. The results obtained by the given method demonstrate that it is convenient and efficient for FROE
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    Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delay
    (Elsevier, 2019) Aimene, Djihad; Baleanu, D.; Seba, Djamila
    We discuss the controllability of semilinear differential equations of fractional order with impulses and delay. We make use of the Atangana-Baleanu derivative. Our main tools are semigroup theory, the fixed point theorem due to Darbo and their combination with the properties of measures of noncompactness. Our abstract results are well supported by an illustrative example.
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    Local and global existence of mild solutions of time-fractional Navier–Stokes system posed on the Heisenberg group
    (Birkhauser, 2021) Kirane, Mokhtar; Aimene, Djihad; Seba, Djamila
    This paper is a development of the results and techniques of the two papers (Carvalho-Neto and Planas in J Differ Equ 259:2948–2980, 2015; Oka in J Math Anal Appl 473:382–407, 2019) for the aim of addressing the existence and uniqueness of local and global mild solutions, on the Heisenberg group Hd, of the time-fractional Navier–Stokes system with time derivative of order α∈ (0 , 1). The proof relies on Schaefer’s fixed point theorem. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
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    Controllability results for nondensely defined impulsive fractional-order functional semilinear differential inclusions in abstract space
    (American Institute of Physics, 2019) Boudjerida, Assia; Seba, Djamila; Laoubi, Karima
    In this work, we prove the controllability result of integral solutions defined on a real compact interval for a class of impulsive functional differential inclusions with fractional order and nonlocal conditions, in the case when the linear part is a non-densely defined operator and satisfies the Hille-Yosida condition. The main tool is an appropriate fixed point theorem, integrated semigroup, and the known facts about fractional calculus