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Permanent URI for this collectionhttps://dspace.univ-boumerdes.dz/handle/123456789/13

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Has files: YesSubject: search.filters.subject.Reproducing kernel Hilbert space 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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    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