Yazar "Nour, Abdelkader" seçeneğine göre listele

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    A novel method for fractal-fractional differential equations
    (Elsevier, 2022) Attia, Nourhane; Akgul, 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. (c) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/
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    An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator
    (Mdpi, 2023) Attia, Nourhane; Akgul, Ali; Seba, Djamila; Nour, Abdelkader; De la Sen, Manuel; 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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    An efficient numerical technique for a biological population model of fractional order
    (Pergamon-Elsevier Science Ltd, 2020) Attia, Nourhane; Akgul, Ali; Seba, Djamila; Nour, Abdelkader
    In the present paper, a biological population model of fractional order (FBPM) with one carrying capacity has been examined with the help of reproducing kernel Hilbert space method (RKHSM). This important fractional model arises in many applications in computational biology. It is worth noting that, the considered FBPM is used to provide the changes that is made on the densities of the predator and prey populations by the fractional derivative. The technique employed to construct new numerical solutions for the FBPM which is considered of a system of two nonlinear fractional ordinary differential equations (FODEs). In the proposed investigation, the utilised fractional derivative is the Caputo derivative. The most valuable advantages of the RKHSM is that it is easily and fast implemented method. The solution methodology is based on the use of two important Hilbert spaces, as well as on the construction of a normal basis through the use of Gram-Schmidt orthogonalization process. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results, which are compared with the homotopy perturbation Sumudu transform method (HPSTM), clearly show: On the one hand, the effect of the fractional derivative in the obtained outcomes, and on the other hand, the great agreement between the mentioned methods, also the superior performance of the RKHSM. The numerical computational are presented in illustrated graphically to show the variations of the predator and prey populations for various fractional order derivatives and with respect to time. (C) 2020 Elsevier Ltd. All rights reserved.
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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. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.
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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. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.
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    On Solutions of Fractional Logistic Differential Equations
    (Natural Sciences Publishing, 2023) Attia, Nourhane; Akgul, Ali; Seba, Djamila; Nour, Abdelkader
    This paper presents an efficient computational technique based on the reproducing kernel theory for approximating the solutions of logistic differential equations of fractional order. The Caputo fractional derivative is utilized in the current approach. The numerical solution can be produced by taking the n-terms of the analytical solution. The convergence of the approximate solution to the analytical solution can be demonstrated with the help of numerical experiments. The numerical comparisons depict that the given method has high effectiveness, accuracy, and feasibility for fractional logistic differential equations. © 2023 NSP Natural Sciences Publishing Cor.
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    On solutions of time-fractional advection-diffusion equation
    (Wiley, 2023) Attia, Nourhane; Akgul, Ali; Seba, Djamila; Nour, Abdelkader
    In this paper, we present an attractive reliable numerical approach to find an approximate solution of the time-fractional advection-diffusion equation (FADE) under the Atangana-Baleanu derivative in Caputo sense (ABC) with Mittag-Leffler kernel. The analytic and approximate solutions of FADE have been determined by using reproducing kernel Hilbert space method (RKHSM). The most valuable advantage of the RKHSM is its ease of use and its quick calculation to obtain the numerical solution of the FADE. Our main tools are reproducing kernel theory, some important Hilbert spaces, and a normal basis. The convergence analysis of the RKHSM is studied. The computational results are compared with other results of an appropriate iterative scheme and also by using specific examples, these results clearly show: On the one hand, the effect of the ABC-fractional derivative with the Mittag-Leffler kernel in the obtained outcomes, and on the other hand, the superior performance of the RKHSM. From a numerical viewpoint, the RKHSM provides the solution's representation in a convergent series. Furthermore, the obtained results elucidate that the proposed approach gives highly accurate outcomes. It is worthy to observe that the numerical results of the specific examples show the efficiency and convenience of the RKHSM for dealing with various fractional problems emerging in the physical environment.
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    Reproducing kernel Hilbert space method for solving fractal fractional differential equations
    (Elsevier B.V., 2022) Attia, Nourhane; Akgül, Ali; Seba, Djamila; Nour, Abdelkader; Riaz, Muhammad Bilal
    Based on reproducing kernel theory, an analytical approach is considered to construct numerical solutions for some basic fractional ordinary differential equations (FODEs, for short) under fractal fractional derivative with the exponential decay kernel. For the first time, the implemented approach, namely reproducing kernel Hilbert space method (RKHSM), is proposed in terms of analytic and numerical fractal fractional solutions. Through the convergence analysis, we illustrate the high competency of the RKHSM. Our results are compared with the exact solutions, and they show us how the fractal-fractional derivative when the kernel is exponential decay affects the obtained outcomes. And, they also confirm the superior performance of the RKHSM. © 2022
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    Reproducing kernel Hilbert space method for the numerical solutions of fractional cancer tumor models
    (Wiley, 2023) Attia, Nourhane; Akgul, Ali; Seba, Djamila; Nour, Abdelkader
    This research work is concerned with the new numerical solutions of some essential fractional cancer tumor models, which are investigated by using reproducing kernel Hilbert space method (RKHSM). The most valuable advantage of the RKHSM is its ease of use and its quick calculation to obtain the numerical solutions of the considered problem. We make use of the Caputo fractional derivative. Our main tools are reproducing kernel theory, some important Hilbert spaces, and a normal basis. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results clearly show the superior performance of the RKHSM.
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    Solving Duffing-Van der Pol Oscillator Equations of Fractional Order by an Accurate Technique
    (Shahid Chamran Univ Ahvaz, Iran, 2021) Attia, Nourhane; Seba, Djamila; Akgul, Ali; Nour, Abdelkader
    In this paper, an accurate technique is used to find an approximate solution to the fractional-order Duffing-Van der Pol (DVP, for short) oscillators equation which is reproducing kernel Hilbert space (RKHS, for short) method. The numerical results show that the n-term approximation is a rapidly convergent series representation and they present also the high accuracy and effectiveness of this method. The efficiency of the proposed method has been proved by the theoretical predictions and confirmed by the numerical experiments.