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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/
A reproducing kernel Hilbert space method for nonlinear partial differential equations: applications to physical equations
(Iop Publishing Ltd, 2022) Attia, Nourhane; Akgul, Ali
The partial differential equations (PDEs) describe several phenomena in wide fields of engineering and physics. The purpose of this paper is to employ the reproducing kernel Hilbert space method (RKHSM) in obtaining effective numerical solutions to nonlinear PDEs, which are arising in acoustic problems for a fluid flow. In this paper, the RKHSM is used to construct numerical solutions for PDEs which are found in physical problems such as sediment waves in plasma, sediment transport in rivers, shock waves, electric signals' transmission along a cable, acoustic problems for a fluid flow, vibrating membrane, and vibrating string. The RKHSM systematically produces analytic and approximate solutions in the form of series. The convergence analysis and error estimations are discussed to prove the applicability theoretically. Three applications are tested to show the performance and efficiency of the used method. Computational results indicated a good agreement between the exact and numerical solutions.
Advancing differential equation solutions: A novel Laplace-reproducing kernel Hilbert space method for nonlinear fractional Riccati equations
(World Scientific Publ Co Pte Ltd, 2024) Akgul, Ali; Attia, Nourhane
Fractional ordinary differential equations are crucial for modeling various phenomena in engineering and physics. This study aims to enhance solution methodologies by combining the Laplace transform with the reproducing kernel Hilbert space method (RKHSM). This integration leads to a more effective approach compared to the classical RKHSM. We apply the Laplace-reproducing kernel Hilbert space method (L-RKHSM) to develop novel numerical solutions for nonlinear fractional Riccati differential equations. The L-RKHSM systematically produces both approximate and analytic solutions in series form. We present detailed results for four illustrative examples, showcasing the superior performance of the L-RKHSM over traditional methods. This innovative approach not only advances our understanding of nonlinear fractional ordinary differential equations but also demonstrates its effectiveness through significantly improved outcomes in various applications.
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.
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.
Analysis of a fractional order Bovine Brucellosis disease model with discrete generalized Mittag-Leffler kernels
(Elsevier, 2023) Farman, Muhammad; Shehzad, Aamir; Akgul, Ali; Baleanu, Dumitru; Attia, Nourhane; Hassan, Ahmed M.
Bovine Brucellosis, a zoonotic disease, can infect cattle in tropical and subtropical areas. It remains a critical issue for both human and animal health in many parts of the world, especially those where livestock is an important source of food and income. An efficient method for monitoring the illness's increasing prevalence and developing low-cost prevention strategies for both its effects and recurrence is brucellosis disease modeling. We create a fractional-order model of Bovine Brucellosis using a discrete modified Atangana-Baleanu fractional difference operator of the Liouville-Caputo type. An analysis of the suggested system's well-posedness and a qualitative investigation are both conducted. The examination of the Volterra-type Lyapunov function for global stability is supported by the first and derivative tests. The Lipschitz condition is also used for the model in order to meet the criterion of the uniqueness of the exact solution. We created an endemic and disease-free equilibrium. Solutions are built in the discrete generalized form of the Mittag-Leffler kernel in order to analyze the effect of the fractional operator with numerical simulations and emphasize the effects of the sickness due to the many factors involved. The capacity of the suggested model to forecast an infectious disease like brucellosis can help researchers and decision-makers take preventive actions.
ANALYSIS OF NEW TRANSFER FUNCTIONS WITH SUM INTEGRAL TRANSFORMATION
(Wilmington Scientific Publisher, Llc, 2024) Akgul, Ali; Baleanu, Dumitru; Ulgul, Enver; Sakar, Necibullah; Attia, Nourhane
We explore the novel SUM integral transform method for solving ordinary and partial differential equations, offering an effective approach beyond conventional Laplace and Sumudu transforms. Using this method, we address various differential equations, deriving transfer functions for classical and fractional derivatives. The resultant transfer functions provide valuable insights into diverse mathematical models.
Analysis of the Fractional Differential Equations Using Two Different Methods
(Mdpi, 2023) Partohaghighi, Mohammad; Akgul, Ali; Akgul, Esra Karatas; Attia, Nourhane; De la Sen, Manuel; Bayram, Mustafa
Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.
ENHANCING SOLUTIONS FOR NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS VIA COMBINED LAPLACE TRANSFORM AND REPRODUCING KERNEL METHOD
(Wilmington Scientific Publisher, 2025) Akgül, Ali; Attia, Nourhane
Ordinary differential equations (ODEs) describe diverse phenomena in engineering and physics, such as electrical networks, oscillating systems, satellite orbits, and chemical reactions. Solving these equations, particularly the non-linear ones, is often challenging due to their complexity. This study aims to innovate by integrating the Laplace transform with the reproducing kernel Hilbert space method (RKHSM), introducing an enhanced approach that surpasses classical RKHSM. The combined Laplace-RKHSM method simplifies the original non-linear ODEs, allowing for the construction of novel numerical solutions. These solutions are systematically obtained in series form, providing both analytic and approximate results. The effectiveness and efficacy of the Laplace-RKHSM are demonstrated through three applications, each showcasing the method’s superior performance in terms of accuracy and computational efficiency. This new approach not only enhances the existing RKHSM but also broadens its applicability to a wider range of non-linear problems in physics and engineering. © 2025, Wilmington Scientific Publisher. All rights reserved.
Extension of the Reproducing Kernel Hilbert Space Method's Application Range to Include Some Important Fractional Differential Equations
(Mdpi, 2023) Attia, Nourhane; Akgul, Ali; Alqahtani, Rubayyi T. T.
Fractional differential equations are becoming more and more indispensable for modeling real-life problems. Modeling and then analyzing these fractional differential equations assists researchers in comprehending and predicting the system they want to study. This is only conceivable when their solutions are available. However, the majority of fractional differential equations lack exact solutions, and even when they do, they cannot be assessed precisely. Therefore, in order to analyze the symmetry analysis and acquire approximate solutions, one must rely on numerical approaches. In order to solve several significant fractional differential equations numerically, this work presents an effective approach. This method's versatility and simplicity are its key benefits. To verify the RKHSM's applicability, the convergence analysis and error estimations related to it are discussed. We also provide the profiles of a variety of representative numerical solutions to the problem at hand. We validated the potential, reliability, and efficacy of the RKHSM by testing some examples.
investigating nonlinear fractional systems: reproducing kernel Hilbert space method
(Springer, 2024) Attia, Nourhane; Akgul, Ali; Alqahtani, Rubayyi T.
The reproducing kernel Hilbert space method (RK-HS method) is used in this research for solving some important nonlinear systems of fractional ordinary differential equations, such as the fractional Susceptible-Infected-Recovered (SIR) model. Nonlinear systems are widely used across various disciplines, including medicine, biology, technology, and numerous other fields. To evaluate the RK-HS method's accuracy and applicability, we compare its numerical solutions with those obtained via Hermite interpolation, the Adomian decomposition method, and the residual power series method. To further support the reliability of the RK-HS method, the convergence analysis is discussed.
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.
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.
On Solutions of Biological Models Using Reproducing Kernel Hilbert Space Method
(Springer Science and Business Media Deutschland GmbH, 2023) Attia, Nourhane; Akgül, Ali
Differential equations (DEs, for short) are becoming more and more indispensable for modeling real-life problems. Modeling and then analyzing these DEs help scientists to understand and make predictions about the system that they want to analyze. And this is possible only in one case when their solutions are available. However, the majority of fractional differential equations lack exact solutions. This chapter’s goal is to introduce the reproducing kernel Hilbert space method (RKHSM) for some systems that have significant applications in biology. The proposed method’s error estimations and convergence analyses are discussed. The assessment of the RKHSM is made by testing some illustrative applications. The results suggest that the RKHSM is an efficient and highly convenient method to solve the fractional systems arising in biology. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023.
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.
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.
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
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.
Reproducing Kernel Method with Global Derivative
(Hindawi Ltd, 2023) Attia, Nourhane; Akguel, Ali; Atangana, Abdon
Ordinary differential equations describe several phenomena in different fields of engineering and physics. Our aim is to use the reproducing kernel Hilbert space method (RKHSM) to find a solution to some ordinary differential equations (ODEs) that are described by using the global derivative. In this research, we used the RKHSM to construct new numerical solutions for nonlinear ODEs with global derivative. The used method systematically produces analytic and approximate solutions in the series's form. We tested three applications for showing the performance of the RKHSM.
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.

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