Article
Mathematics
Mahmoud A. Zaky, Ahmed S. Hendy, Rob H. De Staelen
Summary: A finite difference/Galerkin spectral discretization method for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed, showing high accuracy in both space and time. Numerical experiments confirm the theoretical claims and demonstrate the efficiency of the approach.
Article
Computer Science, Interdisciplinary Applications
Ahmed S. Hendy, Mahmoud A. Zaky
Summary: In this paper, an efficient numerical method is developed to solve a coupled system of nonlinear multi-term time-space fractional diffusion equations. The method is proven to be convergent and the optimal error estimate is obtained. Numerical experiments are conducted to verify the theoretical claims.
ENGINEERING WITH COMPUTERS
(2022)
Article
Engineering, Mechanical
Ahmed S. Hendy, Mahmoud A. Zaky
Summary: This paper presents an efficient numerical method combining Galerkin-Legendre spectral approximation with L1 finite difference formula to solve the generalized nonlinear fractional Schrodinger equation with space- and time-fractional derivatives. The stability and convergence analyses of the scheme show that it is unconditionally stable and has a convergence order of min{kappa theta, 2-theta} in time and spectral accuracy in space. Two numerical test problems are conducted to validate the proposed algorithm with convergence and error analysis.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Mahmoud A. Zaky, Ahmed S. Hendy, Jorge E. Macias-Diaz
Summary: This paper proposes and analyzes a high-order difference/Galerkin spectral scheme for the time-space fractional Ginzburg-Landau equation. The scheme achieves spectral accuracy in space and second-order accuracy in time. Error estimates are established using a fractional Gronwall inequality and its discrete form. Numerical experiments confirm the theoretical claims and discuss the effect of fractional-order parameters on pattern formation.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Mahmoud A. Zaky, Ahmed S. Hendy
Summary: This paper introduces a numerical scheme for solving nonlinear fractional Schrodinger equations, utilizing finite difference and spectral-Galerkin methods to handle Riesz space- and Caputo time-fractional derivatives. The scheme is proven to be unconditionally stable and convergent with accuracy in time and spatial accuracy for smooth solutions. Numerical tests are conducted to verify the theoretical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Engineering, Multidisciplinary
Hassen Arfaoui, A. Ben Makhlouf, Lassaad Mchiri, Mohamed Rhaima
Summary: In this article, the authors study the Finite-Time Stability (FTS) of Linear Stochastic Fractional Differential Equations of Ito-Doob Type with Delay (LSFDEIDTwD) for a derivative order q E (0, 1). The study investigates the stability of the LSFDEIDTwD in a finite-time domain [0, T] using the generalized Gronwall Inequality (GWI) and stochastic calculus theory. The main results are illustrated with two numerical examples.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics, Applied
Minghua Chen, Suzhen Jiang, Weiping Bu
Summary: This paper investigates the convergence rate of the time-fractional Feynman-Kac equation under different conditions, and proposes an implicit-explicit L1 scheme to achieve a better convergence rate on graded meshes. Numerical experiments are conducted to validate the theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Qing Yang, Chuanzhi Bai, Dandan Yang
Summary: This paper studies the finite time stability of stochastic psi-Hilfer fractional-order time-delay systems. By using stochastic analysis techniques and the generalized Gronwall's inequality for psi-fractional derivative, a criterion for finite time stability of the solution for nonlinear stochastic psi-Hilfer fractional systems with time delay is obtained. An example is provided to illustrate the effectiveness of the proposed methods, and some known results in the literature are extended.
Article
Computer Science, Interdisciplinary Applications
Ramy M. Hafez, Magda Hammad, Eid H. Doha
Summary: This paper proposes two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection-diffusion-reaction equations. The algorithms do not require imposition of artificial smoothness assumptions in time direction and the numerical results demonstrate their flexibility and effectiveness.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, I. K. Youssef
Summary: This paper presents a spectral numerical technique for solving multi-order fractional pantograph equations with varying coefficients, demonstrating its superiority in accuracy and efficiency.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Yuxiang Huang, Fanhai Zeng, Ling Guo
Summary: This paper analyzes the convergence of the fast L1 method for the semi-linear time-fractional subdiffusion equations using the generalized discrete Gronwall inequality. It demonstrates that the difference between the solutions of the L1 method and the fast L1 method can be arbitrarily small, regardless of the sizes of the time and/or space grids. The proof is simple and provides a helpful approach to simplifying the convergence analysis of fast methods for time-fractional evolution equations.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Zareen A. Khan, Fahd Jarad, Aziz Khan, Hasib Khan
Summary: In this text, certain discrete fractional nonlinear inequalities are replicated using the sigma fractional sum operator. By considering the methodology of discrete fractional calculus, estimations of Gronwall type inequalities for unknown functions are established. These new form inequalities serve as a supportive strategy to assess the solutions of discrete partial differential equations numerically.
ADVANCES IN DIFFERENCE EQUATIONS
(2021)
Article
Mathematics, Applied
M. A. Zaky, K. Van Bockstal, T. R. Taha, D. Suragan, A. S. Hendy
Summary: A linearized spectral Galerkin/finite difference approach is developed for variable fractional-order nonlinear diffusion-reaction equations with a fixed time delay. The approach utilizes L1-approximation for temporal discretization and Legendre polynomials as basis functions for spatial discretization. The advantage of this approach is the avoidance of implementing the iterative process for the nonlinear term in the variable fractional-order problem. Theoretical proofs of convergence and stability are provided, along with numerical experiments to demonstrate the efficiency and accuracy of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Correction
Mathematics, Applied
William McLean, Kassem Mustapha, Raed Ali, Omar M. Knio
Summary: The note corrects the statement of Theorem 12 from the paper mentioned and fills some gaps in the proof.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Aniruddha Seal, Srinivasan Natesan
Summary: In this manuscript, a nonlinear time-fractional diffusion equation with a generalized memory kernel is studied. The original problem is linearized using Newton's quasilinearization technique. A generalized Caputo derivative is used to approximate the time-fractional term, and a generalized discrete fractional Grönwall inequality is developed to establish stability and analyze error estimates.
NUMERICAL ALGORITHMS
(2023)
Article
Engineering, Multidisciplinary
Ahmed S. Hendy, Mahmoud A. Zaky, Eid H. Doha
Summary: The aim of this paper is to derive a novel discrete form of stochastic fractional Gronwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Gronwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The main challenges of the stated and proved main theorem are the release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < alpha < 1 on the left-hand side. An a priori estimate for a discrete stochastic fractional model is derived using the constructed theorem.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2023)
Article
Mathematics
K. Van Bockstal, A. S. Hendy, M. A. Zaky
Summary: This work focuses on investigating the initial-boundary value problem for a fractional wave equation with space-dependent variable-order and coefficients that depend on both spatial and time variables. This type of operator is commonly used in the modeling of viscoelastic materials. The global existence of a unique weak solution to the model problem has been proven under appropriate data conditions, using Rothe's time discretization method.
QUAESTIONES MATHEMATICAE
(2023)
Article
Mathematics, Applied
Adan J. Serna-Reyes, J. E. Macias-Diaz, Nuria Reguera-Lopez
Summary: In this work, a discrete model is designed and analyzed to approximate the solutions of a parabolic partial differential equation in multiple dimensions. It is found that the proposed discrete model can preserve the positivity and dissipation of the system, and it exhibits second-order consistency and quadratic convergence.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mahmoud A. Zaky, Ahmed S. Hendy, Mehdi Dehghan
Summary: In this paper, a numerical formulation with second-order accuracy in the time direction and spectral accuracy in the space variable is proposed for solving a nonlinear high-dimensional Rosenau-Burgers equation. The spectral element method and the two-grid idea are combined to simulate the equation, and a three-level algorithm is used for the proposed technique. The existence and uniqueness of the solutions to Steps 1, 2, and 3 are investigated, and error analysis is also discussed.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Tao Guo, Mahmoud A. Zaky, Ahmed S. Hendy, Wenlin Qiu
Summary: This paper proposes a third-order backward differentiation formula (BDF3) fourth-order compact difference scheme for approximating the solution of the Burger's equation, which is a useful description for modeling nonlinear acoustics, gas dynamics, fluid mechanics, etc. The proposed method involves two main steps: temporal discretization using the BDF3 approach and spatial discretization using a developed fourth-order operator and the classic compact difference formula combined with the method of order reduction. The theoretical analysis is proved by the discrete energy method, and numerical results validate the effectiveness and accuracy of the proposed method.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Stefania Tomasiello, Jorge E. Macias-Diaz, Joel Alba-Perez
Summary: This work presents an explicit form of the approximate solution using the differential quadrature method, a well-known numerical approach for solving ordinary and partial differential equations. Analogies with Taylor's expansion are discussed and some properties are formally addressed. Additionally, the approach is interpreted from the perspective of neural networks. Comparing the obtained results with known numerical solutions, the method demonstrates good performance.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
Article
Mathematics, Applied
Sidra Ghazanfar, Jorge E. E. Macias-Diaz, Muhammad Sajid Iqbal, Nauman Ahmed
Summary: In this article, the families of solitary waves solutions of a general third-order nonlinear non-Ohmic cable equation in cardio-electro-physiology are obtained using the exp(-phi(xi))-expansion method. The exact soliton-like solutions are derived using this analytic technique and illustrated through surface and contour plots. The existence of solution and their optimal regularity is proven rigorously using Shauder's fixed-point theorem to support the analytical findings presented in this work.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Meihui Zhang, Jinhong Jia, Ahmed S. Hendy, Mahmoud A. Zaky, Xiangcheng Zheng
Summary: We propose a fast numerical technique for a time-fractional option pricing model with asset-price-dependent variable order. The method can handle the complicated variable-order fractional derivative and its fast approximations and overcomes the uncommon difficulties caused by the coupled temporal coefficients and the loss of monotonicity. We demonstrate that the proposed method has error estimates and significantly reduces the computational cost compared to the traditional time-stepping solver.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Briceyda B. Delgado, Aaron Guillen-Villalobos, Jorge E. Macias-Diaz
Summary: In this manuscript, we provide an explicit characterization of the kernel of the parabolic Dirac operator. Specifically, we demonstrate that the members of this kernel satisfy a generalized div-curl system. Additionally, we establish that knowing four scalar solutions of the heat equation is sufficient to construct a Cl1,3 Cℓ1,3-valued function with 16 components belonging to the kernel of the parabolic Dirac operator. This work also derives additional inherited properties of the heat equation and provides concrete examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Physics, Multidisciplinary
Zaky A. Zaky, M. Al-Dossari, Ahmed S. Hendy, Arafa H. Aly
Summary: This study investigates the impact of interface roughness on a layered photonic crystal gas sensor based on Tamm resonance in the terahertz range. The performance of the sensor is observed for different thicknesses of the rough layer between adjacent layers. The results show that an increase in the rough layer between adjacent porous and cavity layers leads to a decrease in sensitivity but enhances the quality factor and figure of merit of the proposed structure. However, changing the thickness of the rough layer between the metallic and the last cavity layer does not affect the performance.
Article
Multidisciplinary Sciences
Zaky A. Zaky, M. A. Mohaseb, Ahmed S. Hendy, Arafa H. Aly
Summary: This paper examines the use of a finite one-dimensional phononic crystal made of branched open resonators with a horizontal defect to detect harmful gas concentrations like CO2. The study investigates the impact of periodic open resonators, a defect duct in the structure's center, and geometrical parameters on the model's performance. This research is unique in the field of sensing, and simulations show that the investigated phononic crystal with a horizontal defect is a promising sensor.
SCIENTIFIC REPORTS
(2023)
Article
Mathematics, Applied
Karel Van Bockstal, Mahmoud A. Zaky, Ahmed Hendy
Summary: In this study, a wave equation with a time-dependent variable-order fractional damping term and a nonlinear source is investigated. The existence and uniqueness of the problem are explored using Rothe's method, avoiding the need for closed-form expressions in terms of special functions. A weak formulation is proposed, and the uniqueness of the solution is established using Gronwall's lemma. A uniform mesh time-discrete scheme is introduced based on a backward discrete convolution approximation, and a priori estimates are obtained for the time-discrete solution. The full discretisation of the problem is achieved using Galerkin spectral techniques, and numerical examples are provided.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2023)
Article
Physics, Multidisciplinary
M. A. Zaky, A. S. Hendy, A. A. Aldraiweesh
Summary: A numerical simulation technique is developed in this paper for the coupled system of variable-order time fractional nonlinear Schr？dinger equations using the finite difference/spectral method. The proposed algorithm discretizes the variable-order Caputo time-fractional derivative using the finite difference method and uses the spectral technique for spatial approximation. The advantage of this algorithm is that it does not require an iterative procedure for the nonlinear term in the coupled system. Numerical experiments are performed to verify the accuracy of the method.
ROMANIAN REPORTS IN PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, Mahmoud A. Zaky, Ahmed S. Hendy, Mehdi Dehghan
Summary: In this paper, a novel numerical solution based on machine learning technique and a generalized moving least squares approximation is developed for solving two-dimensional fractional partial differential equations on irregular domains. The method approximates spatial derivatives on convex and non-convex non-rectangular computational domains and is validated on various specific problems.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2024)
Article
Mathematics, Applied
Mahmoud A. Zaky, Ahmed S. Hendy, Jorge E. Macias-Diaz
Summary: This paper proposes and analyzes a high-order difference/Galerkin spectral scheme for the time-space fractional Ginzburg-Landau equation. The scheme achieves spectral accuracy in space and second-order accuracy in time. Error estimates are established using a fractional Gronwall inequality and its discrete form. Numerical experiments confirm the theoretical claims and discuss the effect of fractional-order parameters on pattern formation.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
