Article
Mathematics
Taher S. Hassan, Ismoil Odinaev, Rasool Shah, Wajaree Weera
Summary: This article presents a technique for solving fractional integro differential equations using the Chebyshev pseudospectral method. The results demonstrate the high accuracy and reliability of the proposed technique, which is compared to other methods and the exact solution. The Chebyshev pseudospectral method is shown to be more accurate and straightforward compared to other methods.
Article
Engineering, Multidisciplinary
Khalid K. Ali, Mohamed A. Abd El Salam, Emad M. H. Mohamed
Summary: This paper presents a numerical technique for solving a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series. The proposed method is extended to study this problem as a discretization scheme. The obtained results show that the proposed method is very effective and convenient.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Applied
Chuanli Wang, Biyun Chen
Summary: We propose a multi-step spectral collocation method for solving Caputo-type fractional integro-differential equations with weakly singular kernels. By reformulating the problem as a second type Volterra integral equation with two different weakly singular kernels, we construct a multi-step Legendre-Gauss spectral collocation scheme and rigorously establish the hp-version convergence. Numerical experiments are conducted to validate the effectiveness of the suggested method and the theoretical results.
Article
Mathematics, Applied
Changqing Yang, Xiaoguang Lv
Summary: In this study, we propose an approximation method for solving pantograph-type fractional-order differential equations using Chebyshev polynomials. We construct operational matrices for pantograph and Caputo fractional derivatives based on Chebyshev interpolation and use them to approximate the fractional derivative. Convergence analysis in terms of the weighted square norm is provided. Numerical experiments confirm the applicability and accuracy of the proposed computational scheme.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Engineering, Multidisciplinary
Khalid K. Ali, Emad M. H. Mohamed, Mohamed A. Abd El Salam, Kottakkaran Sooppy Nisar, M. Motawi Khashan, Mohammed Zakarya
Summary: This article presents a general form of multiterm variable-order fractional delay differential equations (GVOFDDEs), with variable-order multi-terms and integer-order derivatives. A collocation approach using shifted Chebyshev polynomials is applied to solve the GVOFDDEs, transforming all terms into a matrix equation with novel operational matrices. The qualification of the presented scheme is assessed through numerous numerical test examples.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Mathematics, Applied
Yong-Suk Kang, Son-Hyang Jo
Summary: In this paper, we solve multi-term fractional integro-differential equations with variable coefficients and nonlinear integral using spectral collocation method. We construct a spectral collocation algorithm and provide rigorous error analysis for equations with variable coefficients and nonlinear integral. Finally, we demonstrate the convergence rate and efficiency of the proposed method through numerical examples.
MATHEMATICAL SCIENCES
(2022)
Article
Mathematics, Applied
Obaid Algahtani, M. A. Abdelkawy, Antonio M. Lopes
Summary: A new technique is proposed to solve variable order fractional stochastic Volterra integro-differential equations, which shows superiority when dealing with problems with non-smooth solutions.
Article
Mathematics, Applied
Suayip Yuzbasi
Summary: In this study, a collocation method based on the Bell polynomials is introduced for solving linear fractional integro-differential equations. By expressing the fractional Bell functions in matrix forms and using Caputo derivative and equal spacing points, the linear fractional problem is transformed into a system of linear algebraic equations, and its solutions provide the coefficients of the assumed solution.
MATHEMATICAL SCIENCES
(2022)
Article
Mathematics, Interdisciplinary Applications
Zhongshu Wu, Xinxia Zhang, Jihan Wang, Xiaoyan Zeng
Summary: This paper explores the use of Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. The collocation points were determined using shifted Jacobi-Gauss-Lobatto or Jacobi-Gauss-Radau quadrature nodes, and the fractional differentiation matrices for Caputo fractional derivatives were derived. By transforming the fractional differential equations into linear systems using these differentiation matrices, easier solutions were obtained. Numerical simulations of two types of fractional differential equations demonstrated the fast convergence and high accuracy of the proposed methods.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Ampol Duangpan, Ratinan Boonklurb, Matinee Juytai
Summary: In this paper, the finite integration method and the operational matrix of fractional integration were implemented using the shifted Chebyshev polynomial to solve systems of fractional and classical integro-differential equations. The numerical procedures showed significant improvement in accuracy for solving stiff ODE systems, with experimental examples demonstrating efficiency and numerical convergence.
FRACTAL AND FRACTIONAL
(2021)
Article
Computer Science, Interdisciplinary Applications
E. H. Doha, M. A. Abdelkawy, A. Z. M. Amin, Antonio M. Lopes
Summary: This paper presents a spectral collocation technique for solving fractional stochastic Volterra integro-differential equations (FSV-IDEs) based on shifted fractional order Legendre orthogonal functions. The method aims to approximate the FSV-IDEs to obtain a system of algebraic equations, with Brownian motion function discretized by Lagrange interpolation for computational purposes. The proposed technique demonstrates accuracy and applicability, even for non-smooth solutions, through numerical examples.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Xiaojun Zhou, Yue Dai
Summary: This paper analyzes the coupled system of nonlinear fractional differential equations involving the caputo fractional derivatives. It proposes a spectral collocation method based on the Legendre polynomials for discretizing the resulting equation. The theoretical result is validated by numerical tests, and error bounds under the L-2- and L-infinity-norms are provided.
Article
Mathematics, Applied
Khalid K. Ali, Mohamed A. Abd El Salam, Mohamed S. Mohamed
Summary: This paper proposes a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs) using Shifted Chebyshev fifth-kind polynomials with the spectral collocation approach. The proposed method shows high accuracy and simplicity in implementation. It reduces the GFPDEs to a system of differential equations that can be solved numerically, using a combination of the Chebyshev collocation method and the finite difference method. Numerical approximations performed by the method are compared with other numerical methods, and the results demonstrate that the proposed method is effective.
Article
Engineering, Multidisciplinary
Fatima Youbi, Shaher Momani, Shatha Hasan, Mohammed Al-Smadi
Summary: This paper investigates the analytic-approximate solutions for fractional system of Volterra integro-differential equations using the Caputo-Fabrizio operator. The methodology involves creating reproducing kernel functions and constructing orthonormal basis systems to obtain rapidly convergent analytical solutions. Convergence and error analysis are also discussed, and numerical examples are provided to demonstrate the feasibility and efficiency of the proposed method. The results indicate that the methodology is sound, straightforward, and suitable for dealing with physical issues involving Caputo-Fabrizio derivatives.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Mathematics, Applied
M. Mossa Al-Sawalha, Azzh Saad Alshehry, Kamsing Nonlaopon, Rasool Shah, Osama Y. Ababneh
Summary: In this article, the pantograph delay differential equations are solved using the efficient numerical technique known as Chebyshev pseudospectral method, with the Caputo method employed for fractional derivatives. The proposed method is shown to be effective and accurate through the verification with two examples. Furthermore, it can be applied to other linear and nonlinear fractional delay differential equations due to its innovation and scientific significance.
Article
Mathematics, Applied
Sunil Kumar, Ram P. Chauhan, Mohamed S. Osman, S. A. Mohiuddine
Summary: The main aim of this work is to numerically investigate a mathematical model of the HIV/AIDS disease with awareness effect. The study extends the integer-order model to fractional order using the generalized Caputo fractional derivative. A fixed-point approach is utilized to prove the existence and uniqueness of the model. The fractional-order HIV/AIDS transmission model is then solved using a new numerical technique based on the Adams-Bashforth predictor-corrector method, and the results are illustrated through graphs.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Engineering, Multidisciplinary
Khalid K. Ali, Nuha Al-Harbi, Abdel-Haleem Abdel-Aty
Summary: In this article, the authors discuss the analytical and numerical analysis of the three-dimensional conformal time derivative generalized q-deformed Sinh-Gordon equation using the (Gi/G)-expansion approach and finite element method. The analytical method successfully extracts the solutions and identifies the constraint requirements for the existence of solutions. The numerical findings are presented using the extended cubic B-spline technique, and various soliton propagation patterns are demonstrated with figures.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Engineering, Multidisciplinary
Lei Shi, Soumia Tayebi, Omar Abu Arqub, M. S. Osman, Praveen Agarwal, W. Mahamoud, Mahmoud Abdel-Aty, Mohammed Alhodaly
Summary: In this analysis, the high order cubic B-spline method is used to approximate solutions for fractional Painleve' and Bagley-Torvik equations. The approach considers different boundary set conditions. The discretization of the fractional model problems is achieved using a piecewise spline of a 3rd-degree polynomial. The spline method is demonstrated to be cost-efficient and precise in its calculations, making it suitable for various applications.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Engineering, Multidisciplinary
Khalid K. Ali, Mohamed Omri, M. S. Mehanna, Hatem Besbes, Abdel-Haleem Abdel-Aty
Summary: This article aims to obtain families of new traveling wave solutions to the (2+1) dimensional nonlinear evolution equation by utilizing three powerful analytical techniques: the general form of Kudryashov's method, the Bernoulli sub-ODE method, and the extended direct algebraic method. Several graphics are used to illustrate the findings. The resulting solutions are expressed in terms of hyperbolic, trigonometric, rational, and exponential functions, and include bright, dark, periodic, and singular soliton solutions. The obtained solutions play a critical role in understanding wave dynamics in different models, and the article confirms that the used approaches are highly effective for solving the given model and producing diverse solutions. The dynamics of the model's solutions can be controlled by adjusting the model's parameters.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Computer Science, Theory & Methods
Amira Abd-Elall Ibrahim, Afaf A. S. Zaghrout, K. R. Raslan, Khalid K. Ali
Summary: In this paper, a coupled system of nonlocal fractional q-integro-differential equations is introduced. The existence and uniqueness of solutions for this system are proven under certain assumptions. Continuous dependence is also studied. The system is then numerically solved using the finite-Simpson's and cubic spline-Simpson's methods. Finally, the efficacy of the methods employed is demonstrated through three examples.
INTERNATIONAL JOURNAL OF MODELING SIMULATION AND SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Seydi Battal Gazi Karakoc, Khalid K. Ali, Derya Yildirim Sucu
Summary: In this study, an analytical method and a numerical approach based on the finite element method are implemented to obtain approximate soliton solutions for two special forms of the fifth-order KdV equation (fKdV): the Kaup-Kupershmidt (K-K) equation and the Ito equation. The simplest equation method and septic B-spline collocation method are introduced to achieve this goal. The proficiency and accuracy of the method are demonstrated by computing L2 and L infinity error norms for single soliton solutions. The von-Neumann stability analysis shows that the method is unconditionally stable. The obtained numerical results are presented in tables, and the analytical and numerical behaviors of the single soliton are illustrated in 2D and 3D figures. The analytical and numerical results confirm that the methods are more suitable and systematic for handling the solution procedures of nonlinear equations in mathematical physics.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Khalid K. Ali, K. R. Raslan, Amira Abd-Elall Ibrahim, Mohamed S. Mohamed
Summary: In this paper, the existence, uniqueness, and continuous dependence of solutions for a coupled system of fractional q-integro-differential equations are investigated using the definitions of the Caputo-Fabrizio fractional derivative and the Riemann-Liouville fractional q-integral. An overview of the finite-trapezoidal method is provided. Finally, some numerical examples are presented to demonstrate the effectiveness of the method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Engineering, Electrical & Electronic
Khalid K. Ali, Salman A. AlQahtani, M. S. Mehanna, Ahmet Bekir
Summary: In this study, three efficient methods, namely the Sardar sub-equation approach, the Bernoulli sub-ODE method, and the generic Kudryashov's method, are used to provide various families of solutions for the (2+1) Fokas system. Our suggested approaches offer several types of function solutions, including hyperbolic, trigonometric, power, exponential, and rational functions. Numerous graphs are presented at the end of the text to highlight the many solutions found.
OPTICAL AND QUANTUM ELECTRONICS
(2023)
Article
Materials Science, Multidisciplinary
Riaz Ur Rahman, Maysoon Mustafa Mohammad Qousini, Ahmed Alshehri, Sayed M. Eldin, K. El-Rashidy, M. S. Osman
Summary: In this article, the nonlinear fractional Kudryashov's equation and the space-time fractional nonlinear Tzitzeica-Dodd-Bullough (TDB) equation are solved using the new auxiliary equation method, which yields innovative analytical solutions using beta and M-Truncated fractional derivatives. The fractional wave and Painleve transformations are implemented to transform the space and time fractional nonlinear equations into a nonlinear ordinary differential equation. Various soliton solutions, including hyperbolic, trigonometric, rational, exponential, and other forms, are discovered, demonstrating the superiority of the method. The findings highlight the applicability and expertise of fractional derivatives and the proposed approach for evaluating nonlinear fractional partial differential equations.
RESULTS IN PHYSICS
(2023)
Article
Multidisciplinary Sciences
Rajib Mia, M. Mamun Miah, M. S. Osman
Summary: In this study, novel exact traveling wave solutions for the (2 + 1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation are found through a comprehensive analytical study. Using the recently developed (������+PRIME;) -expansion technique, new analytical solutions expressed as trigonometric and exponential functions are obtained. These exact wave solutions are advanced and unique compared to previous literature, and contour simulations, 2D and 3D graphical representations show that the solutions obtained are periodic and solitary wave solutions.
Article
Materials Science, Biomaterials
Ayman A. Gadelhak, Kh. S. Mekheimer, M. A. Seddeek, R. E. Abo-Elkhair, Khalid K. Ali, Ahmed M. Salem
Summary: This paper investigates the flow of a two-dimensional incompressible non-Newtonian fluid (Williamson model) through a special curvilinear coordinate system. The model is designed as a simulation of energy transport in thermal engineering processes using the Buongiorno model for Williamson nanofluid boundary layer flow. The study evaluates the impact of the curvature parameter on energy and mass transfer, including linear thermal radiation effects. The governing equations are solved numerically using the Finite Difference Method, and the results show good agreement with a previous study for the Nusselt number.
Article
Physics, Multidisciplinary
M. Mamun Miah, M. Ashik Iqbal, M. S. Osman
Summary: In this paper, dynamic solitary perturb solutions of a unidirectional stochastic longitudinal wave equation in a magneto-electro-elastic annular bar were obtained using the dual (G'/G, 1/G)-expansion method. Trigonometric, hyperbolic, and rational solutions were classified, which are important in searching for scientific events. The technique employed is an extension of the (G'/G)-expansion technique for finding previously discovered solutions.
COMMUNICATIONS IN THEORETICAL PHYSICS
(2023)
Article
Materials Science, Multidisciplinary
Nura Talaq Alqurashi, Maria Manzoor, Sheikh Zain Majid, Muhammad Imran Asjad, M. S. Osman
Summary: The objective of this research is to investigate the exact traveling wave solutions of the nonlinear Landau-Ginzburg-Higgs equation using an extended direct algebraic technique. Various types of soliton solutions are obtained and presented graphically. The reliability and efficiency of the proposed method are demonstrated through chaotic analysis.
RESULTS IN PHYSICS
(2023)
Article
Engineering, Multidisciplinary
Sania Qureshi, Moses Adebowale Akanbi, Asif Ali Shaikh, Ashiribo Senapon Wusu, Oladotun Matthew Ogunlaran, W. Mahmoud, M. S. Osman
Summary: A new adaptive numerical method is proposed for solving nonlinear, singular, and stiff initial value problems encountered in real life. The method's performance is significantly enhanced by introducing an adaptive step-size approach. The proposed method has been investigated for efficiency and reliability, showing fifth-order accuracy, zero stability, L-stability, consistency, and convergent properties, and its performance is demonstrated to be favorable compared to existing methods through numerical experiments.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics, Applied
Khalid K. Ali, Mohamed S. Osman, Haci Mehmet Baskonus, Nasser S. Elazabb, Esin Ilhan
Summary: This paper presents a numerical and analytical study on the HIV-1 infection of CD4(+) T-cells using a conformable fractional mathematical model. The model is analyzed using Kudryashov and modified Kudryashov methods, and numerically studied using the finite difference method. A comparison between the results obtained from the analytical and numerical methods is conducted, and figures are provided to demonstrate the accuracy of the solutions obtained.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
