Article
Mathematics, Interdisciplinary Applications
Ahmed Z. Amin, Mohamed A. Abdelkawy, Emad Solouma, Ibrahim Al-Dayel
Summary: This paper proposes a numerical solution based on the shifted Legendre Gauss-Lobatto collocation technique for the non-linear distributed-order fractional Bagley-Torvik differential equation, and verifies its accuracy through numerical examples.
FRACTAL AND FRACTIONAL
(2023)
Article
Engineering, Multidisciplinary
Ahmed Z. Amin, Antonio M. Lopes, Ishak Hashim
Summary: A numerical approach based on the shifted Chebyshev-Gauss collocation method is proposed to solve the nonlinear variable-order fractional Bagley-Torvik differential equation (VO-FBTE). The solution of the VO-FBTE is approximated using a truncated series of shifted Chebyshev polynomials, with the interpolation nodes chosen as the shifted fractional Chebyshev-Gauss collocation points. The accuracy of the proposed method is confirmed through numerical examples and compared against other methods.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Mohamed Adel, Hari M. Srivastava, Mohamed M. Khader
Summary: In this study, we propose an accurate numerical procedure to solve the mathematical model of fractional order in electrical circuits. The method is validated by comparing the approximate solutions with the exact solution.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Engineering, Multidisciplinary
Suayip Yuzbasi, Gamze Yildirim
Summary: In this study, a collocation approach is proposed to solve the Bagley-Torvik equation, a type of fractional differential equation. The method utilizes Laguerre polynomials and matrix relations to convert the problem into a system of linear algebraic equations, leading to approximate solutions. Additionally, an error estimation method is provided, and the Laguerre polynomial solution is improved for enhanced accuracy.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Tianyi Pu, Marco Fasondini
Summary: We propose a spectral method for one-sided linear fractional integral equations on a closed interval, achieving exponential convergence even for equations with irrational or multiple fractional orders. The method utilizes Jacobi fractional polynomials, a new orthogonal basis obtained from weighted classical Jacobi polynomials. We introduce new algorithms for constructing the matrices used to represent fractional integration operators. Despite the instability and high-precision computations required, the spectral method produces well-conditioned and efficient linear systems. In comparison to a sparse spectral method, our orthogonal basis-based method demonstrates superior stability for time-fractional heat and wave equations.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Murugesan Meganathan, Thabet Abdeljawad, M. Motawi Khashan, Gnanaprakasam Britto Antony Xavier, Fahd Jarad
Summary: This research introduces a discrete version of the fractional Bagley-Torvik equation and obtains solutions using the nabla discrete Laplace transform. The solutions are expressed in terms of Mittag-Leffler functions with three parameters and are numerically handled for specific examples.
ADVANCES IN DIFFERENCE EQUATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Omar Abu Arqub, Ahlem Ben Rabah, Shaher Momani
Summary: In this paper, the well-known Bagley-Torvik and Painleve models are solved numerically using the cubic B-spline polynomials approximation. Matrix operations and fundamental linear algebra are employed to transform the models into a computational scheme of linear and nonlinear algebraic equations. The accuracy and computational complexity of the scheme are analyzed based on extensive independent runs and statistical analysis.
INTERNATIONAL JOURNAL OF MODERN PHYSICS C
(2023)
Article
Materials Science, Multidisciplinary
Kamran, Muhammad Asif, Kamal Shah, Bahaaeldin Abdalla, Thabet Abdeljawad
Summary: This article aims to solve the Bagley-Torvik equation involving the Atangana-Baleanu derivative using the Laplace transform method. The Laplace transform is an effective tool for solving differential equations, but sometimes leads to solutions in the Laplace domain that cannot be inverted back to the time domain analytically. Therefore, numerical methods are used to convert the solution from the Laplace domain to the time domain. Four numerical inverse Laplace transform methods are utilized, and their accuracy and efficiency are validated through four test problems. The computational results are presented using tables and figures, and compared with other methods in the literature to demonstrate the superiority of the proposed methods.
RESULTS IN PHYSICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Haiyong Xu, Lihong Zhang, Guotao Wang
Summary: This paper investigates the existence of extremal solutions for a nonlinear boundary value problem involving the Caputo-Fabrizio-type fractional differential operator. By utilizing a new inequality and applying a monotone iterative technique with upper and lower solutions, the main result is obtained. The paper concludes with an illustrative example.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Mohammad Izadi, Suayip Yuzbasi, Carlo Cattani
Summary: This paper presents a practical and effective scheme to approximate the solutions of the Bagley-Torvik equations using Bessel polynomials as approximation basis functions. The equations are reduced into an algebraic form and a fast algorithm with linear complexity is introduced to compute the fractional derivatives. The proposed approximation algorithm shows high accuracy and efficiency for long-time computations.
RICERCHE DI MATEMATICA
(2023)
Article
Mathematics, Interdisciplinary Applications
Mohamed Fathy, K. M. Abdelgaber
Summary: This paper presents and applies the Galerkin method to obtain semi-analytical solutions of quadratic Riccati and Bagley-Torvik differential equations in fractional order. New theorems are proposed to optimize the efficiency of the method by minimizing the generated residual. The proposed method is compared with other methods through solving initial value problems of different fractional orders, and the results are illustrated using tables and figures. The Legendre-Galerkin method is found to be efficient and reliable for these problems.
CHAOS SOLITONS & FRACTALS
(2022)
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
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
Mathematics, Applied
Shatha Hasan, Nadir Djeddi, Mohammed Al-Smadi, Shrideh Al-Omari, Shaher Momani, Andreea Fulga
Summary: This paper investigates the generalized Bagley-Torvik equation based on the Caputo-Fabrizio fractional derivative concept using a modified reproducing kernel Hilbert space treatment. By reformulating the problem as a system of fractional differential equations while preserving the second type of these equations, the study establishes reproducing kernel functions to construct an orthogonal system for both analytical and approximate solutions in the appropriate Hilbert spaces. The proposed method shows accuracy, efficiency, and reliability in solving various real-life problems through numerical examples.
ADVANCES IN DIFFERENCE EQUATIONS
(2021)
Article
Mathematics, Applied
Qinxu Ding, Patricia J. Y. Wong
Summary: This paper develops a higher order numerical method for the fractional Bagley-Torvik equation, utilizing a new fourth-order approximation for the fractional derivative and a discrete cubic spline approach. The theoretical convergence order of the method is shown to improve upon previous work. Five examples are provided to demonstrate the efficiency of the method and compare it with others in the literature.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
