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
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)
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
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, 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
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, Applied
Hossein Fazli, Hongguang Sun, Sima Aghchi, Juan J. Nieto
Summary: In this study, the existence of extremal solutions for a class of fractional differential equations in fluid dynamics was investigated. By establishing a new comparison theorem and applying the classical monotone iterative approach, sufficient conditions for the existence of extremal solutions were established and twin convergent monotone explicit iterative schemes were constructed. Special cases of the discussed problem include the generalized nonlinear nonlocal Bagley-Torvik equation and the generalized Basset equation with nonlinear source functions.
CARPATHIAN JOURNAL OF MATHEMATICS
(2021)
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
Hazrat Ali, Md. Kamrujjaman, Afroza Shirin
Summary: In this paper, a new numerical solution scheme for the Bagley-Torvik equation of order (0, 2) based on the finite element method is developed, showing its accuracy, efficiency, and simplicity through numerical examples and results. The method's accuracy is better than existing methods, providing guidance for solving fractional-order boundary value problems.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2021)
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
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
M. Hosseininia, M. H. Heydari, M. Razzaghi
Summary: This paper examines a time fractional version of the 2D Fokker-Planck equation involving the Atangana-Baleanu-Caputo fractional derivative and proposes a fully discretization approach for this equation. By implementing this method, a solution for the problem is obtained and the validity of the method is investigated through solving four examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
S. E. Aitzhanov, U. R. Kusherbayeva, K. S. Bekenayeva
Summary: This paper focuses on the solvability of a pseudo-parabolic equation with a Caputo fractional derivative. The existence of a weak solution is investigated using Galerkin approximations and a priori estimates. The uniqueness of the weak solution is proven using the Sobolev embedding theorem, Rellich-Kondrashov theorem, and Gronwall-Bellman lemma. Additionally, the blow up of the solution in finite time is proved. The global solvability of the initial boundary value problem and the uniqueness of the weak generalized solution are also studied.
CHAOS SOLITONS & FRACTALS
(2022)
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
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, Applied
Zahra Abbasi, Mohammad Izadi, Mohammad Mehdi Hosseini
Summary: This research presents a highly accurate matrix approach for handling strongly nonlinear boundary value problems in the modeling of the human eye's corneal shape. It reduces the nonlinear model into a sequence of linearized problems using quasilinearization technique, and then applies a spectral collocation procedure based on shifted Vieta-Fibonacci to transform each subproblem into a linear algebraic system. The error is bounded, and convergence analysis of the SVF series solution is discussed in the weighted L2 norm. The presented collocation algorithm is validated through various numerical tests and comparison with existing numerical solutions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Mohammad Izadi, Pradip Roul
Summary: A shifted Vieta-Fibonacci (SVF) matrix technique is proposed for solving third-order Emden-Fowler equations with nonlinearity and multi-singularity. The SVF-QLM algorithm combines the quasilinearization method (QLM) and SVF functions to iteratively solve a linear algebraic system. Error analysis shows the convergence of the SVF series solution in weighted L-2 and L-infinity norms. Numerical experiments confirm the theoretical results and demonstrate the efficiency and applicability of the proposed SVF-QLM algorithm.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Materials Science, Multidisciplinary
Mohammad Izadi, Ndolane Sene, Waleed Adel, A. El-Mesady
Summary: In this research paper, the numerical solutions of a nonlinear complex Layla and Majnun fractional mathematical model describing the emotional behavior of two lovers are investigated. The model is defined using the Liouville-Caputo derivative and solved using a spectral collocation matrix method and a quasilinearization method with Schroder polynomials as basis functions. The existence of solutions, stability analysis, convergence analysis, and error bound for the main model are discussed, and the theoretical findings are validated through examples. The proposed techniques demonstrate the ability to find accurate solutions and capture the effect of heredity in the fractional order model, and can be extended to simulate similar complex problems.
RESULTS IN PHYSICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Hari Mohan Srivastava, Waleed Adel, Mohammad Izadi, Adel A. El-Sayed
Summary: In this research, a new computational technique for solving physics problems involving fractional-order differential equations is presented. The technique utilizes a collocation technique with a new operational matrix and the Tau spectral method. The method is proven to be effective and efficient through error analysis and comparison with other techniques from the literature. The technique offers better results with fewer bases and less computational time, highlighting important features of the model.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Mohammad Izadi, Hari Mohan Srivastava
Summary: This paper proposes two accurate and efficient spectral collocation techniques for handling a nonlinear fractional system in financial modeling with chaotic behavior. The techniques are based on a domain-splitting strategy, where the spectral method is applied locally in each subdomain. The methods use the generalized version of modified Bessel polynomials and a combination of quasilinearization method and direct numerical matrix method. Convergence theorem and error bounds are proved, and simulation results demonstrate the utility and applicability of the proposed techniques.
Article
Mathematics, Applied
Mohammad Izadi, Hari Mohan Srivastava
Summary: Two effective and accurate matrix collocation techniques based on novel generalized shifted Chebyshev functions of the third kind (GSCFTK) are presented to examine the approximate solutions of a fractional-order population model considering the impact of carrying capacity. The convergence analysis of the new generalized bases is established. Results show that the presented techniques provide accurate results in comparison to other available numerical models and can be extended to other similar biological problems. Additionally, the numerical schemes are robust.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2023)
Article
Mathematics, Applied
Donya Rezaei, Mohammad Izadi
Summary: This manuscript introduces a novel combination of the complex fractional transform (CFT) and residual power series method (RPSM) to solve the time-space fractional Black-Scholes equation in the financial market. The presented CFT-RPSM achieves an accurate approximation with a few iterations by considering the fractional operator as a local fractional derivative and utilizing the initial conditions. Numerical results and comparisons with existing methods demonstrate the accuracy and straightforward implementation of the proposed approach. This strategy can be easily extended to other fractional models in physical science and engineering.
UNIVERSITY POLITEHNICA OF BUCHAREST SCIENTIFIC BULLETIN-SERIES A-APPLIED MATHEMATICS AND PHYSICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Hari M. M. Srivastava, Mohammad Izadi
Summary: In this manuscript, the author discusses the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM), and a family of linearized equations is obtained. Through numerical simulations, it is shown that the proposed hybrid QLM-SAPSK approach is capable of tackling the inherit singularity at the origin and produces effective numerical solutions to the model problem with different nonlinearity parameters and two fractional order derivatives. The accuracy of the present technique is checked via the technique of residual error functions. The QLM-SAPSK technique is simple and efficient for solving the underlying electrohydrodynamic flow model, and the computational outcomes are accurate in comparison with those of numerical values reported in the literature.
FRACTAL AND FRACTIONAL
(2023)
Article
Chemistry, Multidisciplinary
Mohammad Izadi, Hari M. Srivastava
Summary: This paper presents a numerical treatment method for a class of boundary value problems with singularity and nonlinearity. The proposed technique combines spectral matrix technique and quasilinearization method, using a novel class of polynomials introduced by S.K. Chatterjea. The convergence analysis of the method is proven and numerical examples demonstrate its efficiency and flexibility in solving similar problems in science and engineering.
APPLIED SCIENCES-BASEL
(2023)
Article
Mathematics, Applied
Mohammad Izadi, Suayip Yuzbasi, Waleed Adel
Summary: This paper investigates the treatment of a class of fractional-order delay differential equations. The method discretizes the unknown solution using a truncated series and solves the resulting system to produce highly accurate solutions. Detailed error analysis and comparisons with other techniques demonstrate the method's effectiveness in terms of accuracy and computational cost. Thus, the method is considered a promising and efficient candidate for simulating such problems in science.
MATHEMATICAL SCIENCES
(2023)
Article
Mathematics, Applied
Mohammad Izadi, Suayip Yuzbasi
Summary: Our study proposes a hybrid spectral collocation approach for solving singularly perturbed 1-D parabolic convection-diffusion problems. This approach discretizes in time using Taylor series expansions and applies novel special polynomials to the spatial operator in the time step. A detailed error analysis of the space variable demonstrates the advantages of this method compared to other existing schemes.
MATHEMATICAL COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Mohammad Izadi, Hari M. Srivastava, Waleed Adel
Summary: In this research study, a novel computational algorithm is presented for solving a second-order singular functional differential equation. By using Bessel bases, the problem is transformed into a system of algebraic equations and the unknown Bessel coefficients are determined to obtain the solution. The method proves to provide accurate results and shows potential for application to similar problems.
Article
Engineering, Multidisciplinary
M. Izadi
Summary: The aim of this study is to develop a combined approximation technique to find a numerical solution to the foam drainage equation in various absorption and distillation processes. The solution is obtained through time discretization and space collocation using Bessel polynomials, solving a linear system of algebraic equations. Numerical simulations are conducted to assess the accuracy and efficiency of the algorithm.
Article
Mathematics, Applied
Mohammad Izadi, Suayip Yuzbasi, Dumitru Baleanu
Summary: This paper presents an approximate solution for the Burgers equation using a novel combined approximation algorithm based on linearized Taylor method and spectral Chebyshev collocation method. The efficiency and practicality of the combined scheme are validated through numerical simulations and comparisons with exact solutions and existing methods. The results indicate that the combined approach is effective, practical, and easily extendable to other nonlinear models.
MATHEMATICAL SCIENCES
(2022)
