Browsing by Author "Baleanu, D."

Now showing 1 - 20 of 373

Citation - WoS: 72
Citation - Scopus: 99
A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel
(Springer, 2018) Baleanu, D.; Baleanu, Dumitru; Shiri, B.; Srivastava, H. M.; Al Qurashi, M.; 56389; Matematik
In this paper, we solve a system of fractional differential equations within a fractional derivative involving the Mittag-Leffler kernel by using the spectral methods. We apply the Chebyshev polynomials as a base and obtain the necessary operational matrix of fractional integral using the Clenshaw-Curtis formula. By applying the operational matrix, we obtain a system of linear algebraic equations. The approximate solution is computed by solving this system. The regularity of the solution investigated and a convergence analysis is provided. Numerical examples are provided to show the effectiveness and efficiency of the method.
Citation - WoS: 24
A chebyshev-laguerre-gauss-radau collocation scheme for solving a time fractional sub-diffusion equation on a semi-infinite domain
(Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Abdelkawy, M. A.; Alzahrani, A. A.; Baleanu, D.; Alzahrani, E. O.; Matematik
We propose a new efficient spectral collocation method for solving a time fractional sub-diffusion equation on a semi-infinite domain. The shifted Chebyshev-Gauss-Radau interpolation method is adapted for time discretization along with the Laguerre-Gauss-Radau collocation scheme that is used for space discretization on a semi-infinite domain. The main advantage of the proposed approach is that a spectral method is implemented for both time and space discretizations, which allows us to present a new efficient algorithm for solving time fractional sub-diffusion equations.
Citation - WoS: 11
Citation - Scopus: 11
A Computational Approach Based On The Fractional Euler Functions And Chebyshev Cardinal Functions For Distributed-Order Time Fractional 2D Diffusion Equation
(Elsevier, 2023) Heydari, M. H.; Hosseininia, M.; Baleanu, D.; 56389
In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonal-ity, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative oper-ational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss-Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results.(c) 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).
Citation - WoS: 20
Citation - Scopus: 24
A coupled system of generalized Sturm-Liouville problems and Langevin fractional differential equations in the framework of nonlocal and nonsingular derivatives
(Springer, 2020) Baleanu, D.; Baleanu, Dumitru; Alzabut, J.; Jonnalagadda, J. M.; Adjabi, Y.; Matar, M. M.; 56389; Matematik
In this paper, we study a coupled system of generalized Sturm-Liouville problems and Langevin fractional differential equations described by Atangana-Baleanu-Caputo (ABC for short) derivatives whose formulations are based on the notable Mittag-Leffler kernel. Prior to the main results, the equivalence of the coupled system to a nonlinear system of integral equations is proved. Once that has been done, we show in detail the existence-uniqueness and Ulam stability by the aid of fixed point theorems. Further, the continuous dependence of the solutions is extensively discussed. Some examples are given to illustrate the obtained results.
Citation - Scopus: 15
A Decomposition Method for Solving Q-Difference Equations
(Natural Sciences Publishing Co., 2015) Jafari, H.; Baleanu, Dumitru; Johnston, S.J.; Sani, S.M.; Baleanu, D.; 56389; Matematik
The q-difference equations are important in q-calculus. In this paper, we apply the iterative method which is suggested by Daftardar and Jafari, hereafter called the Daftardar-Jafari method, for solving a type of q-partial differential equations. We discuss the convergency of this method. In the implementation of this technique according to other iterative methods such as Adomian decomposition and homotopy perturbation methods, one does not need the calculation of the Adomian's polynomials for nonlinear terms. It is proven that under a special constraint, the given result by this method converges to exact solution of nonlinear q-ordinary or q-partial differential equations. © 2015 NSP Natural Sciences Publishing Cor.
Citation - WoS: 8
Citation - Scopus: 12
A detailed study on a new (2+1)-dimensional mKdV equation involving the Caputo-Fabrizio time-fractional derivative
(Springer, 2020) Hosseini, K.; Baleanu, Dumitru; Ilie, M.; Mirzazadeh, M.; Baleanu, D.; 56389; Matematik
The present article aims to present a comprehensive study on a nonlinear time-fractional model involving the Caputo-Fabrizio (CF) derivative. More explicitly, a new (2+1)-dimensional mKdV (2D-mKdV) equation involving the Caputo-Fabrizio time-fractional derivative is considered and an analytic approximation for it is retrieved through a systematic technique, called the homotopy analysis transform (HAT) method. Furthermore, after proving the Lipschitz condition for the kernel psi (x,y,t;u), the fixed-point theorem is formally utilized to demonstrate the existence and uniqueness of the solution of the new 2D-mKdV equation involving the CF time-fractional derivative. A detailed study finally is carried out to examine the effect of the Caputo-Fabrizio operator on the dynamics of the obtained analytic approximation.
Citation - Scopus: 11
A Detailed Study on a Tumor Model with Delayed Growth of Pro-Tumor Macrophages
(Springer, 2022) Dehingia, K.; Hosseini, K.; Salahshour, S.; Baleanu, D.; 56389
This paper investigates a tumor-macrophages interaction model with a discrete-time delay in the growth of pro-tumor M2 macrophages. The steady-state analysis of the governing model is performed around the tumor dominant steady-state and the interior steady-state. It is found that the tumor dominant steady-state is locally asymptotically stable under certain conditions, and the stability of the interior steady-state is affected by the discrete-time delay; as a result, the unstable system experiences a Hopf bifurcation and gets stabilized. Furthermore, the transversality conditions for the existence of Hopf bifurcations are derived. Several graphical representations in two and three-dimensional postures are given to examine the validity of the results provided in the current study. © 2022, The Author(s), under exclusive licence to Springer Nature India Private Limited.
Citation - WoS: 57
Citation - Scopus: 63
A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets
(Sage Publications Ltd, 2019) Moradi, L.; Baleanu, Dumitru; Mohammadi, F.; Baleanu, D.; 56389; Matematik
The aim of the present study is to present a numerical algorithm for solving time-delay fractional optimal control problems (TDFOCPs). First, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet functions and their properties are implemented to derive some operational matrices. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by means of the Chelyshkov wavelets. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algebraic system. Moreover, some illustrative examples are considered and the obtained numerical results were compared with those previously published in the literature.
Citation - WoS: 6
A fite type result for sequental fractional differintial equations
(Dynamic Publishers, inc, 2010) Abdeljawad, Thabet; Abdeljawad, T.; Baleanu, Dumitru; Baleanu, D.; Jarad, Fahd; Jarad, Fahd; Mustafa, O. G.; Trujillo, J. J.; Matematik
Given the solution f of the sequential fractional differential equation aD(t)(alpha)(aD(t)(alpha) f) + P(t)f = 0, t is an element of [b, a], where -infinity < a < b < c < + infinity, alpha is an element of (1/2, 1) and P : [a, + infinity) -> [0, P-infinity], P-infinity < + infinity, is continuous. Assume that there exist t(1),t(2) is an element of [b, c] such that f(t(1)) = (aD(t)(alpha))(t(2)) = 0. Then, we establish here a positive lower bound for c - a which depends solely on alpha, P-infinity. Such a result might be useful in discussing disconjugate fractional differential equations and fractional interpolation, similarly to the case of (integer order) ordinary differential equations.
Citation - WoS: 24
Citation - Scopus: 29
A FRACTAL FRACTIONAL MODEL FOR CERVICAL CANCER DUE TO HUMAN PAPILLOMAVIRUS INFECTION
(World Scientific Publ Co Pte Ltd, 2021) Akgul, A.; Baleanu, Dumitru; Ahmed, N.; Raza, A.; Iqbal, Z.; Rafiq, M.; Rehman, M. A.; Baleanu, D.; 56389; Matematik
In this paper, we have investigated women's malignant disease, cervical cancer, by constructing the compartmental model. An extended fractal-fractional model is used to study the disease dynamics. The points of equilibria are computed analytically and verified by numerical simulations. The key role of R-0 in describing the stability of the model is presented. The sensitivity analysis of R-0 for deciding the role of certain parameters altering the disease dynamics is carried out. The numerical simulations of the proposed numerical technique are demonstrated to test the claimed facts.
Citation - WoS: 35
Citation - Scopus: 44
A fractional derivative with non-singular kernel for interval-valued functions under uncertainty
(Elsevier Gmbh, Urban & Fischer verlag, 2017) Salahshour, S.; Baleanu, Dumitru; Ahmadian, A.; Ismail, F.; Baleanu, D.; 56389; Matematik
The purpose of the current investigation is to generalize the concept of fractional derivative in the sense of Caputo Fabrizio derivative (CF-derivative) for interval-valued function under uncertainty. The reason to choose this new approach is originated from the non singularity property of the kernel that is critical to interpret the memory aftermath of the system, which was not precisely illustrated in the previous definitions. We study the properties of CF-derivative for interval-valued functions under generalized Hukuhara-differentiability. Then, the fractional differential equations under this notion are presented in details. We also study three real-world systems such as the falling body problem, Basset and Decay problem under interval-valued CF-differentiability. Our cases involve a demonstration that this new notion is accurately applicable for the mechanical and viscoelastic models based on the interval CF-derivative equations. (C) 2016 Elsevier GmbH. All rights reserved.
Citation - Scopus: 1
A Fractional Lagrangian Approach for Two Masses with Linear and Cubic Nonlinear Stiffness
(Institute of Electrical and Electronics Engineers Inc., 2023) Defterli, O.; Defterli, Özlem; Baleanu, D.; Jajarmi, A.; Wannan, R.; Asad, J.; 31401; 56389; Matematik
In this manuscript, the fractional dynamics of the two-mass spring system with two kinds of stiffness, namely linear and strongly nonlinear, are investigated. The corresponding fractional Euler-Lagrange equations of the system are derived being a system of two-coupled fractional differential equations with strong cubic nonlinear term. The numerical results of the system are obtained using Euler's approximation method and simulated with respect to the different values of the model parameters as mass, stiffness and order of the fractional derivative in use. The interpretation of the approximate results of the so-called generalized two-mass spring system is discussed via the fractional order. © 2023 IEEE.
Citation - Scopus: 11
A fractional order co-infection model between malaria and filariasis epidemic
(Taylor and Francis Ltd., 2024) Kumar, P.; Baleanu, Dumitru; Kumar, A.; Kumar, S.; Baleanu, D.; 56389; Matematik
In this article, we investigate a mathematical malaria-filariasis co-infection model with the assistance of the non-integer order operator. Using the fractal-fractional operator in the Caputo-Fabrizio (CF) sense, it has been possible to understand the dynamical behaviour and complicatedness of the malaria-filariasis model. An investigation of the existence and uniqueness of the solution employs fixed-point theory. Ulam-Hyers stability helps examine the stability analysis of the proposed co-infection model. The malaria-filariasis model has been investigated using the Toufik-Atanagana (TA), a sophisticated numerical method for these biological co-infection models. With the help of numerical procedures, we provide the approximate solutions for the proposed model. A variety of fractal dimension and fractional order options are utilized for the presentation of the results. When we adjust sensitive parameters like τ and γ, the graphical representation illustrates the system’s behaviour and identifies suitable parameter ranges for solutions. In addition, we evaluate the model along with the regarded operators and various β1 values using an exceptional graphical representation. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
Citation - Scopus: 5
A Frational Finite Differene Inclusion
(Eudoxus Press, LLC, 2016) Baleanu, D.; Abdeljawad, Thabet; Rezapour, S.; Salehi, S.; Matematik
In this manuscript, we investigated the fractional finite difference inclusion (formula presented) via the boundary conditions Δx(b+μ)=A and x(μ-2)=B, where 1 < μ ≤ 2, A, B ε ℝ. and (formula presented) is a compact valued multifunction. © 2016 by Eudoxus Press, LLC, All rights reserved.
Citation - Scopus: 0
A General Form of Fractional Derivatives for Modelling Purposes in Practice
(Institute of Electrical and Electronics Engineers Inc., 2023) Jajarmi, A.; Baleanu, Dumitru; Baleanu, D.; 56389; Matematik
In this paper, we propose new mathematical models for the complex dynamics of the world population growth as well as a human body's blood ethanol concentration by using a general formulation in fractional calculus. In these new models, we employ a recently introduced ψ-Caputo fractional derivative whose kernel is defined based on another function. Meanwhile, a number of comparative experiences are carried out in order to verify the models according to some sets of real data. Simulation results indicate that better approximations are achieved when the systems are modeled by using the new general fractional formulation than the other cases of fractional- and integer-order descriptions. © 2023 IEEE.
Citation - Scopus: 5
A Generalized Barycentric Rational Interpolation Method for Generalized Abel Integral Equations
(Springer, 2020) Azin, H.; Baleanu, Dumitru; Mohammadi, F.; Baleanu, D.; 56389; Matematik
The paper is devoted to the numerical solution of generalized Abel integral equation. First, the generalized barycentric rational interpolants have been introduced and their properties investigated thoroughly. Then, a numerical method based on these barycentric rational interpolations and the Legendre–Gauss quadrature rule is developed for solving the generalized Abel integral equation. The main advantages of the presented method is that it provides an infinitely smooth approximate solution with no real poles for the generalized Abel integral equation. © 2020, Springer Nature India Private Limited.
Citation - WoS: 19
Citation - Scopus: 19
A highly accurate Jacobi collocation algorithm for systems of high-order linear differential-difference equations with mixed initial conditions
(Wiley, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Doha, E. H.; Baleanu, D.; Hafez, R. M.; 56389; Matematik
In this paper, a shifted Jacobi-Gauss collocation spectral algorithm is developed for solving numerically systems of high-order linear retarded and advanced differential-difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi-Gauss interpolation nodes as collocation nodes. The system of differential-difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright (C) 2015 John Wiley & Sons, Ltd.
Citation - Scopus: 16
A hybrid fractional COVID-19 model with general population mask use: Numerical treatments
(Elsevier B.V., 2021) Sweilam, N.H.; Baleanu, Dumitru; AL-Mekhlafi, S.M.; Almutairi, A.; Baleanu, D.; 56389; Matematik
In this work, a novel mathematical model of Coronavirus (2019-nCov) with general population mask use with modified parameters. The proposed model consists of fourteen fractional-order nonlinear differential equations. Grünwald-Letnikov approximation is used to approximate the new hybrid fractional operator. Compact finite difference method of six order with a new hybrid fractional operator is developed to study the proposed model. Stability analysis of the used methods are given. Comparative studies with generalized fourth order Runge–Kutta method are given. It is found that, the proposed model can be described well the real data of daily confirmed cases in Egypt. © 2021 THE AUTHORS
Citation - WoS: 36
Citation - Scopus: 36
A hybrid fractional optimal control for a novel Coronavirus (2019-nCov) mathematical model
(Elsevier, 2021) Sweilam, N. H.; Baleanu, Dumitru; AL-Mekhlafi, S. M.; Baleanu, D.; 56389; Matematik
Introduction: Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann-Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo's fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. Objectives: A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. Methods: The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grunwald-Letnikov nonstandard finite difference method. Results: The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin's maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grunwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. Conclusions: The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists. (C) 2021 The Authors. Published by Elsevier B.V. on behalf of Cairo University.
Citation - WoS: 108
Citation - Scopus: 120
A hybrid functions numerical scheme for fractional optimal control problems: Application to nonanalytic dynamic systems
(Sage Publications Ltd, 2018) Mohammadi, F.; Baleanu, Dumitru; Moradi, L.; Baleanu, D.; Jajarmi, A.; 56389; Matematik
In this paper, a numerical scheme based on hybrid Chelyshkov functions (HCFs) is presented to solve a class of fractional optimal control problems (FOCPs). To this end, by using the orthogonal Chelyshkov polynomials, the HCFs are constructed and a general formulation for their operational matrix of the fractional integration, in the Riemann-Liouville sense, is derived. This operational matrix together with HCFs are used to reduce the FOCP to a system of algebraic equations, which can be solved by any standard iterative algorithm. Moreover, the application of presented method to the problems with a nonanalytic dynamic system is investigated. Numerical results confirm that the proposed HCFs method can achieve spectral accuracy to approximate the solution of FOCPs.