WoS İndeksli Yayınlar Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8653

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Now showing 1 - 10 of 26
  • Article
    Citation - WoS: 66
    Citation - Scopus: 63
    An Accurate Numerical Technique for Solving Fractional Optimal Control Problems
    (Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.; Abdelkawy, M. A.; Matematik
    In this article, we propose the shifted Legendre orthonormal polynomials for the numerical solution of the fractional optimal control problems that appear in several branches of physics and engineering. The Rayleigh-Ritz method for the necessary conditions of optimization and the operational matrix of fractional derivatives are used together with the help of the properties of the shifted Legendre orthonormal polynomials to reduce the fractional optimal control problem to solving a system of algebraic equations that greatly simplifies the problem. For confirming the efficiency and accuracy of the proposed technique, an illustrative numerical example is introduced with its approximate solution.
  • Article
    Citation - WoS: 29
    Citation - Scopus: 31
    A Novel Spectral Approximation for the Two-Dimensional Fractional Sub-Diffusion Problems
    (Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Zaky, M. A.; Baleanu, D.; Abdelkawy, M. A.; Matematik
    This paper reports a new numerical method that enables easy and convenient discretization of a two-dimensional sub-diffusion equation with fractional derivatives of any order. The suggested method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional derivatives, described in the Caputo sense. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. The validity and effectiveness of the method are demonstrated by solving two numerical examples, which are presented in the form of tables and graphs to make more easier comparisons with the exact solutions and the results obtained by other methods.
  • Article
    Citation - WoS: 89
    Citation - Scopus: 124
    New Numerical Approximations for Space-Time Fractional Burgers' Equations Via a Legendre Spectral-Collocation Method
    (Editura Acad Romane, 2015) Bhrawy, A. H.; Zaky, M. A.; Baleanu, D.
    Burgers' equation is a fundamental partial differential equation in fluid mechanics. This paper reports a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers' equation (FBE) based on spectral shifted Legendre collocation (SLC) method in combination with the shifted Legendre operational matrix of fractional derivatives. The fractional derivatives are described in the Caputo sense. We propose a spectral shifted Legendre collocation method in both temporal and spatial discretizations for the space-time FBE. The main characteristic behind this approach is that it reduces such problem to that of solving a system of nonlinear algebraic equations that can then be solved using Newton's iterative method. Numerical results with comparisons are given to confirm the reliability of the proposed method for FBE.
  • Article
    Citation - WoS: 3
    Citation - Scopus: 3
    Solving 2d-Integro Problems With Nonlocal Boundary Conditions Via a Matrix Formulated Approach
    (Elsevier, 2023) Borhanifar, A.; Shahmorad, S.; Feizi, E.; Baleanu, D.
    A new operational matrix based approach is studied for numerical solution of 2D-integro-differential equations with non-local (integral) boundary conditions whose arise in some physical problems. Some important theoretical results are presented to reduce complexity and computational costs of the proposed method. We also give an error estimation which will be useful in estimating the error of approximate solution for the problems that we do not have any information about their exact solution. Illustrative numerical examples are also given to clarify the performance and accuracy of the new method.& COPY; 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
  • Article
    Citation - WoS: 11
    Citation - Scopus: 21
    Normalized Lucas Wavelets: an Application To Lane-Emden and Pantograph Differential Equations
    (Springer Heidelberg, 2020) Koundal, Reena; Srivastava, K.; Baleanu, D.; Kumar, Rakesh
    In this paper, a novel normalized Lucas wavelet scheme based on tau approach is proposed for the two classes of second-order differential equations, namely Lane-Emden and pantograph equations. The introduced scheme depends on shifted Lucas polynomials (SLPs) and their operational matrix of derivative (which are developed here). The weight function for the orthogonality of Lucas polynomials, and Rodrigues formula are proposed for the first time, which form the basis for the construction of SLPs. Normalized Lucas wavelets are constructed by utilizing SLPs and their novel properties. Literally, the present scheme transforms the given method to a set of nonlinear algebraic equations with undetermined coefficients which are here tackled by tau method. Meanwhile, new treatment of convergence and error analysis is provided for the established approach. Finally, the accuracy and applicability of present scheme is ensured by considering several examples.
  • Article
    Citation - WoS: 10
    Citation - Scopus: 8
    Derivation of Operational Matrix of Rabotnov Fractional-Exponential Kernel and Its Application To Fractional Lienard Equation
    (Elsevier, 2020) Gomez-Aguilar, J. F.; Lavin-Delgado, J. E.; Baleanu, D.; Kumar, Sachin
    Our motive in this contribution is to find out the operational matrix of fractional derivative having non singular kernel namely Rabotnov fractional-exponential (RFE) kernel which is recently introduced and seeking numerical solution of non-linear Lienard equation which have Rabotnov fractional-exponential kernel fractional derivative. First we derive an approximation formula of the fractional order derivative of polynomial function z(k) in term of RFE kernel. Using this formula and some properties of shifted Legendre polynomials, we find out the operational matrix of fractional order differentiation. In the author of knowledge this operational matrix of RFE kernel fractional derivative is derived first time. We solve a new class of fractional partial differential equation (FPDEs) by implementation of this newly derived operational matrix. We show that our newly derived operational matrix is valid by taking an fractional derivative of a polynomial. Also, we study a new model of Lienard equation with RFE kernel fractional derivative and we can easily predict the feasibility of our numerical method to this new model. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
  • Article
    Citation - WoS: 42
    Citation - Scopus: 44
    A Numerical Approach for Solving Fractional Optimal Control Problems With Mittag-Leffler Kernel
    (Sage Publications Ltd, 2022) Ganji, Roghayeh M.; Sayevand, Khosro; Baleanu, Dumitru; Jafari, Hossein
    In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana-Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme.
  • Article
    Citation - WoS: 42
    Citation - Scopus: 56
    A Novel Jacobi Operational Matrix for Numerical Solution of Multi-Term Variable-Order Fractional Differential Equations
    (Taylor & Francis Ltd, 2020) Baleanu, D.; Agarwal, P.; El-Sayed, A. A.
    In this article, we introduce a numerical technique for solving a class of multi-term variable-order fractional differential equation.The method depends on establishing a shifted Jacobi operational matrix (SJOM) of fractional variable-order derivatives. By using the constructed (SJOM) in combination with the collocation technique, the main problem is reduced to an algebraic system of equations that can be solved numerically. The bound of the error estimate for the suggested method is investigated. Numerical examples are introduced to illustrate the applicability, generality, and accuracy of the proposed technique. Moreover, many physical applications problems that have the multi-term variable-order fractional differential equation formulae such as the damped mechanical oscillator problem and Bagley-Torvik equation can be solved via the presented method. Furthermore, the proposed method will be considered as a generalization of many numerical techniques.
  • Article
    Citation - WoS: 185
    Citation - Scopus: 202
    An Efficient Numerical Method for Fractional Sir Epidemic Model of Infectious Disease by Using Bernstein Wavelets
    (Mdpi, 2020) Ahmadian, Ali; Kumar, Ranbir; Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru; Salimi, Mehdi; Kumar, Sunil
    In this paper, the operational matrix based on Bernstein wavelets is presented for solving fractional SIR model with unknown parameters. The SIR model is a system of differential equations that arises in medical science to study epidemiology and medical care for the injured. Operational matrices merged with the collocation method are used to convert fractional-order problems into algebraic equations. The Adams-Bashforth-Moulton predictor correcter scheme is also discussed for solving the same. We have compared the solutions with the Adams-Bashforth predictor correcter scheme for the accuracy and applicability of the Bernstein wavelet method. The convergence analysis of the Bernstein wavelet has been also discussed for the validity of the method.
  • Article
    Citation - WoS: 19
    Citation - Scopus: 20
    Numerical Solution of Variable Fractional Order Advection-Dispersion Equation Using Bernoulli Wavelet Method and New Operational Matrix of Fractional Order Derivative
    (Wiley, 2020) Arabameri, Maryam; Baleanu, Dumitru; Barfeie, Mahdiar; Soltanpour Moghadam, Abolfazl
    In this article, the Bernoulli wavelet method is used to solve the space-time variable fractional order advection-dispersion equation. The equation contains Coimbra time fractional derivatives with variable order of gamma 1(x) as well as the Riemann-Liouville space fractional derivatives with variable orders of gamma 2(x,t) and gamma 3(x,t). In fact, first, using the new operational matrices, we study the relationship between Bernoulli wavelets and piecewise functions. Then, according to the properties of piecewise functions and computing operational matrices of their fractional derivatives, we obtain operational matrices of the Bernoulli wavelet fractional derivatives. Using new operational matrices furnished from Caputo and Riemann-Liouville and also suitable collocation points, the advection-dispersion equation would be converted to a system of algebraic equations. Then, we would solve the equation numerically by utilizing a common method. Finally, the upper bound of the errors of the defined operational matrices and convergence analysis of the proposed method would be discussed. We would also reveal high accuracy of the method using some numerical samples.