WoS İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8653
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Article Citation - WoS: 66Citation - Scopus: 63An 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.; MatematikIn 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: 29Citation - Scopus: 31A 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.; MatematikThis 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: 89Citation - Scopus: 124New 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: 3Citation - Scopus: 3Solving 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: 11Citation - Scopus: 21Normalized Lucas Wavelets: an Application To Lane-Emden and Pantograph Differential Equations(Springer Heidelberg, 2020) Koundal, Reena; Srivastava, K.; Baleanu, D.; Kumar, RakeshIn 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: 10Citation - Scopus: 8Derivation 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, SachinOur 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: 42Citation - Scopus: 56A 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: 29Citation - Scopus: 36A Tau-Like Numerical Method for Solving Fractional Delay Integro-Differential Equations(Elsevier, 2020) Ostadzad, M. H.; Baleanu, D.; Shahmorad, SedaghatIn this paper, an operational matrix formulation of the Tau method is herein discussed to solve a class of delay fractional integrodifferential equations. The approximate solution is sought by using a suitable matrix representation of fractional and delay integrals. An error bound is herein for the first time discussed. Numerical examples show the effectiveness of the method. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.Article Citation - WoS: 9Citation - Scopus: 11Study on Application of Hybrid Functions To Fractional Differential Equations(Springer international Publishing Ag, 2018) Baleanu, D.; Torkzadeh, L.; Nouri, K.In this study we propose an efficient technique for approximate solution of linear and nonlinear differential equations with fractional order. The operational matrices based upon block-pulse functions and Chebyshev polynomials of the second kind are used for this purpose. Also, we focus on the upper bound of error for performance of the our estimates. The numerical results confirm the convergence of the suggested method. Correspondingly, the obtained results of our method are compared with other approaches in terms of efficiency and accuracy.Article Citation - WoS: 64Citation - Scopus: 69A Direct Numerical Solution of Time-Delay Fractional Optimal Control Problems by Using Chelyshkov Wavelets(Sage Publications Ltd, 2019) Mohammadi, F.; Baleanu, D.; Moradi, L.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.