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: 42Citation - Scopus: 44A Numerical Approach for Solving Fractional Optimal Control Problems With Mittag-Leffler Kernel(Sage Publications Ltd, 2022) Ganji, Roghayeh M.; Sayevand, Khosro; Baleanu, Dumitru; Jafari, HosseinIn 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: 19Citation - Scopus: 20Numerical 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, AbolfazlIn 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.Article Citation - WoS: 21Citation - Scopus: 21Numerical Solution of Two-Dimensional Time Fractional Cable Equation With Mittag-Leffler Kernel(Wiley, 2020) Baleanu, Dumitru; Kumar, SachinThe main motive of this article is to study the recently developed Atangana-Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag-Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two-dimensional cable equation in which time-fractional derivative is of Mittag-Leffler kernel type. First, we derive an approximation formula of the fractional-order ABC derivative of a function t(k) using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two-dimensional model of the time-fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two-dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag-Leffler fractional derivative.Article New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method(Editura ACAD Romane, 2015) Bhrawy, Ali H.; Zaky, Mahmoud A.; Baleanu, DumitruBurgers’ 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. © 2015, Editura Academiei Romane. All rights reserved.Article Citation - WoS: 86Citation - Scopus: 93Solving Multi-Dimensional Fractional Optimal Control Problems With Inequality Constraint by Bernstein Polynomials Operational Matrices(Sage Publications Ltd, 2013) Rostamy, Davood; Baleanu, Dumitru; Alipour, MohsenIn this paper, we present a method for solving multi-dimensional fractional optimal control problems. Firstly, we derive the Bernstein polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been done before. The main characteristic behind the approach using this technique is that it reduces the problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The results obtained are in good agreement with the existing ones in the open literature and it is shown that the solutions converge as the number of approximating terms increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach 1.