Browsing by Author "Ezz-Eldien, S. S."

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A COMPUTATIONALLY EFFICIENT METHOD FOR A CLASS OF FRACTIONAL VARIATIONAL AND OPTIMAL CONTROL PROBLEMS USING FRACTIONAL GEGENBAUER FUNCTIONS
(2018) El-Kalaawy, A. A.; Doha, E. H.; Ezz-Eldien, S. S.; Abdelkawy, M. A.; Hafez, R. M.; Amin, A. Z. M.; Baleanu, D.; Zaky, M. A.
This paper is devoted to investigate, from the numerical point of view, fractional-order Gegenbauer functions to solve fractional variational problems and fractional optimal control problems. We first introduce an orthonormal system of fractional-order Gegenbauer functions. Then, a formulation for the fractional-order Gegenbauer operational matrix of fractional integration is constructed. An error upper bound for the operational matrix of the fractional integration is also given. The properties of the fractional-order Gegenbauer functions are utilized to reduce the given optimization problems to systems of algebraic equations. Some numerical examples are included to demonstrate the efficiency and the accuracy of the proposed approach.
An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Subdiffusion and Reaction Subdiffusion Equations
(ASME, 2015) Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S.
n this work, we discuss an operational matrix approach for introducing an approximate solution of the fractional subdiffusion equation (FSDE) with both Dirichlet boundary conditions (DBCs) and Neumann boundary conditions (NBCs). We propose a spectral method in both temporal and spatial discretizations for this equation. Our approach is based on the space-time shifted Legendre tau-spectral method combined with the operational matrix of fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. In addition, this approach is also investigated for solving the FSDE with the variable coefficients and the fractional reaction subdiffusion equation (FRSDE). For conforming the validity and accuracy of the numerical scheme proposed, four numerical examples with their approximate solutions are presented. Also, comparisons between our numerical results and those obtained by compact finite difference method (CFDM), Box-type scheme (B-TS), and FDM with Fourier analysis (FA) are introduced.
An Efficient Numerical Scheme for Solving Multi-Dimensional Fractional Optimal Control Problems With a Quadratic Performance Index
(WILEY, 2015) Bhrawy, A. H.; Doha, E. H.; Tenreiro Machado, J. A.; Ezz-Eldien, S. S.
The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.
A new Jacobi spectral collocation method for solving 1+1 fractional Schrodinger equations and fractional coupled Schrodinger systems
(SPRINGER HEIDELBERG, 2014) Bhrawy, A. H.; Doha, E. H.; Ezz-Eldien, S. S.; Van Gorder, Robert A.
The Jacobi spectral collocation method (JSCM) is constructed and used in combination with the operational matrix of fractional derivatives (described in the Caputo sense) for the numerical solution of the time-fractional Schrodinger equation (T-FSE) and the space-fractional Schrodinger equation (S-FSE). The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, the presented approach is also applied to solve the time-fractional coupled Schrodinger system (T-FCSS). In order to demonstrate the validity and accuracy of the numerical scheme proposed, several numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.
A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems
(SAGE PUBLICATIONS LTD, 2017) Ezz-Eldien, S. S.; Doha, E. H.; Baleanu, D.; Bhrawy, A. H.
The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes an integer derivative in addition to the fractional derivative. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Based on the shifted Legendre orthonormal polynomials, we employ the operational matrix of fractional derivatives, the Legendre-Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations. The convergence of the proposed method is analyzed. For confirming the validity and accuracy of the proposed numerical method, a numerical example is presented along with a comparison between our numerical results and those obtained using the Legendre spectral-collocation method.
A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations
(SPRINGER-VERLAG ITALIA SRL, 2016) Bhrawy, A. H.; Doha, E. H.; Ezz-Eldien, S. S.; Abdelkawy, M. A.
The time-fractional coupled Korteweg-de Vries (KdV) system is a generalization of the classical coupled KdV system and obtained by replacing the first order time derivatives by fractional derivatives of orders nu(1) and nu(2), (0 < nu(1), nu(2) <= 1). In this paper, an accurate and robust numerical technique is proposed for solving the time-fractional coupled KdV equations. The shifted Legendre polynomials are introduced as basis functions of the collocation spectral method together with the operational matrix of fractional derivatives (described in the Caputo sense) in order to reduce the time-fractional coupled KdV equations into a problem consisting of a system of algebraic equations that greatly simplifies the problem. In order to test the efficiency and validity of the proposed numerical technique, we apply it to solve two numerical examples.
On shifted Jacobi spectral approximations for solving fractional differential equations
(ELSEVIER SCIENCE INC, 2013) Doha, E. H; Bhrawy, AH; Baleanu, D; Ezz-Eldien, S. S.
In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value. problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. (C) 2013 Elsevier Inc. All rights reserved
A PSEUDOSPECTRAL METHOD FOR SOLVING THE TIME-FRACTIONAL GENERALIZED HIROTA-SATSUMA COUPLED KORTEWEG-DE VRIES SYSTEM
(EDITURA ACAD ROMANE, 2017) Saker, M. A.; Ezz-Eldien, S. S.; Bhrawy, A. H.
In this paper, a new space-time spectral algorithm is constructed to solve the generalized Hirota-Satsuma coupled Korteweg-de Vries (GHS-C-KdV) system of time-fractional order. The present algorithm consists of applying the collocation-spectral method in conjunction with the operational matrix of fractional derivative for the double Jacobi polynomials, which will be employed as a basis function for the spectral solution. The main characteristic behind this approach is that such problems will reduce to those of solving algebraic systems of equations that greatly simplifying the problem. For ensuring the accuracy and efficiency of the presented algorithm, we apply it to find the approximate solutions of two specific problems, namely, a homogeneous form of the GHS-C-KdV system and a inhomogeneous GHS-C-KdV system.
A quadrature tau method for fractional differential equations with variable coefficients
(PERGAMON-ELSEVIER SCIENCE LTD, 2011) Bhrawy, A. H.; Alofi, A. S.; Ezz-Eldien, S. S.
In this article, we develop a direct solution technique for solving multi-order fractional differential equations (FDEs) with variable coefficients using a quadrature shifted Legendre tau (Q-SLT) method. The spatial approximation is based on shifted Legendre polynomials. A new formula expressing explicitly any fractional-order derivatives of shifted Legendre polynomials of any degree in terms of shifted Legendre polynomials themselves is proved. Extension of the tau method for FDEs with variable coefficients is treated using the shifted Legendre-Gauss-Lobatto quadrature. Numerical results are given to confirm the reliability of the proposed method for some FDEs with variable coefficients. (C) 2011 Elsevier Ltd. All rights reserved.
A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations
(ACADEMIC PRESS INC ELSEVIER SCIENCE, 2015) Bhrawy, A. H.; E. H., Doha; Baleanu, D; Ezz-Eldien, S. S.
In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five 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. (C) 2014 Elsevier Inc. All rights reserved.