Browsing by Author "Bhrawy, Ali H."

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An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems
(SPRINGEROPEN, 2015) Doha, Eid H.; Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; Hafez, Ramy M.
In this article, we introduce a numerical technique for solving a general form of the fractional optimal control problem. Fractional derivatives are described in the Caputo sense. Using the properties of the shifted Jacobi orthonormal polynomials together with the operational matrix of fractional integrals (described in the Riemann-Liouville sense), we transform the fractional optimal control problem into an equivalent variational problem that can be reduced to a problem consisting of solving a system of algebraic equations by using the Legendre-Gauss quadrature formula with the Rayleigh-Ritz method. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.
A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation
(SCIENDO, 2014) Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohamed A.; Hafez, Ramy M.
This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers' equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nystrom (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers' equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.
A Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithm for solving fractional Fokker-Planck equations
(SPRINGER, 2015) Hafez, Ramy M.; Ezz-Eldien, Samer S.; Bhrawy, Ali H.; Ahmed, Engy A.; Baleanu, Dumitru
In this article, we construct a new numerical approach for solving the time-fractional Fokker-Planck equation. The shifted Jacobi polynomials are used as basis functions, and the fractional derivative is described in the sense of Caputo. The proposed approach is a combination of shifted Jacobi Gauss-Lobatto scheme for the spatial discretization and the shifted Jacobi Gauss-Radau scheme for temporal approximation. The problem is then reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. In addition, our numerical algorithm is also applied for solving the space-fractional Fokker-Planck equation and the time-space-fractional Fokker-Planck equation. Numerical results are consistent with the theoretical analysis, indicating the high accuracy and effectiveness of the proposed algorithm.
Jacobi rational-Gauss collocation method for Lane-Emden equations of astrophysical significance
(INST MATHEMATICS & INFORMATICS, 2014) Doha, Eid H.; Bhrawy, Ali H.; Hafez, Ramy M.; Van Gorder, Robert A.
In this paper, a new spectral collocation method is applied to solve Lane-Emden equations on a semi-infinite domain. The method allows us to overcome difficulty in both the nonlinearity and the singularity inherent in such problems. This Jacobi rational-Gauss method, based on Jacobi rational functions and Gauss quadrature integration, is implemented for the nonlinear Lane-Emden equation. Once we have developed the method, numerical results are provided to demonstrate the method. Physically interesting examples include Lane-Emden equations of both first and second kind. In the examples given, by selecting relatively few Jacobi rational-Gauss collocation points, we are able to get very accurate approximations, and we are thus able to demonstrate the utility of our approach over other analytical or numerical methods. In this way, the numerical examples provided demonstrate the accuracy, efficiency, and versatility of the method.
New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials
(SPRINGER/PLENUM PUBLISHERS, 2017) Ezz-Eldien, Samer S.; Hafez, Ramy M.; Bhrawy, Ali H.; Baleanu, Dumitru; El-Kalaawy, Ahmed A.
This paper reports a new numerical approach for numerically solving types of fractional variational problems. In our approach, we use the fractional integrals operational matrix, described in the sense of Riemann-Liouville, with the help of the Lagrange multiplier technique for converting the fractional variational problem into an easier problem that consisting of solving an algebraic equations system in the unknown coefficients. Several numerical examples are introduced, combined with their approximate solutions and comparisons with other numerical approaches, for confirming the accuracy and applicability of the proposed approach.
Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method
(SCIENDO, 2013) Doha, Eid H.; Bhrawy, Ali H.; Ezz-Eldien, Samer S.
n this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. 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. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.
The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation
(SPRINGEROPEN, 2014) Doha, Eid H.; Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.
In this article, an accurate and efficient numerical method is presented for solving the space-fractional order diffusion equation (SFDE). Jacobi polynomials are used to approximate the solution of the equation as a base of the tau spectral method which is based on the Jacobi operational matrices of fractional derivative and integration. The main advantage of this method is based upon reducing the nonlinear partial differential equation into a system of algebraic equations in the expansion coefficient of the solution. In order to test the accuracy and efficiency of our method, the solutions of the examples presented are introduced in the form of tables to make a comparison with those obtained by other methods and with the exact solutions easy.