Browsing by Author "Bhrawy, Ali H."

Now showing 1 - 13 of 13

Citation Count: Hafez, R.M...et al. (2015). A Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithm for solving fractional Fokker-Planck equations. Nonlinear Dynamics, 82(3), 1431-1440. http://dx.doi.org/10.1007/s11071-015-2250-7
A Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithm for solving fractional Fokker-Planck equations
(Springer, 2015) Hafez, R. 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
Citation Count: Bhrawy, A.H., Baleanu, D., Mallawi, F. (2015). A new numerical technique for solving fractional sub-diffusion and reaction sub-diffusion equations with a non-linear source term. Thermal Science, 19, 25-34. http://dx.doi.org/10.2298/TSCI15S1S25B
A new numerical technique for solving fractional sub-diffusion and reaction sub-diffusion equations with a non-linear source term
(Vinca Inst Nuclear Sci., 2015) Bhrawy, Ali H.; Baleanu, Dumitru; Mallawi, Fouad
In this paper, we are concerned with the fractional sub-diffusion equation with a non-linear source term. The Legendre spectral collocation method is introduced together with the operational matrix of fractional derivatives (described in the Caputo sense) to solve the fractional sub-diffusion equation with a non-linear source term. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. In addition, the Legendre spectral collocation methods applied also for a solution of the fractional reaction sub-diffusion equation with a non-linear source term. For confirming the validity and accuracy of the numerical scheme proposed, two numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.
Citation Count: Bhrawy, Ali H...et al. (2018). "A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel", Hacettepe Journal of Mathematics and Statistics, Vol. 47, No. 3, pp, 553-566.
A Spectral Technique for Solving Two-Dimensional Fractional Integral Equations With Weakly Singular Kernel
(Hacettepe Univ, Fac Sci, 2018) Bhrawy, Ali H.; Abdelkawy, M. A.; Baleanu, Dumitru; Amin, A. Z. M.; 56389
This paper adapts a new numerical technique for solving twodimensional fractional integral equations with weakly singular. Using the spectral collocation method, the fractional operators of Legendre and Chebyshev polynomials, and Gauss-quadrature formula, we achieve a reduction of given problems into those of a system of algebraic equations. We apply the reported numerical method to solve several numerical examples in order to test the accuracy and validity. Thus, the novel algorithm is more responsible for solving two-dimensional fractional integral equations with weakly singular.
Citation Count: Doha, E.H...et al. (2015). An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Advance in Difference Equations. http://dx.doi.org/10.1186/s13662-014-0344-z
An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems
(Springer International Publishing, 2015) Doha, Eid H.; Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; Hafez, R. 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.
Citation Count: Baleanu, Dumitru...et al. (2014). "Fractional and Time-Scales Differential Equations", Abstract and Applied Analysis.
Fractional and Time-Scales Differential Equations
(Hindawi LTD, 2014) Baleanu, Dumitru; Bhrawy, Ali H.; Torres, Delfim F. M.; Salahshour, Soheil; 56389
Citation Count: Ezz-Eldien, Samer S... et.al. (2017). "New numerical approach for fractional variational problems using shifted legendre orthonormal polynomials", Journal Of Optimization Theory And Applications, Vol.174, No.1, pp.295-320.
New numerical approach for fractional variational problems using shifted legendre orthonormal polynomials
(Springer/Plenum Publishers, 2017) Ezz-Eldien, Samer S.; Hafez, R. M.; Bhrawy, Ali H.; Baleanu, Dumitru; El-Kalaawy, Ahmed A.; 56389
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.
Citation Count: Bhrawy, Ali H.; Zaky, Mahmoud A.; Baleanu, Dumitru (2015). "New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method", Romanian Reports in Physics, Vol. 67, No. 2, pp. 340-349.
New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method
(2015) Bhrawy, Ali H.; Zaky, Mahmoud A.; Baleanu, Dumitru; 56389
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. © 2015, Editura Academiei Romane. All rights reserved.
Citation Count: Bhrawy, Ali H...et.al. (2015). "New operational matrices for solving fractional differential equations on the half-line", Plos One, Vol.10, No.9, pp.1-23.
New operational matrices for solving fractional differential equations on the half-line
(Public Library Science, 2015) Bhrawy, Ali H.; Taha, Taha M.; Alzahrani, Ebraheem; Baleanu, Dumitru; Alzahrani, Abdulrahim A.; 56389
In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.
Citation Count: Baleanu, Dumitru; Bhrawy, Ali H.; Van Gorder, Robert A. (2015). "New trends on fractional and functional differential equations", Abstract and Applied Analysis, Vol. 2015.
New trends on fractional and functional differential equations
(2015) Baleanu, Dumitru; Bhrawy, Ali H.; Van Gorder, Robert A.; 56389
Citation Count: Abdelkawy, M. A... et al. (2015). "Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model", Romanian Reports in Physics, Vol. 67, No. 3, pp. 773-791.
Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model
(2015) Abdelkawy, M. A.; Zaky, Mahmoud A.; Bhrawy, Ali H.; Baleanu, Dumitru; 56389
This paper reports a novel numerical technique for solving the time variable fractional order mobile-immobile advection-dispersion (TVFO-MIAD) model with the Coimbra variable time fractional derivative, which is preferable for modeling dynamical systems. The main advantage of the proposed method is that two different collocation schemes are investigated for both temporal and spatial discretizations of the TVFO-MIAD model. The problem with its boundary and initial conditions is then reduced to a system of algebraic equations that is far easier to be solved. Numerical results are consistent with the theoretical analysis and indicate the high accuracy and effectiveness of this algorithm. © 2015, Editura Academiei Romane. All rights reserved.
Citation Count: Doha, Eid Hassan... et al. (2014). "Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations", Romanian Journal of Physics, Vol. 59, No. 3-4, pp. 247-264.
Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations
(2014) Doha, Eid Hassan; Bhrawy, Ali H.; Baleanu, Dumitru; Abdelkawy, M. A.; 56389
A semi-analytical solution based on a Jacobi-Gauss-Lobatto collocation (J-GLC) method is proposed and developed for the numerical solution of the spatial variable for two nonlinear coupled Klein-Gordon (KG) partial differential equations. The general Jacobi-Gauss-Lobatto points are used as collocation nodes in this approach. The main characteristic behind the J-GL-C approach is that it reduces such problems to solve a system of ordinary differential equations (SODEs) in time. This system is solved by diagonally-implicit Runge-Kutta-Nyström scheme. Numerical results show that the proposed algorithm is efficient, accurate, and compare favorably with the analytical solutions.
Citation Count: Bhrawy, Ali H. ..et al. (2015). "Recent theory and applications on numerical algorithms and special functions", Abstract and Applied Analysis, Vol. 2015.
Recent theory and applications on numerical algorithms and special functions
(2015) Bhrawy, Ali H.; Van Gorder, Robert A.; Baleanu, Dumitru; Wu, Guo-Cheng; 56389
The Operational Matrix Formulation of The Jacobi Tau Approximation For Space Fractional Diffusion Equation
(Springer Open, 2014) Bhrawy, Ali H.; Doha, Eid H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; 56389
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.