Browsing by Author "Bhrawy, Ali H."

Now showing 1 - 11 of 11

Citation - WoS: 28
Citation - Scopus: 32
A Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithm for solving fractional Fokker-Planck equations
(Springer, 2015) Hafez, Ramy M.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; Bhrawy, Ali H.; Ahmed, Engy A.; Baleanu, Dumitru; Matematik
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 - WoS: 5
Citation - Scopus: 6
A Modified Generalized Laguerre-Gauss Collocation Method for Fractional Neutral Functional-Differential Equations on the Half-Line
(Hindawi Ltd, 2014) Bhrawy, Ali H.; Baleanu, Dumitru; AlZahrani, Abdulrahim; Baleanu, Dumitru; Alhamed, Yahia; 56389; Matematik
The modified generalized Laguerre-Gauss collocation (MGLC) method is applied to obtain an approximate solution of fractional neutral functional-differential equations with proportional delays on the half-line. The proposed technique is based on modified generalized Laguerre polynomials and Gauss quadrature integration of such polynomials. The main advantage of the present method is to reduce the solution of fractional neutral functional-differential equations into a system of algebraic equations. Reasonable numerical results are achieved by choosing few modified generalized Laguerre-Gauss collocation points. Numerical results demonstrate the accuracy, efficiency, and versatility of the proposed method on the half-line.
Citation - WoS: 16
Citation - Scopus: 15
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; Baleanu, Dumitru; Mallawi, Fouad; Matematik
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 - WoS: 15
Citation - Scopus: 14
A Spectral Technique for Solving Two-Dimensional Fractional Integral Equations With Weakly Singular Kernel
(Hacettepe Univ, Fac Sci, 2018) Bhrawy, Ali H.; Baleanu, Dumitru; Abdelkawy, Mohamed A.; Baleanu, Dumitru; Amin, Ahmed Z. M.; 56389; Matematik
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 - WoS: 58
Citation - Scopus: 62
An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems
(Springeropen, 2015) Doha, Eid H.; Baleanu, Dumitru; Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; Hafez, Ramy M.; Matematik
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 - WoS: 2
Citation - Scopus: 3
Fractional and Time-Scales Differential Equations
(Hindawi Publishing Corporation, 2014) Baleanu, Dumitru; Baleanu, Dumitru; Bhrawy, Ali H.; Torres, Delfim F. M.; Salahshour, Soheil; 56389; Matematik
Citation - WoS: 41
Citation - Scopus: 49
New numerical approach for fractional variational problems using shifted legendre orthonormal polynomials
(Springer/plenum Publishers, 2017) Ezz-Eldien, Samer S.; Baleanu, Dumitru; Hafez, Ramy M.; Bhrawy, Ali H.; Baleanu, Dumitru; El-Kalaawy, Ahmed A.; 56389; Matematik
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.
New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method
(2015) Baleanu, Dumitru; Zaky, Mahmoud A.; Baleanu, Dumitru; 56389; Matematik
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 - WoS: 12
Citation - Scopus: 14
New operational matrices for solving fractional differential equations on the half-line
(Public Library Science, 2015) Bhrawy, Ali H.; Baleanu, Dumitru; Taha, Taha M.; Alzahrani, Ebrahim O.; Baleanu, Dumitru; Alzahrani, Abdulrahim A.; 56389; Matematik
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. (0 < v < 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.. 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 - WoS: 91
Citation - Scopus: 96
New spectral techniques for systems of fractional differential equations using fractional-order generalized laguerre orthogonal functions
(Walter de Gruyter Gmbh, 2014) Bhrawy, Ali H.; Baleanu, Dumitru; Alhamed, Yahia A.; Baleanu, Dumitru; Al-Zahrani, Abdulrahim A.; 56389; Matematik
Fractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order nu (0 < nu < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order nu. 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 FGLFs and compare them with other methods. Several numerical example 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 - WoS: 32
Citation - Scopus: 42
The Operational Matrix Formulation of The Jacobi Tau Approximation For Space Fractional Diffusion Equation
(Springer, 2014) Doha, Eid H.; Baleanu, Dumitru; Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; 56389; Matematik
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.