Browsing by Author "Bhrawy, A. H."

Now showing 1 - 20 of 28

Citation - WoS: 202
Citation - Scopus: 217
A Spectral Tau Algorithm Based on Jacobi Operational Matrix for Numerical Solution of Time Fractional Diffusion-Wave Equations
(Academic Press inc Elsevier Science, 2015) Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.; Bhrawy, A. H.
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.
Citation - WoS: 26
A Chebyshev-Laguerre Collocation Scheme for Solving A Time Fractional Sub-Diffusion Equation on A Semi-Infinite Domain
(Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Abdelkawy, M. A.; Alzahrani, A. A.; Baleanu, D.; Alzahrani, E. O.; Matematik
We propose a new efficient spectral collocation method for solving a time fractional sub-diffusion equation on a semi-infinite domain. The shifted Chebyshev-Gauss-Radau interpolation method is adapted for time discretization along with the Laguerre-Gauss-Radau collocation scheme that is used for space discretization on a semi-infinite domain. The main advantage of the proposed approach is that a spectral method is implemented for both time and space discretizations, which allows us to present a new efficient algorithm for solving time fractional sub-diffusion equations.
Citation - WoS: 48
Citation - Scopus: 54
Solving Fractional Optimal Control Problems Within a Chebyshev-Legendre Operational Technique
(Taylor & Francis Ltd, 2017) Ezz-Eldien, S. S.; Doha, E. H.; Abdelkawy, M. A.; Baleanu, D.; Bhrawy, A. H.
In this manuscript, we report a new operational technique for approximating the numerical solution of fractional optimal control (FOC) problems. The operational matrix of the Caputo fractional derivative of the orthonormal Chebyshev polynomial and the Legendre-Gauss quadrature formula are used, and then the Lagrange multiplier scheme is employed for reducing such problems into those consisting of systems of easily solvable algebraic equations. We compare the approximate solutions achieved using our approach with the exact solutions and with those presented in other techniques and we show the accuracy and applicability of the new numerical approach, through two numerical examples.
Citation - WoS: 51
Citation - Scopus: 66
Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems
(Hindawi Ltd, 2013) Bhrawy, A. H.; Taha, T. M.; Baleanu, D.
We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multitermFDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.
Citation - WoS: 4
Citation - Scopus: 5
Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method
(Hindawi Ltd, 2013) Alghamdi, M. A.; Baleanu, D.; Bhrawy, A. H.
The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-leg theta-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.
Citation - WoS: 13
Citation - Scopus: 16
Generalized Laguerre-Gauss Scheme for First Order Hyperbolic Equations on Semi-Infinite Domains
(Editura Acad Romane, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Hafez, R. M.; Alzahrani, E. O.; Baleanu, D.; Alzahrani, A. A.; Matematik
In this article, we develop a numerical approximation for first-order hyperbolic equations on semi-infinite domains by using a spectral collocation scheme. First, we propose the generalized Laguerre-Gauss-Radau collocation scheme for both spatial and temporal discretizations. This in turn reduces the problem to the obtaining of a system of algebraic equations. Second, we use a Newton iteration technique to solve it. Finally, the obtained results are compared with the exact solutions, highlighting the performance of the proposed numerical method.
Citation - WoS: 88
Citation - Scopus: 124
New 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.
Citation - WoS: 9
Citation - Scopus: 10
On a Generalized Laguerre Operational Matrix of Fractional Integration
(Hindawi Ltd, 2013) Baleanu, D.; Assas, L. M.; Tenreiro Machado, J. A.; Bhrawy, A. H.
A new operational matrix of fractional integration of arbitrary order for generalized Laguerre polynomials is derived. The fractional integration is described in the Riemann-Liouville sense. This operational matrix is applied together with generalized Laguerre tau method for solving general linear multiterm fractional differential equations (FDEs). The method has the advantage of obtaining the solution in terms of the generalized Laguerre parameter. In addition, only a small dimension of generalized Laguerre operational matrix is needed to obtain a satisfactory result. Illustrative examples reveal that the proposed method is very effective and convenient for linear multiterm FDEs on a semi-infinite interval.
Citation - WoS: 2
Citation - Scopus: 4
A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions
(Hindawi Ltd, 2014) Baleanu, D.; Bhrawy, A. H.; Hafez, R. M.; Doha, E. H.
A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational- Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational- Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.
Citation - WoS: 20
Citation - Scopus: 19
A Highly Accurate Jacobi Collocation Algorithm for Systems of High-Order Linear Differential-Difference Equations With Mixed Initial Conditions
(Wiley, 2015) Doha, E. H.; Baleanu, D.; Hafez, R. M.; Bhrawy, A. H.
In this paper, a shifted Jacobi-Gauss collocation spectral algorithm is developed for solving numerically systems of high-order linear retarded and advanced differential-difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi-Gauss interpolation nodes as collocation nodes. The system of differential-difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several 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. Copyright (C) 2015 John Wiley & Sons, Ltd.
Citation - WoS: 8
Citation - Scopus: 35
A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line
(Hindawi Ltd, 2013) Baleanu, D.; Bhrawy, A. H.; Taha, T. M.
This paper deals with modified generalized Laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations (FDEs) on the half line. A new formula expressing the Caputo fractional derivatives of modified generalized Laguerre polynomials of any degree and for any fractional order in terms of the modified generalized Laguerre polynomials themselves is derived. An efficient direct solver technique is proposed for solving the linear multiterm FDEs with constant coefficients on the half line using a modified generalized Laguerre tau method. The spatial approximation with its Caputo fractional derivatives is based on modified generalized Laguerre polynomials L-i((alpha,beta)) (x) with x is an element of Lambda = (0, infinity), alpha > -1, and beta > 0, and i is the polynomial degree. We implement and develop the modified generalized Laguerre collocation method based on the modified generalized Laguerre-Gauss points which is used as collocation nodes for solving nonlinear multiterm FDEs on the half line.
Citation - WoS: 6
Citation - Scopus: 10
A Jacobi Collocation Method for Solving Nonlinear Burgers-Type Equations
(Hindawi Ltd, 2013) Baleanu, D.; Bhrawy, A. H.; Abdelkawy, M. A.; Doha, E. H.
We solve three versions of nonlinear time-dependent Burgers-type equations. The Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. This approach has the advantage of obtaining the solution in terms of the Jacobi parameters alpha and beta In addition, the problem is reduced to the solution of the system of ordinary differential equations (SODEs) in time. This system may be solved by any standard numerical techniques. Numerical solutions obtained by this method when compared with the exact solutions reveal that the obtained solutions produce high-accurate results. Numerical results show that the proposed method is of high accuracy and is efficient to solve the Burgers-type equation. Also the results demonstrate that the proposed method is a powerful algorithm to solve the nonlinear partial differential equations.
Citation - WoS: 29
Citation - Scopus: 31
A 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.; Matematik
This 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.
Citation - WoS: 36
Citation - Scopus: 41
Numerical Treatment of Coupled Nonlinear Hyperbolic Klein-Gordon Equations
(Editura Acad Romane, 2014) Doha, E. H.; Baleanu, Dumitru; Bhrawy, A. H.; Baleanu, D.; Abdelkawy, M. A.; Matematik
A semi-analytical solution based on a Jacobi-Gauss-Lobatto collocation (J-GL-C) 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-Nystrom scheme. Numerical results show that the proposed algorithm is efficient, accurate, and compare favorably with the analytical solutions.
Citation - WoS: 76
Citation - Scopus: 82
On Shifted Jacobi Spectral Approximations for Solving Fractional Differential Equations
(Elsevier Science inc, 2013) Bhrawy, A. H.; Baleanu, D.; Ezz-Eldien, S. S.; Doha, E. H.
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.
Citation - WoS: 13
Citation - Scopus: 14
An Accurate Legendre Collocation Scheme for Coupled Hyperbolic Equations With Variable Coefficients
(Editura Acad Romane, 2014) Doha, E. H.; Baleanu, Dumitru; Bhrawy, A. H.; Baleanu, D.; Abdelkawy, M. A.; Matematik
The study of numerical solutions of nonlinear coupled hyperbolic partial differential equations (PDEs) with variable coefficients subject to initial-boundary conditions continues to be a major research area with widespread applications in modern physics and technology. One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (NPDEs) as well as PDEs with variable coefficients. A numerical solution based on a Legendre collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients. This approach, which is based on Legendre polynomials and Gauss-Lobatto quadrature integration, reduces the solving of nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equations that is far easier to solve. The obtained results show that the proposed numerical algorithm is efficient and very accurate.
Citation - WoS: 4
Citation - Scopus: 6
Modified Jacobi-Bernstein Basis Transformation and Its Application To Multi-Degree Reduction of Bezier Curves
(Elsevier Science Bv, 2016) Doha, E. H.; Saker, M. A.; Baleanu, D.; Bhrawy, A. H.
This paper reports new modified Jacobi polynomials (MJPs). We derive the basis transformation between MJPs and Bernstein polynomials and vice versa. This transformation is merging the perfect Least-square performance of the new polynomials together with the geometrical insight of Bernstein polynomials. The MJPs with indexes corresponding to the number of endpoints constraints are the natural basis functions for Least-square approximation of Bezier curves. Using MJPs leads us to deal with the constrained Jacobi polynomials and the unconstrained Jacobi polynomials as orthogonal polynomials. The MJPs are automatically satisfying the homogeneous boundary conditions. Thereby, the main advantage of using MJPs, in multi-degree reduction of Bezier curves on computer aided geometric design (CAGD), is that the constraints in CAGD are also satisfied and that decreases the steps of multi-degree reduction algorithm. Several numerical results for the multi-degree reduction of Bezier curves on CAGD are given. (C) 2016 Elsevier B.V. All rights reserved.
Citation - WoS: 32
Citation - Scopus: 37
Numerical Solutions of Two-Dimensional Mixed Volterra-Fredholm Integral Equations Via Bernoulli Collocation Method
(Editura Acad Romane, 2017) Hafez, R. M.; Baleanu, Dumitru; Doha, E. H.; Bhrawy, A. H.; Baleanu, D.; Matematik
The mixed Volterra-Fredholm integral equations (VFIEs) arise in various physical and biological models. The main purpose of this article is to propose and analyze efficient Bernoulli collocation techniques for numerically solving classes of two-dimensional linear and nonlinear mixed VFIEs. The novel aspect of the technique is that it reduces the problem under consideration to a system of algebraic equations by using the Gauss-Bernoulli nodes. One of the main advantages of the present approach is its superior accuracy. Consequently, good results can be obtained even by using a relatively small number of collocation nodes. In addition, several numerical results are given to illustrate the features of the proposed technique.
Citation - WoS: 27
Citation - Scopus: 28
An Efficient Collocation Technique for Solving Generalized Fokker-Planck Type Equations With Variable Coefficients
(Editura Acad Romane, 2014) Bhrawy, A. H.; Baleanu, Dumitru; Ahmed, Engy A.; Baleanu, D.; Matematik
This paper proposes an efficient numerical integration process for the generalized Fokker-Planck equation with variable coefficients. For spatial discretization the Jacobi-Gauss-Lobatto collocation (J-GL-C) method is implemented in which the Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. This approach has the advantage of obtaining the solution in terms of the Jacobi parameters alpha and beta. Using the above technique, the problem is reduced to the solution of a system of ordinary differential equations in tithe. This system can be also solved by standard numerical techniques. Our results demonstrate that the proposed method is a powerful algorithm for solving nonlinear partial differential equations.
Citation - WoS: 67
Citation - Scopus: 82
A Numerical Approach Based on Legendre Orthonormal Polynomials for Numerical Solutions of Fractional Optimal Control Problems
(Sage Publications Ltd, 2017) Doha, E. H.; Baleanu, D.; Bhrawy, A. H.; Ezz-Eldien, S. S.
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.