Browsing by Author "Hafez, R. M."

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A Chebyshev-Gauss-Radau Scheme For Nonlinear Hyperbolic System Of First Order
(NATURAL SCIENCES PUBLISHING CORP-NSP, 2014) Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.; Abdelkawy, M. A.
A numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral collocation method is presented in this article. A Chebyshev-Gauss-Radau collocation (C-GR-C) method in combination with the implicit Runge-Kutta scheme are employed to obtain highly accurate approximations to the mentioned problem. The collocation points are the Chebyshev interpolation nodes. This approach reduces this problem to solve system of nonlinear ordinary differential equations which are far easier to be solved. Indeed, by selecting a limited number of collocation nodes, we obtain an accurate results. The numerical examples demonstrate the accuracy, efficiency, and versatility of the method.
COMPOSITE BERNOULLI-LAGUERRE COLLOCATION METHOD FOR A CLASS OF HYPERBOLIC TELEGRAPH-TYPE EQUATIONS
(EDITURA ACAD ROMANE, 2017) Baleanu, D; El-Kalaawy, A. A; Amin, A. Z. M; Zaky, M. A; Taha, T. M; Ezz-Eldien, S. S; Abdelkawy, M. A; Hafez, R. M.; Doha, E. H
In this work, we introduce an efficient Bernoulli-Laguerre collocation method for solving a class of hyperbolic telegraph-type equations in one dimension. Bernoulli and Laguerre polynomials and their properties are utilized to reduce the aforementioned problems to systems of algebraic equations. The proposed collocation method, both in spatial and temporal discretizations, is successfully developed to handle the two-dimensional case. In order to highlight the effectiveness of our approachs, several numerical examples are given. The approximation techniques and results developed in this paper are appropriate for many other problems on multiple-dimensional domains, which are not of standard types.
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 collocation algorithm for multidimensional wave type equations with nonlocal conservation conditions
(ELSEVIER SCIENCE INC, 2015) Bhrawy, A. H.; Doha, E. H.; Abdelkawy, M. A.; Hafez, R. M.
In this paper, we derive and analyze an efficient spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. The Legendre pseudospectral approximation is investigated for spatial approximation of the wave equations. The Legendre-Gauss-Lobatto quadrature rule is established to treat the nonlocal conservation conditions, and then the problem with its nonlocal conservation conditions are reduced to a system of ODEs in time. As a theoretical result, we study the convergence of the solution for the one-dimensional case. In addition, the proposed method is extended successfully to the two-dimensional case. Several numerical examples with comparisons are given. The computational results indicate that the proposed method is more accurate than finite difference method, the method of lines and spline collocation approach. (C) 2015 Elsevier Inc. All rights reserved.
Efficient Jacobi-Gauss Collocation Method for Solving Initial Value Problems of Bratu Type
(PLEIADES PUBLISHING INC, 2013) Doha, E. H.; Bhrawy, A. H.; Baleanu, D.; Hafez, R. M.
In this paper, we propose the shifted Jacobi-Gauss collocation spectral method for solving initial value problems of Bratu type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The spatial approximation is based on shifted Jacobi polynomials J(n)((alpha, beta))(x) with alpha, beta is an element of (-1, infinity), x is an element of [0, 1] and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes. Illustrative examples have been discussed to demonstrate the validity and applicability of the proposed technique. Comparing the numerical results of the proposed method with some well-known results show that the method is efficient and gives excellent numerical results.
GENERALIZED LAGUERRE-GAUSS-RADAU SCHEME FOR FIRST ORDER HYPERBOLIC EQUATIONS ON SEMI-INFINITE DOMAINS
(EDITURA ACAD ROMANE, 2015) Bhrawy, A. H.; Hafez, R. M.; Alzahrani, E. O.; Baleanu, D.
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
A JACOBI COLLOCATION METHOD FOR TROESCH'S PROBLEM IN PLASMA PHYSICS
(EDITURA ACAD ROMANE,, 2014) Doha, E. H.; Baleanu, D.; Bhrawi, A. H.; Hafez, R. M.
In this paper, we propose a numerical approach for solving Troesch's problem which arises in the confinement of a plasma column by radiation pressure. It is also an inherently unstable two-point boundary value problem. The spatial approximation is based on shifted Jacobi-Gauss collocation method in which the shifted Jacobi-Gauss points are used as collocation nodes. The results presented here demonstrate reliability and efficiency of the method.
Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation
(SPRINGER HEIDELBERG, TIERGARTENSTRASSE 17, D-69121 HEIDELBERG, GERMANY, 2018-09) Hafez, R. M.; Youssri, Y. H.
We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and nonlinear reaction-subdiffusion equations using the shifted Jacobi collocation method. Basically, a time-space collocation approximation for temporal and spatial discretizations is employed efficiently to tackle these equations. The convergence and stability analyses of the suggested basis functions are presented in-depth. The validity and efficiency of the proposed method are investigated and verified through numerical examples.
A Jacobi Dual-Petrov-Galerkin Method for Solving Some Odd-Order Ordinary Differential Equations
(HINDAWI PUBLISHING CORPORATION, 2011) Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.
A Jacobi dual-Petrov-Galerkin (JDPG) method is introduced and used for solving fully integrated reformulations of third- and fifth-order ordinary differential equations (ODEs) with constant coefficients. The reformulated equation for the Jth order ODE involves n-fold indefinite integrals for n = 1, ... , J. Extension of the JDPG for ODEs with polynomial coefficients is treated using the Jacobi-Gauss-Lobatto quadrature. Numerical results with comparisons are given to confirm the reliability of the proposed method for some constant and polynomial coefficients ODEs.
A Jacobi rational pseudospectral method for Lane-Emden initial value problems arising in astrophysics on a semi-infinite interval
(SPRINGER HEIDELBERG, 2014) Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.; Van Gorder, Robert A.
We derive an operational matrix representation for the differentiation of Jacobi rational functions, which is used to create a new Jacobi rational pseudo spectral method based on the operational matrix of Jacobi rational functions. This Jacobi rational pseudospectral method is implemented to approximate solutions to Lane-Emden type equations on semi-infinite intervals. The advantages of using the Jacobi rational pseudospectral method over other techniques are discussed. Indeed, through several numerical examples, including the Lane-Emden problems of first and second kind, we evaluate the accuracy and performance of the proposed method. We also compare our method to other approaches in the literature. The results suggest that the Jacobi rational pseudospectral method is a useful tool for studying Lane-Emden initial value problems, as well as related problems which have regular singular points and are nonlinear.
A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations
(PERGAMON-ELSEVIER SCIENCE LTD, 2011) Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.; https://cutt.ly/hePtIOE
his paper analyzes a method for solving the third-and fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov-Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P-n((alpha,beta)) with alpha,beta is an element of (-1,infinity) and n is the polynomial degree. By choosing appropriate base functions, the resulting system is sparse and the method can be implemented efficiently. A Jacobi-Jacobi dual-Petrov-Galerkin method for the differential equations with variable coefficients is developed. This method is based on the Petrov-Galerkin variational form of one Jacobi polynomial class, but the variable coefficients and the right-hand terms are treated by using the Gauss-Lobatto quadrature form of another Jacobi class. Numerical results illustrate the theory and constitute a convincing argument for the feasibility of the proposed numerical methods. (C) 2011 Elsevier Ltd. All rights reserved.
A NEW COLLOCATION SCHEME FOR SOLVING HYPERBOLIC EQUATIONS OF SECOND ORDER IN A SEMI-INFINITE DOMAIN
(EDITURA ACAD ROMANE, 2016) Hafez, R. M.; Abdelkawy, M. A.; Doha, E. H.; Bhrawy, A. H.
This paper reports a new fully collocation algorithm for the numerical solution of hyperbolic partial differential equations of second order in a semi-infinite domain, using Jacobi rational Gauss-Radau collocation method. The widely applicable, efficiency, and high accuracy are the key advantages of the collocation method. The series expansion in Jacobi rational functions is the main step for solving the mentioned problems. The expansion coefficients are then determined by reducing the hyperbolic equations with their boundary and initial conditions to a system of algebraic equations for these coefficients. This system may be solved analytically or numerically in a step-by-step manner by using Newton's iterative method. Numerical results are consistent with the theoretical analysis and indicate the high accuracy and effectiveness of this algorithm.
A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations
(ELSEVIER SCIENCE BV, 2014) Bhrawy, A. H.; Doha, E. H.; Baleanu, D.; Hafez, R. M.
his paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational-Gauss collocation points. The proposed Jacobi rational-Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line. (C) 2013 IMACS. Published by Elsevier B.V. All rights reserved.
Numerical solution of initial-boundary system of nonlinear hyperbolic equations
(INDIAN NAT SCI ACAD, 2015) Doha, EH; Bhrawy, A. H.; Abdelkawy, M. A.; Hafez, R. M.
In this article, we present a numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral Jacobi-Gauss-Radau collocation (J-GR-C) method. A J-GR-C method in combination with the implicit Runge-Kutta scheme are employed to obtain a highly accurate approximation to the mentioned problem. J-GR-C method, based on Jacobi polynomials and Gauss-Radau quadrature integration, reduces solving the system of nonlinear hyperbolic equations to solve a system of nonlinear ordinary differential equations (SNODEs). In the examples given, numerical results by the J-GR-C method are compared with the exact solutions. In fact, by selecting relatively few J-GR-C points, we are able to get very accurate approximations. In this way, the results show that this method has a good accuracy and efficiency for solving coupled partial differential equations.
NUMERICAL SOLUTIONS OF TWO-DIMENSIONAL MIXED VOLTERRA-FREDHOLM INTEGRAL EQUATIONS VIA BERNOULLI COLLOCATION METHOD
(EDITURA ACAD ROMANE, 2017) Hafez, R. M.; Doha, E. H.; Bhrawy, A. H.; Baleanu, D.
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.
On Numerical Methods for Fractional Differential Equation on a Semi-infinite Interval
(DE GRUYTER OPEN LTD, 2015) Hafez, R. M.; Abdelkawy, M. A; Taha, T. M; Bhrawy, A. H
Chapter 11 is devoted to numerical solutions of fractional differential equations (FDEs) on a semi-infinite interval. This chapter presents a broad discussion of spectral techniques based on operational matrices of fractional derivatives and integration methods for solving several kinds of linear and nonlinear FDEs. We present the operational matrices of fractional derivatives and integrals for some orthogonal polynomials/functions on a semi-infinite interval, and use them together with different spectral techniques for solving the aforementioned equations on a semi-infinite interval. Numerous examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on a semi-infinite interval.
On shifted Jacobi spectral method for high-order multi-point boundary value problems
(ELSEVIER, 2012) Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.
This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss-Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved. (C) 2012 Elsevier B.V. All rights reserved.
A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions
(HINDAWI PUBLISHING CORPORATION, 2014) Doha, E. H.; Baleanu, D.; Bhrawy, A. H.; Hafez, R. M.
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.