Browsing by Author "Abdelkawy, M. A."

Now showing 1 - 6 of 6

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.
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.
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.
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.
A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations
(SPRINGER-VERLAG ITALIA SRL, 2016) Bhrawy, A. H.; Doha, E. H.; Ezz-Eldien, S. S.; Abdelkawy, M. A.
The time-fractional coupled Korteweg-de Vries (KdV) system is a generalization of the classical coupled KdV system and obtained by replacing the first order time derivatives by fractional derivatives of orders nu(1) and nu(2), (0 < nu(1), nu(2) <= 1). In this paper, an accurate and robust numerical technique is proposed for solving the time-fractional coupled KdV equations. The shifted Legendre polynomials are introduced as basis functions of the collocation spectral method together with the operational matrix of fractional derivatives (described in the Caputo sense) in order to reduce the time-fractional coupled KdV equations into a problem consisting of a system of algebraic equations that greatly simplifies the problem. In order to test the efficiency and validity of the proposed numerical technique, we apply it to solve two numerical examples.