Browsing by Author "Doha, E. H."

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Citation Count: Bhrawy, AH...et.al. (2015). "A highly accurate Jacobi collocation algorithm for systems of high-order linear differential-difference equations with mixed initial conditions" Mathematical Methods In The Applied Sciences, Vol.38, No.14, pp.3022-3032.
A highly accurate Jacobi collocation algorithm for systems of high-order linear differential-difference equations with mixed initial conditions
(Wiley, 2015) Bhrawy, A. H.; Doha, E. H.; Baleanu, Dumitru; Hafez, R. M.; 56389
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.
A Jacobi Collocation Method for Solving Nonlinear Burgers-Type Equations
(Hindawi LTD, 2013) Doha, E. H.; Baleanu, Dumitru; Bhrawy, A. H.; Abdelkawy, M. A.; 56389
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 Count: Baleanu, Dumitru...et al. (2013). "A k-Dimensional System of Fractional Finite Difference Equations", Abstract and Applied Analysis.
A k-Dimensional System of Fractional Finite Difference Equations
(Hindawi LTD, 2013) Baleanu, Dumitru; Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; 56389
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.
A New Jacobi Rational-Gauss Collocation Method For Numerical Solution of Generalized Pantograph Equations
(Elsevier, 2014) Doha, E. H.; Bhrawy, A. H.; Baleanu, Dumitru; Hafez, R. M.; 56389
This 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.
Citation Count: Ezz-Eldien, S. S...et al. (2017). "A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems", Journal Of Vibration And Control, Vol. 23, No.1, pp.16-30.
A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems
(Sage Publications LTD, 2017) Ezz-Eldien, S. S.; Doha, E. H.; Baleanu, Dumitru; Bhrawy, A. H.; 56389
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.
A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions
(Hindawi LTD, 2014) Doha, E. H.; Baleanu, Dumitru; Bhrawy, A. H.; Hafez, R. M.; 56389
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 Count: Bhrawy, A.H...et al. (2015). A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. Journal Of The Computational Physics, 293, 142-156. http://dx.doi.org/10.1016/j.jcp.2014.03.039
A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations
(Academic Press INC Elsevier Science, 2015) Bhrawy, A. H.; Doha, E. H.; Baleanu, Dumitru; Ezz-Eldien, S. S.
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.
Citation Count: Doha, E.H...et al. (2014). "An Accurate Legendre Collocation Scheme for Coupled Hyperbolic Equations With Variable Coefficients", Romanian Journal of Physics, Vol. 59, No. 5-6.
An Accurate Legendre Collocation Scheme for Coupled Hyperbolic Equations With Variable Coefficients
(Editura Academiei Romane, 2014) Doha, E. H.; Bhrawy, A. H.; Baleanu, Dumitru; Abdelkawy, M. A.; 56389
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 Count: Bhrawy, A.H...et al. (2015). "An Accurate Numerical Technique for Solving Fractional Optimal Control Problems", Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science, Vol. 16, No. 1, pp. 47-54.
An Accurate Numerical Technique for Solving Fractional Optimal Control Problems
(Editura Academiei Romane, 2015) Bhrawy, A. H.; Doha, E. H.; Baleanu, Dumitru; Abdelkawy, M. A.; Ezz-Eldien, S. S.; 56389
In this article, we propose the shifted Legendre orthonormal polynomials for the numerical solution of the fractional optimal control problems that appear in several branches of physics and engineering. The Rayleigh-Ritz method for the necessary conditions of optimization and the operational matrix of fractional derivatives are used together with the help of the properties of the shifted Legendre orthonormal polynomials to reduce the fractional optimal control problem to solving a system of algebraic equations that greatly simplifies the problem. For confirming the efficiency and accuracy of the proposed technique, an illustrative numerical example is introduced with its approximate solution.
Citation Count: Doha, E. H...et.al. (2017). "Composite Bernoulli-Laguerre collocation method for a class of hyperbolic telegraph-type equations", Romanian Reports In Physics, Vol.69, No.4.
Composite Bernoulli-Laguerre collocation method for a class of hyperbolic telegraph-type equations
(Editura Academiei Romane, 2017) Doha, E. H.; Hafez, R. M.; Abdelkawy, M. A.; Ezz-Eldien, S. S.; Taha, T. M.; Zaky, M. A.; Amin, A. Z. M.; El-Kalaawy, A. A.; Baleanu, Dumitru; 56389
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.
Efficient Jacobi-Gauss Collocation Method for Solving Initial Value Problems of Bratu Type
(Pleiades Publishing INC, 2013) Doha, E. H.; Bhrawy, A. H.; Baleanu, Dumitru; Hafez, R. M.; 56389
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.
Citation Count: Bhrawy, A.H...et al. (2016). Modified Jacobi-Bernstein basis transformation and its application to multi-degree reduction of Bezier curves. Journal of Computational and Applied Mathematics, 302, 369-384. http://dx.doi.org/10.1016/j.cam.2016.01.009
Modified Jacobi-Bernstein basis transformation and its application to multi-degree reduction of Bezier curves
(Elsevier Science BV, 2016) Bhrawy, A. H.; Doha, E. H.; Saker, M. A.; Baleanu, Dumitru
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.
Citation Count: Hafez, R. M...et al. (20179. "Numerical solutions of two-dimensional mixed volterra-fredholm integral equations via bernoulli collocation method", Romanian Journal Of Physics, Vol. 62, No. 3-4.
Numerical solutions of two-dimensional mixed volterra-fredholm integral equations via bernoulli collocation method
(Editura Academiei Romane, 2017) Hafez, R. M.; Doha, E. H.; Bhrawy, A. H.; Baleanu, Dumitru; 56389
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 Shifted Jacobi Spectral Approximations For Solving Fractional Differential Equations
(Elsevier Science, 2013) Doha, E. H.; Bhrawy, A. H.; Baleanu, Dumitru; Ezz-Eldien, S. S.; 56389
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 Count: Bhrawy, A. H...et al. (2017). "Solving fractional optimal control problems within a Chebyshev-Legendre operational technique", International Journal Of Control, Vol. 90, No.6, pp. 1230-1244.
Solving fractional optimal control problems within a Chebyshev-Legendre operational technique
(Taylor&Francis, 2017) Bhrawy, A. H.; Ezz-Eldien, S. S.; Doha, E. H.; Abdelkawy, M. A.; Baleanu, Dumitru; 56389
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 Count: Doha, E. H. (2018). "Spectral technique for solving variable-order fractional Volterra integro-differential equations" Vol.34, No.5, pp. 1659-1677.
Spectral Technique for Solving Variable-Order Fractional Volterra Integro-Differential Equations
(Wiley, 2018) Doha, E. H.; Abdelkawy, M. A.; Amin, A. Z. M.; Baleanu, Dumitru; 56389
This article, presented a shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method which is introduced for solving variable-order fractional Volterra integro-differential equation (VO-FVIDEs) subject to initial or nonlocal conditions. Based on shifted Legendre Gauss-Lobatto (SL-GL) quadrature, we treat with integral term in the aforementioned problems. Via the current approach, we convert such problem into a system of algebraic equations. After that we obtain the spectral solution directly for the proposed problem. The high accuracy of the method was proved by several illustrative examples.
Citation Count: Doha, E. H...et al. (2018). "Spectral technique for solving variable-order fractional Volterra integro-differential equations", Numerical Methods for Partial Differential Equations, Vol. 34, No. 5, pp. 1659-1677.
Spectral technique for solving variable-order fractional Volterra integro-differential equations
(Wiley, 2018) Doha, E. H.; Abdelkawy, M. A.; Amin, A. Z. M.; Baleanu, Dumitru; 56389
This article, presented a shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method which is introduced for solving variable-order fractional Volterra integro-differential equation (VO-FVIDEs) subject to initial or nonlocal conditions. Based on shifted Legendre Gauss-Lobatto (SL-GL) quadrature, we treat with integral term in the aforementioned problems. Via the current approach, we convert such problem into a system of algebraic equations. After that we obtain the spectral solution directly for the proposed problem. The high accuracy of the method was proved by several illustrative examples.