Browsing by Author "Doha, Eid H."

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A Computationally Efficient Method For a Class of Fractional Variational and Optimal Control Problems Using Fractional Gegenbauer Functions
(Editura Academiei Romane, 2018) El-Kalaawy, Ahmed A.; Doha, Eid H.; Ezz-Eldien, Samer S.; Abdelkawy, M. A.; Hafez, R. M.; Amin, A. Z. M.; Baleanu, Dumitru; Zaky, M. A.; 56389
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.
Citation Count: Doha, E.H...et al. (2015). An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Advance in Difference Equations. http://dx.doi.org/10.1186/s13662-014-0344-z
An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems
(Springer International Publishing, 2015) Doha, Eid H.; Bhrawy, Ali H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; Hafez, R. M.
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 Count: Alsuyuti, Muhammad M...et al. (2019). "Modified Galerkin algorithm for solving multitype fractional differential equations", Mathematical Methods in the Applied Sciences, Vol. 42, No. 5, pp. 1389-1412.
Modified Galerkin algorithm for solving multitype fractional differential equations
(Wiley, 2019) Alsuyuti, Muhammad M.; Doha, Eid H.; Ezz-Eldien, Samer S.; Bayoumi, Bayoumi I.; Baleanu, Dumitru; 56389
The primary point of this manuscript is to dissect and execute a new modified Galerkin algorithm based on the shifted Jacobi polynomials for solving fractional differential equations (FDEs) and system of FDEs (SFDEs) governed by homogeneous and nonhomogeneous initial and boundary conditions. In addition, we apply the new algorithm for solving fractional partial differential equations (FPDEs) with Robin boundary conditions and time-fractional telegraph equation. The key thought for obtaining such algorithm depends on choosing trial functions satisfying the underlying initial and boundary conditions of such problems. Some illustrative examples are discussed to ascertain the validity and efficiency of the proposed algorithm. Also, some comparisons with some other existing spectral methods in the literature are made to highlight the superiority of the new algorithm.
Citation Count: Zaky, Mahmoud A...et al. (2018). "New Recursive Approximations for Variable-Order Fractional Operators with Applications", Mathematical Modelling and Analysis, Vol. 23, No. 2, pp. 227-239.
New Recursive Approximations for Variable-Order Fractional Operators with Applications
(Vilnius Gediminas Tech Univ, 2018) Zaky, Mahmoud A.; Doha, Eid H.; Taha, Taha M.; Baleanu, Dumitru; 56389
To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods.
Citation Count: Doha, Eid H...et al. (2019). "Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations", Nonlinear Analysis-Modelling and Control, Vol. 24, No. 3, pp. 332-352.
Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations
(Inst Mathematics & Informatics, 2019) Doha, Eid H.; Abdelkawy, M. A.; Amin, A. Z. M.; Baleanu, Dumitru; 56389
This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE.
The Operational Matrix Formulation of The Jacobi Tau Approximation For Space Fractional Diffusion Equation
(Springer Open, 2014) Bhrawy, Ali H.; Doha, Eid H.; Baleanu, Dumitru; Ezz-Eldien, Samer S.; 56389
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.