Browsing by Author "Abdelhakem, Mohamed"

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Citation - WoS: 28
Citation - Scopus: 34
Approximating Real-Life Bvps Via Chebyshev Polynomials' First Derivative Pseudo-Galerkin Method
(Mdpi, 2021) Alaa-Eldeen, Toqa; Baleanu, Dumitru; Alshehri, Maryam G.; El-Kady, Mamdouh; Abdelhakem, Mohamed
An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems.
Citation - WoS: 33
Citation - Scopus: 36
Shifted Ultraspherical Pseudo-Galerkin Method for Approximating the Solutions of Some Types of Ordinary Fractional Problems
(Springer, 2021) Mahmoud, Doha; Baleanu, Dumitru; El-kady, Mamdouh; Abdelhakem, Mohamed
In this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.