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Browsing by Author "Zaky, Mahmoud A."

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    New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method
    (2015) Baleanu, Dumitru; Zaky, Mahmoud A.; Baleanu, Dumitru; 56389; Matematik
    Burgers’ equation is a fundamental partial differential equation in fluid mechanics. This paper reports a new space-time spectral algorithm for obtaining an approximate solution for the space-time fractional Burgers’ equation (FBE) based on spectral shifted Legendre collocation (SLC) method in combination with the shifted Legendre operational matrix of fractional derivatives. The fractional derivatives are described in the Caputo sense. We propose a spectral shifted Legendre collocation method in both temporal and spatial discretizations for the space-time FBE. The main characteristic behind this approach is that it reduces such problem to that of solving a system of nonlinear algebraic equations that can then be solved using Newton’s iterative method. Numerical results with comparisons are given to confirm the reliability of the proposed method for FBE. © 2015, Editura Academiei Romane. All rights reserved.
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    Citation - WoS: 22
    Citation - Scopus: 26
    New Recursive Approximations for Variable-Order Fractional Operators with Applications
    (Vilnius Gediminas Tech Univ, 2018) Zaky, Mahmoud A.; Baleanu, Dumitru; Doha, Eid H.; Taha, Taha M.; Baleanu, Dumitru; 56389; Matematik
    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.