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Keywords = chebyshev polynomials of sixth kind

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17 pages, 1895 KB  
Article
A Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials
by Esraa Magdy Abdelghany, Waleed Mohamed Abd-Elhameed, Galal Mahrous Moatimid, Youssri Hassan Youssri and Ahmed Gamal Atta
Symmetry 2023, 15(3), 594; https://doi.org/10.3390/sym15030594 - 25 Feb 2023
Cited by 25 | Viewed by 3118
Abstract
The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which is based on the spectral tau method. There are two sets of basis functions used. The first set is the set of non-symmetric polynomials, [...] Read more.
The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which is based on the spectral tau method. There are two sets of basis functions used. The first set is the set of non-symmetric polynomials, namely, the shifted Chebyshev polynomials of the sixth-kind (CPs6), and the second set is a set of modified shifted CPs6. The approximation of the solution is written as a product of the two chosen basis function sets. For this method, the key concept is to transform the problem governed by the underlying conditions into a set of linear algebraic equations that can be solved by means of an appropriate numerical scheme. The error analysis of the proposed extension is also thoroughly investigated. Finally, a number of examples are shown to illustrate the reliability and accuracy of the suggested tau method. Full article
(This article belongs to the Special Issue Numerical Methods for Differential Problems and Symmetry)
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16 pages, 644 KB  
Article
A Fast Galerkin Approach for Solving the Fractional Rayleigh–Stokes Problem via Sixth-Kind Chebyshev Polynomials
by Ahmed Gamal Atta, Waleed Mohamed Abd-Elhameed, Galal Mahrous Moatimid and Youssri Hassan Youssri
Mathematics 2022, 10(11), 1843; https://doi.org/10.3390/math10111843 - 27 May 2022
Cited by 25 | Viewed by 1969
Abstract
Herein, a spectral Galerkin method for solving the fractional Rayleigh–Stokes problem involving a nonlinear source term is analyzed. Two kinds of basis functions that are related to the shifted sixth-kind Chebyshev polynomials are selected and utilized in the numerical treatment of the problem. [...] Read more.
Herein, a spectral Galerkin method for solving the fractional Rayleigh–Stokes problem involving a nonlinear source term is analyzed. Two kinds of basis functions that are related to the shifted sixth-kind Chebyshev polynomials are selected and utilized in the numerical treatment of the problem. Some specific integer and fractional derivative formulas are used to introduce our proposed numerical algorithm. Moreover, the stability and convergence accuracy are derived in detail. As a final validation of our theoretical results, we present a few numerical examples. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)
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13 pages, 1057 KB  
Article
A Chebyshev Collocation Approach to Solve Fractional Fisher–Kolmogorov–Petrovskii–Piskunov Equation with Nonlocal Condition
by Dapeng Zhou, Afshin Babaei, Seddigheh Banihashemi, Hossein Jafari, Jehad Alzabut and Seithuti P. Moshokoa
Fractal Fract. 2022, 6(3), 160; https://doi.org/10.3390/fractalfract6030160 - 15 Mar 2022
Cited by 5 | Viewed by 2853
Abstract
We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. [...] Read more.
We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. We write our approximate solution in the product form, which consists of unknown coefficients and shifted Chebyshev polynomials. To compute the numerical values of coefficients, we use the initial and boundary conditions and the collocation technique to create a system of equations whose number matches the unknowns. We test the applicability and accuracy of this numerical approach using two examples. Full article
(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)
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20 pages, 587 KB  
Article
Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation
by Waleed Mohamed Abd-Elhameed
Fractal Fract. 2021, 5(2), 53; https://doi.org/10.3390/fractalfract5020053 - 6 Jun 2021
Cited by 55 | Viewed by 3454
Abstract
This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions [...] Read more.
This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm. Full article
(This article belongs to the Section Numerical and Computational Methods)
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15 pages, 1020 KB  
Article
A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise
by Afshin Babaei, Hossein Jafari and S. Banihashemi
Symmetry 2020, 12(6), 904; https://doi.org/10.3390/sym12060904 - 1 Jun 2020
Cited by 29 | Viewed by 3431
Abstract
A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding [...] Read more.
A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method. Full article
(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)
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