Generalized shifted Chebyshev polynomials: Solving a general class of nonlinear variable order fractional PDE

Hassani, Hossein; Machado, J. A. Tenreiro; Avazzadeh, Z.; Naraghirad, Eskandar

doi:10.1016/j.cnsns.2020.105229

Cited by 27 publications

(12 citation statements)

References 22 publications

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“…The displacements of polybutadiene beam and butyl B252 beam under different external loads are calculated, and the properties of the two materials are compared and analyzed. Hassani et al 27 used a novel class of basis functions based on the shifted Chebyshev polynomials to solve the nonlinear variable order fractional derivative equations. The accuracy and efficiency of the method were confirmed by the convergence analysis and several numerical examples.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Cao

Chen

Wang

et al. 2021

Math Methods in App Sciences

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An innovative numerical procedure for solving the viscoelastic arch problem based on variable fractional rheological models, directly in time domain, is proposed and investigated. First, the nonlinear integral‐differential governing equation is established according to the variable fractional constitutive relation and geometrical relationship. Second, the nonlinear integral‐differential governing equation is transformed into algebraic equations and solved by using the shifted Legendre polynomials. Furthermore, the accuracy and effectiveness of the algorithm are verified according to the mathematical example. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic arch is investigated to determine the displacement at different times and positions. The displacement of the viscoelastic arch is compared under various loading (uniformly distributed load and linear load). The displacement of the viscoelastic arch of different materials under the same load conditions is also investigated. The results in the paper show the efficiency of the proposed numerical algorithm in the dynamical analysis of the viscoelastic arch.

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Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Cao

Chen

Wang

et al. 2021

Math Methods in App Sciences

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“…These moments have been used to reconstruct an image. Also, we have compared our results with fractional order Chebyshev orthogonal moments, which is recently introduced by Hassani et al (2020). This set of moments mathematically is able to represent any two-dimension image, and in practice, we noticed the numerical result perfectly represents an image as shown in Fig.…”

Section: Resultsmentioning

confidence: 82%

“…Shifted Chebyshev polynomials are developed to the new family of basis functions, namely generalized shifted Chebyshev polynomials (Fernández et al, 2011). The bivariate orthogonal polynomials are used to define continuous and discrete orthogonal moments are discussed in Hassani et al (2020). Only a few papers have used bivariate or multivariate orthogonal polynomials for image analysis and pattern recognition (Xu, 2004;Fernández, 2007).…”

Section: Introductionmentioning

confidence: 99%

Image representation based on fractional order Legendre and Laguerre orthogonal moments

Farouk¹,

Awad²

2021

Int. j. adv. appl. sci.

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In this paper, we have introduced new sets of fractional order orthogonal basis moments based on Fractional order Legendre orthogonal Functions (FLeFs) and Fractional order Laguerre orthogonal Functions (FLaFs) for image representation. We have generated a novel set of Fractional order Legendre orthogonal Moments (FLeMs) from fractional order Legendre orthogonal functions and a new set of Fractional order Laguerre orthogonal Moments (FLaMs) from the fractional order Laguerre orthogonal functions. The new presented sets of (FLeMs) and (FLaMs) are tested with the recently introduced Fractional order Chebyshev orthogonal Moments (FCMs). This edge detection filter can be used successfully in the gray level image and color images. The new sets of fractional moments are used to reconstruct the gray level image. The numerical results show FLeMs and FLaMs are promised techniques for image representation. The computational time of the proposed techniques is compared with the computational time of Chebyshev orthogonal Moments techniques and gives better results. Also, the fractional parameters give the flexibility of studying global features of the image at different positions of moments.

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“…Moreover, in [10], the authors solved a class of non-linear variable-order fractional reaction-diffusion equation based on using the shifted Chebyshev polynomials of the fifth kind. For some other articles concerned with the different kinds of Chebyshev polynomials, see for example [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

Abd-Elhameed¹

2021

Fractal Fract

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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.

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Generalized shifted Chebyshev polynomials: Solving a general class of nonlinear variable order fractional PDE

Cited by 27 publications

References 22 publications

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Image representation based on fractional order Legendre and Laguerre orthogonal moments

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

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