Meshless Petrov–Galerkin Method for Rotating Rayleigh Beam Using Laguerre and Gegenbauer Polynomials

Panchore, Vijay

doi:10.1007/s42417-022-00719-1

Cited by 4 publications

(3 citation statements)

References 43 publications

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“…For example, the authors of [1] used shifted Gegenbauer polynomials to solve the space fractional diffusion equation problems. Furthermore, as the authors of [2] show, the Laguerre and Gegenbauer polynomials together provided the basis for a meshless Petrov-Galerkin method that was especially used for spinning Rayleigh beams. Gegenbauer polynomials are also used in harmonic analysis and potential theory, where they easily emerge as Legendre polynomial extensions.…”

Section: Introductionmentioning

confidence: 99%

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

Sayed,

Mohamed,

El-Dahab

et al. 2024

Contemp. Math.

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This article introduces two numerical methods to address boundary value problems associated with secondorder and fractional differential equations. These methods employ two parameters related to shifted Gegenbauer polynomials as their basis functions. The process involves establishing a differentiation operational matrix for the shifted Gegenbauer polynomials. Subsequently, the initial/boundary value problems for ordinary and fractional differential equations are transformed into a system of equations through the Galerkin, collocation, and tau methods. The convergence analysis is ensured by leveraging theorems pertaining to the shifted Gegenbauer polynomials. To validate the accuracy of the approach, numerous numerical examples are presented.

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Section: Introductionmentioning

confidence: 99%

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

Sayed,

Mohamed,

El-Dahab

et al. 2024

Contemp. Math.

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“…For example, in [13], the authors used SGPs to solve the space fractional diffusion equation. In [14], the authors presented a meshless Petrov-Galerkin method for rotating Rayleigh beams using Laguerre and Gegenbauer polynomials. In [15], the authors proposed a novel operational matrices-based method for solving fractional-order delay differential equations via SGPs.…”

Section: Introductionmentioning

confidence: 99%

Two spectral Gegenbauer methods for solving linear and nonlinear time fractional Cable problems

Atta

2023

Int. J. Mod. Phys. C

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This research aims to analyze and implement two numerical spectral schemes for solving linear and nonlinear time fractional Cable equations (TFCEs). Two modified sets of shifted Gegenbauer polynomials (SGPs) are used as basis functions. The approximation of the solution is written as a product of the two chosen basis function sets. For these methods, the principle idea is to convert the problem governed by the underlying conditions into a set of linear and nonlinear algebraic equations that can be solved using appropriate numerical techniques. The upper estimate of the truncation error for the proposed expansion is investigated. In the end, four examples are presented to illustrate the accuracy of the suggested schemes.

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“…The method can be explored with other functions as well, where there is a possibility of increasing the number of nodes within the sub-domain of the trial function [6,7]. The Gaussian radial basis functions are generally used where higher-order derivatives are possible [8,9].…”

Section: Introductionmentioning

confidence: 99%

Archive of Mechanical Engineering

Panchore

2022

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The numerical solutions are obtained for rotating beams; the inclusion of centrifugal force term makes it difficult to get the analytical solutions. In this paper, we solve the free vibration problem of rotating Rayleigh beam using Chebyshev and Legendre polynomials where weak form of meshless local Petrov-Galerkin method is used. The equations which are derived for rotating beams result in stiffness matrices and the mass matrix. The orthogonal polynomials are used and results obtained with Chebyshev polynomials and Legendre polynomials are exactly the same. The results are compared with the literature and the conventional finite element method where only first seven terms of both the polynomials are considered. The first five natural frequencies and respective mode shapes are calculated. The results are accurate when compared to literature.

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Meshless Petrov–Galerkin Method for Rotating Rayleigh Beam Using Laguerre and Gegenbauer Polynomials

Cited by 4 publications

References 43 publications

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

Alleviated Shifted Gegenbauer Spectral Method for Ordinary and Fractional Differential Equations

Two spectral Gegenbauer methods for solving linear and nonlinear time fractional Cable problems

Archive of Mechanical Engineering

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