Computation of Chebyshev Polynomials for Union of Intervals

Foucart, Simon; Lasserre, Jean-Bernard

doi:10.1007/s40315-019-00285-w

Cited by 9 publications

(5 citation statements)

References 11 publications

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“…Its advantages are high‐accuracy, fewer iterations, and less computation time 17 . Legendre, 18 Bernstein, 19,20 and Chebyshev polynomials 21‐23 are often employed to construct wavel et algorithms. Chen et al 24 employed wavelet method for solving the nonlinear fractional differential equations.…”

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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“…In this section, we uncover situations where, using techniques from Robust Optimization [2], it is possible to exactly compute the Chebyshev center of the set Q(K E (y)) defined by the observational data y ∈ R m . It is assumed here that the quantity of interest Q takes values in Z = ℓ K ∞ and we write (8) Q…”

Section: Computation Of Chebyshev Centersmentioning

confidence: 99%

Instances of Computational Optimal Recovery: Dealing with Observation Errors

Ettehad¹,

Foucart²

2020

Preprint

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When attempting to recover functions from observational data, one naturally seeks to do so in an optimal manner with respect to some modeling assumption. With a focus put on the worst-case setting, this is the standard goal of Optimal Recovery. The distinctive twists here are the consideration of inaccurate data through some boundedness models and the emphasis on computational realizability. Several scenarios are unraveled through the efficient constructions of optimal recovery maps: local optimality under linearly or semidefinitely describable models, global optimality for the estimation of linear functionals under approximability models, and global near-optimality under approximability models in the space of continuous functions.

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“…Chen et al (2015) applied the wavelet method to solve nonlinear fractional differential equations, offering a different perspective on solving complex problems. Legendre (Boonrod and Mohsen, 2019), Bernstein (Bourne et al, 2017; Wang et al, 2018), and Chebyshev polynomials (Foucart and Lasserre, 2019; Sakran, 2019; Sweilam et al, 2015) are commonly used to construct wavelet algorithms.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of the dynamic characteristics of a fractional-order viscoelastic beam with fixed supports at both ends

Li,

Chen,

Chen

et al. 2023

Journal of Intelligent Material Systems and Structures

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In this paper, we present a generalized model of a viscoelastic beam, which takes into account the influence of axial forces and incorporates a fractional constitutive relationship. In addition, we propose a novel numerical calculation method for analyzing fractional-order viscoelastic beams. This method takes the transverse displacement and bending moment of the beam as state variables, transforms the beam model into a discrete state space equation through the application of the central difference method, and utilizes an improved precise integration algorithm to solve the equation. To evaluate the performance of the method, the responses of the beam under two types of excitations, namely uniformly distributed transverse load and the motion of the support at both ends, are calculated under fixed hinge conditions. The results demonstrate that the present method has excellent accuracy and convergence, and also reveal some nonlinear phenomena of the system.

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Computation of Chebyshev Polynomials for Union of Intervals

Cited by 9 publications

References 11 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

Instances of Computational Optimal Recovery: Dealing with Observation Errors

Numerical analysis of the dynamic characteristics of a fractional-order viscoelastic beam with fixed supports at both ends

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