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Structural Reliability Theory
Published in Srinivasan Chandrasekaran, Offshore Structural Engineering, 2017
In the first stage, polynomial order ki is determined by numerically testing the significance of polynomial coefficients along the coordinate axes xi. The Chebyshev polynomial is selected as the basis function, which is orthogonal and bounded within the range of [-1,1]. A Chebyshev polynomial of degree M is given by () TM(x)=cos(Marccos(x))
Insertion Loss Filters
Published in Herbert J. Carlin, Pier Paolo Civalleri, Wideband Circuit Design, 2018
Herbert J. Carlin, Pier Paolo Civalleri
The Chebyshev polynomials have special properties which are important in the theory of approximation and make them optimum for the transducer gain of polynomial filters. Define the monic polynomial tn(ω) = (2n−1)−1Tn(ω), with the coefficient of ωn equal to unity, see eq. (6.6.4). Then tn is that monic polynomial among all monic polynomials of degree n, whose maximum deviation from zero in the interval −1 ≤ ω ≤ 1, is minimum (minimax property). Evidently the minimax deviation is (2n−1)−1.
Digital Filters
Published in Jerry C. Whitaker, Microelectronics, 2018
Jonathon A. Chambers, Sawasd Tantaratana, Bruce W. Bomar
where TN(x) is the Nth-degree Chebyshev polynomial of the first kind that is given recursively by T0(x) = 1, T1(x) = x, and Tn+1(x) = 2xTn(x) − Tn−1(x) for n ≥ 1. Figure 12.13(b) shows an example of the magnitude response square. Notice that there are equiripples in the pass band. The filter order required to satisfy Eq. (12.8) is () N≥log{[(δ−2−1)12/ε]+[(δ−2−1)/ε2−1]12}log{(λs/λp)+[(λs/λp)2−1]12
Vibrations of graphene platelet reinforced composite doubly curved shells subjected to thermal shock
Published in Mechanics Based Design of Structures and Machines, 2022
In above and are constant coefficients yet to be determined, and are i-th and j-th admissible function given by the Chebyshev-type polynomials of the first kind which have better numerical stability than the algebraic polynomial series. Also and are the total number of the Chebyshev-type polynomial series. It should be noted that Chebyshev polynomials are defined in [-1 1] and are also nonzero at the ends of the interval.
Stability of magnetohydrodynamic heat transfer free convection of vertical surface with OHMIC heating by radiation and viscous dissipation
Published in Waves in Random and Complex Media, 2021
Mohamed H. Hendy, Sayed I. El-Attar
Chebyshev polynomials are used widely in numerical computations. Chebyshev polynomials have proven success in the numerical solution of various boundary value problems [25–27] and in computational fluid dynamics [14,28,29]. The present work deals with the application of a Chebyshev spectral collocation approach to the computation of the boundary-layer equations in MHD flows. This approach requires the definition of a grid point and it is applied to satisfy the differential equations and the boundary conditions at these grid points. It can be regarded as a non-uniform finite difference scheme. The derivatives of the function at a point is a linear combination of the values of the function at the Gauss-Lobatto points where and is an integer [2,5].
Use of the Third-kind Chebyshev Polynomials in Generating New Filter Functions
Published in IETE Journal of Research, 2021
Chebyshev polynomials have been in the focus of many studies and have drawn a wide attention due to their frequent appearance in various applications in polynomial approximation, integral and differential equations, etc. There are four kinds of these polynomials as projected by Mason and Handscomb [7] which leads to an extended range of new filter functions. The Chebyshev polynomials of the first-kind [8–10] are used to generate integer coefficients of Coleman filter [9]. Filter functions given in [11] and [12] prove the use of the first- and second-kind Chebyshev polynomials in filter design, respectively. The principle idea behind obtaining these filter functions is based on Coleman filter described in [9].