Combination Of Chebyshev Polynomials Research Articles

Spectral method solutions of a variable aspect ratio duct flow in a uniform transverse magnetic field with two different thermal boundary conditions

Abstract A laminar electrically-conducting flow inside an electrically-insulated rectangular duct in a uniform transverse magnetic field with both uniform surface temperature and uniform heat flux boundary conditions are considered numerically. The problem with aspect ratios of the range from 1:10 to 10:1 and Hartmann number M up to 1000 is solved using a highly accurate technique which is spectral method. The flow variables are expanded in terms of linear combinations of Chebyshev polynomials chosen to satisfy the boundary conditions implicitly. The resulting equations are collocated using the Gauss points to produce a system of nonlinear algebraic equations which is solved iteratively using Gauss elimination. Convergence properties of the numerical method reveals that for aspect ratio of less than 4 to 1, the flow is well resolved with as small as 29 by 29 Chebyshev polynomials; however, as the aspect ratio increases to more than 4 to 1, the number of polynomials required for an adequate resolution can be as high as 49 by 49 polynomials. It is found when the magnetic field is turned on, the pressure drop, in general, increases with the field for different aspect ratios. However, the pressure drop increase will be slower near the aspect ratio of 10. Also, the heat transfer increases with the field for most of the cases, but for some cases the field will have adverse effect on the heat transfer, particularly, for constant surface temperature boundary condition at aspect ratio > 4 and M 4 and M

A Numerical Solver for First Order Initial Value Problems of Ordinary Differential Equation Via the Combination of Chebyshev Polynomial and Exponential Function

Purpose: The purpose of this study is to derive a numerical solver for first order initial value problems of ordinary differential equation via the combination of Chebyshev polynomial and exponential function.Methodology: A new numerical method for solving Initial Value Problems of first order ordinary differential equation is developed. The method is based on finite difference method with a combination of Chebyshev polynomials and exponential function as interpolant. The accuracy, stability, consistency and convergence of the derived scheme were investigated. Numerical experiment was carried out by solving some test problems using the derived scheme.Findings: Results of the numerical experiment revealed that the derived method compared favourably with exact solutions and also performs better than some existing methods for solving initial value problems of first order. Unique Contribution to theory, practice and policy: The study therefore concludes that the method solves problems to expected level of accuracy and can thus be considered among the numerous methods suitable for solving IVPs of first order.

RESONANT CHARACTERISTICS OF CIRCULAR HTC SUPERCONDUCTING PRINTED ANTENNA COVERED WITH A DIELECTRIC LAYER

—Effects of a superstrate layer on the resonant frequency and bandwidth of a high Tc superconducting (HTS) circular printed patch are investigated in this paper. For that, a rigorous full-wave spectral analysis of superconducting patch in multilayer configuration is described. In such an approach, the spectral dyadic Green's function which relates the tangential electric field and currents at various conductor planes should be determined. Integral equations are solved by a Galerkin's moment method procedure, and the complex resonance frequencies are studied with basis functions involving Chebyshev polynomials in conjunction with the complex resistive boundary condition. To include the superconductivity of the disc, its complex surface impedance is determined by using London's equation and the model of Gorter and Casimir. Numerical results are compared with experimental results of literature as well as with the most recent published calculations using different methods. A very good agreement is obtained. Finally, superstrate loading effects are presented and discussed showing interesting enhancement on the resonant characteristics of the superconducting antenna using combinations of Chebyshev polynomials as set of basis functions.

On the spectrum of banded Hankel matrices

We derive a weak limit law for the distribution of eigenvalues of a tridiagonal Hankel matrix. The result is given in terms of the push-forward of an arcsine density under a combination of Chebyshev polynomials. We also advance a conjecture concerning Hankel matrices with more than three non-zero skew diagonals.

Vector Hankel transform analysis of a tunable circular microstrip patch

AbstractIn this paper, a rigorous analysis of the tunable circular microstrip patch is performed using a dyadic Green's function formulation. To make the theoretical formulation more general and hence valid for various antennas structures (not only limited to tunable microstrip patch); the dyadic Green's function is derived when the patch is assumed to be embedded in a multilayered dielectric substrate. A very efficient technique to derive the dyadic Green's function in the vector Hankel transform domain is proposed. Using the vector Hankel transform, the mixed boundary value problem is reduced to a set of vector dual integral equations. Galerkin's method is then applied to solve the integral equation where two sets of disk current expansions are used. One set is based on the complete set of orthogonal modes of the magnetic cavity, and the other consists of combinations of Chebyshev polynomials with weighting factors to incorporate the edge condition. Convergent results for these two sets of disk current expansions are obtained with a small number of basis functions. The calculated resonant frequencies and quality factors are compared with experimental data and shown to be in good agreement. Finally, numerical results for the air gap tuning effect on the resonant frequency and half‐power bandwidth are also presented. Copyright © 2005 John Wiley & Sons, Ltd.

The Chebyshev polynomial fitting properties of discrete cosine transform

Abstract In this paper, the least-squares approximation/fitting aspects of discrete cosine transform (DCT) are identified and analyzed. Through constructing a monovariant function skillfully defined on a discrete set, we obtain a specified series expansion which is a combination of Chebyshev polynomials. It will be shown that the corresponding Chebyshev polynomial expansion is exactly a mathematical equivalence of DCT.

Determination of the Zeros of a Linear Combination of Chebyshev Polynomials

Determination of the Zeros of a Linear Combination of Chebyshev Polynomials

Zolotarev problem in the metric of L1([?1, 1

In this paper we solve the problem of the determination of a polynomial of degree n with given two leading coefficients which has the least deviation from zero in the metric of L1 ([−1, 1]). The extremal polynomial is expressed in the form of some linear combination of Chebyshev polynomials of the second kind.