The Kidder problem is uxx + 2x(1 − αu)−1/2ux = 0 with u(0) = 1 and u(∞) = 0 where α ∈ [0, 1]. This looks challenging because of the square root singularity. We prove, however, that |u(x ;α) − erfc(x)| ≤ 0.046 for all x, α. Other very simple but very accurate curve fits and bounds are given in the text; |u(x ;α) − erfc(x + 0.15076x/(1 + 1.55607x2))| ≤ 0.0019. Maple code for a rational Chebyshev ...

The Chebyshev polynomials have many beautiful properties and countless applications, arising in a variety of continuous settings. They are a sequence of orthogonal polynomials appearing in approximation theory, numerical integration, and differential equations. In this paper we approach them instead as discrete objects, counting the sum of weighted tilings. Using this combinatorial approach, on...

We examine to what extent finite-dimensional spaces defined on locally compact subsets of the line and possessing various weak Chebyshev properties (involving sign changes, zeros, alternation of best approximations, and peak points) can be uniformly approximated by a sequence of spaces having related properties. r 2003 Elsevier Science (USA). All rights reserved.

in this paper, we introduce a family of fractional-order chebyshev functions based on the classical chebyshev polynomials. we calculate and derive the operational matrix of derivative of fractional order $gamma$ in the caputo sense using the fractional-order chebyshev functions. this matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...

in this paper, the chebyshev spectral collocation method(cscm) for one-dimensional linear hyperbolic telegraph equation is presented. chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. a straightforward implementation of these methods involves the use of spectral differentiation matrices. firstly, we transform ...

Abstract Trace estimators allow us to approximate thermodynamic equilibrium observables with astonishing accuracy. A prominent representative is the finite-temperature Lanczos method (FTLM) which relies on a Krylov space expansion of exponential describing Boltzmann weights. Here we report investigations an alternative approach employs Chebyshev polynomials. This turns out be also very accurate...

This paper is concerned with establishing novel expressions that express the derivative of any order orthogonal polynomials, namely, Chebyshev polynomials sixth kind in terms themselves. We will prove these involve certain terminating hypergeometric functions type 4F3(1) can be reduced some specific cases. The derived along linearization formula serve obtaining a numerical solution non-linear o...

In this research, we analyze the acoustic–vibration coupling of liquid-filled cylindrical shells under complex excitations. A calculation model to determine acoustic characteristics and steady-state response a shell coupled an cavity is proposed. The displacement sound pressure are described by Chebyshev–Fourier series in three dimensions. uncertain expansion coefficient determined with Rayleig...