New Runge–Kutta methods for method of lines solution of systems of ordinary differential equations arising from discretizations of spatial derivatives in hyperbolic equations, by Chebyshev or modified Chebyshev methods, are introduced. These Runge–Kutta methods optimize the time step necessary for stable solutions, while holding dispersion and dissipation fixed. It is found that maximizing disp...

The computation of spectral expansion coefficients is an important aspect in the implementation of spectral methods. In this paper, we explore two strategies for computing the coefficients of polynomial expansions of analytic functions, including Chebyshev, Legendre, ultraspherical and Jacobi coefficients, in the complex plane. The first strategy maximizes computational efficiency and results i...

In this paper, by the Chebyshev-type inequalities we define three mappings, investigate their main properties, give some refinements for Chebyshev-type inequalities, obtain some applications.

We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x = T3(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials and b + degC = 3N . If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for a ≤ 3. Most results are derived from continued fractions and their matrix represe...

A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(logN)2/ log logN) operations is derived. The basis of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the di...

The Falkner-Skan equation is a nonlinear third-order boundary value problem defined on the semi-infinite interval [0,∞). This equation plays an important role to illustrate the main physical features of boundary layer phenomena. This paper presents a new collocation method for solving the Falkner-Skan equation. The proposed approach is equipped by the orthogonal Chebyshev polynomials that have ...

A semi-spectral Chebyshev method for solving numerically singular integral equations is presented and applied in the quarkonium bound-state problem in momentum space. The integrals containing both, logarithmic and Cauchy singular kernels, can be evaluated without subtractions by dedicated automatic quadratures. By introducing a Chebyshev mesh and using the Nystrom algorithm the singular integra...

Abstract. It is proven that a class of Gegenbauer tau approximations to a 4th order differential eigenvalue problem of hydrodynamic type provide real, negative, and distinct eigenvalues as for the exact solutions. This class of Gegenbauer tau methods includes Chebyshev and Legendre Galerkin and ‘inviscid’ Galerkin but does not include Chebyshev and Legendre tau. Rigorous and numerical results s...

This paper presents a constructive method to (sub)optimal finite impulse response (FIR) approximation of a given infinite impulse response (IIR) model. The method minimizes the Hankel norm of approximation error by using the explicit solution of norm−preserve dilation problem. It has the advantage over the existing methods that it provides an explicitly constructive solution and allows the trad...

in this paper we introduce a type of fractional-order polynomials basedon the classical chebyshev polynomials of the second kind (fcss). also we construct the operationalmatrix of fractional derivative of order $ gamma $ in the caputo for fcss and show that this matrix with the tau method are utilized to reduce the solution of some fractional-order differential equations.