In this paper, an Adomian decomposition method using Chebyshev orthogonal polynomials is proposed to solve a well-known class of weakly singular Volterra integral equations. Comparison with the collocation method using polynomial spline approximation with Legendre Radau points reveals that the Adomian decomposition method using Chebyshev orthogonal polynomials is of high accuracy and reduces th...

We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials.

We construct a simple closed-form representation of degree-ordered system of bivariate Chebyshev-I orthogonal polynomials Tn,r(u, v, w) on simplicial domains. We show that these polynomials Tn,r(u, v, w), r = 0, 1, . . . , n; n ≥ 0 form an orthogonal system with respect to the Chebyshev-I weight function.

The nonlinear integral equation P (x) = ∫ β α dy w(y)P (y)P (x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P (x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial sol...

This paper reviews the notion of interpolation of a smooth function by means of Chebyshev polynomials, and the well-known associated results of spectral accuracy when the function is analytic. The rate of decay of the error is proportional to ρ−N , where ρ is a bound on the elliptical radius of the ellipse in which the function has a holomorphic extension. An additional theorem is provided to c...

A Chebyshev polynomial of a square matrix A is a monic polynomial p of specified degree that minimizes ‖p(A)‖2. The study of such polynomials is motivated by the analysis of Krylov subspace iterations in numerical linear algebra. An algorithm is presented for computing these polynomials based on reduction to a semidefinite program which is then solved by a primaldual interior point method. Exam...

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 we show how polynomial mappings of degree K from a union of disjoint intervals onto [−1, 1] generate a countable number of special cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus g, from which the coefficients of xn can be found explicitly in terms of the branch points and the recurrence c...

This article considers the extension of V. A. Markov’s theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality...