The use of Chebyshev polynomials in solving finite horizon optimal control problems associated with general linear time-varying systems with constant delay is well known in the literature. The technique is modified in the present paper for the finite horizon control of dynamical systems with time periodic coefficients and constant delay. The governing differential equations of motion are conver...

Tthis paper, is concerned with obtaining numerical solutions for a class of convection-diffusion equations (CDEs) with variable coefficients. Our approaches are based on collocation methods. These approaches implementing all four kinds of shifted Chebyshev polynomials in combination with Sinc functions to introduce an approximate solution for CDEs . This approximate solution can be expressed as...

We study a new method in reducing the roundo error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Koslo and Tal-Ezer, and the proper choice of the parameter , the roundo error of the k-th derivative can be reduced from O(N2k) to O((N jln j)k), where is the machine precision and N is the number of collocation points. This drastic reduction of rou...

We study a new method in reducing the roundo error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Koslo and Tal-Ezer, and the proper choice of the parameter , the roundo error of the k-th derivative can be reduced from O(N) to O((N jln j)k), where is the machine precision and N is the number of collocation points. This drastic reduction of round...

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 ...

The IBM Power4 processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of non-stand...

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...

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.