In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix 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.

We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the pap...

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebys...

It is an important fact that general families of Chebyshev and L-splines can be locally represented, i.e. there exists a basis of B-splines which spans the entire space. We develop a special technique to calculate with 4 order Chebyshev splines of minimum deficiency on nonuniform meshes, which leads to a numerically stable algorithm, at least in case one special Hermite interpolant can be const...

Since properly normalized Chebyshev polynomials of the first kind T„(z) satisfy (?„, ?„) = [ f,(z)7ÜÖ |1 z2\~TM\dz\ = Smn for ellipses ep with foci at ± 1, any function analytic in ep has an expansion,/(z) = J3 anfn{z) with a„ = (/, Tn). Applying the integration error operator E yields E(J) = 2~Z a„E(Tn)Applying the Cauchy-Schwarz inequality to E(J) leads to the inequality |£CDIsá Z W¿2\E(Tn)\2...