In this paper, a new approach for solving the system of fractional integro-differential equation with weakly singular kernels is introduced. The method based on class symmetric orthogonal polynomials called shifted sixth-kind Chebyshev polynomials. First, operational matrices are constructed, and after that, described. This reduces equations (WSFIDEs) by collocation into algebraic equations. Th...

We provide a detailed description of numerical approach that makes use the shifted Chebyshev polynomials sixth kind to approximate solution some fractional order differential equations. Specifically, we choose Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) describe this method. write our in product form, which consists unknown coefficients and polynomials. To compute values coefficient...

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of p(A) over all monic polynomials p(z) of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well known propertie...

Odd degree Chebyshev polynomials over a ring of modulo 2 have two kinds of period. One is an “orbital period”. Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate affecting a factor of the ring, we can observe an orbit over the ring. The “ orbital period ” is a period of the orbit. The other is a “degree period”. It is observe...

We determine the infinite sequences (ak) of integers that can be generated by polynomials with integral coefficients, in the sense that for each finite initial segment of length n there is an integral polynomial fn(x) of degree < n such that ak = fn(k) for k = 0, 1, . . . , n − 1. Let P be the set of such sequences and Π the additive group of all infinite sequences of integers. Then P is a subg...

In this study, a spectral tau solution to the heat conduction equation is introduced. As basis functions, orthogonal polynomials, namely, shifted fifth-kind Chebyshev polynomials (5CPs), are used. The proposed method’s derivation based on solving integral that corresponds original problem. approach and some theoretical findings serve transform problem with its underlying conditions into suitabl...

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.

We give two recursive expressions for both MacWilliams and Chebyshev matrices. The expressions give rise to simple recursive algorithms for constructing the matrices. In order to derive the second recursion for the Chebyshev matrices we find out the Krawtchouk coefficients of the Discrete Chebyshev polynomials, a task interesting on its own.

We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all poly...