In paper [4], transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is rem...

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelets approximations. The continues Legendre wavelets constructed on [0, 1] are utilized as a basis in collocation method. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In preparation for computational work with molecular orbitals written as linear combinaions of atomic orbitals, for first and second row atoms, we have developed a general procedure to parametrize linear combinations of Gaussian orbitals. We used a Gaussian transform, and a Gauss-Legendre quadrature, to express hydrogenic orbitals (with the correct long-range dependence for collisions) as contr...

An incompressible variational ideal ballooning mode equation is discretized with the COOL finite element discretization scheme using basis functions composed of variable order Legendre polynomials. This reduces the second order ordinary differential equation to a special block pentadiagonal matrix equation that is solved using an inverse vector iteration method. A benchmark test of BECOOL (Ball...

in this paper, an effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (rbfs). we present an algorithm based on interpolation by radial basis functions including multiquadratics (mqs), using legendre-gauss-lobatto nodes and weights. also a theorem is proved for convergence of the algorithm. some numerical examples are presented...

A novel higher order large-domain hierarchical finite-element technique using curl-conforming vector basis functions constructed from standard Legendre polynomials on generalized curvilinear hexahedral elements is proposed for electromagnetic modeling. The technique combines the inherent modeling flexibility of hierarchical elements with excellent orthogonality and conditioning properties of Le...

We consider the problem of recovering polynomials that are sparse with respect to the basis of Legendre polynomials from a small number of random samples. In particular, we show that a Legendre s-sparse polynomial of maximal degree N can be recovered fromm s log(N) random samples that are chosen independently according to the Chebyshev probability measure dν(x) = π−1(1 − x2)−1/2dx. As an effici...

In this paper, a new stochastic operational matrix for the Legendre wavelets is presented and a general procedure for forming this matrix is given. A computational method based on this stochastic operational matrix is proposed for solving stochastic Itô-Voltera integral equations. Convergence and error analysis of the Legendre wavelets basis are investigated. To reveal the accuracy and efficien...

We derive a new deterministic algorithm for the computation of a sparse Legendre expansion f of degree N with M N nonzero terms from only 2M function resp. derivative values f (1), j = 0, . . . , 2M − 1 of this expansion. For this purpose we apply a special annihilating filter method that allows us to separate the computation of the indices of the active Legendre basis polynomials and the evalu...

In this paper, continuous Legendre multi-wavelets are utilized as a basis in a practical direct method to approximate the solutions of the Fredholm integral equations system. To begin with we describe the characteristic of Legendre multi-wavelets and will go on to indicate that through this method a system of Fredholm integral equations can be reduced to an algebraic equation. Finally, numerica...