This is the first part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. The main result of this part is a constructibility result for the analytic Gröbner fan of a parametric ideal in the ring of analytic differential operators. In this part, the main tool is the notion of generic reduced standard basis.

In this paper, we improve the generalized Bernstein basis functions introduced by Han, et al. The new basis functions not only inherit the most properties of the classical Bernstein basis functions, but also reserve the shape parameters that are similar to the shape parameters of the generalized Bernstein basis functions. The degree elevation algorithm and the conversion formulae between the ne...

In this paper, the Ritz-Galerkin method in Bernstein polynomial basis is applied for solving the nonlinear problem of the magnetohydrodynamic (MHD) flow of third grade fluid between the two plates. The properties of the Bernstein  polynomials together with the Ritz-Galerkin method are used to reduce the solution of the MHD Couette flow of non-Newtonian fluid in a porous medium to the solution o...

A new application of Bernstein–Bezoutian matrices, a type of resultant matrices constructed when the polynomials are given in the Bernstein basis, is presented. In particular, the approach to curve implicitization through Sylvester and Bézout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational paramet...

In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subd...

The Bernstein operator Bn for a simplex in Rd is naturally defined via the Bernstein basis obtained from the barycentric coordinates given by its vertices. Here we consider a generalisation of this basis and the Bernstein operator, which is obtained from generalised barycentric coordinates that are given for more general configurations of points in Rd . We call the associated polynomials a Bern...

A Bernstein-Bézier basis is developed for H(div)-conforming finite elements that gives a clear separation between the curls of the Bernstein basis for the polynomial discretisation of the space H1, and the non-curls that characterize the specific H(div) finite element space (Raviart-Thomas in our case). The resulting basis has two distinct components reflecting this separation with the basis fu...

We compare approaches to location of the algebraic curve f(x, y) = 0 in a rectangular region of the plane, based on recursive use of conservative estimates of the range of the function over a rectangle. Previous work showed that performing interval arithmetic in the Bernstein basis is more accurate than using the power basis, and that affine arithmetic in the power basis is better than using in...

In order to investigate the fundamental properties of q-Bernstein basis functions, we give generating functions for these basis functions and their functional and differential equations. In [16], [15] and [17], we construct a novel collection of generating functions to derive many known and some new identities for the classical Bernstein basis functions. The main purpose of this paper is to con...

The mathematical modeling of the large deflections for the cantilever beams leads to a nonlinear differential equation with the mixed boundary conditions. Different numerical methods have been implemented by various authors for such problems. In this paper, two novel numerical techniques are investigated for the numerical simulation of the problem. The first is based on a spectral method utiliz...