A new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The Bernstein–Petrov–Galerkinmethod (BPG) is applied to co...

The conventional representation of Zernike polynomials Rn (ρ) gives unacceptable numerical results for large values of the degree n. We present an algorithm for the computation of Zernike polynomials of arbitrary degree n. The algorithm has the form of a discrete Fourier (cosine) transform which comes with advantages over other methods in terms of computation time, accuracy and ease of implemen...

In this part of the work, a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable. This is accomplished by either multiplying enrichment functions with the original global partition polynomials, or increasing the order of global p...

We present three different approaches to solving PDEs based Taylor-models. In each approach we stress a high accuracy which can be achieved with a grid of relatively low-order Taylor polynomials. In particular, in solving Dirichlet boundary problem for the Laplace equation with the grid of step h of quadratic Taylor polynomials we get the discretization error of order h^10. Similar results are ...