This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling Kawahara partial differential equation. The double basis of fifth-kind Chebyshev polynomials and their shifted ones are used as functions. Some theoretical results in deriving our proposed algorithm. nonlinear term equation is linearized using new product formula with first derivative polynomi...

Both the quality of the results of TEX s formula layout algorithm and the complexity of its description in the TEXbook are hard to beat The algorithm is verbally described as an imperative program with very complex control ow and complicated manipulations of the data structures representing formulae In a forthcoming textbook we describe TEX s formula layout algorithm as a functional program tra...

In this paper we consider the discrete Allen-Cahn equation posed on a two-dimensional rectangular lattice. We analyze large-time behaviour of solutions that start as bounded perturbations to well-known planar front solution travels in horizontal direction. In particular, construct an asymptotic phase function \begin{document}$ \gamm...

The generalized Bernstein basis in the space Πn of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced recently by G.M. Phillips, is given by the formula (see S. Lewanowicz & P. Woźny, BIT 44 (2004), 63–78) Bn i (x;ω| q) := 1 (ω; q)n [ n i ] q x (ωx−1; q)i (x; q)n−i (i = 0, 1, . . . , n). We give explicitly the dual basis functions Dn k (x; a, b, ω| q) for th...

BACKGROUND Previous studies reported an increase of upper body mass index (BMI) quantiles for formula fed infants compared to breastfed infants, while corresponding mean differences were low. The aim of this study was to assess the impact of known risk factors for childhood obesity on the BMI distribution. METHODS Data on 4,884 children were obtained at obligatory school entry health examinat...