The error function in the study of singularly perturbed convection-diffusion problems with discontinuous boundary data

We show the importance of the error function in the approximation of the solution of singularly perturbed convection-diffusion problems with discontinuous boundary conditions. It is observed that the error function (or a combination of them) provides an excellent approximation and reproduces accurately the effect of the discontinuities on the behaviour of the solution at the boundary and interior layers

First order approximation of an elliptic 3D singular perturbation problem

A three-dimensional elliptic singular perturbation problem with discontinuous boundary values is considered. The solution of the problem is written in terms of a double integral. A saddle point analysis is used to obtain a first approximation, which is expressed in terms of a function that can be viewed as a generalization of the complementary error functio

Multi-point Taylor approximations in one-dimensional linear boundary value problems

We consider second-order linear differential equations in a real interval $I$ with mixed Dirichlet and Neumann boundary data. We consider a representation of its solution by a multi-point Taylor expansion. The number and location of the base points of that expansion are conveniently chosen to guarantee that the expansion is uniformly convergent $\forall x\in I$. We propose several algorithms to approximate the multi-point Taylor polynomials of the solution based on the power series method for initial value problems

A three-point Taylor algorithm for three-point boundary value problems

We consider second-order linear differential equations $\varphi(x)y''+f(x)y'+g(x)y=h(x)$ in the interval $(-1,1)$ with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions given at three points of the interval: the two extreme points $x=\pm 1$ and an interior point $x=s\in(-1,1)$. We consider $\varphi(x)$, $f(x)$, $g(x)$ and $h(x)$ analytic in a Cassini disk with foci at $x=\pm 1$ and $x=s$ containing the interval $[-1,1]$. The three-point Taylor expansion of the solution $y(x)$ at the extreme points $\pm 1$ and at $x=s$ is used to give a criterion for the existence and uniqueness of the solution of the boundary value problem. This method is constructive and provides the three-point Taylor approximation of the solution when it exists. We give several examples to illustrate the application of this technique

Numerical calculation of Bessel, Hankel and Airy functions

The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument is to be evaluated. The coefficients in the well-known uniform asymptotic expansions have a complex mathematical structure which involves Airy functions. For Bessel and Hankel functions, we present an adapted algorithm which relies on a combination of three methods: (i) numerical evaluation of Debye polynomials, (ii) calculation of Airy functions with special emphasis on their Stokes lines, and (iii) resummation of the entire uniform asymptotic expansion of the Bessel and Hankel functions by nonlinear sequence transformations. In general, for an evaluation of a special function, we advocate the use of nonlinear sequence transformations in order to bridge the gap between the asymptotic expansion for large argument and the Taylor expansion for small argument ("principle of asymptotic overlap"). This general principle needs to be strongly adapted to the current case, taking into account the complex phase of the argument. Combining the indicated techniques, we observe that it possible to extend the range of applicability of existing algorithms. Numerical examples and reference values are given.Comment: 18 pages; 7 figures; RevTe

Reproducibility and validity of a diet quality index for children assessed using a FFQ

The diet quality index (DQI) for preschool children is a new index developed to reflect compliance with four main food-based dietary guidelines for preschool children in Flanders. The present study investigates: (1) the validity of this index by comparing DQI scores for preschool children with nutrient intakes, both of which were derived from 3d estimated diet records; (2) the reproducibility of the DQI for preschoolers based on a parentally reported forty-seven-item FFQ DQI, which was repeated after 5 weeks; (3) the relative validity of the FFQ DQI with 3d record DQI scores as reference. The study sample included 510 and 58 preschoolers (2-5-6.5 years) for validity and reproducibility analyses, respectively. Increasing 3d record DQI scores were associated with decreasing consumption of added sugars, and increasing intakes of fibre, water, Ca and many micronutrients. Mean FFQ DQI test-retest scores were not significantly different: 72 (so 11) v. 71 (Si) 10) (P-=0-218) out of a maximum of 100. Mean 3d record DQI score (66 (so 10)) was significantly lower than mean FFQ DQI (71 (so 10);

Епідигматичний модус англійських числівників

У статті розглядаються англійські числівники, денумеративи та їх епідигматичний модус. Семантична структура порівнюваних груп розглядаються на векторах їх походження та модифікації. Частиномовна представленість досліджується на просторах їх граматичного, семантичного і словотворчого буття. Визначаються перспективи подальшого дослідження. При цитуванні документа, використовуйте посилання http://essuir.sumdu.edu.ua/handle/123456789/30551В статье исследуются английские числительные, денумеративы, а также их эпидигматичекие аспекты. Семантическая структура сравниваемых единиц изучается на векторах семантики, грамматики и словотворчества. Намечаются перспективы дальнейшего исследования. При цитировании документа, используйте ссылку http://essuir.sumdu.edu.ua/handle/123456789/30551The article in question deals with the English numerals, denumerals and their epidigmatic tendencies. The semantic structures of both groups are being focused upon. The words under analysis are viewed in terms of their initial and secondary grounds. The item of part of speech charge is being considered in terms of grammatical, semantic and epidigmatic aspects. Thus, the modus of the paradigmatic groups have been considered wide perspectives for further investigation. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3055