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We investigate the internal observability of the wave equation with Dirichlet boundary conditions in tilings. The paper includes a general result relating internal observability problems in general domains to their tiles, and a discussion of the case in which the domain is the 30-60-90 triangle.

The goal of this study is to develop and apply an approximate method for calculating integrals that are part of models using Riemann-Liouville integrals, and to create a software product that allows such calculations for given functions. The main results of the study consist in the construction of a quadrature formula for an integral, and the cases where the density of the integral is a function from the spaces of continuous functions with generalized derivatives with weight and the Helder classes of functions with weight were considered. For the proposed quadrature formula we further investigated the error of its approximation in the spaces of continuous functions and quadratic-summing functions with weight. As a result of the study, effective error estimates of the approximating apparatus in the proposed classes of functions have been established. In addition, the approximated method has been implemented on the computer in the form of a program in the C language. The significance of the obtained results for the construction industry consists in the fact that when solving problems, including problems on finding the shapes of structures, taking into account the properties of materials, environmental changes, in the models of which the Riemann-Liouville integrals are used, it will be possible to apply an approximate approach, the quadrature formula proposed in the article.

In the present paper, two quite new reults on the degree of approximation of a function f belonging to the class Lip(α,r), 1≤ r

For the complete higher order differential equation u(n)(t)=∑k=0n-1Aku(k)(t)+f(t), 0 ≤ t ≤ T, on a Banach space E, we give necessary and sufficient conditions for the periodicity of mild solutions. The results, which are proved in a simple manner, generalize some well-known ones.

A good amount of work has been done on degree of approximation of functions belonging to Lipα, Lip(α,r), Lip(ξ(t),r) and W(Lr, ξ(t)) classes using Cesàro and (generalized) Nörlund single summability methods by a number of researchers like Alexits, Sahney, Goel, Qureshi, Neha, Chandra, Khan, Leindler and Rhoades. But till now no work seems to have been done so far in the direction of present work. Therefore, in present paper, two quite new results on degree of approximation of functions f∈ Lipα and f∈ W(Lr,ξ(t)) class by (E,1)(C,1) product summability means of Fourier series have been obtained.

Henry Problem (HP) still plays an important role in benchmarking numerical models of seawater intrusion (SWI) as well as being applied to practical and managerial purposes. The popularity of this problem is due to having a closed-form semi-analytical (SA) solution. The early SA solutions obtained for HP were limited to extensive assumptions that restrict its application in practical works. Several further studies expended the generality of the solution by assuming lower diffusion coefficients or including velocity-dependent dispersion in the results. However, all these studies are limited to homogeneous and isotropic domains. The present work made an attempt to improve the reality of the SA solution obtained for dispersive HP by considering anisotropic and stratified heterogeneous coastal aquifers. The solution is obtained by defining Fourier series for both stream function and salt concentration, applying a Galerkin treatment using the Fourier modes as trial functions and solving the flow and the salt transport equations simultaneously in the spectral space. In order to include stratified heterogeneity, a special depth-hydraulic conductivity model is applied that can be solved analytically without significant mathematical complexity. Several examples are proposed and studied. The results show excellent agreement between the SA and numerical solutions obtained with an in-house advanced finite element code.

In this note it is shown that functions of C^{(i)}(−π, π), i = 1, 2 as well as their relevant Fourier series are the natural basis to get generating functions of particular infinite-variable generalized Bessel functions which can be exploited in radiation emission problems.

We show embedding theorems of Sobolev type for Sobolev-Besicovitch spaces Wk,q ap of almost periodic functions with q ∈ [1, 2]. The fundamental tool for the proof of the main theorem is the Hausdorff-Young theorem for a.p. functions.

We give the extension of the classical H-Y theorem for periodic Lpfunctions to Besicovitch almost periodic functions, defined on IRs and with values in a vector-valued Hilbert Space IH. The generalization of the Riesz-Fischer theorem is also given.

A new class of infinite-variable Bessel functions is discussed and their basic properties are given. The relevant analogy with the ordinary case and the link with Fourier series of proper smooth functions are also presented and discussed.

We give the extension of the classical Hausdorff-Young theorem for periodic functions to the space Bq ap(IR,IH) of the Besicovitch almost periodic functions with values in a complex Hilbert space.

Assembled bladed disks may have many contact interfaces. Because of relative displacements at these interfaces, fretting-wear occurs, which affects negatively the lifetime of the structure. Methods exist to predict fretting-wear by quasi-static analysis. However they do not predict all the phenomena observed in blade attachments on real industrial plants. A numerical treatment of fretting-wear under vibratory loading is proposed. The method is based on the Dynamical Lagrangian Frequency Time method. It models unilateral contact through Coulomb's friction law. The basic idea is to separate time in two scales, slow scale for tribological phenomena and fast scale for dynamics. For a chosen number of periods of vibration, a steady state is assumed and the variables are decomposed in Fourier series. Numerical investigations on a bladed-disk with friction contact interfaces illustrate the performances of this method and show the coupling between dynamical and tribological phenomena. [ABSTRACT FROM AUTHOR]

The transient dynamics of evacuated and fluid-filled circular elastic shells, submerged in an infinite fluid medium and subjected to an external weak shock wave, is considered in this paper. This circular shell/acoustic medium interaction problem has already been tackled with simplified thin shell models, based on the Love-Kirchhoff hypotheses for the structural dynamics. In this case, the resulting radiated pressure field displays some discrepancies related to the A0/S0 waves when compared to the experimental data available in the literature for the evacuated case. These drawbacks are overcome here by the use of an isotropic elastic model for the structural dynamics and an inviscid acoustic flow for the fluid dynamics, in a two-dimensional framework. The approach is based on the methods of Laplace transform in time, Fourier series expansions and separation of variables in space. For the fluid-filled case, the transient thick shell-weak shock wave interaction problem is explored and the radiated acoustic field described. [ABSTRACT FROM AUTHOR]

Abstract: We present a parameterized family of finite difference schemes for the exact controllability of the 1-D wave equation. These schemes differ from the usual centered ones by additional terms of order to , where h denotes the discretization step in space. Using a discrete version of Ingham''s inequality for nonharmonic Fourier series, the spectral properties and dispersion diagrams of the schemes, we determine the parameters leading to a uniform controllability property with respect to h and an optimal stability CFL condition. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 339 (2004). [Copyright &y& Elsevier]

The coupling between yarns in a piece of fabric has been analysed at the mesoscopic scale, in terms of its impact on the macroscopic unidirectional behaviour. Starting from a discrete model of a woven structure associated to a variational formulation of the equilibrium of the structure, the coupling between both yarns is introduced, the potential energy of which is calculated. The initial shape of the yarn, represented by a planar undulated beam supposed to be periodic, is described by a Fourier series. The coefficients of the series are expressed vs. the contact force exerted at the top of the undulations, and vs. the mechanical properties of the solicited yarn. The contact force is then expressed vs. the mechanical properties of the transverse yarn and vs. the vertical displacement of the contact point. The potential energy of the coupling is then built, assuming the continuity of the displacement at the contact points. The equilibrium shape of the yarn submitted to unidirectional traction is obtained numerically as the minimum of the total potential energy. The simulated traction curve reproduces in a satisfactorily manner the observed behaviour. The respective contributions of the flexional and extensional effects of the yarn are analysed. The consideration of the coupling enhances the rigidity of the response of the yarn; one demonstrates the effect of the geometrical and mechanical parameters of the transverse yarn. To cite this article: B. Ben Boubaker et al., C. R. Mecanique 331 (2003). [Copyright &y& Elsevier]

A method for the solution of cyclic thermomechanical nonlinear problems is presented: it is a Direct Cyclic Method which is an alternative approach to classical methods. It consists in directly seeking the asymptotic response of an inelastic structure subjected to a periodic thermomechanical loading without following the loading path. It is based on the Large Time Increment Method, the periodicity of the limit state and the Fourier transform. To cite this article: M.H. Maitournam et al., C. R. Mecanique 330 (2002) 703–708. [Copyright &y& Elsevier]

In this study, a quadrupole model for solving the conductive transfer in a one-dimensional periodic contact is presented. The modeling is achieved through Fourier series development of periodically time-varying variables. This method could be used for predicting the thermal behavior of the contact under any periodic condition. In the case of intermittent contact, the solution of the model presents an harmonic oscillation and a Gibbs'' phenomenon at the discontinuities of the contact conductance due to the imperfection of Fourier series in describing discontinuous functions. On the other hand, this model presents a very fast convergence when the conductance varies continuously. Finally, an experiment of intermittent contact (contact–separation) is performed and the results are compared with the theoretical solutions. A good agreement is found when the dimensionless periods are greater than 1. For the periods less than 1, the experiment does not correspond perfectly to the assumptions of the model, and the corrections should be introduced. [Copyright &y& Elsevier]