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In this work the author gathers several methods and techniques to construct systematically Stieltjes classes for densities defined on R+.

The author uses complex integration to obtain integrable functions with vanishing moments sequence, and then the author considers some operators defined on the vanishing moments subspace.

The author gather several methods and techniques to construct systematically Stieltjes classes for densities defined on R+. The author constructs explicitly Stieltjes classes with center at well-known probability densities. The author gives a lot of examples, including old cases and new ones.

The author computes the Hilbert transform of powers of |ln⁡x| to construct Stieltjes classes by using a recent result connecting the Krein condition and the Hilbert transform.

The purpose of this study is an application of the multi-wavelets Galerkin method to delay differential equations with vanishing delay known as Pantograph equation.

The method consists of expanding the required approximate solution at the elements of the Alpert multi-wavelets. Using the operational matrices of integration and wavelet transform matrix, the authors reduce the problem to a set of algebraic equations.

Because of the large size of the system, thresholding is used to obtain a new sparse system, and then this new system is solved to reduce the computational effort and related computer run time. The authors demonstrate that the solutions may be efficiently represented in a multi-wavelets basis because of flexible vanishing moments property of this type of multi-wavelets.

The L₂ convergence of the scheme for the proposed equation has been investigated. A series of numerical tests is provided to demonstrate the validity and applicability of the technique.

In a previous paper [COMPEL, Vol. 2, No. 3, pp. 117–139], an analysis was presented of a discretization procedure for a class of elliptic problems, including the Scharfetter‐Gummel method for the continuity equations of stationary semiconductor device models. Here, the previous results are extended in two directions: firstly, to such problems in an arbitrary number of dimensions and, secondly, to include the computation of suitable moments, as opposed to point‐values, of the carrier density distributions.

An explanation is given for the apparent accuracy of the most commonly used method for discretization of the stationary continuity equations in semiconductor device models. The accuracy of this method does not depend on relatively small changes in the electrostatic potential or the quasi‐Fermi potentials between neighboring mesh points, or on the flow of current essentially along the mesh lines as has been previously suggested. It is obtained because implicit in this procedure is a consistent, and fairly accurate, discretization of the associated systems for the stream functions and recombination potential. Our analysis indicates suitable choices for various parameters appearing in the discrete system, and conditions on the construction and refinement of a mesh so as to obtain reasonable or optimal accuracy. In addition, it is determined that given the electrostatic potential distribution, the values of the device terminal currents (but not the point values of the carrier densities or the local current densities) can be computed with an accuracy independent of some of the bias voltages by this procedure.

The real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem.

To generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh.

Adaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS.

The proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes.

Although many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh.

Model order reduction (MOR) has been widely used in the electric networks but little has been done to reduce higher index differential algebraic equations (DAEs). The paper aims to discuss these issues.

Most methods first do an index reduction before reducing a higher DAE but this can lead to a loss of physical properties of the system.

The paper presents a MOR method for DAEs called the index-aware MOR (IMOR) which can reduce a DAE while preserving its physical properties such as the index. The feasibility of this method is tested on real-life electric networks.

MOR has been widely used to reduce large systems from electric networks but little has been done to reduce higher index DAEs. Most methods first do an index reduction before reducing a large system of DAEs but this can lead to a loss of physical properties of the system. The paper presents a MOR method for DAEs called the IMOR which can reduce a DAE while preserving its physical properties such as the index. The feasibility of this method is tested on real-life electric networks.

The purpose of this paper is to apply the wavelet thresholding technique in order to analyze economic socio-political situations in Tunisia using textual data sets. This technique is used to remove noise from contingency table. A comparative study is done on correspondence analysis and classification results (using k-means algorithm) before and after denoising.

Textual data set is collected from an electronic newspaper that offers actual economic news about Tunisia. Both the hard and the soft-thresholding techniques are applied based on various Daubechies wavelets with different vanishing moments.

The results obtained have proved the effectiveness of wavelet denoising method in textual data analysis. On one hand, this technique allowed reducing the loss of information generated by correspondence analysis, ensured a better quality of representation of the factorial plan, neglected the interest of lemmatization in textual analysis and improved the results of classification by k-means algorithm. On the other hand, the proximities provided by the factorial visualization validate the economic situation of Tunisia during the studied period showing mainly a stable situation before the revolution and a deteriorated one after the revolution.

The results are the first to analyze economic socio-political relations using textual data. The originality of this paper comes also from the joint use of correspondence analysis and wavelet thresholding in textual data analysis.

The purpose of this paper is to present a new hierarchical finite element formulation for approximation in time.

The present approach using wavelets as basis functions provides a global control over the solution error as the equation of motion is satisfied for the entire duration in the weighted integral sense. This approach reduces the semi‐discrete system of equations in time to be solved to a single algebraic problem, in contrast to step‐by‐step time integration methods, where a sequence of algebraic problems are to be solved to compute the solution.

The proposed formulation has been validated for both inertial and wave propagation types of problems. The stability and accuracy characteristics of the proposed formulation has been examined and is found to be energy conserving.

The paper presents a new hierarchical finite element formulation for the solution of structural dynamics problems. This formulation uses wavelets as the analyzing basis for the desired transient solution. It is found to be very well behaved in solution of wave‐propagation problems.

The purpose of this paper is to describe a variational multiscale finite element approximation for the incompressible Navier‐Stokes equations using the Boussinesq approximation to model thermal coupling.

The main feature of the formulation, in contrast to other stabilized methods, is that the subscales are considered as transient and orthogonal to the finite element space. These subscales are solution of a differential equation in time that needs to be integrated. Likewise, the effect of the subscales is kept, both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation.

This strategy allows the approaching of the problem of dealing with thermal turbulence from a strictly numerical point of view and discussion important issues, such as the relationship between the turbulent mechanical dissipation and the turbulent thermal dissipation.

The treatment of thermal turbulence from a strictly numerical point of view is the main originality of the work.

The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an adaptive generalized finite element method (GFEM). The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames.

The variational problem of free vibration is formulated and the main aspects of the adaptive GFEM are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform Euler-Bernoulli beams are checked. The application of this technique in a frame is also presented.

The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. The GFEM, which was conceived on the basis of the partition of unity method, allows the inclusion of enrichment functions that contain a priori knowledge about the fundamental solution of the governing differential equation. The proposed enrichment functions are dependent on the geometric and mechanical properties of the element. This approach converges very fast and is able to approximate the frequency related to any vibration mode.

The main contribution of the present study consisted in proposing an adaptive GFEM for vibration analysis of Euler-Bernoulli uniform and non-uniform beams and frames. The GFEM results were compared with those obtained by the h and p-versions of FEM and the c-version of the CEM. The adaptive GFEM has shown to be efficient in the vibration analysis of beams and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems.