Haar Series Research Articles

On the convergence of multiple Haar series

We prove that the rectangular and spherical partial sums of the multiple Fourier–Haar series of an individual summable function may behave differently at almost every point, although it is known that they behave in the same way from the point of view of almost-everywhere convergence in the scale of integral classes: is the best class in both cases. We also find optimal additional conditions under which the spherical convergence of a multiple Fourier–Haar series (general Haar series, lacunary series) follows from its convergence by rectangles, and prove that these conditions are indeed optimal.

Numerical solution of first order linear fuzzy differential equations using Leapfrog method

This paper presents numerical solutions of first order linear fuzzy differential equations using Leapfrog method. The obtained discrete solutions are compared with single-term Haar wavelet series (STHWS) method (1). Error graphs and error calculation tables are presented to highlight the efficiency of the Leapfrog method. This method can be implemented in the digital computers and take any time of solutions.   h t y h t y h t y / 2     

A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures

This paper focuses on the free vibration analysis of composite laminated conical, cylindrical shells and annular plates with various boundary conditions based on the first order shear deformation theory, using the Haar wavelet discretization method. The equations of motion are derived by applying the Hamilton’s principle. The displacement and rotation fields are expressed as products of Fourier series for the circumferential direction and Haar wavelet series and their integral along the meridional direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations. Then natural frequencies of the laminated shells are obtained by solving algebraic equations. Accuracy, stability and reliability of the current method are validated by comparing the present results with those in the literature and very good agreement is observed. Effects of some geometrical and material parameters on the natural frequencies of composite shells are discussed and some representative mode shapes are given for illustrative purposes. Some new results for laminated shells are presented, which may serve as benchmark solutions.

Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory Van der Pol system

In this paper, an efficient numerical method for the solution of nonlinear damped Van der Pol equation based on the Haar wavelets approach is proposed. The proposed scheme can be used for solving one of the widely studied and challenging equation in nonlinear dynamics like the damped Van der Pol equation. We have applied a numerical procedure involving haar series for the highest order of derivative. Moreover the numerical result shows that the proposed mechanism is quite reasonable when compare to exact solution.

Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics

This paper provides a technique to investigate the solutions of generalized nonlinear singular Lane–Emden equations of first and second kinds by using a Haar wavelet quasi-linearization approach. The Lane–Emden equation is widely studied and is treated as a challenging equation in the theory of stellar structure for the gravitational potential of a self gravitating, spherically symmetric polytropic fluid which models the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics. The proposed method is based on the quasi-linearization approximation and replacement of an unknown function through a truncated series of Haar wavelet series of the function. The method is shown to be very reliable and easy to capture the solutions of generalized nonlinear singular Lane–Emden equations. The applicability of the method is shown by numerical tests on various cases of the generalized Lane–Emden equation and solutions are also reported in the neighborhood of a singular point.

Spaces defined by the Paley function

The paper is concerned with Haar and Rademacher series in symmetric spaces, and also with the properties of spaces defined by the Paley function. In particular, the symmetric hull of the space of functions with uniformly bounded Paley function is found.Bibliography: 27 titles.

Feedback Control Method Using Haar Wavelet Operational Matrices for Solving Optimal Control Problems

Most of the direct methods solve optimal control problems with nonlinear programming solver. In this paper we propose a novel feedback control method for solving for solving affine control system, with quadratic cost functional, which makes use of only linear systems. This method is a numerical technique, which is based on the combination of Haar wavelet collocation method and successive Generalized Hamilton-Jacobi-Bellman equation. We formulate some new Haar wavelet operational matrices in order to manipulate Haar wavelet series. The proposed method has been applied to solve linear and nonlinear optimal control problems with infinite time horizon. The simulation results indicate that the accuracy of the control and cost can be improved by increasing the wavelet resolution.

Empirical recovery of input nonlinearity in distributed element models

Abstract Two algorithms recovering an input nonlinearity in a nonlinear distributed element modeled as a Hammerstein system are proposed. The first is based on the empirical distribution function while the other on the empirical Haar orthogonal series. Both algorithms self-adjust their accuracy to a local density of the input measurements.

A study on linear and nonlinear stiff problems using single-term Haar wavelet series technique

A study on linear and nonlinear stiff problems using single-term Haar wavelet series technique

Haar wavelet approximation for magnetohydrodynamic flow equations

This study proposes Haar wavelet (HW) approximation method for solving magnetohydrodynamic flow equations in a rectangular duct in presence of transverse external oblique magnetic field. The method is based on approximating the truncated double Haar wavelets series. Numerical solution of velocity and induced magnetic field is obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The calculations show that the accuracy of the Haar wavelet solutions is quite good even in the case of a small number of grid points. The HW approximation method may be used in a wide variety of high-order linear partial differential equations. Application of the HW approximation method showed that it is reliable, simple, fast, least computation at costs and flexible.

Haar series and spaces determined by the Paley function

Haar series and spaces determined by the Paley function

Multipliers of the Haar series

We calculate the norms of multipliers for the Haar system in some rearrangement invariant spaces for which the Haar system is not an absolute basis.

Coefficients of convergent multiple Haar series

It is well-known that if a multiple trigonometric series almost everywhere converges in the square or restricted rectangular sense to a finite function, then its coefficients grow slower than any exponential function. In this paper we prove the existence of a multiple Haar series that converges in the square or restricted rectangular sense to a finite function and contains a subsequence of coefficients that grows faster than any sequence defined in advance. Moreover, we show that for such series conditions of the Arutyunyan-Talalyan type can be violated at some points.

Traveltime seismic tomography with adaptive wavelet parameterization

An algorithm for solving the inverse kinematic problem of traveltime seismic tomography is developed and tested. The algorithm is intended for imaging the three-dimensional (3D) velocity model composed of a layer underlain by a half-space. This algorithm considers the bottom boundary of the layer as a first-order seismic velocity discontinuity with unknown position that has to be determined in the inversion together with the velocity variations inside the overlying layer and the sub-interface boundary velocities. The inversion can be applied to the travel times of refracted, head and reflected waves. The main idea behind the algorithm is the adaptive parameterization of the medium by the sparse Haar wavelet series expansion. In order to throw off the poorly resolved coefficients of expansion, we suggest using two empirical local resolution measures: the number of seismic rays crossing the support of the corresponding wavelet support area and their angular coverage, i.e., the spread in the azimuths of these rays. The adequacy of these measures is tested by their comparison with the estimation of the diagonal elements of the resolution matrix on the synthetic examples. This comparison proved that the proposed measures can be successfully applied for statistical estimation of the resolution and for constructing the adaptive parameterization. It was shown also that the best results are achieved while using the number of rays normalized to the size of the wavelet support together with their angular coverage. An automated procedure for throwing off poorly resolved unknowns is developed. The parameters of this procedure can be tuned to provide the desired level of detail of the model to be reconstructed. The synthetic checkerboard testing proved the efficiency of the algorithm. The proposed algorithm can be applied to solve different types of problems, including regional seismic studies, as well as exploration and engineering seismology. The use of this algorithm is especially convenient when the medium is essentially three-dimensional and when the conventional seismic methods implying regular network measurements directly above the studied structure (such as the common depth point method) are inapplicable, e.g., in the seismic studies of the foundations of buildings and in rugged terrains.

Computationally efficient delamination detection in composite beams using Haar wavelets

The paper presents an integrated vibration-based method for delaminations detection in homogeneous and composite beams. The method is based on Haar wavelets and artificial neural networks (ANNs). Firstly, scaled modal responses of the structure are expanded into Haar series by Chen–Hsiao method (CHM), and a delamination feature index is constructed. The database of 68 datasets built on Haar wavelet and frequency-based approaches was utilized by different ANNs to establish the mapping relationship between the delamination status and the delamination feature index or frequencies. The results are compared to each other. The simulations show the proposed complex method with delamination index detects the location of delaminations and identifies the delamination extent with high precision ( > 90 % ); the approach requires less computations and processing time than the frequency-based approach.

Quasi-measures, Hausdorff -measures and Walsh and Haar series

We study the classes of multiple Haar and Walsh series with at most polynomial growth of the rectangular partial sums. In terms of the Hausdorff -measure, we find a sufficient condition (a criterion for the multiple Haar series) for a given set to be a -set for series in the given class. We solve the recovery problem for the coefficients of the series in this class converging outside a uniqueness set. A Bari-type theorem is proved for the relative uniqueness sets for multiple Haar series. For one-dimensional Haar series, we get a criterion for a given set to be a -set under certain assumptions that generalize the Arutyunyan-Talalyan conditions. We study the problem of describing those Cantor-type sets that are relative uniqueness sets for Haar series.

Numerical evaluation of Hankel transforms using Haar wavelets

The aim of the article is to find an efficient algorithm for local numerical evaluation of F n (p) by expressing the corresponding finite Hankel transform [Fcirc] n (p) in Haar series and truncating it.

Haar wavelet Method for Solving Initial and Boundary Value Problems of Bratu-type

In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution. Keywords—Haar wavelet method; Bratu's problem; Boundary value problems; Initial value problems; Adomain decomposition method

Generalized Riemann-type integrals on the plane and an example of double Haar series

We prove that du Bois-Reymond-type theorems do not hold for the double Haar series, for the dyadic cubic Henstock integral, and for regular convergence over rectangles.

О рядах Хаара на компактной нуль-мерной группе

О рядах Хаара на компактной нуль-мерной группе