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The present research considers and explains the application of the Haar wavelets as basis functions in solution of the propagation of perturbations in low-dimensional anisotropic media. The computations of the relaxation problem in the form of traveling waves have shown that the present approach possesses several advantages over regular methods.

Not to mention that the study of such non-Newtonian fluids as liquid crystals (LC) has lead to the commercial production of widely used electronic devices, the theory of LC still has a few problems, whose solution presents a difficulty. The problem of a director relaxation in twisted nematic cells (TNCs) in the form of nonlinear waves still remains unsolved. TNCs are integral parts of liquid crystal displays, which are used in laptops and personal computers. TNC is a drop of LC placed between two parallel surfaces, separated by a spacer, and arranged in such a way that the director in the upper surface

The use of TNC in production of LC monitors provided a strong encouraging stimulus for intensive research in this field. The present paper shows that angular moments acting on a unit element of nematic LC can generate traveling waves along the

The goal of the paper is to present the director relaxation problem using the Haar wavelet method [

Methods of wavelet analysis adapted for the solution of partial differential equations have been actively developing in the past 15 years. The advantages of the application of wavelets for the solution of differential problems over regular numerical methods consist of linearly increasing computational costs because of the sparsity of intermediate matrices. Moreover, the quality of solutions obtained by analytically defined basis functions always exceeds classical numerical methods.

The dynamic equation based on the balance of elastic, electric, and hydrodynamic torques [

In order to study the problem of propagation of the traveling wave from the top plate to the bottom plate, it is convenient to present (

The purpose of our approach consists of the study of the generation process of a traveling wave running between bounded plates by means of wavelet basis. In the coordinate system, connected with the motion of disturbance in the media, we can introduce

From all families of wavelets, we can distinguish a few easily defined functions, which represent a powerful tool in the solution of partial differential equations. Such wavelets are: Shannon wavelets [

The family of Haar wavelets for

The unknown function in (

It is obvious to present a question about our choice of basis functions. As we know, many other bases exist, which could be suitable for the solution of this problem. The answer to this question should be originated from the limited opportunities of disclosure of the angle between two directors located in parallel planes, and separated by distance

Since the Haar wavelets are defined for

The solution of (

Propagation of the traveling wave

The simulations have shown that this approach for the problem of relaxation of a director toward its equilibrium position allows its the computation of different dynamic modes in TNC. The relaxation of a director with respect to the laboratory coordinate system is shown in Figure

Propagation of the traveling wave with respect to the laboratory coordinate system

The criterion for relaxation of a director was chosen in the form:

The present paper describe the phenomenon of spatial relaxation in a TNC for the case of strong anchoring of LC molecules with the bounded plates. It was suggested to employ the Haar wavelet method for getting a function, which describes the evolution of the azimuthal angle. The advantage of such an approach over regular methods is that the unknown function is searched as a superposition of analytically defined functions, which constitute a complete basis in the

The work of A. A. Kudreyko is supported by the Istituto Nazionale di Alta Matematica Francesco Severi (Rome-IT) under scholarship U 2010/000139, 1 October 2010. A. A. Kudreyko is thankful to Professor Carlo Cattani (University of Salerno) for many valuable discussions.