Method for finding approximate solutions of elasticity equations using spline wavelets


DOI: 10.34759/trd-2021-121-24

Аuthors

Deniskina G. Y.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: dega17@yandex.ru

Abstract

In the technique being proposed the spline wavelets, built on the basis of an uneven subdivision scheme and a lifting scheme, are being employed for solving problems of the elasticity theory. The choice of such basic functions is justified by the fact that wavelets have a number of advantages compared to other basic functions. For example, the lifting scheme application allows composing wavelets with the given properties, such as smoothness, compactness of the carrier, symmetry, the required number of zero moments, vanishing at the boundary of the domain of functions corresponding to non-boundary mesh vertices. Lately, the wavelet analysis is attracting a lot of attention from the researchers, scientists and specialists in various disciplines. There are several reasons why wavelets are being successfully applied in signal processing, information compression, methods for finding approximate solutions of differential and integral equations [1, 2], computer geometry [3-5]. Such reasons include the following. Firstly, the high rate of wavelet coefficients decay. This allows obtaining rather accurate function approximations employing only a small number of summands in the expansion. Secondly, availability of fast cascade algorithms for finding coefficients of the function wavelet expansion. Thirdly, many commonly used wavelets (such as spline wavelets and Daubechies wavelets) have a compact carrier. In the proposed technique, a wavelet-system, consisted of smooth functions with the compact carrier, was built using the lifting scheme and a mask.

Such wavelet systems may be employed for finding approximate solutions of partial differential equations and, as a consequence, they may be applied to solving problems of the elasticity theory. This application presupposes the use of the least squares method for finding approximate solutions of boundary value problems of mathematical physics [7].

Keywords:

wavelet analysis, spline wavelets, elasticity, lifting scheme, filter, scaling function

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