In recent times the wavelet methods have obtained a great popularity for solving differential and integral equations. From different wavelet families we consider here the Haar wavelets. Since the Haar wavelets are mathematically most simple to be compared with other wavelets, then interest to them is rapidly increasing and there is a great number of papers,where thesewavelets are used tor solving problems of calculus. An overview of such works can be found in the survey paper by Hariharan and Kannan [1] and also in the text-book by Lepik and Hein [2]. The aim of the present paper is more narrow: we want to popularize our method of solution, which is published in 19 papers and presented in the text-book [2]. This method is quite universal, since a large group of problems can be solved by a unit approach. The paper is organised as follows. In Section 1 fundamentals of the wavelet method are described. In Section 2 the Haar wavelet method and solution algorithms are presented. In Sections 3-9 different problems of calculus and structural mechanics are solved. In Section 10 the advantageous features of the Haar wavelet method are summed up.
@article{bwmeta1.element.doi-10_1515_wwfaa-2015-0001,
author = {\"U. Lepik and H. Hein},
title = {Application of the Haar wavelet method for solution the problems of mathematical calculus},
journal = {Waves, Wavelets and Fractals},
volume = {1},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_wwfaa-2015-0001}
}
Ü. Lepik; H. Hein. Application of the Haar wavelet method for solution the problems of mathematical calculus. Waves, Wavelets and Fractals, Tome 1 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_wwfaa-2015-0001/
[1] Hariharan G., Kannan K., An overview of Haar wavelet method for solving differential and integral equations. World Applied Sciences Journal, 25, (2013 ) 1 – 14.
[2] Lepik Ü, Hein H., HaarWaveletswith Applications, Springer, 207 pp., 2014.
[3] Daubechies I., Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41, (1988) 909 – 996. [Crossref] | Zbl 0644.42026
[4] Chen C., Hsiao C., Haar wavelet method for solving lumped and distributed-parameter systems, IEEE, Proc. Control Theory Appl. 144, (1997) 87-94. | Zbl 0880.93014
[5] Basdevant C., Devillle M., Haldenwang P., Lacroix J., Quazzani J., Peyret R., Orlandi P., Patera A., Spectral and finite difference solutions of the Burgers equation, Comput. Fluids, 14, (1986) 23-41. [Crossref] | Zbl 0612.76031
[6] Timoshenko S., Theory of Elastic Stability, McGraw-Hill Book Company, New-York, 1936.
[7] Orhan S., Anaysis of free and forced vibration of a cracked cantilever beam, NDT&E Int. 40, 443-450, 2007. [WoS]
[8] Filipich C.P., Cortinez M.B.R.H., Natural Frequencies of a Timoshenko Beam: Exact values by means of a generalized solution, Mecanica Computadonal, 14, (1994) 134-143.
[9] Shahba A., Attarnejad R., Marvi M.T., Hajilar S., Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites: Part B, 42, (2011) 801-808. [WoS][Crossref]
[10] Majak J., Shvartsman B., Karjust K., Mikola M.,Haavajõe A., Pohlak M, On the accuracy of the Haar wavelet discretization method, Composites: Part B, 80, (2015) 321-327. [WoS][Crossref]