Convergence theorem for the Haar wavelet based discretization method J. Majak a,⇑ , B.S. Shvartsman b , M. Kirs a , M. Pohlak a , H. Herranen a a Dept. of Machinery, Tallinn University of Technology, 19086 Tallinn, Estonia b Estonian Entrepreneurship University of Applied Sciences, 11415 Tallinn, Estonia article info Article history: Available online 25 February 2015 Keywords: Haar wavelet method Accuracy issues Convergence theorem Numerical evaluation of the order of convergence Extrapolation abstract The accuracy issues of Haar wavelet method are studied. The order of convergence as well as error bound of the Haar wavelet method is derived for general nth order ODE. The Richardson extrapolation method is utilized for improving the accuracy of the solution. A number of model problems are examined. The numerically estimated order of convergence has been found in agreement with convergence theorem results in the case of all model problems considered. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Nowadays, the Haar wavelets are most widely used wavelets for solving differential and integro-differential equations, outperforming Legendre, Daubechie, etc. wavelets (Elsevier scientific publication statistics). Prevalent attention on Haar wavelet discretization methods (HWDM) can be explained by their simplicity. The Haar wavelets are generated from pairs of piecewise constant functions and can be simply integrated. Furthermore, the Haar functions are orthogonal and form a good transform basis. Obviously, the Haar functions are not differentiable due to dis- continuities in breaking points. As pointed out in [1] there are two main possibilities to overcome latter shortcomings. First, the quad- ratic waves can be regularized (‘‘smoofed’’) with interpolating splines, etc. [2,3]. Secondly, an approach proposed by Chen and Hsiao in [4,5], according to which the highest order derivative included in the differential equation is expanded into the series of Haar functions, can be applied. Latter approach is applied suc- cessfully for solving differential and integro-differential equations in most research papers covering HWDM [1,4–28]. Following the pioneering works Chen and Hsiao in [4,5] Lepik developed the HWDM for solving wide class of differential, fractional differential and integro-differential equations covering problems from elasto- statics, mathematical physics, nonlinear oscillations, evolution equations [1,6–10]. The results are summarized in monograph [11]. It is pointed out by Lepik in [1,11] that the HWDM is conve- nient for solving boundary value problems, since the boundary conditions can be satisfied automatically (simple analytical approach). Composite structures are examined by use of wavelets first in [12,2]. In [12] the free vibration analysis of the multilayer compos- ite plate is performed by adapting HWDM. The static analysis of sandwich plates using a layerwise theory and Daubechies wavelets is presented in [2]. The delamination of the composite beam is studied in [13]. During last year Xiang et al. adapted HWDM for free vibration analysis of functionally graded composite structures [14–18]. In [14–18] a general approach for handling boundary conditions has been proposed. In all above listed studies the Haar wavelet direct method is applied. The weak form based HWDM has been developed in [19], where the complexity analysis of the HWDM has been performed. Recent studies in area of wavelet based discretization methods cover solving fractional partial differential equations by use of Haar, Legendre and Chebyshev wavelets [20–25]. In [26–28] the Haar wavelets are utilized for solving nuclear reactor dynamics equations. The neutron point kinetics equation with sinusoidal and pulse reactivity is studied in [26]. In [27,28] are solved neutron particle transport equations. In [29–31] the HWDM is employed with success for solving nonlinear integral and integro-differential equations. Most of papers overviewed above found that the implementation of the HWDM is simple. Also, the HWDM is characterized most commonly with terms ,,simple’’, ‘‘easy’’ and effective‘‘ (see [1,14– 18,25–28] and others). The review paper [32] concludes that the HWDM is efficient and powerful in solving wide class of linear and nonlinear reaction–diffusion equations. However, no convergence rate proof found in literature for this method. It is shown in several papers [33–35] that in the case of http://dx.doi.org/10.1016/j.compstruct.2015.02.050 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: juri.majak@ttu.ee (J. Majak). Composite Structures 126 (2015) 227–232 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct