Abstract—The use of wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs), especially for problems with local high gradient. In this work, the Galerkin Method has been adapted for the direct solution of differential equations in a meshless formulation using Daubechies wavelets and Deslauriers-Dubuc interpolating functions (Interpolets). This approach takes advantage of wavelet properties like compact support, orthogonality and exact polynomial representation, which allow the use of a multiresolution analysis. Several examples based on typical differential equations for beams and thin plates were studied successfully. Index Terms—Wavelets, interpolets, wavelet-galerkin method, beam on elastic foundation, thin plates. I. INTRODUCTION The use of wavelet-based numerical schemes has become popular in the last two decades. Wavelets have several properties that are especially useful for representing solutions of partial differential equations (PDEs), such as orthogonality, compact support and exact representation of polynomials of a certain degree. Their capability of representing data at different levels of resolution allows the efficient and stable calculation of functions with high gradients or singularities [1]. Compactly supported wavelets have a finite number of derivatives which can be highly oscillatory. This makes the numerical evaluation of integrals of their inner products difficult and unstable. Those integrals are called connection coefficients and they appear naturally when applying a numerical method for the solution of a PDE. Due to some properties of wavelet functions, these coefficients can be obtained by solving an eigenvalue problem. Working with dyadically refined grids, Deslauriers and Dubuc (1989) obtained a new family of wavelets with interpolating properties, later called Interpolets [2]. Unlike Daubechies’ wavelets [3], Interpolets are symmetric, which is especially interesting in numerical analysis. The use of wavelets as interpolating functions in numerical schemes holds some promise due to their multiresolution properties. The approximation of the solution can be Manuscript received May 14, 2014; revised July 20, 2014. R. B. Burgos is with the State University of Rio de Janeiro (UERJ), Department of Structures and Foundations, RJ, Brazil (e-mail: rburgos@ig.com.br). M. A. Cetale Santos is with Fluminense Federal University (UFF), ISIS Group, Department of Geology and Geophysics, RJ, Brazil (e-mail: marcocetale@id.uff.br). R. R. Silva is with Pontifical Catholic University of Rio de Janeiro, Department of Civil Engineering, RJ, Brazil (e-mail: raul@puc-rio.br). improved by increasing either the level resolution or the order of the wavelet used. Several examples were used for validating the proposed method. First, a beam with a concentrated load was used to test the method’s ability to capture discontinuities. As a second example, critical buckling loads for axially loaded beams on elastic foundation with several boundary conditions were obtained. The method was then applied for the static analysis of thin plates showing excellent performance in 2-D models. Results are presented and compared with analytical values, when available. II. WAVELET THEORY Multiresolution analysis using orthogonal, compactly supported wavelets has been successfully applied in numerical simulation. Wavelet basis are composed of two kinds of functions: scaling functions (ϕ) and wavelet functions (ψ). The two combined form a complete Hilbert space of square integrable functions. The spaces generated by scaling and wavelet functions are complementary and both are based on the same mother function [4]. A. Daubechies Wavelets In the following expressions, known as the two-scale relation, a k are the scaling function filter coefficients and N is the Daubechies wavelet order. 1 1 1 0 0 () (2 ), () ( 1) (2 ) N N k k N k k k x a x k x a x k ϕ ϕ ψ ϕ - - -- = = = - = - - ∑ ∑ (1) In general, there are no analytical expressions for wavelet functions, which can be obtained using iterative procedures like (1). In order to comply with the requirements of orthogonality and compact support, wavelets present, in general, an irregular fractal-like shape. Fig. 1 shows Daubechies scaling function of order N = 4. Fig. 1. Daubechies scaling function of order N = 4. Analysis of Beams and Thin Plates Using the Wavelet-Galerkin Method Rodrigo Bird Burgos, Marco Antonio Cetale Santos, and Raul Rosas e Silva 261 IACSIT International Journal of Engineering and Technology, Vol. 7, No. 4, August 2015 DOI: 10.7763/IJET.2015.V7.802