Solution of plane Dirichlet problem using compactly supported 2D wavelet scaling functions

Abstract

The paper presents the solution of the homogeneous plane Dirichlet problem using the wavelet-Galerkin method with various 2D compactly supported wavelet scaling functions. An analysis of approximation accuracy was performed with respect to the orders of investigated wavelet scaling functions and the level of approximation. The most effective scaling functions for solving the Dirichlet problem were indicated and discussed. The wavelets theory is relatively new, but an effective mathematical tool for solving many theoretical and engineering problems. The wide range of its applica� tions covers developed numerical algorithms for approximation of ordinary differ� ential equations (ODE) and partial differential equations (PDE), image processing, signal analysis etc. The algorithms based on wavelet theory for solving ODE and PDE offer several advantages in comparison with the algorithms based on finite difference and finite element methods, e.g. low timeconsumption and good accu� racy (1). The two most popular waveletbased methods for solvi ng differential equations (DE) are (2): the collocation method and the waveletGalerkin method (WGM). According to some good properties, e.g. the possibility of multiresolution analysis and stability of decomposition and reconstruction in spaces of squareintegrable functions L 2 (R n ), solution stability, WGM gain great popularity. The crucial task in application of waveletbased num erical approximation algo� rithms is the choice of appropriate wavelet scaling functions. In the function ap� proximation problems, the wavelets with infinite support results in poor effects, therefore the orthogonal wavelets were primarily chosen for solving DE (3). One of the most popular wavelets is the Daubechies' ( db ) wavelet, which is often used for solving DE (4, 5). However, several authors used other wavelet scaling func� tions for solving DE, e.g. symlets ( sym ), coiflets ( coif ), Bspline wavelets ( bsp ), biorthogonal ( bior ) and reversed biorthogonal ( rbio ) wavelets (6�8). Due to the separability property of the abovementio ned compactly supported wavelets, they could be simply generalized to the higher dimensions by tensor