Direct Solution of Differential Equations Using the Wavelet-Galerkin Method

Abstract

The use of compactly supported wavelet functions has become increasingly popular in the development of numerical solutions for differential equations, especially for problems with local high gradient. Daubechies wavelets have been successfully used as base functions in several schemes like the Wavelet-Galerkin Method, due to their compact support, orthogonality, and multi-resolution properties. Another advantage of wavelet-based methods is the fact that the calculation of the inner products of wavelet basis functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of approximating the integral by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness, mass and geometry matrices. In this work, the Wavelet-Galerkin Method has been adapted for the direct solution of differential equations in a meshless formulation. This approach enables the use of a multiresolution analysis. Several examples based on differential equations for beams and plates were studied successfully.