Treatment of Boundary Conditions in One-Dimensional Wavelet-Galerkin Method

Abstract

One of the main problems of the Wavelet-Galerkin Method is the treatment of boundary conditions. To deal with this difficulty, the boundaries of wavelet series expansion are assumed to be the analytic boundaries of the problem. The boundary condition equations are replaced by end equations in the Galerkin system. The manipulation discussed here enables us to use classical wavelets and to tackle the problem more simply. However, we find that the end equations are a necessary part of the Galerkin equation system within the boundaries. To maintain the integrity of the system, the boundaries of wavelet series expansion are shifted until the end equations do not depend on any expansion coefficients c_k of φ(2^jx-k)that affect the solution within the real boundaries. Therefore replacing the end equations gives a good result in comparison to the exact solution.