Spline Wavelets for Ordinary Differential Equations

摘要

Wavelets have proved to be very useful in signal processing and for the numerical solution of integral equations. They have also been used to discretize partial differential equations after transforming the latter into integral equations. The authors propose a direct approach to discretizing ordinary and partial differential equations using wavelet expansions. The main new idea lies in using functions which are wavelets with respect to the energy norm determined by the differential operator in question (in contradistinction to the standard L(2)-norm). For simple O.D.E.'s, they use spline wavelets. In particular, the compactly supported spline wavelets introduced by Chui and Wang are used. This leads to banded, diagonally-dominant matrices. In extreme cases, one obtains just the identity matrix. In all cases, they reduce the condition number of the stiffness matrix from O(N(Sup 2)) to a constant independent of N. (Copyright (c) GMD 1991.)