Cubic Spline Wavelets Satisfying Homogeneous Boundary Conditions for Fourth-Order Problems

摘要

The paper is concerned with a construction of cubic spline wavelet bases on the interval which are adapted to homogeneous Dirichlet boundary conditions for fourth-order problems. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. The construction of boundary scaling functions and wavelets is a delicate task, because they may significantly worsen conditions of resulting bases as well as condition numbers of corresponding stiffness matrices. We present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator.