Wavelet-based Methods for Numerical Solutions of Differential Equations

Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing and computational mathematics. This paper primarily intends to shed some light on the advantages of using wavelets in the context of numerical differential equations. We shall identify a few prominent problems in this field and recapitulate some important results along these directions. Wavelet-based methods for numerical differential equations offer the advantages of sparse matrices with uniformly bounded small condition numbers. We shall demonstrate wavelets' ability in solving some one-dimensional differential equations: the biharmonic equation and the Helmholtz equation with high wave numbers (of magnitude $O(10^4)$ or larger).