| n this work, we study the use of wavelets for numerical solutions of differential equations. Wavelets constitute unconditional bases for a variety of function spaces, thus they can provide accurate approximations of functions in such spaces. Wavelets have certain orthogonality, and may have compact support in time or frequency domain. The corresponding scaling function generates a multiresolution analysis with associated fast wavelet transform. These properties give wavelets potential advantages over some traditional bases for numerical solution of differential equations.;In this thesis, we first use directly two classes of compactly supported wavelets, namely, Daubechies' orthonormal wavelets and Chui-Wang's semi-orthogonal B-spline wavelets, with Galerkin's method for numerical solution of differential equations. A variation relying on Laplace transform (LT) and its numerical inversion techniques is also examined to solve time-dependent linear partial differential equations. Use of LT eliminates time coordinate and possible stability problems associated with time-discretization are avoided. Numerical inversion of LT allows us to obtain an approximate solution at any desired time point in only one step. Wavelet-Galerkin and wavelet-collocation methods are used for the Laplace domain solution. Comparison between these two methods show that the Laplace transform wavelet-Galerkin method usually gives better results than Laplace transform wavelet-collocation method. Our simulation results also show that LT wavelet methods give more accurate approximation than the implicit finite difference method.;Based on the multiresolution analysis properties of wavelet bases and wavelet-Galerkin method, general two-scale methods are proposed and studied. Using these methods, approximate solution at higher resolution level can be obtained by lower level calculations. These two-scale methods can be used to solve general linear or nonlinear, ordinary or partial differential equations. The basic idea of the two-scale methods can be used recursively, and local grid adaptation can also be incorporated, resulting in a multi-level adaptive algorithm.;For certain linear differential equations, the basic idea of the general two-scale method, combined with wavelet Petrov-Galerkin method, results in a fast multi-level adaptive algorithm, when used with adapted wavelets which are biorthogonal with respect to the differential operator. Discretization using the adapted wavelet bases results in a diagonal stiffness matrix and the overall computational complexity is of order... |
