Ten lectures on wavelets
Connection coefficients on an interval and wavelet solutions of Burgers equation
Journal of Computational and Applied Mathematics
Wavelet-Galerkin method for solving parabolic equations in finite domains
Finite Elements in Analysis and Design
An adaptive multiresolution scheme with local time stepping for evolutionary PDEs
Journal of Computational Physics
Harmonic wavelets towards the solution of nonlinear PDE
Computers & Mathematics with Applications
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This article is dedicated to harmonic wavelet Galerkin methods for the solution of partial differential equations. Several variants of the method are proposed and analyzed, using the Burgers equation as a test model. The computational complexity can be reduced when the localization properties of the wavelets and restricted interactions between different scales are exploited. The resulting variants of the method have computational complexities ranging from O(N^3) to O(N) (N being the space dimension) per time step. A pseudo-spectral wavelet scheme is also described and compared to the methods based on connection coefficients. The harmonic wavelet Galerkin scheme is applied to a nonlinear model for the propagation of precipitation fronts, with the front locations being exposed in the sizes of the localized wavelet coefficients.