On the Adaptive Numerical Solution of Nonlinear Partial Differential Equations in Wavelet Bases

requires the development of new algorithms, which are introduced in this paper. This work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the form ut 5 Any wavelet-expansion approach to solving differential Lu 1 N f (u), where L and N are linear differential operators and equations is essentially a projection method. In a projection f (u) is a nonlinear function. These equations are adaptively solved method the goal is to use the fewest number of expansion by projecting the solution u and the operators L and N into a coefficients to represent the solution since this leads to wavelet basis. Vanishing moments of the basis functions permit a sparse representation of the solution and operators. Using these efficient numerical computations. The number of coeffisparse representations fast and adaptive algorithms that apply opercients required to represent a function expanded in a Fouators to functions and evaluate nonlinear functions, are developed rier series (or similar expansions based on the eigenfuncfor solving evolution equations. For a wavelet representation of the tions of a differential operator) depends on the most solution u that contains Ns significant coefficients, the algorithms singular behavior of the function. We are interested in update the solution using O(Ns) operations. The approach is applied to a number of examples and numerical results are given. Q 1997 solutions of partial differential equations that have regions