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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 u t ϭ Any wavelet-expansion approach to solving differential L u ϩ 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 of smooth, nonoscillatory behavior interrupted by a number of well-defined localized shocks or shock-like structures. Therefore, expansions of these solutions, based upon

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