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We devise a class of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin 1 which they applied to general Calderon-Zygmund type integral operators. We apply a modification of their idea to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions. wavelets; hyperbolic; parabolic; numerical methods.