Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs

We have designed a cubic spline wavelet-like decomposition for the Sobolev space H-0(2)(I) where I is a bounded interval. Based on a special point value vanishing property of the wavelet basis functions, a fast discrete wavelet transform (DWT) is constructed. This DWT will map discrete samples of a function to its wavelet expansion coefficients in at most 7N log N operations. Using this transform, we propose a collocation method for the initial boundary value problem 0:2 nonlinear partial differential equations (PDEs). Then, we test the efficiency of the DWT and apply the collocation method to solve linear and nonlinear PDEs.