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Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs. (English) Zbl 0856.65115
A cubic spline wavelet-like decomposition for the Sobolev space \(H^2_0(I)\) where \(I\) is a bounded interval is designed. Based on a special point value vanishing property of the wavelet basis functions, a fast discrete wavelet transform is constructed. It maps discrete samples of a function to its wavelet expansion coefficients in at most \(7N\log N\) operations. Using this transform a collocation method for the initial-boundary value problem of nonlinear partial differential equations (PDEs) is proposed. Numerical examples testing the efficiency of the discrete wavelet transform and the collocation method are presented.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
42C15 General harmonic expansions, frames
65T50 Numerical methods for discrete and fast Fourier transforms
35K55 Nonlinear parabolic equations
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