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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.

##### MSC:
 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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