The compression property of wavelets in the analysis of an evolution problem (with unsmooth initial conditions) is investigated. The effectiveness of wavelets both in the reduction of complexity (number of coefficients) and in better approximation is shown. Haar wavelets, having the simplest interpretation of the wavelet coefficients, are used for defining the wavelet solution of an evolution (parabolic-hyperbolic) problem. The approximate solution, at a given fixed scale (resolution), results from the superimposition of (a small set of) fundamental wavelets, thus giving (also) a physical interpretation to wavelets. Since Haar wavelets are not smooth enough, a numerical derivative algorithm, which allows the scale approximation of partial differential evolution operators, is also defined. As application, the heat propagation (of an initial square wave) is explicitly given in terms of wavelets.

### Haar wavelets based technique in evolution problems

#### Abstract

The compression property of wavelets in the analysis of an evolution problem (with unsmooth initial conditions) is investigated. The effectiveness of wavelets both in the reduction of complexity (number of coefficients) and in better approximation is shown. Haar wavelets, having the simplest interpretation of the wavelet coefficients, are used for defining the wavelet solution of an evolution (parabolic-hyperbolic) problem. The approximate solution, at a given fixed scale (resolution), results from the superimposition of (a small set of) fundamental wavelets, thus giving (also) a physical interpretation to wavelets. Since Haar wavelets are not smooth enough, a numerical derivative algorithm, which allows the scale approximation of partial differential evolution operators, is also defined. As application, the heat propagation (of an initial square wave) is explicitly given in terms of wavelets.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11386/1067210`
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