Abstract
Shannon wavelets are used to define a method for the solution of integrodifferential equations. This method is based on (1) the Galerking method, (2) the Shannon wavelet representation, (3) the decorrelation of the generalized Shannon sampling theorem, and (4) the definition of connection coefficients. The Shannon sampling theorem is considered in a more general approach suitable for analysing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction of L2(ℝ) functions. Shannon wavelets are C∞‐functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series (connection coefficients).
Highlights
In recent years wavelets have been successfully applied to the wavelet representation of integro-differential operators, giving rise to the so-called wavelet solutions of PDE and integral equations
While wavelet solutions of PDEs can be find in a large specific literature, the wavelet representation of integro-differential operators cannot be considered completely achieved and only few papers discuss in depth this question with particular regards to methods for the integral equations
The Shannon wavelet solution of an integrodifferential equation with functions localized in space and slow decay in frequency will be computed by using the PetrovGalerkin method and the connection coefficients
Summary
In recent years wavelets have been successfully applied to the wavelet representation of integro-differential operators, giving rise to the so-called wavelet solutions of PDE and integral equations. The Shannon sampling theorem 27 , which plays a fundamental role in signal analysis and applications, will be generalized, so that under suitable hypotheses a few set of values samples and a preliminary chosen Shannon wavelet basis enable us to completely represent, by the wavelet coefficients, the continuous signal and its frequencies. The Shannon wavelet solution of an integrodifferential equation with functions localized in space and slow decay in frequency will be computed by using the PetrovGalerkin method and the connection coefficients.
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