Abstract
In this paper, we present an ecient modication of the wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel. This -analytical method based on orthonormal wavelet basis, as a consequence three systems are obtained, a Toeplitz system and two systems with condition number close to 1. Since the preconditioned conjugate gradient normal equation residual (CGNR) and preconditioned conjugate gradient normal equation error (CGNE) methods are applicable, we can solve the systems in O(2n log(n)) operations, by using the fast wavelet transform and the fast Fourier transform.
Highlights
[21] and wavelets [10,12,13,16,17,19,20,26]
A considerable part of this proposal is based on a study by [Jin and Yuan, 1998], in which the authors focused on new class the first kind with symmetric kernel
In contrast to their work, we focused on the second kind with non symmetric kernel and as we know that the symmetric property is necessary condition to apply conjugate gradient method and in our case we don’t have this property so we dealt with the equivalent two systems that have the symmetric property
Summary
For the convenience of the reader, we recall here some basic concepts and well-known results concerning the multiresolution analysis (MRA for short). Vj = L2(R) and Vj = {0} ; j∈Z (iv) The sequence (2.1) forms a Riesz basis of V0. The function ψ is called the mother wavelet. A wavelet φ ∈ L2(R) is called orthonormal if the family of functions generated from φ by φj,k(s) = 2j/2φ(2js − k), j, k ∈ Z, is orthonormal, that is, φj,k, φm,n = δj,mδk,n. Let us introduce the following two wavelet sequences: and We recall that φj,k(s) = 2j/2φ(2js − k), j, k ∈ Z, ψj,k(s) = 2j/2ψ(2js − k), j, k ∈ Z. The wavelet sequence {ψj,k} forms a Riesz basis of Hs(R) for s ≥ 0. We note that B1 and B2 follow from the father wavelet φ and the mother wavelet ψ, respectively
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