A quasi-wavelet algorithm for second kind boundary integral equations
The Galerkin method for the solution of the integral equation \[ u(x)= \int^{2\pi}_0 u(y)\Biggl(a(x- y)\log\Biggl|2\sin{x-y\over 2}\Biggr|+ b(x, y)\Biggr) dy+ g(x),\;x\in [0,2\pi],\tag{1} \] where \(a(t)= a_0+ a_1(t)\sin^2{t\over 2}\), \(a_0\) is a constant, \(b(x, y)\) is a continuous periodic function of variables \(x\) and \(y\), \(a_1\) and \(g\) are continuous periodic functions, is investigated. As base functions for the Galerkin method the authors select periodic quasi-wavelet functions, some properties of which are studied in the paper. A detailed description of the algorithm and its modifications and simplifications are given. If \(a_1\), \(g\in W^r\), \(b\in W^{rr}\) the accuracy of the Galerkin method for equation (1) is equal to \(O(n^{-r})\) where \(n\) is the number of quasi-wavelet functions used in the construction of the method. The complexity of the offered method is equal to \(O(nlnn)\).
scientific article; zbMATH DE number 1421728