A Fast Fourier-Collocation Methods For Elliptic Pseudodifferential Equations

We consider in this paper developing a fast Fourier-Collocation method for a class of periodic, elliptic pseudodifferential equations. Those periodic pseudodif-ferential operators have always the form A+B, where B is a smoothing opera-tor and the principal part A has a homogeneous symbol σ(x,l), that is, Au(x)= Σl∈Zσ(x,l)u(l)eilx This class of equations includes boundary integral equations of various types, such as Cauchy singular integral equations, hypersingular integral equations and elliptic integro-differential equations. In order to solve this kind of problems, we choose suitable trial function spaces and collocation functionals to construct fast trigonometric collocation scheme.In this paper, we focus on developing a fast Fourier-Collocation methods for periodic, elliptic pseudodifferential equations. First, we introduce the trigonometric collocation method for solving a class of singular integral equations or pseudodif-ferential equations by choosing suitable collocation functionals and trial function spaces. Secondly, we respectively propose the rule of matrix generation and trun-cation strategy for the coefficient matrixes by the property of operators A and B. In the end, we construct the framework of fast trigonometric collocation scheme and prove this algorithm is convergence.According to the results of pseudodifferential equations, we solve the first kind of weakly singular integral equations, the first kind of singular integral equations and the first kind of hypersingular integral equations by using the fast trigono-metric collocation method. Similarly, we first propose a truncation strategy which reduce the number of nonzero entries of the matrix from O(n2) to O(nlogn), but our trigonometric collocation method still preserves the convergence property of the trigonometric collocation method. We give the quadrature formula by the fast Fouri-er transform and illustrate the theoretical estimates by some numerical examples.This paper is organized as follows:In chapter 2, we introduce the fast trigono-metric collocation method for solving" a class of pseudodifferential equations. In chapter 3, we apply our fast theory to the first kind of weakly singular integral equations, the first kind of singular integral equations and the first kind of hyper-singular integral equations.