High Accuracy Algorithms For Solving Singular Integrals And Singular Integral Equations

Singular integrals and singular integral equations have been greatly appeared in mathematical models of a great number of subjects and engineering,such as mathematical physics,fluid flow,fracture mechanics,electromagnetic mechanics,chemistry,biological engineering and petroleum engineering.As most of these mathematical models are derived from practical problems,to calculate the singular integrals and to solve the corresponding singular integral equations become the important content of the research on mathematical models,to achieve the purpose of estimating the actual problems.In this paper,we mainly introduce numerical methods for evaluating the singular integrals and solving singular integral equations.The main contents of the paper are arranged as follows:1.The contents of three aspects are introduced in briefly,which are the boundary element method and its advantages,the research background and significance of singular integrals and singular integral equations,as well as the integral equations whose integral operators contain Volterra type.2.Numerical methods for evaluating hyper-singular integrals are studied.In the case of singular points of integral operators are any interior point of integrand interval,asymptotic expansions?Euler-Maclaurin expansion?of the errors for hyper-singular integrals and quadrature formulas are derived.Extrapolation formulas of corresponding quadrature formulas are deduced,and error estimate formulas of the quadrature formulas are presented,on the basis of the characteristic of the asymptotic error expansions and the quadrature formulas.3.Numerical algorithms for calculating mixed singular integrals are studied.The mixed singular integrals are the kind of integrals that contains two kind of singularities which are hyper-singularity and logarithmic singularity.According to the analyticity of the Euler-Maclaurin expansions of the hyper-singular integrals about parameter,we achieve the Euler-Maclaurin expansions of the mixed singular integrals,and deduce the asymptotic error expansions of singular integrals with logarithmic singularity.4.Numerical methods for solving planar unsteady Stokes equations are presented.By using single potential theory and fundamental solutions of Stokes equations,boundary integral equations of the first kind are converted from planar unsteady Stokes equations.This kind of boundary integral equations is a kind of singular integral equations with logarithmic singularity.The numerical methods for solving this kind of integral equations are discussed in two different cases as follows.One case is that boundary of the boundary integral equation is a closed smooth curve ?.The mechanical quadrature method?MQM?for solving singular integral equation,asymptotic error expansion,and Richardson extrapolation formula of the MQM are derived.The error estimate formula of MQM is deduced from the asymptotic error expansion.The other case is that the boundary of the integral equation is a closed piecewise smooth curve ?=?_m=1^d?_m?namely,the area enclosed by the curve is polygonal domain?.The mechanical quadrature method,multivariate approximation error expansion,and the splitting extrapolation formula of singular integral equation are presented.At the same time,the posteriori error estimate of the mechanical quadrature method is received,based on the multivariate approximation error expansion.By using Anselone's collective compact and asymptotical compact theory,the convergence of the mechanical quadrature method is proved.5.Numerical methods are introduced to solve the nonlinear and weakly singular Volterra integral equations of the second kind.As the nonlinear singular Volterra integral equations satisfy the Lipschitz conditions,existence and uniqueness of the solutions of this kind of singular integral equations is porved,by using Gronwall inequality,Laplace transformation,and inverse Laplace transformation.Numerical method?modified trapezoidal quadrature method?and it's extrapolation method for solving this kind of integral equations are achieved.At the same time,formulas of error estimates of the numerical method are given.