Numerical Algorithms For Several Kinds Of Multi-dimensional Integral Equations/Singular Integrals

The mathematical models in many scientific and engineering problems are always described by multi-dimensional integrals or multi-dimensional integral equations such as the fine 3D graph drawing of the earth's interior in geology,elastic mechanical problem,electromagnetic scattering problem and so on.Calculating multi-dimensional singular integrals and solving multi-dimensional integral equations efficiently become research hotspot as the applications of these mathematical models increases.It becomes more difficult to study the numerical algorithms for multi-dimensional singular integrals and integral equations because the factors of dimension effect and singularity.This paper aims at improving the calculation accuracy and convergence speed,and studies the numerical algorithms for some multi-dimensional singular integrals and multi-dimensional integral equations with continuous kernel.The main contents are as follows:1.We explore the multi-parameter error asymptotic expansions of multi-dimensional weakly singular integrals.Firstly,in order to overcome the singularity of the mentioned integrals,Duffy transformation is utilized.Then,the quadrature rules of the mentioned in-tegrals are constructed by iteration technique,and the error asymptotic expansions are de-duced.Finally,based on the error asymptotic expansions,we design an accelerated con-vergence algorithm using the extrapolation and splitting extrapolation techniques,which removes the low order terms of the obtained error expansions and improves accuracy.The proposed algorithm has advantages of high parallelism,and avoids the dimensional effect.Numerical simulations demonstrate the effectiveness of the proposed algorithm.2.We study the multi-parameter error asymptotic expansions of multi-dimensional hyper-singular integrals.Considering that hyper-singular integrals do not exist,we prove the existence of hyper-singular integrals in the sense of Hadamard.Employing the itera-tion technique,the numerical quadrature rules of the proposed integrals are constructed and the error asymptotic expansions are obtained.Based on the error asymptotic expan-sions,we propose the splitting extrapolation algorithm,which is beneficial to parallel computing and improving accuracy.In addition,the quadrature rules do not involve any partial derivatives of integrand computing.Numerical experiments illustrate that the con-vergence speed can be improved effectively by the proposed algorithm.3.To solve multi-dimensional Fredholm integral equation,we introduce a novel Nystr(?)m method.Using the numerical quadrature rule obtained from the integral mean value theory,the present equation is reduced to a system of linear algebraic equations.The approximate solution of the proposed integral equation is derived by means of the Nystr(?)m method.The convergence of the proposed method is proved based on the com-pact operator theory framework.The new approach has the advantages of simple struc-ture and easy implementation.Specially,we explore the error asymptotic expansion of the proposed method with?_i_j=1/2.Moreover,the periodic transformation and splitting extrapolation techniques are applied to improve the convergence rate.The effectiveness of the proposed method is verified by several numerical experiments.4.This study mainly focuses on two Sinc Nystr(?)m methods to solve multi-dimensio-nal Urysohn integral equation.Firstly,two Sinc quadrature rules for multi-dimensional integrals are constructed by the single exponential transformation,double exponential transformation and Sinc approximate,then the error estimations are given.Incorporating two Sinc quadrature rules,the present equation is converted into a system of nonlinear algebraic equations.Combining Netwon iterative process with Nystr(?)m interpolation,the approximate solution of the present equation is derived.We prove the convergence of two proposed methods,it is shown that both of these two methods achieve the exponential convergence rates,which are superior to the existing methods.The effectiveness of two proposed methods are illustrated by several numerical experiments.5.Aiming at solution to two-dimensional fuzzy Hammerstein integral equation,we explore an iterative algorithm.Based upon the Gauss quadrature rule for multiple fuzzy integrals and collocation method,the present fuzzy integral equation is reduced to the nonlinear approximate equation that can be solved by a successive approximation iterative algorithm.Furthermore,the convergence of the proposed algorithm is analyzed.The proposed algorithm is simple in implementation,low in calculation cost and high in computation accuracy.Numerical experiments demonstrate that the proposed method is superior to the existing iterative methods.