High Precision Quadrature Formulae Research For Singular Integrals

Some fundamental research on numerical computation for the singular integrals with Cauchy kernel is discussed in the present thesis,and two kinds of high precision quadrature formulae for the singular integrals are established.The gradual properties of the quadrature formulae are consistent with the experimental data,which are verified by the numerical experiments with some function examples.The present thesis consists of five chapters.In the first chapter,the current research situations of high precision numerical quadrature for the singular integrals with Cauchy kernel and high precision quadrature formulae for the proper integrals are introduced.Then,This thesis main innovative work are briefly summarized.In the second chapter,the problem of high precision numerical quadrature for the proper integrals is considered.Firstly,some basic concepts and properties of orthogonal and Stieltjes polynomials are introduced,that provide necessary fittings to derive the high precision quadrature formula for the proper integral;Next,we establish the Gauss-Kronrod quadrature formula for the proper integral,and the nodes of quadrature formula are composed of the zeros of Legendre polynomial and Stieltjes polynomial,we explain that the selection of the quadrature nodes ensures the present quadrature formula with the highest algebraic precision;Lastly,the remainder term expression of the quadrature formula for the proper integrals is given,which is one of the new results of this thesis.In the third chapter,the problem of high precision numerical computation for the singular integrals with Cauchy kernel is discussed.By use of the method of separating singularity,the quadrature problem for the singular integrals is transformed into the numerical computation of the improper integrals,then taking advantage of the Gauss-Kronrod quadrature formula for the proper integral,we derive Gauss-Kronrod quadrature formula for the singular integrals discussed above;Especially,the restriction of the density function being analytical is weakened,so the Rabinowitz’s work is generalized in this thesis,and the simpler method for computing the quadrature coefficients of the quadrature formula is derived,the remainder term expression of the quadrature formula for the singular integrals is given too;Then,the another kind of Gauss-Kronrod quadrature formula for the singular integrals is also established,the remainder term expression and the problem on algebraic precision of this quadrature formula are discussed,it is the original work;Besides,we present the remainder term expressions of the quadrature formulae for the singular integrals that are established by Rabinowitz.What have being discussed in this chaper are all the main innovations of the thesis.In the fourth chapter,one Gauss-Kronrod quadrature formula for the proper integral and two kinds of Gauss-Kronrod quadrature formulae for the singular integrals established in front are implemented by Matlab programs on the computer.According to the error results and numerical integral images of the examples in the experiments,we illustrate that the superiority of the quadrature formulae compared with some classical quadrature formulae and their feasibility.The experimental results are according with our theoretical analysis.