Error Analysis Of The Efficient Quadrature Formulas For Cauchy Principal Value Integrals

With the rapid development of society and technology,the relationship between computer and computational mathematics is becoming closer,and people are more and more dependent on computer.Scientific computing is a combination of computer and computational methods.The numerical calculation method is the basis of modern science and engineering calculation.With the development of its research,the research on numerical computation of(super)singular integrals in boundary element method(BEM)seems particularly important.The application of this method is mainly reflected in the fields of science and engineering.With the development of the effective quadrature formulas,the applicability of the Cauchy principal value integral in the boundary element method has attracted many researchers' attention.Therefore,it is of great theoretical value and practical significance to study the numerical calculation of Cauchy integral.In this paper,we mainly study the error estimation of the effective quadrature formula for the Cauchy integral.In this paper,we do some research on the Cauchy principal value integrals.Through the interval selection,the mesh partition,the appropriate interpolation function selection,the application of various related definitions and properties,containing linear transformation,Clausen function,Taylor series expansion method and so on,the superconvergence of the effect of quadrature formula for computing the Cauchy principal value integral is studied.The main contents of this paper are summarized as follows:Firstly,as the Newton-Cotes method is one of the most popular methods in many fields,and it is a typical method.Based on the generalization of the integral of application,we study the superconvergence of the composite Newton-Cotes rule for Cauchy principal value integral on circle and obtain the new criterion of the error expansion.Later,a numerical example is given to illustrate the significance and practicability of the proposed method by selecting appropriate parameter values.Secondly,the extended error expansion of classical mid-point rectangle rule for Cauchy principal value integrals on an interval is studied.Then,the computational formula of the composite mid-point rectangle and the corresponding superconvergence phenomenon are given.Numerical examples are given to verify the correctness of the theory at last.Finally,we study the superconvergence of the Hermite rule for the Cauchy principal value integral.By constructing proper interpolation function,a new theoretical criterion for the error estimation of the composite Hermite rule and the detailed theoretical analysis are given.