The Nodal-type Newton-Cotes Rules For Hadamard Finite-part Integrals And Its Application

Hadamard finite-part integral equations have been widely used in physi-cal and engineering.Approximations for Hadamard finite-part integrals are often needed in order to construct the numerical algorithms for solving Hadamard finite-part integral equations.Because of the singularity of singular integral,the classical quadrature rules,such as the Newton-Cotes rules,Gauss rules,the trapezoidal rule and Simpson's rule can not be used directly.In recent 30 years,with the study for Hadamard finite-part integrals,and the corresponding numerical quadrature rules appear gradually,which mainly includes the composite trapezoidal rule,composite Newton-Cotes type method and Gaussian quadrature rules,the sig-moidal transformation method,etc.Among various kinds of numerical integration method,due to the ease of implementation,the lower requirements of regularity for density function and the flexibility of constructing a mesh,the(composite)Newton-Cotes rules are often used in practice.The main work of this paper can be divided into four parts.In the 2nd part,we summary the quadrature rules for Hadamard finite-part integrals,include the Gaussian quadrature rules,the sigmoidal transformation method and the Newton-Cotes type method.By comparing these methods,we analyzed their ad-vantages and disadvantages.The 3th part of the article,we propose the nodal-type Newton-Cotes rules for evaluating a general form of fractional Hadamard finite-part integrals based on the piecewise k-th order Newton interpolations,and a general error estimate is obtained on the quasi-uniform meshes.Furthermore,through some subtle analysis we find the rule with even k exhibits the superconvergence phenomenons,i.e.,its convergence order on the uniform mesh is one higher order than the general estimate when the singular point,is far away from the endpoints.We also give some reasons to explain why the rule with odd k does not have this phenomenon.Meanwhile,we obtain a simple error expansion of the trapezoidal rule(k = 1),from which we can modify it to a new one,whose accuracy is remarkably improved,compared to the original one.In the 4th part,as an application,we use the modified trapezoidal rule to construct a collocation scheme for solving the corresponding fractional Hadamard finite-part integrals equations.Optimal error estimate for the proposed scheme is also rigorously obtained.In the 5th part,we give some numerical experiments,the corresponding nu-merical results agree with the theoretical analysis.