| Many physical problem,such as acoustics,electromagnetic scattering and fracture mechanics require an efficient discrete scheme for the Hadamard finite-part integral operator.Hadamard singular integral is divergence in common sense and principal value sense,which increase the difficulty of the research.A related topic is the study quadrature rule for hypersingular integral.Numerous work has been devoted to this area,such as the Gaussian method,the Newton-Cotes type method,the transformation and some other methods.When functions are smooth,Gaussian qudratures are the approach of choice.The Newton-Cotes rule is a commonly used one in many areas due to its ease of implementation and flexibility of mesh.It is need to explore different approximate calculation method in view of the practical problems.In this paper,the background of the Hadamard singular integral and some numerical computation methods are introduced.On this basis we focus on cubic spline rule of Hadamard finite-part integral.We prove both theoretically and numerically cubic spline rule reaching the superconvergence rate based on the literature.This article is divided into four chapters.In the first chapter,it introduces the research status,the research significance and the development trends of domestic and international of hyper-singular integral equation.In the second chapter,it presents the definition of Hadamard and method,and introduces the hyper-singular integral.And it introduces some calculation method of common.In the third chapter,we present cubic spline rule for the Hadamard finite-part integral operator.The superconvergence of cubic spline rule for Hadamard finite-part integral is presented,and we proved that both theoretical and numerical method could reach higher rate of convergence.The examples are presented to confirm our theoretical analysis,and we gave the analysis of data.The last chapter makes a summary and discussion of the full text study,and pointed out the direction for future work. |
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