Numerical Quadrature Research Of Generalized Closed Form Singular Integrals

Some fundamental research on numerical computation for singular integral with Cauchy kernel is discussed in the present thesis.Several closed form quadrature formulae with multiple endpoints for singular integral are established,whose algebraic degree of precision is highest.The theoretical analyses coincide with the experimental data,which is verified by use of the numerical experiments with some functional examples.All the quadrature formulae for singular integral are original,which are the innovative points.And the present thesis consists of five chapters.In the first chapter,the research status are introduced,which is about the numerical computation for singular integral with Cauchy kernel and the generalized closed form quadrature formulae for normal integral.Then,we briefly discussed the organization of this thesis.In the second chapter,firstly,the properties of orthogonal polynomials are introduced,which provide necessary methods on deriving the generalized closed form quadrature formulae for normal integrals.Then,the quadrature nodes are consists of the zeros of orthogonal polynomials with respect to converted weight functions and the endpoints of the integrate interval that need to be multiple nodes,and the quadrature nodes which are belong to the interior of the interval are simple nodes.Next,we have established the generalized Gauss-Radau and generalized Gauss-Lobatto quadrature formulas with the general weight functions for normal integral,whose algebraic degree of precision are highest.Lastly,all the quadrature formulae are verified by the numerical experiments with the function examples.However,a lot of work belongs to Gautschi in this chapter.In the third chapter,we put forward to the concept of the associated function,by using the method of separating singularity,the singular integral with Cauchy kernel on the general weight is transformed into a normal generalized integral and a singular integral.The generalized closed form quadrature formulae for singular integral is shown,which takes advantage of the generalized closed form quadrature formula for normal integral.The quadrature formulae are constructive and the precision of quadrature formulae is thehighest.All numerical examples of these quadrature formulae prove that the theoretical analysis is consistent with the experimental results.In the fourth chapter,the quadrature formulae for singular integral with Cauchy kernel on Jacobi weight is discussed,in the generalized closed form Gauss quadrature formulae for singular integral on Jacobi weight,the coefficient expressions of quadrature formulae and quadrature nodes are computed more specifically by using the Gamma special function,it provides a tool for practical engineering applications.It's the same as above that the experiments are also carried out with numerical examples.