Study On Superconvergence Phenomenon Of Singular Integral Equations

The study of singular integral equations provides a theoretical foundation for many engineering problems such as unbounded region problems and fracture problems.For the approximate calculation of singular integrals,the quadrature formulas of Hermite interpolation and Cubic Spline interpolation are constructed to approximate singular integrals.The superconvergence phenomenon of approximate singular integrals is studied.As for solving the singular integral equations,the midpoint of each subinterval is selected as the configuration scheme based on the trapezoidal formula.The convergence order of the collocation method for solving the supersingular integral equations on circle is proved.Firstly,the Hermite integral rule is constructed to approximate singular integrals on interval.For the calculation of Cauchy principal value integrals,the order of convergence isO(h~4).For the calculation of the supersingular integral,the global convergence order isO(h~2).There exists the phenomenon of superconvergence when the special function is zero,the superconvergence order isO(h~3).Secondly,the Cubic Spline integral rule is studied to approximate singular integrals on interval.The Cubic Spline interpolation rule is smooth inherently,it does not need the derivative value of the function at the interpolation node.For the calculation of Cauchy principal value integrals,the global convergence order isO(h~4).For the calculation of the supersingular integral,the Cubic Spline quadrature rule has the superconvergence phenomenon,the global convergence order isO(h~2).The superconvergence point is the zero of the special function,the order of superconvergence isO(h~3).Finally,the supersingular integral equations on circle based on the trapezoidal formula is solved.The superconvergence point is taken as collocation point,the collocation method is studied to solve the supersingular integral equations.The expression of inverse matrix elements is obtained through the study of its coefficient matrix.The properties of inverse matrix are studied and the error estimation is gained by combining the superconvergence.The results of numerical examples illustrate the feasibility of Hermite integral formula,Cubic Spline integral formula,collocation method and the accuracy of theoretical analysis.Figure 2;Table 17;Reference 46...