Iterative Solution Of Nonlinear Singular Integral Equation

In this article,we study the numerical solution ofnonlinear singular integral equationon the unit disc.We give out a complete method of solving the equation by using rational interpolation.We also give the error estimation and fitting analysis about the given method to make this theory complete and meaningful.Firstly,we give the research background and significance of the numerical solution of singular integral equation and describe the basic knowledge,including some definitions and theorems about the numerical computation and the basic boundary value problem on the unit disc.Secondly,we mainly discuss the rational interpolation method in solving nonlinear singular integral equation.We give out the derivation and analysis of the theory and the innovations of the rational interpolation method,compared with the previous methods.Then the rational interpolation method is applied and extended from lower dimension to higher dimension.We use the rational interpolation method turn solving singular integral equation on the unit disc into solving nonlinear interpolation equations.Finally,we solve the nonlinear interpolation equations which are transformed from nonlinear singular integral equation by using Newton iteration method.Allsolutions of the singular integral equations involved in the paper are given,and graphical representations of the interpolating functions in the complex two-dimensionalare presented.We also give out basic error analysis and fitting representation of this rational interpolation method.In addition,we present two-dimensional fitting representations of the interpolation function to complete the whole theory of this paper.