The Study Of Numerical Algorithms For A Class Of Hypersingular Integral Equation

In recent years, numerical algorithm of Strongly singular integral equation has developed a new numerical method. Although there is few works about this method, it still shows unique advantages and strong vitality at the beginning. this paper is to study a class of strongly singular integral equation numerical method,purpose to solve the Singularity difficulties existed in the natural boundary element method, reducing the computation to improve accuracy. The basic idea is to use interpolation and numerical approximation, and then take advantage of our strongly singular given to the weakly singular integral transformation method, which can greatly reduce the computation.Firstly, we briefly review the boundary element method, links between Cauchy singular integral and hypersingular integral ,the history and development about hypersingular integrals, and gives research significance.Secondly, we give the necessary theoretical knowledge, the first integration by parts of the strongly singular, hyperingular and Cauchy singular integral relationship, Chebyshev orthogonal polynomials and Lagrange interpolation polynomials used in this article.Then, through numerical function approximation and interpolation function, discussed two kinds of numerical method for hypersingular integral equation: (1) Through Chebyshev polynomials approximation,we d numerical algorithm aboutαorder hypersingular integral, and then by the two given theorems, give the numerical solution of two order hypersingular integral equations. (2) by the transformation theorem about hypersingular integral to the Cauchy singular integral, we proposed any order hypersingular integral is transformed into solving Cauchy singular integral, by used Lagrange interpolation, given the numerical solution of second-order hypersingular integral equations. Then, we summary the arbitrary steps ofαorder hypersingular integral equations .Finally, through an example, fully shows that the mehtod effectively improves numerical method of hypersingular equations, indicating the feasibility and effectiveness.