基于超收敛点配点法求解圆周上超奇异积分方程

Collocation method based on the superconvergence point to solvehypersingular integral equations on a circle

超奇异积分方程的求解可用于解决科学工程中的许多问题,如无界区域、断裂、计算生物等。文章基于梯形公式近似计算圆周上三阶超奇异积分方程,在误差泛函特殊函数为0时具有超收敛现象(零点即为超收敛点)基础上,围绕超奇异积分方程的数值计算,选取超收敛点作为配点,研究了求解超奇异积分的配点法。针对奇异线性方程组的求解,引入正则化因子,将奇异线性方程组转化为正定线性方程组,并通过对系数矩阵性质的研究得到其逆矩阵元素的显式表达式,探索了逆矩阵的相关性质,结合超收敛性建立误差估计理论。结果表明:该配点法求解圆周上三阶超奇异积分方程的收敛阶为O(h^(2)|lnh|),数值算例验证了理论分析的正确性。

There are lots of problems in scientific engineering such as unbounded domain,fracture and computational biology problems,which can be deduced into hypersingular integral equation on interval or on a circle.By using the trapezoidal rule to approximate the third order hypersingular integral on a circle,the local coordinate point of the special function equals to zero(The zero point is the superconvergence point).In order to solve the hypersingular integral equations,the middle point of each subinterval is chosen as the collocation point to construct the collocation methods to solve the hypersingular integral equation.In order to find the solution of the singular system of linear equations,by introducing regularization factors,the linear system of singular equations into positive definite linear system of equations has been transformed.With the help of properties of coefficient matrix,the explicit expression of its inverse matrix has been obtained,and the correlation property of inverse matrix has been proved,the theory of error estimation by using the superconvergence has been established.The results show that the convergence order of this collocation method is O(h^(2)|lnh|).Theorem analysis is confirmed by numerical example.