偏微分方程边值反问题的数值方法研究
NUMERICAL METHODS FOR SOLVING INVERSE BOUNDARY VALUE PROBLEM OF PARTIAL DIFFERENTIAL EQUATION

本文研究了奇异积分方程在反边值问题中的应用问题.利用圆周上的自然积分方程及其反演公式，把Laplace方程的边值反问题转化为一对超奇异积分方程和弱奇异积分方程的组合，通过选取三角插值近似奇异积分的计算并构造相应的配置格式，并使用Tikhonov正则化方法求解所得到的线性方程组.数值实验表明了该方法的有效性.
In this paper, we study the application problem of singular integral equation in the inverse boundary value problem. Using the natural integral equation and its inversion formula on a circle, we transformed the Laplace equation inverse boundary value problem into a combination of a hypersingular integral equation and a weakly singular integral equation, then construct the corresponding collocation scheme based on the trigonometric interpolation, and use the Tikhonov regularization to solve the resulting linear equations. Numerical experiments show the effectiveness of the method