基于边界加扰动的Helmhotz方程柯西问题的正则化

BASED ON THE BOUNDARY MODIFICATION REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF HELMHOLTZ EQUATIONS

Helmhotz方程的柯西问题是一类典型的反问题而且是不适定的,也就是说其解不连续依赖于柯西数据,即小的扰动都会导致解的爆破。文章给出了边界加扰动的正则化方法,恢复了解对数据的连续依赖性,并给出了收敛性估计。最后用数值例子说明我们的方法是有效可行的。

The Cauchy problems for the Helrnholtz equations are considered. The problem is ill-posed in the sense that the solution （if exists） does not depend continuously on the given data. In order to obtain a stability approximation solution of the problem, it is necessary to employ some regularized techniques. Furthermore, we use the boundary modification regularized method to solve the Cauchy problems for Helmholtz equations and give the convergence estimates. Finally, the numerical examples show that the proposed numerical method works effectively.