Study on Integral Equation Method for the Cauchy Problem of the Helmholtz Equation

本文研究了在环状区域上利用积分方程方法求解Helmholtz方程Cauchy问题。首先利用Green公式将方程解用积分方程表示，然后利用跳跃关系将区域内点趋于边界，得到两条边界上的积分方程组，采用单双层位势算子表示积分方程组，采用核裂解的方法离散积分方程组中的奇异积分，并使用Tikhonov正则化结合Morozov偏差原理求解病态方程。两种添加了噪声的数值算例验证了算法的有效性。In this paper, we study the problem of solving the Cauchy problem of Helmholtz’s equation on a toroidal region using the integral equation method. Firstly, the solution of the equation is expressed in terms of integral equations using Green’s formula, then the jump relation is used to converge the points in the region to the boundary to get the set of integral equations on the two boundaries, the single and double-layer potential operator is used to express the set of integral equations, the kernel cleavage method is used to discretize the singular integrals in the set of integral equations, and the pathological equation is solved by using the Tikhonov regularization combined with the Morozov’s deviation principle. Two numerical examples with added noise verify the effectiveness of the algorithm.