Several Solutions To The Cauchy Problem Of Elliptic Equation

The Cauchy problem of the elliptic equation is ill-posed in the sense that its solution does not depend continuously on the Cauchy data,i.e.a small perturbation of the Cauchy data may cause a huge deviation in the solution.Therefore,it is necessary to find some effective ways to solve such problem.In this paper,we will introduce two kinds of Cauchy problems of the elliptic equations: the Cauchy problem of the Laplace equation and the Cauchy problem of the Helmholtz equation.The structure of this paper is as follows:For solving the Cauchy problem of the Laplace equation,we provide two methods: a mollification method and an alternating iteration method.The error estimate between the exact and numerical solutions is given for each method.In addition,the numerical experiments are proposed to show the effectiveness of the mollification method.For Helmholtz equation,we consider to solve the Cauchy problem on the simply connected domain and the Cauchy problem on the multi-connected domain,respectively.To solve the Cauchy problem in a simply connected domain,we propose a method based on a combination of the boundary integral equation and Tikhonov regularization technique.Numerical examples are presented to show the validity of the method.To solve the Cauchy problem in a multiply connected domain,an alternating iterative method is proposed.We also prove the convergence of the method.The boundary integral equation method is employed to solve the two boundary value problems of partial differential equations in the alternating iterative process.The viability of the method is verified by the numerical results.