Computational Methods For Determining A Moving Boundary From Cauchy Data In One Dimensional Heat Equation

In this paper, we propose a regularized integral equation method to deter-mine a moving boundary from Cauchy data in one-dimensional heat equation. In the first chapter, we introduce the basic concepts of the ill-posed problem, regularization methods and the identification of the moving boundary in heat equation. In the first section, we introduce the concept of ill-posed problem and its relation with inverse problem. In the second section, we talk about the reg-ularization’s concept and its main methods. In the third section, we describe the identification of the moving boundary from heat conduct equation and its background, and its main methods.During the second chapter, we present several main concepts and theorems for the next chapter’s discussion.In the third chapter, we study how to determine a moving boundary from Cauchy data in one-dimensional heat equation. The first section, we introduce the inverse problem of identification of the moving boundary and extending it to a regular region. In the second section, the problem can be transformed to a Volterra integral equation of first kind by the Fourier method. And then in the third and fourth section we use a singular perturbation regularization approach and the Tikhonov regularization method instead of solving the instable Volterra integral equation of the first kind. The stable boundary identification can be obtained through a half-interval method. The convergence analysis is provided in the fourth section. In fifth section, the numerical experiments are given to show the effects of the proposed method.The last chapter, we give the conclusion of this paper.