The dual techniques have been widely used in many engineering papers, to deal with singularity and ill-conditioning of the boundary element method (BEM). In this paper, we consider Laplace's equation with circular domains with one circular hole. The explicit algebraic equations of the rst and second kinds of the null eld method (NFM) are provided for applications. Traditionally, the rst and the second kinds of the NFM are used for the Dirichlet and the Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, however, the second and the rst kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual NFM (DNFM) in this paper. The excellent stability and the optimal convergence rates are explored in this paper. By using the simple Gaussian elimination or the iteration methods, numerical solutions can be easily obtained. Recently, the study on degenerate scales is active, many removal techniques are proposed, where the advanced solution methods may be needed, such as the truncated singular value decomposition (TSVD) and the overdetermined systems. In contrast, the solution methods of the DNFM in this paper are much simpler, with a little risk of the algorithm singularity from degenerate scales.

In this paper, we consider the reconstruction of heat field in one-dimensional quasiperiodic media with an unknown source from the interior measurement. The innovation of this paper is solving the inverse problem by means of two different homotopy iteration processes. The first kind of homotopy iteration process is not convergent. For the second kind of homotopy iteration process, a convergent result is proved. Based on the uniqueness of this inverse problem and convergence results of the second kind of homotopy iteration process with exact data, the results of two numerical examples show that the proposed method is efficient, and the error of the inversion solution r t is given.