%0 Journal Article
%T A Regularization Method for the Elliptic Equation with Inhomogeneous Source
%A Tuan H. Nguyen
%A Binh Thanh Tran
%J ISRN Mathematical Analysis
%D 2014
%R 10.1155/2014/525636
%X We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at and . The problem is ill-posed. The main aim of this paper is to introduce a regularization method and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively. 1. Introduction The Cauchy problem for the elliptic equation has been extensively investigated in many practical areas. For example, some problems relating to geophysics [1], plasma physics [2], and bioelectric field problems [3] are equivalent to solving the Cauchy problem for the elliptic equation. In this paper, we consider the following Cauchy problem for elliptic equation with nonhomogeneous source: where , are given. Problem (1)¨C(4) is well known to be ill-posed in the sense of Hadamard: a small perturbation in the data may cause dramatically large errors in the solution for . An explicit example to emphasize this fact is given in [4]. In the past, there were many studies on the homogeneous problem, that is in (1). Using the boundary element method, the homogeneous problems were considered in [5¨C7] and the references therein. Similarly, many authors have investigated the Cauchy problem for linear homogeneous elliptic equation, for example, the quasireversibility method [4, 8¨C10], fourth-order modified method [11, 12], Fourier truncation regularized method (or spectral regularized method) [13¨C15], the Backus-Gilbert algorithm [16] and so forth. Some other authors also considered the homogeneous problem such as Beskos [5], Eld¨¦n et al. [17, 18], Marin and Lesnic [19], Qin and Wei [20], Regi¨½ska and Tautenhahn [21], Tautenhahn [22]. Very recently, in 2009, H¨¤o and his group [23] applied the nonlocal-boundary value method to regularize the abstract homogeneous elliptic problem. This method is also given in [24] for solving an elliptic problem with homogeneous source in a cylindrical domain. A mollification regularization method for the Cauchy problem in a multidimensional case has been considered in the recent paper of Cheng and his group [25]. Although there are many papers on the homogeneous elliptic equation, the result on the inhomogeneous case is very scarce, while the inhomogeneous case is, of course, more general and nearer to practical application than the homogeneous one. Shortly, it allows the occurrence of some elliptic source which is inevitable in nature. The main aim of this paper is to
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2014/525636/