THE INVERSE PROBLEM OF KLEIN-GORDON EQUATION BOUNDARY VALUE PROBLEM AND ITS APPLICATION IN DATA ASSIMILATION

Since the solution of elliptic partial differential equations continuously depends on the boundary condition,the Euler equation derived from variational method cannot be solved without boundary condition. It is often difficult to provide the exact boundary condition in the practical use of variational method. However, in some application problems such as the remote sensing data assimilation, the values can be easily obtained in the inner region of the domain. In this paper, the boundary condition is tried to be retrieved by using part solutions in the inner area. Firstly, the variational problem of remote sensing data assimilation within a circular area is established. The Klein-Gordon elliptic equation is derived from the Euler method of variational problems with assumed boundary condition. Secondly, a computer-friendly Green function is constructed for the Dirichlet problem of two-dimensional Klein-Gordon equation, with the formal solution according to Green formula. Thirdly, boundary values are retrieved by solving the optimal problem which is constructed according to the best approximation between formal solutions and high-accuracy measurements in the interior of the domain. Finally, the assimilation problem is solved on substituting the retrieved boundary values into the Klein-Gordon equation. It is a type of inverse problem in mathematics. The advantage of this method lies in that it overcomes the inherent instability of the inverse problem of Fredholm integral equation and alleviates the error introduced by artificial boundary condition in data fusion using variational method in the past.