Completion of right-hand side in the frame of inverse Cauchy problem of elliptic type equation through homogenization meshless collocation method

Abstract

In this study, the well-known Cauchy problem of elliptic type equation possibly with variable coefficient is contemplated while a part of right-hand side source is unknown as well whereas overspecified boundary data is imposed on boundary. It is a supposition that the right-hand side source can be observed as a sum of two-parts which are independent of each other and at the same time, each of them being in terms of own one-variable. It is proved that such an inverse problem possesses unique solution. To approximate this unique solution, a kind of domain type meshless collocation method is proposed so that the boundary data are imposed directly. This is not troublesome because the original problem, by variable transforming through homogenization function, is converted to an inverse problem with homogeneous Cauchy boundary conditions. This surprisingly diminishes the ill-posedness of the right-hand side construction of Cauchy problem. As a result, it does not require any regularization algorithms and therefore reduces the computational time. The convergence and error analysis of the proposed approximation method is fully discussed. It is worth-mentioning that the considered domain is of arbitrary shape and discussed in the polar coordinate for simplicity and, it does not matter how scattered points are chosen, therefore the method is truly meshless one. The accuracy and robustness of this homogenization meshless collocation method (HMCM) is tested on several numerical examples.