Reciprocity based identification of unknown mixed point-distributed sources in coupled elliptic equations

Abstract The paper deals with a nonlinear inverse source problem in a system of two coupled elliptic 2D advection-dispersion-reaction partial differential equations. In such system, we address the identification of multiple unknown mixed point and distributed sources defining the right-hand side of its first equation using some local observations related to the state solution of its second coupled equation. We develop appropriate adjoint functions leading to establish reciprocity gaps fulfilled by the unknown elements defining the sought sources. These adjoint functions are defined by scalar potentials derived from fields collinear to the orthogonal directions pointed by the eigenvectors of the symmetric dispersion tensor. From some interior measuring interfaces suitably set up within the monitored domain, we establish an identifiability result and develop a detection-identification method. Some numerical experiments on the coupled surface water BOD-OD model are presented.