A non-iterative method for identifying multiple unknown time-dependent sources compactly supported occurring in a 2D parabolic equation

We address the non-linear inverse source problem of identifying multiple unknown time-dependent sources spatially supported in some subdomains of a two-dimensional bounded domain subject to an evolution advection-dispersion-reaction equation. Provided to be available within the monitored domain interfaces for recording the state and its flux crossing each suspected zone where a source could occur, we establish an identifiability theorem that yields uniqueness of the unknown elements defining all occurring sources. We develop a non-iterative detection-identification method that goes throughout the monitored domain to detect in each suspected zone whether there exists or not an occurring source. Once a source is detected, the developed method determines lower and upper bounds of the total amount discharged by the occurring source and localizes the geometric centre of its unknown spatial support. Then, given two desired reference geometries for example, squares/discs centred at the already localized geometric centre, the method determines the biggest domain defined by a first reference geometry included in the sought source spatial support as well as the smallest domain defined by a second reference geometry containing this unknown support. In addition, the developed method gives an approximation of the surface area of this latest and identifies its time-dependent intensity function. Some numerical experiments on a variant of the surface water Biological Oxygen Demand pollution model are presented.