Detection and identification of multiple unknown time-dependent point sources occurring in 1D evolution transport equations

Abstract

This paper addresses the nonlinear inverse source problem of identifying multiple unknown time-dependent point sources occurring in a 1D evolution advection–diffusion–reaction equation. We prove that time records of the associated state taken only upstream and downstream all involved source positions do not uniquely identify the unknown elements defining the sought time-dependent point sources if their number is bigger or equal to 2. Then, based on some impulse response techniques we establish an identifiability theorem that under some reasonable assumptions on the time state recording positions yields uniqueness of those unknown elements. This theorem together with a data assimilation result led to develop a source detection-identification method that goes throughout the monitored domain to detect the presence of each unknown active source occurring between two state recording points. Once a source is detected, the established identification procedure localizes its position, determines the total amount loaded by this source and identifies the historic of its time-dependent source intensity function. Some numerical experiments on a variant of the surface water biological oxygen demand pollution model are presented.