Inverse source problem in a one-dimensional evolution linear transport equation with spatially varying coefficients: application to surface water pollution

Abstract

This paper deals with the identification of a time-dependent point source occurring in the right-hand side of a one-dimensional evolution linear advection–dispersion–reaction equation. The originality of this study consists in considering the general case of transport equations with spatially varying dispersion, velocity and reaction coefficients which enables to extend the applicability of the obtained results to various areas of science and engineering. We derive a main condition on the involved spatially varying coefficients that yields identifiability of the sought source, provided its time-dependent intensity function vanishes before reaching the final monitoring time, from recording the generated state at two observation points framing the source region. Then, we establish an identification method that uses those records to determine the elements defining the sought source. Some numerical experiments on a variant of the surface water pollution model are presented.

Keywords:

• inverse source problem
• optimization
• partial differential equations
• advection–dispersion–reaction equations
• environmental monitoring: surface water pollution