Determining time-dependent convection and density terms in convection-diffusion equation using partial data

In this article, we study an inverse boundary value problem for the time-dependent convection-diffusion equation. We use the nonlinear Carleman weight to recover the time-dependent convection term and time-dependent density coefficient uniquely. Nonlinear weight allows us to prove the uniqueness of the coefficients by making measurements on a possibly very small subset of the boundary. We proved that the convection term can be recovered up to the natural gauge, and the density coefficient can be recovered fully from the knowledge of the Dirichlet to Neumann map measured on a very small open subset of the boundary.