On the Inverse Problem for the Burgers' Equation With Smooth Initial Data in a Bounded Domain

In this paper, we study the inverse problem of recovering smooth initial data for the one-dimensional Burgers' equation on a bounded spatial domain. Specifically, this problem focuses on recovering the initial condition from measurements at a part of the boundary over a given time interval. We therefore prove the uniqueness of the solution to the direct problem and establish, under certain assumptions, a stability estimate for the inverse problem based on a backward method in time and space, showing that the initial data can be reconstructed in this smooth case. To validate our theoretical results, we perform several numerical simulations and demonstrate the effectiveness of the proposed approaches.