On Recovering the Initial State of the Transport Equation

Abstract

Widely used and studied, the transport equation is a partial differential equation that describes how a mass is transported (or translated) through time and space. The goal of this thesis is to recover the initial state of the transport equation (at time t = 0) given the measurement of the solution at some end time T . To this end, we carry a thorough study on the direct problem, both theoretically and numerically, for the linear and non-linear transport equations. This helped us then develop robust numerical schemes to accurately approximate the exact solution of the direct transport problem. In turn these schemes are used to solve the inverse problem and recover the initial state through optimization algorithms provided by the MATLAB platform.