Analysis and Numerical Solvability of Backward-Forward Conservation Laws

In this paper, we study the problem of initial data identification for weak-entropy solutions one-dimensional Burgers equation. This consists in identifying set evolving to a given target at final time. Due time-irreversibility equation, some functions are unattainable from making under consideration ill-posed. To get around issue, introduce nonsmooth optimization problem, which minimizing difference between predictions equation and observations system time norm. Here, characterize minimizers aforementioned problem. One is backward entropy solution, constructed using backward-forward method. Some simulations wave-front tracking algorithm.