Abstarct: The Burgers equation is a non-linear partial differential equation of great importance marked by‎, ‎among others‎, ‎its applications to physics and engineering‎. ‎In this thesis‎, ‎we first solve the equation numerically using the Chebyshev Pseudo-Spectral (CPS) method‎. ‎Then‎, ‎we consider its optimal control problem and express the optimality conditions‎. ‎We apply the CPS method in order to obtain an approximate optimal solution‎. ‎At this point‎, ‎we also prove the convergence of the method‎. ‎Subsequently‎, ‎we approximate the solution to the fractional Burgers equation with the fractional CPS method‎. ‎We also show the uniqueness of this solution and the existence of the solution to the optimal control problem over the fractional Burgers equation‎.