In this thesis, we present three projects. First, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite-difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.
Also, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem.
Finally, we study a particle approximation for one-dimensional first-order Mean-Field-Games with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we are dealing with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of the semi-discrete variational problem. Next, we show that our discretization preserves some conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. All results for the discrete problem are illustrated with numerical examples.
Serikbolsyn Duisembay has been a MS/PhD student in the AMCS Program under the supervision of Prof. Diogo Gomes at KAUST since 2016. He received his MS degree in AMCS at KAUST and Bachelor of Science in Electrical and Electronic Engineering at Nazarbayev University, Kazakhstan, in 2016.