In this thesis, we present three projects. First, we investigate the numerical approximation of HamiltonJacobi equations with the Caputo timefractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finitedifference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the HamiltonJacobi equation.
Also, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the timefractional FokkerPlanck equation with timedependent drift. The system is composed of a backward timefractional HamiltonJacobiBellman equation and a forward timefractional FokkerPlanck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the FokkerPlanck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the wellposedness of the scheme and the convergence to the solution of the continuous problem.
Finally, we study a particle approximation for onedimensional firstorder MeanFieldGames with local interactions with planning conditions. Our problem comprises a system of a HamiltonJacobi 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 semidiscrete variational problem. First, we address the existence and uniqueness of the semidiscrete 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.
Date of Award  Nov 2022 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Diogo Gomes (Supervisor) 

 meanfieldgames
 numerical approximations
 fractional derivatives