Discretization of some parabolic problems

Abstract

In this thesis we present two fractional parabolic problems. In the first one, we treat the partial diffusion equation with an unknown boundary condition and a non-local coefficient. Using the Rothe method combined with the finite element method and an additional integral measure, we reconstruct the missing Dirichlet state. The presence of the non-local component leads to a large consuming of time when solving the equation numerically using the Newton method because the obtained Jacobian matrix is complete. In order to resolve these problems, we develop a method inspired from Gudi’s idea. The numerical experiment demonstrate the efficiency of the proposed approach. Secondly, the well-posedness( existence, uniqueness and some stability results) of the problem concerning the reconstruction of the unknown time-dependent boundary function for ractional integro- ifferential equation, for this purpose, we use an additional integral measurement with Rothe time discretization. Finally, we give some numerical examples to illustrate the results.