Analysis of mixed and Cauchy problems associated to a partially-fractional Laplace equation

In this paper, we consider the analysis of both mixed and Cauchy problems associated to a partially-fractional Laplace equation, where the fractional derivative is given in the Caputo sense. This study is divided into three parts. In the first part, we consider the direct problem with mixed boundary conditions. Using the method of separation of variables, we prove a first well-posedness result. A second well-posedness result is provided using a new equivalent problem consisting of a fractional integro-differential equation with Dirichlet boundary conditions. In the second part, we establish a gradient scheme (GS) based on the approximation of the previously-stated equivalent form of the fractional integro-differential equation problem. An error estimate is derived for the approximate solution of the GS. This allows to derive an approximation for the solution of the original continuous problem. In the third part, we consider the same fractional-order equation, but we overspecify a part of the boundary of the solution domain by prescribing both Dirichlet and Neumann data, i.e. Cauchy data, on it. The stability of this inverse and ill-posed problem is restored using a boundary-value problem regularization method. An estimate, in the L²-norm, for the accuracy error between the exact solution and the solution of the boundary-value regularization problem is proved as well.