A novel analysis of an alternating algorithm for solving the Cauchy problem for a fractional-order partial differential equation

We consider the solution of the Cauchy ill-posed problem associated to a fractional-order partial differential equation (PDE) using an alternating algorithm which preserves the governing PDE and changes the type of conditions on the over- and under-specified portions of the boundary from Neumann to Dirichlet and vice versa at each iteration. This encompasses two types of mixed problems for which we present and prove several new stability results. New weak formulations are also developed for these two types of mixed problems. From these weak formulations, existence, uniqueness, and stability results are derived and proved under some appropriate assumptions on the data. The convergence of solutions of the well-posed problems involved in the alternating algorithm toward the solution of the considered ill-posed problem is rigorously proved. Our analysis uses the method of separation of variables to express the solutions in eigenfunction expansions along some recent properties and asymptotic behavior results of Mittag-Leffler functions. The analysis in this paper contributes to the new field of inverse and ill-posed problems for fractional-order PDEs.