An inverse problem for a one-dimensional time-fractional diffusion problem

Over the last two decades, anomalous diusion processes in which the mean squares variance grows or faster than that in a Gaussian process have found many applications. At a macroscopic level, these are adequately described by fractional dierential equations, which involves fractional derivatives in or/and space. The fractional derivatives describe either history mechanism or long range interactions particle motions at a microscopic level. The new physics can change dramatically the behavior of the problems. For example, the solution operator of the time fractional diusion diusion equation has limited smoothing property, whereas the solution for the space fractional diusion equation may contain singularity. Naturally one expects that the new physics will impact related inverse problems in terms uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical of view, i.e., stably reconstructing the quantities of interest. this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leer and singular value decomposition, to examine the degree of ill-posedness of several \" inverse for fractional dierential equations involving a Djrbashian-Caputo fractional derivative in either time space, which represent the fractional analogues of that for classical integral order dierential equations. We four inverse problems, i.e., backward fractional diusion, sideways problem, inverse source problem and potential problem for time fractional diusion, and inverse Sturm-Liouville problem, Cauchy problem, fractional diusion and sideways problem for space fractional diusion. It is found that contrary the wide belief, the in of anomalous diusion on the degree of ill-posedness is not denitive: it can signicantly improve or worsen the conditioning of related inverse problems, depending crucially on specic type of given data and quantity of interest. Further, the study exhibits distinct new features of \" inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diusion is exponentially ill-posed, whereas time fractional backward diusion is only ill-posed in the sense of norms on the domain and range spaces. However, this does not imply the latter always allows a more eective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but can be nearly well-posed. (c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general , but in the fractional case, one single Dirichlet spectrum may suce. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, on the location of the lateral Cauchy data. many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical numerical exploration, which requires new mathematical tools and ingenuities. Further, our ndings fractional diusion inverse problems also provide an excellent case study in the dierences between ill-conditioning involving domain and range norms and the numerical analysis of a nite-dimensional procedure. Throughout we will also describe known analytical and numerical results in the literature.