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

Abstract

Over the last two decades, anomalous diusion processes in which the mean squares variance grows
\nslower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these
\nprocesses are adequately described by fractional dierential equations, which involves fractional derivatives in
\ntime or/and space. The fractional derivatives describe either history mechanism or long range interactions
\nof particle motions at a microscopic level. The new physics can change dramatically the behavior of the
\nforward problems. For example, the solution operator of the time fractional diusion diusion equation has
\nonly limited smoothing property, whereas the solution for the space fractional diusion equation may contain
\nweakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms
\nof uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical
\npoint of view, i.e., stably reconstructing the quantities of interest.
\nIn this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leer
\nfunction and singular value decomposition, to examine the degree of ill-posedness of several \\classical" inverse
\nproblems for fractional dierential equations involving a Djrbashian-Caputo fractional derivative in either time
\nor space, which represent the fractional analogues of that for classical integral order dierential equations. We
\ndiscuss four inverse problems, i.e., backward fractional diusion, sideways problem, inverse source problem and
\ninverse potential problem for time fractional diusion, and inverse Sturm-Liouville problem, Cauchy problem,
\nbackward fractional diusion and sideways problem for space fractional diusion. It is found that contrary
\nto the wide belief, the in
\nuence of anomalous diusion on the degree of ill-posedness is not denitive: it can
\neither signicantly improve or worsen the conditioning of related inverse problems, depending crucially on
\nthe specic type of given data and quantity of interest. Further, the study exhibits distinct new features of
\n\\fractional" inverse problems, and a partial list of surprising observations is given below.
\n(a) Classical backward diusion is exponentially ill-posed, whereas time fractional backward diusion is only
\nmildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply
\nthat the latter always allows a more eective reconstruction.
\n(b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but
\nnumerically can be nearly well-posed.
\n(c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general
\npotential, but in the fractional case, one single Dirichlet spectrum may suce.
\n(d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart,
\ndepending on the location of the lateral Cauchy data.
\nIn many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical
\nand numerical exploration, which requires new mathematical tools and ingenuities. Further, our ndings
\nindicate fractional diusion inverse problems also provide an excellent case study in the dierences between
\ntheoretical ill-conditioning involving domain and range norms and the numerical analysis of a nite-dimensional
\nreconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.