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

Abstract

Abstract. Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest.In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several "classical" inverse problems for fractional differential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of "fractional" inverse problems, and a partial list of surprising observations is given below.(a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the preci...

Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

“…For diffusion equations corresponding to the case α = 1 with time independent source terms, several authors investigated the conditional stability (e.g. [5,37,38] [15,16,17,19,23,31] where some inverse coefficient problems and some related results have been considered.…”

“…Moreover, we can define an operator H ∈ B(L 2 ((0, T ) × ω)), such that f solves the equation 15) which is well-posed. Finally, for every (h, f ) ∈ L 2 ((0, T ) × ω) × L 2 ((0, T ) × ω) satisfying (1.15) the solution u of (1.2)-(1.4) satisfies (1.14).…”

Inverse problems of determining coefficients of the fractional partial differential equations

“…Theorem 4 (Jin and Rundell [19]). Let y(p i , f j )(x, t) be the solution of (9) with {f j } j∈N and…”

“…Theorem 9 (Kian, Soccorsi and Yamamoto [26]). Let t n , n ∈ N fulfill (19). We assume Case (I) or (II).…”

An iterative method for backward time‐fractional diffusion problem

“…However, for some practical problems, a part of boundary data, or initial data, or diffusion coefficients, or source term may not be given and we want to find them from additional measurement data which will give rise to some fractional diffusion inverse problems [12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”