Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations

When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model, for example, the orders of the fractional derivative or the source term, are often unknown, which requires one to discuss inverse problems to identify these physical quantities from some additional information that can be observed or measured practically. This chapter investigates several kinds of inverse coefficient problems for the fractional diffusion equation

Well-posedness for the backward problems in time for general time-fractional diffusion equation

In this article, we consider a partial differential equation with Caputo time-derivative: $\partial_t^\alpha u + Au = F$ where $0< \alpha < 1$ and $u$ satisfies the zero Dirichlet boundary condition. For a non-symmetric elliptic operator $-A$ of the second order and given $F$, we prove the well-posedness for the backward problem in time and our result generalizes the existing results assuming that $A$ is symmetric. The key is the perturbation argument and the completeness of the generalized eigenfunctions of the elliptic operator $A$

An extension of the landweber regularization for a backward time fractional wave problem

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.Méthode des champs : algorithmes et simulations de phénomènes complexesInitiative d'excellence de l'Université de Bordeau

Data regularization for a backward time-fractional diffusion problem

AbstractWe investigate a backward problem for a time-fractional diffusion process in inhomogeneous media, which aims to determine the initial status of some physical field such as temperature for slow diffusion from its present measurement data. This problem is well-known to be ill-posed due to the rapid decay of the forward process. By using the eigenfunction expansion, we construct a new regularizing scheme with an explicit solution for the noisy input data with the number of truncation terms as a regularizing parameter. The convergence rate depending on the choice of strategy of the regularizing parameter is given based on the asymptotic behavior of the Mittag-Leffler function. Numerical implementations are presented to show the validity of the proposed scheme for several models