This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D timefractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem, and the Lipschitz continuity of the solution operator is established. A modified optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function is put forward for the discretization of the minimization problem. Numerical inversions are performed for the diffusion coefficient taking on different functional forms and the additional data having random noise. Several factors which have important influences on the realization of the algorithm are discussed, including the approximate space of the diffusion coefficient, the regularization parameter and the initial iteration. The inversion solutions are good approximations to the exact solutions with stability and adaptivity demonstrating that the optimal perturbation algorithm with the sigmoid-type regularization parameter is efficient for the simultaneous inversion.

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Abstract. This paper deals with numerical inversion for space-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation with additional boundary measurement. An implicit difference scheme for the forward problem is presented based on discretization of Caputo fractional derivative, and numerical stability and convergence of the linear system are proved with the help of matrix analysis. An optimal perturbation regularization algorithm is introduced to determine the space-dependent diffusion coefficient numerically in different approximate space, and numerical inversions are carried out by several numerical examples. Furthermore, impacts of the fractional order, approximate space, regularization parameter, numerical differential step, and initial iterations on the inversion algorithm are analyzed showing that the inversion is of numerical uniqueness, and the inversion algorithm is efficient at least to the inverse problem here. The order of fractional derivative not only represents a global property of the forward problem, but also indicates ill-posedness of the corresponding inverse problem.

Nanoscale materials (such as TiO2, hydroxyapatite nanoparticles) have gained much concern in the coating of implants for cell adhesion and growth to improve the osteoconductivity. However, due to attrition and corrosion, the wear particles would be generated from the joint in living organism, and influence the physiological function of synovial membranes in joint cavity. In this study, the potential cytotoxicity of anatase TiO2 nanoparticles (TiO2 NPs) on rat synovial cell line 364 (RSC-364) was investigated. After treatment with different concentrations of TiO2 NPs (0, 3, 30, 300 microg/ml), the viability of RSC-364 cells were decreased in a dose-dependent manner. TiO2 NPs exposure could disrupt the integrity of cell plasma membrane, leading to the increased leakage of lactate dehydrogenase (LDH) into the culture medium. TiO2 NPs were uptaken by RSC-364 cells. The ultrastructure of RSC-364 cells was changed such as nuclear shrinkage and mitochondrial swelling. The reactive oxygen species (ROS) was over-produced especially in the cells exposed to 30 and 300 microg/ml TiO2 NPs. The activities of endogeneous antioxidant enzymes, superoxide dismutase (SOD) and catalase (CAT), were significantly decreased. The increased lipid peroxidation product (malondialdehyde, MDA) suggests the oxidative damage in cells. The flow cytometry detected that the cell cycle was blocked in G0/G1 phase, inhibiting the cell proliferation. These preliminary results indicate the oxidative stress injury and cytotoxicity of anatase TiO2 NPs on rat synovial cells. The reasonable and safe application of nanomaterials in artificial implants needs further study.