Inverse Problems of Determining Sources of the Fractional Partial Differential Equations

In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1. determination of time-dependent functions in interior source terms 2. determination of space-dependent functions in interior source terms 3. determination of time-dependent functions appearing in boundary condition

Identification of Chemotaxis Models with Volume Filling

Chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant. A crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant. In this paper, we investigate the identification of these non-linear parameter functions in a chemotaxis model with volume filling. We also discuss the numerical realization of Tikhonov regularization for the stable solution of the inverse problem. Our theoretical findings are supported by numerical tests.Comment: Added bibfile missing in v2, no changes on conten