On an inverse potential problem for a fractional reaction-diffusion equation

This paper considers an inverse problem for a reaction diffusion equation from overposed final time data. Specifically, we assume that the reaction term f(u) is known but modified by a space-dependent coefficient q(x) to obtain q(x)f(u). Thus the strength of the reaction can vary with location. The inverse problem is to recover this coefficient. Our technique is to use iterative Newton-type methods although we also use and analyse higher order schemes of Halley type. We show that such schemes are well defined and prove convergence results. Our assumption about the diffusion process is also more general in that we will extend the traditional parabolic equation paradigm to include the subdiffusion case based on non-local fractional order operators in time. The final section of the paper shows numerical reconstructions based on the above methods and compares our methodology to previous work based on the linear model with f(u) = u as well as to the nonlinear case. We also show the interdependence between effective reconstruction of q and the coupling between the value of the final time of measurement and the subdiffusion parameter.