Reconstruction of space-dependence and nonlinearity of a reaction term in a subdiffusion equation

Abstract

In this paper we study the simultaneous reconstruction of two coefficients in a reaction-subdiffusion equation, namely a nonlinearity and a space dependent factor. The fact that these are coupled in a multiplicative matter makes the reconstruction particularly challenging. Several situations of overposed data are considered: boundary observations over a time interval, interior observations at final time, as well as a combination thereof. We devise fixed point schemes and also describe application of a frozen Newton method. In the final time data case we prove convergence of the fixed point scheme as well as uniqueness of both coefficients. Numerical experiments illustrate performance of the reconstruction methods, in particular dependence on the differentiation order in the subdiffusion equation.