Reconstruction of space-dependence and nonlinearity of a reaction term in a subdiffusion equation
Barbara Kaltenbacher, William Rundell
TL;DR
The paper tackles the simultaneous identification of a space-dependent coefficient $q(x)$ and a nonlinear reaction term $f(u)$ in a fractional subdiffusion model $\partial_t^\alpha u - \Delta u + q(x) f(u) = r$. It develops forward-problem analysis establishing well-posedness and Lipschitz differentiability, and introduces iterative reconstruction schemes—fixed-point projections and a frozen Newton method—tailored to time-trace, final-time, and mixed data. In the final-time case with $\alpha=1$ it proves contractivity and uniqueness of the fixed-point scheme in 1D, with stability results for noisy data and demonstrated numerical performance showing fixed-point methods can outperform frozen Newton in speed and robustness. The work advances identifiability for coupled spatial and nonlinear components in subdiffusion equations, enabling practical reconstruction from diverse overposed data in applications such as biology and physics.
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.