Quasi-optimality approach in the reconstruction of singularities in multi-term subdiffusion equations

In this paper, we propose a (very effective in practice) technique to reconstruct simultaneously the order $\nu$ of the leading derivative in the fractional operator and the order $\gamma$ of a singularity in the memory kernel arising in multi-term subdiffusion equations with memory terms. Particular cases of these equations model the transport of oxygen through capillaries to tissue. Analyzing the corresponding inverse problem with an additional smooth measurement for small times in fractional Hölder spaces, we derive explicit formulas for $\nu$ and $\gamma$. Moreover, the paper addresses the issues of uniqueness and stability estimates of the recognizing parameters via small-time observation. Utilizing the Tikhonov regularization scheme and the quasi-optimality approach, we describe the computational algorithm to build $\nu$ and $\gamma$ from discrete noisy measurements. We give several numerical examples demonstrating the effectiveness of the proposed technique in practice.