Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation

TL;DR

This paper develops an analytical and numerical method to identify scalar parameters in a fractional subdiffusion equation with memory effects, using nonlocal observations and Tikhonov regularization for noisy data.

Contribution

It introduces explicit formulas for unknown parameters in a fractional subdiffusion model and proposes a regularization-based computational algorithm with demonstrated numerical effectiveness.

Findings

Explicit formulas for unknown parameters derived.

Regularization scheme effectively recovers parameters from noisy data.

Numerical tests confirm practical applicability.

Abstract

In the paper, we propose an analytical and numerical approach to identify scalar parameters (coefficients, orders of fractional derivatives) in the multi-term fractional differential operator in time, $\mathbf{D}_t$. To this end, we analyze inverse problems with an additional nonlocal observation related to a linear subdiffusion equation $\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u=g(x,t),$ where $\mathcal{L}_{i}$ are the second order elliptic operators with time-dependent coefficients, $\mathcal{K}$ is a summable memory kernel, and $g$ is an external force. Under certain assumptions on the given data in the model, we derive explicit formulas for unknown parameters. Moreover, we discuss the issues concerning to the uniqueness and the stability in these inverse problems. At last, by employing the Tikhonov regularization scheme with the quasi-optimality approach, we give a computational algorithm to recover the scalar parameters from a noisy discrete measurement and demonstrate the effectiveness (in practice) of the proposed technique via several numerical tests.