Inverse problem of determining a time-dependent coefficient in the time-fractional subdiffusion equation

TL;DR

This work studies forward and inverse problems for a time-fractional subdiffusion equation with time-dependent diffusion and reaction coefficients, using a Gerasimov–Caputo derivative of order $\rho\in(0,1)$. A spectral (Fourier sine) decomposition and Mittag-Leffler kernel representations yield existence and uniqueness for the forward problem via Banach contraction, while the inverse problem uses boundary data $\psi(t)$ to recover $q(t)$ through a contraction mapping, under Assumption 2. The results provide a constructive framework and regularity guarantees for identifying time-dependent parameters in fractional diffusion models, filling a gap for problems with both $\sigma(t)$ and $q(t)$ varying in time. The analysis offers rigorous contraction estimates and explicit representations for $q(t)$ in terms of mode solutions, with potential implications for parameter identification in anomalous transport in heterogeneous media.

Abstract

This paper explores the forward and inverse problems for a fractional subdiffusion equation characterized by time-dependent diffusion and reaction coefficients. Initially, the forward problem is examined, and its unique solvability is established. Subsequently, the inverse problem of identifying an unknown time-dependent reaction coefficient is addressed, with rigorous proofs of the existence and uniqueness of its solution. Both problems' existence and uniqueness are demonstrated using Banach's contraction mapping theorem. Notably, this work is the first to investigate direct and inverse problems for such equations with time-dependent coefficients.

Theorems & Definitions (22)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 2.7
• Definition 3.1
• Theorem 3.2
• Theorem 3.3