On backward problem for a time-fractional fourth order parabolic equation

TL;DR

This work investigates the backward in time problem for a time-fractional ($0<\alpha<1$) fourth-order parabolic equation with Caputo derivative, aiming to recover the initial data from final-time observation and a source term. It develops and analyzes three regularization approaches—quasi-boundary value method (QBVM), modified QBVM (MQBVM), and Fourier truncation method (FTM)—providing both apriori and aposteriori parameter choice strategies under Sobolev source conditions, with provable convergence rates. The Fourier truncation method achieves rate $O(\delta^{p/(p+2)})$ for all $p>0$ and is free from saturation, while QBVM and MQBVM exhibit saturation phenomena, yielding slower or capped rates in some regimes. The results leverage spectral representations, Mittag-Leffler function estimates, and operator-theoretic formulations to establish order-optimal, discrepancy-principle-based rates, advancing stable reconstruction in fractional-derivative inverse problems for higher-order spatial operators.

Abstract

This paper is concerned with the inverse problem of retrieving the initial value of a time-fractional fourth order parabolic equation from source and final time observation. The considered problem is an {\it ill-posed problem.} We obtain regularized approximations for the sought initial value by employing the quasi-boundary value method, its modified version and by Fourier truncation method(FTM). We provide both the apriori and aposteriori parameter choice strategies and derive the error estimates for all these methods under some {\it source conditions} involving some Sobolev smoothness. As an important implication of the obtained rates, we observe that for both the apriori and aposteriori cases, the rates obtained by all these three methods are same for some source sets. Moreover, we observe that in both the apriori and aposteriori cases, the FTM is free from the so-called {\it saturation effect}, whereas both the quasi-boundary value method and its generalizations possesses the saturation effect for both the cases. Further, we observe that the rates obtained by the FTM is always order optimal for all the considered source sets.

Theorems & Definitions (50)

• Theorem 2.1
• Theorem 2.2
• LEMMA 2.3
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Remark 3.5