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

TL;DR

This paper addresses the inverse problem of recovering initial data for a time-fractional fourth order parabolic equation, proposing regularization methods and analyzing their error rates under source conditions.

Contribution

It introduces regularized approximation techniques including quasi-boundary value and Fourier truncation methods, with error analysis and parameter strategies for this ill-posed problem.

Findings

FTM is free from saturation effect and order optimal for all source sets.

All three methods yield the same convergence rates under certain source conditions.

Error estimates depend on Sobolev smoothness assumptions.

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.