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<p>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 <i>ill-posed problem,</i> that is, a small perturbation in the data may lead to a large deviation in the sought solution. Thus, we obtain regularized approximations for the sought initial value by employing the quasi-boundary value method, its modified version and by the Fourier truncation method (FTM). We provide both the a priori and a posteriori parameter choice strategies and derive the error estimates for all these methods under some <i>source conditions</i> involving some Sobolev smoothness. As an important implication of the obtained rates, we observe that for both the a priori and a posteriori cases, the rate obtained by all these three methods are the same and also order optimal for some source sets. However, for higher smoothness the rates obtained by the FTM are better than the other two. Further, we observe that the rates obtained by the FTM are always order optimal for the considered source set.</p>

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Initial Value Identification Problem for a Time-Fractional Fourth Order Equation: Convergence Rates

  • 期刊论文

  • 发表日期：

    2026年4月7日

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  • Subhankar Mondal

摘要

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 ill-posed problem, that is, a small perturbation in the data may lead to a large deviation in the sought solution. Thus, we obtain regularized approximations for the sought initial value by employing the quasi-boundary value method, its modified version and by the Fourier truncation method (FTM). We provide both the a priori and a posteriori parameter choice strategies and derive the error estimates for all these methods under some source conditions involving some Sobolev smoothness. As an important implication of the obtained rates, we observe that for both the a priori and a posteriori cases, the rate obtained by all these three methods are the same and also order optimal for some source sets. However, for higher smoothness the rates obtained by the FTM are better than the other two. Further, we observe that the rates obtained by the FTM are always order optimal for the considered source set.