Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source ^*

Abstract

This paper is concerned with an inverse problem of recovering a potential term and fractional order in a one-dimensional subdiffusion problem, which involves a Djrbashian-Caputo fractional derivative of order α ∈ (0, 1) in time, from the lateral Cauchy data. In the model, we do not assume a full knowledge of the initial data and the source term, since they might be unavailable in some practical applications. We prove the unique recovery of the spatiallydependent potential coefficient and the order α of the derivation simultaneously from the measured trace data at one end point, when the model is equipped with a boundary excitation with a compact support away from t = 0. One of the initial data and the source can also be uniquely determined, provided that the other is known. The analysis employs a representation of the solution and the time analyticity of the associated function. Further, we discuss a two-stage procedure, directly inspired by the analysis, for the numerical identification of the order and potential coefficient, and illustrate the feasibility of the recovery with several numerical experiments.

Reconstruction and stability analysis of potential appearing in time‐fractional subdiffusion

“…Jing and Yamamoto [12] showed that two boundary measurement data can uniquely determine the order of the fractional derivative, a spatially varying potential, initial values, and Robin coefficients simultaneously in subdiffusion/diffusion-wave (i.e., 0 < 𝛼 < 2) equation. Recently, Jin and Zhou [13] proved the unique recovery of the spatially dependent potential coefficient and the order of the derivation simultaneously from the measured trace data at one endpoint, when the model is equipped with a boundary excitation with a compact support away from t = 0. Whereas Kaltenbacher and Rundell [14] showed the invertibility of the linearized map of the forward problem from the finite energy space L 2 (Ω) to H 2 (Ω) under the condition u(., 0) > 0 in the domain Ω and the potential q * ∈ L ∞ (Ω) using a Paley-Wiener-type result and a type of strong maximum principle.…”

“…Jing and Yamamoto [12] showed that two boundary measurement data can uniquely determine the order of the fractional derivative, a spatially varying potential, initial values, and Robin coefficients simultaneously in subdiffusion/diffusion‐wave (i.e., $$$ 0&amp;amp;amp;lt;\alpha &amp;amp;amp;lt;2 $$$) equation. Recently, Jin and Zhou [13] proved the unique recovery of the spatially dependent potential coefficient and the order of the derivation simultaneously from the measured trace data at one endpoint, when the model is equipped with a boundary excitation with a compact support away from $$$ t&amp;amp;amp;amp;#x0003D;0. $$$ Whereas Kaltenbacher and Rundell [14] showed the invertibility of the linearized map of the forward problem from the finite energy space $$$ {L}&amp;amp;amp;amp;#x0005E;2\left(\Omega \right) $$$ to $$$ {H}&amp;amp;amp;amp;#x0005E;2\left(\Omega \right) $$$ under the condition $$$ u\left(.,0\right)&amp;amp;amp;gt;0 $$$ in the domain $$$ \Omega $$$ and the potential $$$ {q}&amp;amp;amp;amp;#x0005E;{\ast}\in {L}&amp;amp;amp;amp;#x0005E;{\infty}\left(\Omega \right) $$$ using a Paley–Wiener‐type result and a type of strong maximum principle.…”

Multiple Terms Identification of Time Fractional Diffusion Equation with Symmetric Potential from Nonlocal Observation

“…In [16,17], the Levenberg-Marquardt regularization method was utilized to reconstruct the temporal potential term in a multi-term time-fractional diffusion equation. The work in [18] investigated the inverse problem of recovering the spatial potential term and the fractional order from the transversal Cauchy data without assuming full knowledge of the initial data and source term, where the authors proposed a two-stage numerical identification method for reconstructing the fractional order and the potential coefficient. For the time-fractional diffusion-wave equation, the authors in [19] studied the nonlinear inverse problem of identifying the fractional orders and the temporal potential term from the integral data.…”

Parameter Estimation for a Type of Fractional Diffusion Equation Based on Compact Difference Scheme

“…Most of the literature on the inverse problem of the fractional differential equations exploited deterministic techniques, such as exact matching, least squares optimizations, without considering measurement error and numerical error. In general, very roughly speaking, one may split the corresponding approaches of recovering the order into the following two categories: solving the corresponding inverse problems analytically or using numerical-analytical methods [25][26][27][28][29][30][31], or using some more soft, metaheuristic, or statistical optimization and regularization techniques [32][33][34][35][36][37]. However, the observations are usually contaminated with measurement error, and the forward problem of the models will bring the numerical error, so the uncertainties are non-ignorable.…”