Recovering multiple fractional orders in time-fractional diffusion in an unknown medium

Abstract

In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g. diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Furthermore, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach for small time t .

The enclosure method for the detection of variable order in fractional diffusion equations

“…Without being exhaustive we can mention the works of [2,7,11,21,22,26,34,30,32,33,41] devoted to the determination of single or multiple constant fractional orders, sometimes together with other parameters (coefficients or internal sources), from several class of observational data. We mention also the recent works [23,24] where the determination of constant fractional order have been studied in the context of an unknown medium (unknown source, coefficients, domain...). All the above mentioned results have been devoted to the determination of constant fractional order.…”

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.…”

The enclosure method for the detection of variable order in fractional diffusion equations

“…Without being exhaustive we can mention the works of [2,7,11,21,22,26,31,33,34,32,41] devoted to the determination of single or multiple constant fractional orders, sometimes together with other parameters (coefficients or internal sources), from several class of observational data. We mention also the recent works [23,24] where the determination of constant fractional order have been studied in the context of an unknown medium (unknown source, coefficients, domain...). All the above mentioned results have been devoted to the determination of constant fractional order.…”