Regularization for 2-D Fractional Sideways Heat Equations

Abstract

In this article, an inverse problem of Caputo-time-fractional sideways heat equations is considered. The aim is to find the inaccessible boundary data of some heterogeneous materials through interior measurements. Instead of the standard Tikhonov approach, we propose a series of fast filters for reconstructing the missing boundary data. Theoretical error bounds are provided for both boundary and (near-boundary) interior reconstructions. Several numerical examples are included to verify the proven convergence results.

Regularization Error Analysis for a Sideways Problem of the 2D Nonhomogeneous Time-Fractional Diffusion Equation

“…Compared to the 1D setting, the literature of fractional diffusion equation in 2D or higher dimensional setting is much more scarce. Some articles [25][26][27][28] study the following 2D homogeneous fractional diffusion equation:…”

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

“…For the theoretical results of the stability and the uniqueness of the solution, a backward problem and inverse source problem for a multi-dimensional time-fractional diffusion equation were included in [28]. The authors of [29,30] managed the inverse problem for the two-dimensional time-fractional sideways heat equation in the infinite domain. Inverse problems for two-dimensional time-fractional diffusion equation were also considered in [29,[31][32][33].…”

“…The authors of [29,30] managed the inverse problem for the two-dimensional time-fractional sideways heat equation in the infinite domain. Inverse problems for two-dimensional time-fractional diffusion equation were also considered in [29,[31][32][33]. The study [34] considered the inverse problem for two-dimensional fractional partial differential equations.…”

An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation

“…In the past decades, fractional differential equations have attracted wide attention. Various models using fractional partial differential equations have been successfully applied to describe a range of problems in mechanical engineering [1], viscoelasticity [2], electron transport [3], dissipation [4], heat conduction [5,6], and high-frequency financial data [7]. e time-fractional diffusion equation is deduced by replacing the standard time derivative with a time-fractional derivative and can be used to describe the superdiffusion and subdiffusion phenomena [8][9][10][11][12][13].…”