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

Abstract

The article presents a method for solving the inverse problem of a two-dimensional anomalous diffusion equation with a Riemann–Liouville fractional-order derivative. In the first part of the present study, the authors present a numerical solution of the direct problem. For this purpose, a differential scheme was developed based on the alternating direction implicit method. The presented method was accompanied by examples illustrating its accuracy. The second part of the study concerned the inverse problem of recreating the model parameters, including the orders of the fractional derivative, in the anomalous diffusion equation. Equations of this type can be used to describe, inter alia, the heat conductivity in porous materials. The ant colony optimization algorithm was used to solve this problem. The authors investigated the impact of the distribution of measurement points, the use of different mesh sizes, and the input data errors on the obtained results.

Computational Methods for Parameter Identification in 2D Fractional System with Riemann–Liouville Derivative

“…The matrices from the difference scheme ( 16) are large, so the obtainedsystem of equations is time-consuming to solve. Hence, we applied the alternating direction implicit method (ADIM) to the difference scheme (11), which significantly reduces the computation time (details can be found in [27]). This is an important issue in the case of inverse problems, where a direct problem should be solved many times.…”

“…The Bi-CGSTAB [28,29] method is used to solve the equation systems, which has significance influences on the computation time. More implementation details and a comparison of times for the described method can be found in the papers [27,30].…”

“…The used differential scheme is unconditionally stable and has the approximation order equal to O((∆x) 2 + (∆y) 2 + (∆t) 2 ) [26]. The convergence of the differential scheme is fast; already for sparse meshes, the approximation errors for the solution of the direct problem are small [27]. In addition, in the case of the inverse problem considered in this paper, it is enough to use a relatively sparse mesh to very well reconstruct the searched parameters.…”

Parameter Estimation for Nonlinear Diffusion Problems by the Constrained Homotopy Method

“…For example, it can help reservoir engineers make important decisions about the type of the recovery method, fluid production and injection rates, and well locations. From then on, a variety of effective numerical methods have appeared in the literatures of the inverse problem for non-linear diffusion problems [1][2][3][4][5][6]. This inverse problem can be viewed as a parametric data-fitting problem.…”

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

“…In these works, the problems of finding the solution and the right side of the equation in the case of one spatial variable were considered. Inverse problems in the case of two spatial variables for differential equations of fractional order were studied in [11][12][13][14][15][16].…”