Restoration of the order of fractional derivative

D. A. Tverdyi

News of the Kabardin-Balkar scientific center of RAS, no. 6 (2023), pp. 83-94

Résumé

The paper researches the issues related to the nonlinear transport of radon gas through the geosphere, in particular, when describing the variation of volumetric activity (RVA) in an accumulation chamber with recording sensors. RVA is considered to be an informative and operational precursor to earthquakes. Based on the assumption that the radon transport process takes place in a permeable geosphere, an ereditary RVA model based on the Riccati equation with fractional Gerasimov–Kaputo derivative is used for modelling. The model has been previously validated at the geodynamic test site in PetropavlovskKamchatsky. In the study the identification of the order value of the fractional derivative, which is associated with such geo-environmental characteristics as porosity and permeability is of most interest. However, we do not have information about some parameters of the process under consideration to determine this value accurately enough. But we know additional information obtained from the experiment. This information can be used to reconstruct the values of interest. Which leads us to the inverse problems. To reconstruct the order of the fractional derivative, we solve the one-dimensional optimisation problem using the iterative Levenberg–Marquardt method of Newtonian type. It is shown that this method can be used to reconstruct some parameters of such a dynamic system as radon transport through geo-environment. It is shown that the solution of the inverse problem by the Levenberg–Marquardt method gives a more accurate result in a shorter time than the manual selection of parameter values and types of functions for the model equations.

Détail

Keywords: mathematical modelling, Gerasimov–Kaputo fractional derivative, inverse problems,Levenberg–Marquardt method, stress-strain state, geo-environment, volumetric radon activity, RVA,earthquake precursors