Some aspects of the implementation of the PRPHMM 1.0 software package for refining the parameters of hereditary mathematical models of radon transfer in a storage chamber

D. A. Tvyordyj ; E. O. Makarov

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 49 (2024) no. 4, pp. 135-156

Résumé

Mathematical models of some dynamic processes can be significantly enhanced by using derivatives and integrals of non-integer order in them, taking into account effects that cannot be described by ordinary derivatives. For example, by using fractional Gerasimov- Caputo derivatives of constant and variable order, it is possible to take into account the memory effect in the process model, and the order of the derivative will be related to the intensity of the process. In particular, the authors have previously developed an hereditary $\alpha$-model of the volumetric activity of radon, where the parameter $\alpha$ is related to the permeability of the medium. However, the question arises about determination of optimal values of both $\alpha$ and other parameters of the model. To solve the problem, it is possible to solve the inverse problem, a common type of problem in many scientific fields, where it is necessary to determine the values of model parameters from observed data, but it is impossible to make direct measurements of these parameters. The need for such an approach often arises when working with geological data. The article describes the software implementation of the PRPHMM 1.0 software package which can clarifying optimal values of hereditary mathematical models based on the Gerasimov–Caputo derivative. The Levenberg-Marquardt unconditional Newtonian optimisation algorithm is adapted and implemented in MATLAB language. Subroutines for reading, processing and visualisation of experimental and model data are implemented. A test case solving the inverse problem for the hereditary $\alpha$-model for the parameters $\alpha$ and $\lambda_0$-air exchange coefficient on the basis of experimental radon monitoring data is presented. It is shown that PRPHMM 1.0 allows for the clarify of parameter values close to the optimum values for the hereditary mathematical models.

Détail

Keywords: мathematical modeling, fractional derivatives, Gerasimov-Caputo, memory effect, nonlocality, nonlinear equations, inverse problems, unconditional optimization, MATLAB.
Mots-clés : Levenberg-Marquardt algorithm