Solution of the Inverse Problem of Identification of Parameters of the Mathematical Model Based on the Fractional Riccati Equation from Experimental Data of Solar Activity Dynamics

The research of solar-terrestrial relations is of great importance both for solving practical problems and is an important problem of fundamental science. In the study, based on experimental data, the order of the fractional derivative in the previously proposed mathematical model of the dynamics of solar activity at the ascent stage is refined by mathematical optimization methods. The order of the fractional derivative \(\alpha \) in the presented model reflects the intensity of the solar activity process. The fractional derivative is understood in the Gerasimov-Caputo sense of constant order and describes the memory effect of the system in the model. The experimental data, the Wolf number, represents the number of spots on the surface of the Sun, which is considered a clear indicator of solar activity, a dynamic process characterized by alternating periods of growth and decline. Data on solar activity are preprocessed for use in the task. The analysis of experimental data is carried out in order to identify areas corresponding to the increase in solar activity. The Cauchy problemCauchy problem is posed for a nonlinear fractional model equation, which is solved numerically using a non-local implicit finite difference scheme. Previously, the parameters of the mathematical model, including \(\alpha \) , were selected manually to obtain the best similarity coefficients, comparing data and solving direct problems. Now, to clarify the values of the order of the fractional derivative, we will resort to methods for solving inverse problems, namely, the unconditional iterative Levenberg-Marquardt method of the Newtonian type. The results show that it is possible to improve the known model of solar activity by restoring some parameters of the mathematical model. The results obtained in the course of solving inverse problems are in better agreement with experimental data.