Solution of the inverse problem of identifying the order of the fractional derivative in a mathematical model of the dynamics of solar activitythe at rising phase

The article refines the mathematical model of solar activity dynamics by solving the inverse problem. Experimental data on the observation of Wolf number values are used as additional information. This parameter of solar activity reflects the number of spots on the surface of the sun, and is considered an indicator of its activity. This process is characterized by observable cyclicality, periods of growth and decline. The analysis and processing of the initial data is carried out in order to isolate from the time series areas corresponding to an increase in solar activity. To describe this dynamic process, a previously proposed mathematical model for describing cycles 23 and 24 is used. The model is a Cauchy problem for a fractional analogue of the nonlinear Riccati equation, where the first-order derivative is replaced by the Gerasimov-Caputo fractional differentiation operator with an order from 0 to 1. The order of the fractional derivative is associated with the intensity of the process. This model equation is solved numerically using a nonlocal implicit finite-difference scheme. To clarify the values of the order of the fractional derivative, the one-dimensional optimization problem was solved using the second-order Levenberg-Marquardt iterative method, based on processed experimental data. It is shown that it is possible to refine the order of the fractional derivative in the solar activity model by solving the corresponding inverse problem, and the results obtained are in better agreement with the data.