Fractional Model of the Deformation Process

• Published in: Fractal and fractional 2022-07, Vol.6 (7), p.372
• Main Authors: Sheremetyeva, Olga, Shevtsov, Boris
• Format: Article
• Language: English
• Subjects: approximation; Deformation; Distribution functions; Earthquakes; Fractals; fractional model; fractional Poisson process; Mathematical models; Mittag–Leffler function; non-local effects; Parameters; Poisson density functions; relaxation processes; Seismic activity
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract6070372

The article considers the fractional Poisson process as a mathematical model of deformation activity in a seismically active region. The dislocation approach is used to describe five modes of the deformation process. The change in modes is determined by the change in the intensity of the event stream, the regrouping of dislocations, and the change in and the appearance of stable connections between dislocations. Modeling of the change of deformation modes is carried out by changing three parameters of the proposed model. The background mode with independent events is described by a standard Poisson process. To describe variations from the background mode of seismic activity, when connections are formed between dislocations, the fractional Poisson process and the Mittag–Leffler function characterizing it are used. An approximation of the empirical cumulative distribution function of waiting time of the foreshocks obtained as a result of processing the seismic catalog data was carried out on the basis of the proposed model. It is shown that the model curves, with an appropriate choice of the Mittag–Leffler function’s parameters, gives results close to the experimental ones and can be allowed to characterize the deformation process in the seismically active region under consideration.