Igor Podlubny’s research while affiliated with Technical University of Košice and other places

In this work we explore the Sibuya discrete probability distribution, which serves as the basis and the main instrument for numerical simulations of Grunwald--Letnikov fractional derivatives by the Monte Carlo method. We provide three methods for simulating the Sibuya distribution. We also introduce the Sibuya-like sieved probability distributions, and apply them to numerical fractional-order differentiation. Additionally, we use the Monte Carlo method for evaluating fractional-order integrals, and suggest the notion of the continuous Sibuya probability distribution. The developed methods and tools are illustrated by examples of computation. We provide the MATLAB toolboxes for simulation of the Sibuya probability distribution, and for the numerical examples.

This discussion paper presents some parts of the work in progress. It is shown that G.W. Leibniz was the first who raised the question about geometric interpretation of fractional-order operators. Geometric interpretations of the Riemann--Liouville fractional integral and the Stieltjes integral are explained. Then, for the first time, a geometric interpretation of the Stieltjes derivatives is introduced, which holds also for so-called ``fractal derivatives'', which are a particular case of Stieltjes derivatives.

A new approach to dividing the mathematical content into partial modules is presented. This allows to compose subjects with mathematical content from such partial modules and flexibly adapt these subjects to the needs of specific study programs at technical universities. The consistent and systematic implementation of this approach in a typical learning management system is described in detail. This approach means significant changes in the massive (or bulk) delivery of knowledge using available information technologies. The main benefits of the presented system consist in the increase the resulting level of knowledge of students along with their satisfaction with the results and the form of their study. The most important changes arising from our approach are the following. First, the study process became distributed in space and in time. Second, it can be piecewise continuous in time, and, since all students can study at their own pace, it runs in multiple individual time scales. The most important change, however, is the shift of the paradigm the educational process from transmissive “teach–learn” to active “study”.

In this work the Monte Carlo method, introduced recently by the authors for orders of differentiation between zero and one, is further extended to differentiation of orders higher than one. Two approaches have been developed on this way. The first approach is based on interpreting the coefficients of the Grünwald–Letnikov fractional differences as so called signed probabilities, which in the case of orders higher than one can be negative or positive. We demonstrate how this situation can be processed and used for computations. The second approach uses the Monte Carlo method for orders between zero and one and the semi-group property of fractional-order differences. Both methods have been implemented in MATLAB and illustrated by several examples of fractional-order differentiation of several functions that typically appear in applications. Computational results of both methods were in mutual agreement and conform with the exact fractional-order derivatives of the functions used in the examples.

In this work the Monte Carlo method is introduced for numerical evaluation of fractional-order derivatives. A general framework for using this method is presented and illustrated by several examples. The proposed method can be used for numerical evaluation of the Grünwald-Letnikov fractional derivatives, the Riemann-Liouville fractional derivatives, and also of the Caputo fractional derivatives, when they are equivalent to the Riemann-Liouville derivatives. The proposed method can be enhanced using standard approaches for the classic Monte Carlo method, and it also allows easy parallelization, which means that it is of high potential for applications of the fractional calculus.

A "lost chapter" from Antoine de Saint-Exupéry's Le Petite Prince about the Little Prince visiting a mathematician, written in French in the style of the original work, is presented along with several translations.

In this paper, the fractional-order modeling of multiple groups of lithium-ion batteries with different states is discussed referring to electrochemical impedance spectroscopy (EIS) analysis and iterative learning identification method. The structure and parameters of the presented fractional-order equivalent circuit model (FO-ECM) are determined by EIS from electrochemical test. Based on the working condition test, a P-type iterative learning algorithm is applied to optimize certain selected model parameters in FO-ECM affected by polarization effect. What's more, considering the reliability of structure and adaptiveness of parameters in FO-ECM, a pre-tested nondestructive 1 / f noise is superimposed to the input current, and the correlative information criterion (CIC) is proposed by means of multiple correlations of each parameter and confidence eigen-voltages from weighted co-expression network analysis method. The tested batteries with different state of health (SOH) can be successfully simulated by FO-ECM with rarely need of calibration when excluding polarization effect. Particularly, the small value of CIC α indicates that the fractional-order α is constant over time for the purpose of SOH estimation. Meanwhile, the time-varying ohmic resistance R 0 in FO-ECM can be regarded as a wind vane of SOH due to the large value of CIC R 0 . The above analytically found parameter-state relations are highly consistent with the existing literature and empirical conclusions, which indicates the broad application prospects of this paper.

A new mathematical tool, porous functions, has been suggested recently. In this paper operations with porous functions are further discussed alongside with computational tools, which are necessary for evaluation of porous functions, their visualization, and basic operations with such functions in two- and three-dimensional cases.