FRACTAL RADIOPHYSICS. PART 3. FRACTIONAL CALCULUS IN ELECTRODYNAMICS - Lazorenko, O.; Chernogor, L. (Radio physics and radio astronomy (Print), 2024)
Subject and Purpose. At the beginning of the 21st century, a fundamentally new scientific direction was formed, currently known as fractal radiophysics. The present work is an overview of the principal theoretical and practical ideas concerning "fractalization" in radio physics. The purpose is a systematic presentation of the main practical results suitable for application of the fractional calculus in modern theoretical radiophysics. Methods and Methodology. The basic theoretical principles of fractional calculus are outlined in a structured form. Results of applying fractional calculus methods in electrodynamics are systematized. Essential features, advantages and disadvantages of the technique are demonstrated and the problems still remaining discussed. Results. The basics of fractional (or fractal) calculus have been considered with emphasis on practical application to problems of radiophysics. A variety of approaches to constructing fractional integrals and Riemann–Liouville, etc. fractional derivatives have been presented. Using the Newton-Leibnitz formula and fundamental theorems of fractional calculus, principles of generalization of the classic vector calculus to fractal problems have been discussed, suggesting the examples of fractional vector-differential and vector-integral operators, Green’s and Stokes’ fractional formulas, etc. With the use of Gauss’s fractional formula the basics of fractal electrodynamics are expounded. Some different types of fractal Maxwellian equations has been induced and analyzed. Also, the main approaches to solving radio wave propagation problems in fractal media are discussed. Conclusions. As a practical example of applying fractals in modern theoretical radiophysics, results have been presented of the use of fractional calculus in electrodynamics. These results signify appearance of a fundamentally new direction in radiophysics, namely fractal electrodynamics.