Introduction to fractional analysis of singular («rigid») systems of differential equations representing material objects of railway transport

Abstract

Quite often in science, as well as in transport mechanics, one faces a phenomenon in which the argument of some function is extremely small, while its value is very large, but their product is still limited in size. Such mathematical models or systems of differential equations are called «rigid». Their numerical integration requires the use of special methods, like a Gear method; otherwise, it may take too much «machine» time. Although in this case the correct result is not guaranteed either. Academician of the Russian Academy of Sciences A.N. Tikhonov proved a theorem on separating the motion of a dynamic system into «fast" and «slow» components, which, if applied correctly, guarantees a given accuracy of the solution, its basis being made up of only 5 conditions. According to the figurative and apt expression of the professor of Moscow State University I.V. Novozhilov, it’s as if the researcher first looks through a telescope  seeing a change in «slow” variables and then, looking through a microscope, obviously sees «fast» components. Unfortunately, the theoretical material devoted to this theorem is mainly published by mathematicians in the corresponding journals at the appropriate level. Engineers graduating from technical universities do not have the necessary body of mathematical knowledge to understand the Tikhonov’s theorem, although in life they often face problems requiring its application. Great efforts to introduce this theorem into applications were spent by Professor of Moscow State University I.V. Novozhilov, who published several monographs on this issue. In this article, an attempt is made from an engineering, methodological point of view to consider the solution of the well-known, and, of course, solved, Stokes problem. Exact and approximate solutions are obtained and compared.