Special Issue Editorial Asymptotic Methods in the Mechanics and Nonlinear Dynamics

The idea of asymptotic approximation is one of the most important and profound in mathematics, especially in the parts of it those are in close contact with physics, mechanics, and engineering. The fact is that almost any physical theory formulated in a general way is very complex from a mathematical point of view. Therefore, both in creating a theory and in its further development, the simplest limiting cases, which admit analytical solutions, are of particular importance.

In these limiting cases, the number of equations usually decreases, their order decreases, the transition from a discrete system to a continuous medium or from an inhomogeneous medium to a homogeneous one becomes possible, the process is localized near the boundary of the region under consideration, etc. However, behind all these idealizations, no matter how different they may seem, there is a high degree of symmetry inherent in the mathematical model of the phenomenon under consideration in the limiting situation. An asymptotic approach to a complex problem consists in treating the original (insufficiently symmetric) system as close to some symmetric one. It is fundamentally important that the determination of corrections that take into account deviations from the limiting case is much simpler than a direct study of the original system. At first glance, the possibilities of this approach are limited by a narrow range of system parameters.

However, the experience of studying various physical problems shows that with a significant change in the parameters of the system and its removal from one limiting symmetric case, as a rule, there is another limiting system, often with less obvious symmetry, and a solution can already be constructed for it. This makes it possible to describe the behavior of the system over the entire range of parameter changes, based on a small number of limiting cases.

The increased interest in asymptotic methods in recent years is all the more remarkable because the advances in computational mathematics would seem to lead to the opposite trend. The fact is that asymptotic methods develop our intuition in every possible way, and therefore play an important role in shaping the thinking of a modern natural scientist or engineer. Even in those cases where the main goal remains to obtain numerical results, a preliminary asymptotic analysis allows one to choose the best computational method and understand the vast but disordered numerical material. In addition, such an analysis is especially effective in those ranges of parameter values where direct computer calculations encounter serious difficulties (rigid systems). Asymptotic solutions are useful in terms of numerical results far beyond their formal domain of applicability and can often be used directly.

I hope this volume will be of interest to scientists from many fields of applied mathematics, mechanics and physics. Here, researchers can find examples of application of asymptotic methods for various mathematical objects, in particular, delay DUs [1], nonlinear ODEs and PDEs [2,3], and fractional DEs [4].

Various physical objects are also studied in this volume by asymptotic methods: corrugated rings [5]; shells [6]; and multirobot systems [7].

Considerable attention is paid to such physical phenomena as nonlinear waves [3,4,6,8] and oscillations [9]; buckling [5]; and heat transfer [10].

Along with well-known asymptotic approaches, the reader will find in great volume descriptions and applications of the methods of Padé approximants and other summation and interpolation procedures [2,3,11]; non-standard discretization and continualization methods [2]; non-smooth temporal transformations [9]; and perturbation iteration method [4].