We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The method is based on the Chernoff approximation procedure applied to a specially constructed shift operator. It is proven that approximations converge uniformly to the exact solution.
The current state of methods of the solution of boundary problems of mechanics of continuous environments is characterized. It is noted that packages of applied programs applied in engineering practice are based on the methods leading to solutions of boundary problems in the form of massifs of numbers. As a shortcoming the im-possibility of a reliable assessment of an error of such decisions for the majority of complex engineering chal-lenges is noted. As the alternative is stated an essence of a fictitious canonic regions method. It is shown that its application leads to solutions of boundary problems not in the form of massifs of numbers, but in the form of the functions which are identically satisfying with to the differential equations of boundary problems. The main ad-vantage of the fictitious canonic regions method - high precision of received results and possibility of a reliable assessment of their error. The review of stages of development of the fictitious canonic regions method is executed. The review of the works devoted to its application for the solution of scientific and engineering problems is executed.
The volume contains extended abstracts of reports at the International Conference «Shell and Membrane Theories in Mechanics and Biology: From Macro- to Nanoscale Structures» (SMT in MB-2013) held in the Belarusian State University. The papers deal with modern problems of the shell and membrane theories and their applications in Mechanics, Biology, Medicine, Industry and Nanotechnology. The book is of interest for researches, post-graduate and master students working in different branches of Mechanics, Biomechanics and Nanomechanics.
A motion problem for material points embedded in a standard three-dimensional sphere $S^3$ is considered in terms of classical mechanics. In particular, spherical analogs of Newton’s laws are discussed.