GENERALIZED TRIGONOMETRIC SYSTEMS AND SPECTRAL TASKS WITH ADDITIONAL INTERNAL BOUNDARY CONDITIONS
Abstract
When solving problems related to the study of the strength properties of various structures,
some sets of trigonometric (sine or cosine), as well as hyperbolic functions are often used, which
cyclically pass into each other when taking derivatives. These sets consist of two functions, and the
last of these functions, when differentiating, passes into the first, taken respectively with a plus sign (a
trigonometric system of the first type) or a minus sign (a trigonometric system of the second type).
Trigonometric and hyperbolic functions are also used in solving many applied problems, whose
mathematical models contain second derivatives in spatial variables. If the mathematical model contains
fourth-order derivatives with respect to spatial variables, then when solving the corresponding
problems, it is possible to use functions whose fourth derivatives are proportional to these functions.
There are a number of works on the general theory of systems of functions, where generalized trigonometric
systems (GTS) of functions are described, the derivatives of a certain order of which are
proportional to these functions. In this paper, this theory is developed in the direction of studying the
quadratic forms of the functions that make up the GTS. It is shown that the quadratic forms of GTS
functions can themselves be GTS functions of the same order (of the first or second types). The obtained
identities and the created theory are used to solve spectral problems for a fourth-order operator
for functions with certain conditions. The specificity of the problems under consideration is that in
addition to the standard boundary conditions, there are additional conditions on the inner boundary.
These conditions are not sufficient to independently solve the problem in each separate domain in
which the functions under study are specified. The use of the GTS properties identified herein allows
us to solve such problems in the entire area under consideration.
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