Batyrev O. Singular integral equations method in mathematical models of linear systems.

Objects of study are the physical processes of signal propagation in media with variable characteristics, which are modeled by means of integral equations. The aims of study are development of mathematical models of linear systems, which are based on singular integral equations, the study of solvability of such equations and construction of algorithms for computing solutions.The research use methods of mathematical physics, integral and differential equations, linear algebra and numerical methods. The estimate of the size of kernel of singular integral operator with non-Carleman shift has been obtained for the first time and the case of linear-fractional shift has been investigated. Numerical methods have been developed. Mathematical models of complex systems with commutation were formed; the method of modeling of linear systems with variable parameters has been developed on basis of singular integral equations. The solution of impulse characteristics decomposition problem has been obtained in the explicit form for the first time for convolution equations with a special kind of sounding signals. The approximate method for numeric solutions has been developed. Using of such mathematical models can solve some actual problems related to the, remote methods, predicting physical phenomena and processes in problems of underwater acoustics, sounding, fracture mechanics, etc. The results are planned for using in work projects.