Asymptotic Representations for the Solutions of Nonautonomous Ordinary Differential Equations

Ukrainian Mathematical Journal - The conditions of existence and asymptotic representations as t ↑ �� (�� ≤ + ∞) are obtained for one class of...     

Asymptotic representations of the solutions of essentially nonlinear nonautonomous second-order differential equations

We establish asymptotic representations for the solutions of a class of nonlinear nonautonomous second-order differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 310–331, March, 2008.     

Asymptotic behavior of unbounded solutions of essentially nonlinear second-order differential equations. II

We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 901–921, July, 2006.     

Conditions of oscillatory or nonoscillatory nature of solutions for a class of second-order semilinear differential equations

For a class of second-order semilinear differential equations, we prove the theorems on oscillatory or nonoscillatory nature of all proper solutions. These theorems are analogs of the well-known Kneser theorems for linear differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 458–466, April, 2007.     

Acoustic tomography of the nonlinear parameter by a small number of transducers

A wave approach to solving the problem of reconstructing the acoustic nonlinear parameter distribution by a simple transmitting-receiving system with broadband modulation of primary waves is proposed. The approach uses the effect of sound scattering by sound, which makes it possible to employ a complicated time processing of signals instead of spatial processing and, consequently, to reduce the number of transmitting and receiving transducers. The possibility of practical implementation of a two-dimensional tomographic scheme based on coherent correlation processing of broadband quasi-random signals at combination frequencies is analyzed. Results of model numerical experiments on reconstructing complicated spatial distributions of the nonlinear parameter are presented.     

Experimental modeling of the processes of active-passive thermoacoustic tomography

Experiments confirming the major results of the theoretical study of an active-passive mode of acoustic thermotomography are described. Experimental results are obtained with a setup intended for physical modeling of the processes of correlation reconstruction of the temperature, absorption, and phase velocity of sound in the object under investigation in the presence of an additionally introduced “illuminating” acoustic noise field. A possibility of reconstructing the local values of absorption and inhomogeneity of sound velocity from analyzing the correlation dependences based on difference and summary delays and also with the help of a controlled anisotropic acoustic irradiation is demonstrated.