On the Modeling Process of Ultrasonic Wave Propagation in a Relaxation Medium by the Three-Point in Time Problem

A mathematical model of the process of ultrasonic oscillations in a relaxation medium with known acoustic wave profiles at three points in time is proposed. The model is reduced to the study of a three-point problem for a hyperbolic equation of third order, which is widely used in ultrasound diagnostics. A differential-symbol method for constructing a solution of the three-point problem is proposed and a class of quasipolynomials as the class of uniqueness solvability of the problem is found. The technique which specified in the work allows to investigate in detail the main parameters of acoustic oscillations in problems of ultrasonic diagnostics. The method is demonstrated on specific examples of three-point problems.

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