Open AccessMathematical Model of the Process of Ultrasonic wave Propagation in a Relax Environment with its Given Profiles at three Time Moments
Author(s) -
Zinovii Nytrebych,
В. С. Ільків,
Оксана Маланчук
Publication year - 2021
Publication title -
the open bioinformatics journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.259
H-Index - 8
ISSN - 1875-0362
DOI - 10.2174/1875036202114010087
Subject(s) - partial differential equation , ultrasonic sensor , inverse problem , point (geometry) , acoustic wave equation , wave equation , hyperbolic partial differential equation , uniqueness , process (computing) , mathematical analysis , differential equation , computer science , wave propagation , mathematics , mathematical optimization , acoustics , acoustic wave , physics , optics , geometry , operating system
Objective: The process of ultrasound oscillations in a relaxed environment, provided that the profiles of the acoustic wave at three time moments are known, is modeled by a three-point problem for the partial differential equation of the third order in time. This equation as a partial case contains a hyperbolic equation of the third order, which is widely used in ultrasound diagnostics. Methods: The differential-symbol method is applied to study a three-point in-time problem. The advantage of this method is the possibility to obtain a solution of the problem only through operations of differentiation. Results: We propose the formula to construct the analytic solution of the problem, which describes the process of ultrasound oscillations propagation in a relax environment. Due to this, the profile of the ultrasonic wave is known at any time and at an arbitrary point of space. The class of quasi-polynomials is distinguished as a class of uniqueness solvability of a three-point problem. Conclusion: Using the proposed method, it is possible to analyze the influence of the main parameters of ultrasound diagnostics problems on the propagation of acoustic oscillations in a relaxed environment. The research example of a specific three-point problem is given.