Abstract
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.
Highlights
To describe the processes of various natures, there are many models which are considered basic parameters of the process and they can effectively be investigated by mathematical methods.the same type of models is often used in completely different fields of knowledge
Eq [1] of ultrasound oscillations by replacement x =(c1τ)−1y, α= (с2)−1c1 is transformed into the one-parameter (parameter α Î (0, 1)) hyperbolic equation
The mathematical model of the process of ultrasound wave propagation which contains Eq [2] and the profile of oscillation is given at three equidistant moments of time t = jh, where j Î J = {0,1, 2}, h > 0 (Eq 3): u( jh, x) f j (x), j J, x R3, [3]
Summary
To describe the processes of various natures, there are many models which are considered basic parameters of the process and they can effectively be investigated by mathematical methods.the same type of models is often used in completely different fields of knowledge. One of the important areas of mathematical modeling is the simulation of processes by differential equations, which is used, in particular, in modeling the processes of wave propagation. The property of ultrasonic waves to change the speed of their propagation and absorption with any changes in the environment is taken into account. This property, as well as the reflection of the ultrasonic wave at the boundaries of different environments in the human body, are the basis of the ultrasound diagnostics, which is one of the most informative methods of non-invasive diagnosis in medicine. It is widely used to diagnose the work of organs and obtain their threedimensional images, accelerate metabolic processes in the body and destroy various tumors
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