=Paper=
{{Paper
|id=Vol-2753/paper10
|storemode=property
|title=On the Modeling Process of Ultrasonic Wave Propagation in a Relaxation Medium by the Three-Point in Time Problem
|pdfUrl=https://ceur-ws.org/Vol-2753/paper6.pdf
|volume=Vol-2753
|authors=Zinovii Nytrebych,Volodymyr Il’kiv,Oksana Malanchuk
|dblpUrl=https://dblp.org/rec/conf/iddm/NytrebychIM20
}}
==On the Modeling Process of Ultrasonic Wave Propagation in a Relaxation Medium by the Three-Point in Time Problem==
https://ceur-ws.org/Vol-2753/paper6.pdf
On the Modeling Process of Ultrasonic Wave Propagation in a
Relaxation Medium by the Three-Point in Time Problem
Zinovii Nytrebycha, Volodymyr Il’kiva and Oksana Malanchukb
a
Lviv Polytechnic National University, Bandery str., 12, Lviv, 79013, Ukraine
b
Danylo Halytsky Lviv National Medical University, Pekarska str., 69, Lviv, 79017, Ukraine
Abstract
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.
Keywords 1
Mathematical model, acoustic oscillations, three-point problem, differential-symbol method,
ultrasound diagnostics
1. Introduction
In the theory of mathematical modeling there are many models of processes of various nature.
Increasingly, these models from some areas of knowledge are used in other areas. In particular,
modeling of hydromechanics and gas dynamics problems [1, 2] is successfully used in modeling
biomechanical and medical processes [3-5].
Modern mathematical models increasingly contain partial differential equations, both linear and
nonlinear. Therefore, the research of such models is quite complex and their study involves powerful
numerical, qualitative and asymptotic methods (in particular, see [6, 7]). In addition to traditional
partial differential equations of the second order, which are actively studied in the equations of
mathematical physics, there are often partial differential equations of the third order in time in
mechanical, biomedical and geophysical models [8-11]. Among the problems of ultrasonic
diagnostics in [12-14] the Cauchy problem for the hyperbolic equation of the third order of the form
3t c12 t t2 c22 u(t , x ) 0, x ( x1 , x2 , x3 ) 3 , (1)
is investigated. In equation (1), is relaxation time, u(t , x ) is dynamic pressure,
2
2 2
2 2 is three-dimensional Laplace operator, t / t , constants c1 and c2 are
x1 x2 x3
2
limiting phase speeds of sound.
In addition to the Cauchy problem for partial differential equations, multipoint in time problems
with the given values of the unknown solution not at only one time point, but at several moments of
time are intensively studied. In particular, papers [15–18] and [19, 20] are devoted to problems with
multipoint in time conditions in bounded and unbounded domains respectively. Problems with n -
point time conditions have a simple physical interpretation, namely in these problems the state of the
IDDM’2020: 3rd International Conference on Informatics & Data-Driven Medicine, November 19-21, 2020, Växjö, Sweden
EMAIL: zinovii.m.nytrebych@lpnu.ua (Z. Nytrebych); ilkivv@i.ua (V. Ilkiv); oksana.malan@gmail.com (O. Malanchuk)
ORCID: 0000-0002-9599-8517 (Z. Nytrebych); 0000-0002-0359-5025 (V. Ilkiv); 0000-0001-7518-7824 (O. Malanchuk)
2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
research process at n different time points are given. However, despite of the simplicity of physical
interpretation, multipoint problems are not easy to study. In contrast to the Cauchy problem, the
kernel of multipoint in time problems is nontrivial [21, 22]. Therefore, the study of the corresponding
incorrect multipoint problems for partial differential equations requires new research methods, among
of them, the differential-symbol method is especially effective [23–26].
The aim of this work is:
study of a mathematical model describing the motion of an ultrasonic wave in a relaxation
medium with given profiles of the wave at three time points;
establish the class of existence and uniqueness of the solution of the corresponding three-
point in time problem;
recommend the method for constructing the solution of the problem;
development of a method for determining the influence of the wave process parameters under
the condition of specific initial data of the three-point problem.
2. Posing of the problem and main results
x c
By replacing x , 1 , ultrasonic wave equation (1) is transformed in the one-parameter
c1 c2
hyperbolic equation
3t t t2 u(t , x) 0, (t , x) (0, ) 3 . (2)
Let’s note that the spatial variable x and the parameter in equation (2) are dimensionless,
moreover belongs to the interval (0,1) .
We consider the mathematical model of the process of an ultrasonic wave propagation which is
described by equation (2), if the profiles of wave are given at three equidistant moments of time
t jh, j J {0,1,2}, h 0 :
u( jh, x) f j ( x), j J , x 3
. (3)
To study the three-point problem (2), (3), we use the differential-symbol method which was
previously [23] used to solve the two-point in time problem. Based on the differential equation (2), for
the unknown function (t , ) we write the corresponding ordinary differential equation with the
parameter
dt3 dt2 dt (t , ) 0 , (4)
in which | |2 12 22 32 , (1 , 2 , 3 ) 3 , dt d / dt .
Taking into account the dependence , let us denote the roots of the characteristic equation
2
for (4)
3 2 0 , (5)
by 1 1 ( ) , 2 2 ( ) , 3 3 ( ) . These roots have the following form
1 3
2(1 3 ) A
1 1 ( ) 3 ,
3 3A 3 2
1 (1 i 3)(1 3 ) (1 i 3) A
2 2 ( ) , (6)
3 3А 3 4 63 2
1 (1 i 3)(1 3 ) (1 i 3) A
3 3 ( ) ,
3 3А 3 4 63 2
where i 2 1 , A 3 B 4(3 1)3 B 2 , B 27 9 2 .
1 1 1
Remark 1. The roots (6) coincide only in the case and . Then 1 2 3 . If
9 3 3
1 1
( , ) , , there are at least two different roots among the roots of (6). If
9 3
(4 27 2 18 4 ) 0 there are simple roots.
2
Remark 2. For 0 equation (5) has the form
3 2 0
and the roots 1 1 , 2 3 0 do not depend on .
For the ordinary differential equation of third order (4) we construct a fundamental system of
solutions 0 (t , ), 1 (t , ), 2 (t , ) , which satisfies local three-point conditions
1, k j,
k jh, k, j J . (7)
0, k j ,
This system can be formed only for vectors 3
which fulfill the condition
( ) det gk ( jh, ) k , jJ 0 . (8)
We get:
gk (t , ) ek 1 ( )t for simple roots 1 2 3 1 ,
gk (t , ) t k e1 ( )t for triple root 1 ( ) ,
g0 (t , ) e1 ( )t , g1 (t , ) e2 ( )t , g2 (t , ) t e2 ( )t for 1 2 3 .
Let’s denote E1 e1 ( ) h , E2 e2 ( ) h , E3 e3 ( ) h for 1 2 3 1 . For condition (8) the
elements of system 0 (t , ), 1 (t , ), 2 (t , ) have the following form
E3 E2 ( E3 E2 )e1 ( )t E3 E1 ( E3 E1 )e 2 ( ) t E2 E1 ( E2 E1 )e 3 ( ) t
0 (t , ) ,
( )
( E32 E22 )e1 ( )t ( E32 E12 )e2 ( ) t ( E22 E12 )e3 ( )t
1 (t , ) , (9)
( )
( E3 E2 )e1 ( ) t ( E3 E1 )e2 ( ) t ( E2 E1 )e3 ( )t
2 (t , ) ,
( )
where ( ) E3 E1 E3 E2 E2 E1 .
If 1 2 3 , then ( ) hE2 E2 E1 . The functions 0 (t , ) , 1 (t , ) , 2 (t , ) take the
2
form
hE 3e1 ( )t hE1 E2 (2 E2 E1 )e2 ( ) t E1 E2 ( E2 E1 ) t e 2 ( ) t
0 (t , ) 2 ,
( )
2hE22 e1 ( ) t 2hE22 e2 ( ) t ( E22 E12 ) t e2 ( ) t
1 (t , ) , (10)
( )
hE2 e1 ( ) t hE2 e2 ( ) t ( E2 E1 ) t e2 ( ) t
2 (t , ) .
( )
In the case of triple root of equation (5), that is 1 2 3 , we have ( ) 2h3 E13 0 for each
1
3 , for which | |2 . The functions 0 (t , ), 1 (t , ), 2 (t, ) get the following form
3
3 1
0 (t , ) 1 t 2 t 2 e1 ( ) t ,
2 h 2 h
tt
1 (t , ) 2 e1 ( )( t h ) , (11)
h h
t
2 (t , ) h t 2 e1 ( )( t 2 h ) .
2h
1 1 1
According to Remark 1, all roots of equation (5) are equal to for , . Then
3 3 9
h
E1 e 3 , ( ) 2h3e h 0 and functions (11) can be written as follows
t
1
3 1
0 (t , ) 2 1 1 t 2 t 2 e 3 ,
3 2h 2h
t t 1 (t h )
1 (t , ) 2 1 2 e 3 , (12)
3 hh
t 13 ( t 2 h )
2 (t , ) 2 1 h t
e .
3 2h 2
According to Remark 2, for 0 we get 1 1 , 2 3 0 , E1 e h , E2 1 . Then
( ) h 1 e h 0 . The functions (8) take the form
2
he t he h (2 e h ) e h (1 e h ) t
g 0 (t ) 0 (t , ) 2 0 ,
h(1 e h ) 2
2he t 2h (1 e 2 h ) t
g1 (t ) 1 (t , ) 2 0 , (13)
h(1 e h ) 2
he t h (1 e h ) t
g 2 (t ) 2 (t , ) 2 0
h(1 e h ) 2
and do not depend on parameter . The graphical representations of these functions for h 1 are
given in Figure 1. Dashed lines on Figure 1 indicate asymptotes.
Figure 1: The graphs of the functions g0 (t ), g1 (t ), g2 (t )
Let us consider nontrivial quasipolynomials of the form
f ( x) Q( x)e x , x 1 x1 2 x2 3 x3 , (14)
in which Q( x) is nonzero polynomial of variables x1 , x2 , x3 with complex coefficients, 1 , 2 , 3 are
complex parameters, the vector (1 , 2 , 3 ) satisfies condition (8), that is M , where the set M
is determined by the formula
M 3 : ( ) 0 .
(15)
1
Note that the set (15) is nonempty, since the vectors in the case 0 and for
3 2
,
9
1
2
belong to this set.
3
For the set (15), let K M is the class of functions which can be represented as a finite sum of
quasipolynomials of the form (14) that differ from each other by different vectors . So, K M is the
class of quasipolynomials of the variables x1 , x2 , x3 and the zero quasipolynomial belongs to K M .
Let the right-hand sides of conditions (3), namely the functions f0 ( x), f1 ( x), f 2 ( x) belong to the
class K M . Then there is an unique solution of problem (2), (3) in the class of quasipolynomials of
variables t , x1 , x2 , x3 which belong to K M for each fixed t . This solution can be represented by the
formula
u (t , x) f k k (t , ) e x
2
, (16)
k 0 O
where x 1 x1 2 x2 3 x3 , O (0,0,0) , ( 1 , 2 , 3 ) .
The differential expressions f 0 , f1 , f 2 are obtained from the functions f 0 ( x) ,
f1 ( x) , f 2 ( x) by replacing the vector x by vector-derivative . For each summand of the form (14)
in the quasipolynomials f 0 ( x) , f1 ( x) , f 2 ( x) we put in correspondence the differential expression
1 2 3
Q( ) e 1 2 3
, which acts on the function k (t , ) e x by formula
(t, ) e Q( ) k (t , ) e x
1 2 3 x
Q ( ) e 1 2 3
k
O
e Q( x ) k (t , )
x
,
that is, the differential polynomial Q( ) acts onto the function k (t , ) e x , then we set the vector-
parameter equals to (1 , 2 , 3 ) . If among the functions f 0 ( x) , f1 ( x) , f 2 ( x) is zero, then the
corresponding summand in formula (16) is zero. Therefore, formula (16) for finding the solution of
problem (2), (3) assumes the implementation of a finite number of differentiation of functions
0 (t , ), 1 (t , ) , 2 (t , ) by parameters 1 , 2 , 3 .
The fact that function (16) is the solution of problem (2), (3) follows from the commutativity of
the differentiation operators t , x , , taking into account that the functions 0 (t , ) , 1 (t , ) ,
2 (t , ) satisfy equations (4) and conditions (7).
The fact that the found solution of three-point problem (2), (3) is unique in the specified class of
quasipolynomials can be proved by contradiction method (see, for example, [23]). The choice of
quasipolynomials f 0 ( x) , f1 ( x) , f 2 ( x) exactly from the class K M is significant.
Thus, if the functions f0 ( x), f1 ( x), f 2 ( x) in conditions (3) belong to the set K M , then
quasipolynomial solutions of problem (2), (3) are constructed by formula (16).
Note that numerous studies have been devoted to the construction of quasipolynomial solutions of
partial differential equations and boundary value problems for these equations [27–31].
Main result. The process of propagation of an ultrasonic wave in a relax medium with given wave
profiles at three time points is modeled by problem (2), (3) for the hyperbolic partial differential
equation of the third order in time (2) with three-point time conditions (3). The method of
constructing the solution of problem (2), (3) is proposed. The class of quasipolynomials as a class of
uniqueness solvability of problem (2), (3) is indicated. Equation (2) belongs to the important partial
differential equations which are used in the problems of ultrasound diagnostics.
3. The examples of application of the method to constructing solution of the
problem with given profiles of the ultrasonic wave at three moments of
time
Let us investigate the process of acoustic oscillations for specifically given right-hand sides of
three-point conditions and parameters of the differential equation. We use the method which is
proposed in the previous section to construct the solution of problem (2), (3).
Example 1. Let us consider problem (2), (3) for h 1 , f 0 ( x) x12 , f1 ( x) x2 , f 2 ( x) 2 . The
functions f 0 ( x) , f1 ( x) , f 2 ( x) are polynomials, therefore they have the form (14) and O . Since
(O) 1 e1 0 , then these functions belong to K M . So, the unique solution of the problem
2
exists in indicated above class of quasipolynomials (in particular, in a subclass of polynomials). The
solution we can find by formula (16):
u(t , x) 21 0 (t , ) e x 2 1 (t , ) e x 22 (t , ) e x
O O O
1 0 (t , )
2
2 x1 1 0 (t , ) x 0 (t , O)
2
1
O O
x21 (t , O) 2 1 (t , ) 22 (t , O)
O
et e1 (2 e1 ) e1 (1 e1 ) t
21 0 (t , ) 0 x12
O (1 e1 )2
2et 2 (1 e2 ) t et 1 (1 e1 ) t
x2 0 2 .
(1 e1 )2 (1 e1 )2
Therefore,
u (t , x) 21 0 (t , )
O
e e (2 e 1 ) e 1 (1 e 1 ) t
t 1
x12 (17)
(1 e1 ) 2
2e t 2 (1 e 2 ) t e t 1 (1 e 1 ) t
x2 1 2
2 .
(1 e ) (1 e 1 ) 2
In formula (17), the function (t ) 21 0 (t , ) is the solution of three-point problem for
O
nonhomogeneous ordinary differential equation
dt3 dt2 (t ) 2dt 0 (t , O) ,
(0) (h) (2h) 0 .
This function is obtained by differentiating the problem (4), (5) for 0 (t , ) by the parameter 1
at the point O . Calculations show that the function (t ) is a quasipolynomial of the form
(t ) c1 et 1 c2t Atet Bt 3 Ct 2 ,
2(1 )e2 e e 1 3e 2
where A , B , C , constants c1 and c2 satisfy a
(e 1) 2
3(e 1) (e 1) 2
nondegenerate system of algebraic equations
c1 e1 1 c2 Ae1 B C ,
c1 e 1 2c2 2 Ae 8B 4C.
2 2
Note that found solution (17) of the problem is a linear function of the parameters A , B , C and
.
Example 2. We consider the process of propagation of an ultrasonic wave in the relaxation
medium which is described by the problem (2), (3), where
1 x x x
, f 0 ( x) cos 1 2 3 , f1 ( x) f 2 ( x) 0 , h 1 .
9 3
The function f 0 ( x) is a sum of two quasipolynomials of form (14), namely
x x x x x x
x x x3 1 1 32 3 i 1 1 32 3 i
cos 1 2 e e ,
3 2 2
i 1 1
in which the vectors of parameters equal to (1,1,1) and to . Since and ,
2
3 3 9
then ( ) 2e 0 . Therefore, M .
1
The solution of problem (2), (3) for these data is found by formulas (16), (12):
1
3
u (t , x) cos 1 2 3 0 (t , ) e x
O
1
2
0 (t , ) e x
i
0 (t , ) e x
1
2 i
,
(1,1,1) (1,1,1)
3 3
where
0 (t , ) i (1,1,1) 0 (t , ) i (1,1,1)
3 3
1 .
3 1 t
1 t t 2 e 3
2 2
Finally, we obtain the following solution of the problem
1 3 1 2
x1 x2 x3 t x1 x2 x3 t
u (t , x) t t e e
3 3
2 4 4
3 1 t
1
x x x3
1 t t 2 e 3 cos 1 2 .
2 2 3
For graphically illustrating the process of acoustic oscillations, we consider the solution on parallel
planes x1 x2 x3 3 0 of the space 3 , where , 3 is the distance from the origin of
coordinates to the plane.
The solution of problem in variables t and is 2 -periodical function by and has the such
analytical factorized form
(t 1)(t 2) 13 t
u (t , ) e cos . (18)
2
The graph of function (18) of two variables t and is depicted in Figure 2.
Figure: 2. Graphical dependence of the solution u(t , ) on time t and distance .
Therefore, function (18) describes the periodic oscillations of the ultrasonic wave with the period
T 2 by the variable . The amplitudes of these oscillations for 0 and are determined
3
by the corresponding formulas
1 1
3 1 t 1 3 1 t
A1 (t ) 1 t t 2 e 3 , A2 (t ) 1 t t 2 e 3 .
2 2 2 2 2
These amplitudes are depicted in Figure 3 by a top and bottom lines.
Figure 3: Graphs of amplitudes of the oscillating process
As noted above, the found solution is unique in the class of quasipolynomials which for the fixed
t belong to K M .
Note that the ultrasonic wave oscillates in a limited range at arbitrary time moment and goes
exponentially to zero for t on planes which are parallel to the plane x1 x2 x3 0 .
4. Conclusions
The mathematical model of the process of ultrasonic wave propagation in a relaxation medium
under the condition of setting the wave profile at three time points is investigated. The model is
reduced to a problem with three-point time conditions for a hyperbolic partial differential equation of
the third order.
The class of quasipolynomials as a class of uniqueness solvability of the problem is established
and a practically effective method of constructing the solution in this class is proposed. The examples
of application of the specified technique are given.
The proposed method is important in mathematical modeling of acoustic oscillatory processes in
relaxation environments. The results of the research can be used in medicine, in particular, in the
theory of ultrasound diagnostics.
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