LONGITUDINAL WAVES IN THE WALLS OF AN ANNULAR CHANNEL FILLED WITH LIQUID AND MADE OF A MATERIAL WITH FRACTIONAL NONLINEARITY Текст научной статьи по специальности «Физика»

Нелинейные волны.

Солитоны. Автоволны. Самоорганизация

Известия высших учебных заведений. Прикладная нелинейная динамика. 2023. Т. 31, № 3 Izvestiya Vysshikh Uchebnykh Zavedeniy. Applied Nonlinear Dynamics. 2023;31(3)

Article DOI: 10.18500/0869-6632-003040

Longitudinal waves in the walls of an annular channel filled with liquid and made of a material with fractional nonlinearity

L.I. MogilevichE. V. Popova

Yuri Gagarin State Technical University of Saratov, Russia E-mail: ЁЗmogilevichli@gmail.com, elizaveta.popova.97@bk.ru Received 26.02.2023, accepted 14.03.2023, available online 11.05.2023, published 31.05.2023

Abstract. Purpose of this paper is to study the evolution of longitudinal strain waves in the walls of an annular channel filled with a viscous incompressible fluid. The walls of the channel were represented as coaxial shells with fractional physical nonlinearity. The viscosity of the fluid and its influence on the wave process was taken into account within the study. Metods. The system of two evolutionary equations, which are generalized Schamel equations, was obtained by the two-scale asymptotic expansion method. The fractional nonlinearity of the channel wall material leads to the necessity to use a computational experiment to study the wave dynamics in them. The computational experiment was conducted based on obtaining new difference schemes for the governing equations. These schemes are analogous to the Crank-Nicholson scheme for modeling heat propagation. Results. Numerical simulation showed that over time, the velocity and amplitude of the deformation waves remain unchanged, and the wave propagation direction concurs with the positive direction of the longitudinal axis. The latter specifies that the velocity of the waves is supersonic. For a particular case, the coincidence of the computational experiment with the exact solution is shown. This substantiates the adequacy of the proposed difference scheme for the generalized Schamel equations. In addition, it was shown that solitary deformation waves in the channel walls are solitons.

Keywords: wave dynamics, annular channel, viscous fluid, fractional nonlinearity, computational experiment. Acknowledgements. The study was funded by Russian Science Foundation (RSF) according to the project No. 23-29-00140.

For citation: Mogilevich LI, Popova EV. Longitudinal waves in the walls of an annular channel filled with liquid and made of a material with fractional nonlinearity. Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(3):365-376. DOI: 10.18500/ 0869-6632-003040

This is an open access article distributed under the terms of Creative Commons Attribution License (CC-BY 4.0).

Научная статья УДК 530.182

DOI: 10.18500/0869-6632-003040 EDN: RKTVVT

Продольные волны в стенках кольцевого канала из материала с дробной нелинейностью, заполненного жидкостью

Л. И. Могилевич^, Е. В. Попова

Саратовский государственный технический университет имени Гагарина Ю. А., Россия E-mail: E3mogilevichli@gmail.com, elizaveta.popova.97@bk.ru Поступила в редакцию 26.02.2023, принята к публикации 14.03.2023, опубликована онлайн 11.05.2023, опубликована 31.05.2023

Аннотация. Целью данной статьи является исследование эволюции продольных волн деформации в стенках кольцевого канала, заполненного вязкой несжимаемой жидкостью. Стенки канала представлялись коаксиальными оболочками с дробной физической нелинейностью. В ходе исследования учитывалась вязкость жидкости и ее влияние на волновой процесс. Методы. Используя метод двухмасштабных разложений получена разрешающая система двух эволюционных уравнений, которые представляют собой обобщенные уравнения Шамеля. Дробная нелинейность материала стенок канала приводит к необходимости использования вычислительного эксперимента для исследования волновой динамики в них. Вычислительный эксперимент проводился на основе получения новых разностных схем для системы эволюционных уравнений. Эти схемы получены с использованием техники базиса Грёбнера и аналогичны схеме Кранка-Николсона для моделирования распространения тепла. Результаты. Численное моделирование показало, что скорость и амплитуда волн деформации остаются неизменными, а направление распространения волн совпадает с положительным направлением продольной оси. Последнее указывает на то, что скорость волн сверхзвуковая. Для частного случая показано совпадение вычислительного эксперимента с точным решением. Это обосновывает адекватность предложенной разностной схемы для обобщенных уравнений Шамеля. Кроме того, показано, что уединенные волны деформации в стенках канала являются солитонами.

Ключевые слова: волновая динамика, кольцевой канал, вязкая жидкость, дробная нелинейность, вычислительный эксперимент.

Благодарности. Работа выполнена при поддержке Российского научного фонда (РНФ) в соответствии с проектом № 23-29-00140.

Для цитирования: Могилевич Л. И., Попова Е.В. Продольные волны в стенках кольцевого канала из материала с дробной нелинейностью, заполненного жидкостью//Известия вузов. ПНД. 2023. T. 31, № 3. С. 365-376. DOI: 10.18500/0869-6632-003040. EDN: RKTVVT

Статья опубликована на условиях Creative Commons Attribution License (CC-BY 4.0).

Introduction

The wave technologies of nondestructive testing are increasingly being used in different industries, for example, for pipeline diagnostics. Papers [1-5] are devoted to various aspects of wave propagation in rods, plates and shells. In the above papers, the effect on the wave process in the shell of a viscous liquid located inside it was not studied. The shell-fluid interaction was studied in [6] outside the consideration of wave processes. On the other hand, in [7] the wave process in an annular channel is studied considering the fluid inertia forces and cubic nonlinearity of shells forming the channel. In the cases of shells containing a viscous fluid, the use of qualitative analysis methods for the analytical study of nonlinear models for deformation waves causes significant difficulties [7-9]. Consequently, for these cases it is necessary to carry out computational experiments [10]. In our study, the system of governing

equations for wave processes in two cylindrical shells forming an annular channel are obtained by the perturbations’ method with respect to a small parameter of the problem, as well as considering the fractional physical nonlinearity of the shell material. This system is two generalized Schamel equations and allows us to estimate the wiggle of dissipative properties of the fluid on longitudinal strain waves in the channel walls. For a particular case, an exact solution of this system is found, for the general case, difference schemes are developed, and its numerical solution is performed.

1. Mathematical statement of the problem

Let us consider an annular channel formed by two cylindrical shells. Further in the paper we denote by the index i = 1 the parameters for the outer shell and i = 2 the parameters for the inner shell. We assume that the Cartesian coordinate system xyz associated with the channel symmetry axis and the corresponding cylindrical coordinate system rQx are introduced. The ж-axis coincides with the symmetry axis and the z-axis is directed along the normal to the unperturbed middle surface of the shells. For the shells material, relationship between the stress and strain tensors, as well as deformation intensity in the frames of the plasticity theory of A. A. Ilyushin [11,12] is

4 = E ( poe^ + eft

A*) p /AW + и A®)

°e _ E + цо£ж

1+ E

£(г)1 m 1+ E

/ (i - m0) / (i - m0)

4° _2 (hi (4)2+4°2) - и24г)4г))2 /v/3,

Ц1

1 +

Ho

(1 - Ho) -

2 , H2 _ 3

1

2po

(1 - Ho)2 J

1

1

3

(1)

here po is the shell material Poisson’s ratio, E is the shell material Young’s modulus, m is the constant determined from tensile-compression experiments [13], ox, oe are the normal stresses along the x and e axes; ex, ee are the tensile-compression strains along the x and e axes; eu is the strain intensity. Note that the relation of stresses ox, oe with strains ex, ee and strain intensity eu on the basis of the physical law with nonlinearity in the form of power function with a fractional value of the exponent for the case of incompressible material, i.e. when po _ 1/2, is considered in [5,14].

The relationship between the components of the strain tensor and i-th shell displacements has the form

£

(0

e

r(i) _

dU(i) d2W(i)

dx W(i)

— 2-

dx2

W(i)

- ZR(i)2

at

К

w

К

w

X

2

(2)

Here z is the local coordinate along the axis normal to the shell middle surface (z _ 0 corresponds to the shell middle surface), x is coordinate along the longitudinal axis of the shell median surface, W(г) is the deflection of the г-th shell, the positive direction of which is taken to the shell curvature center,

и « is the longitudinal displacement of i-th shell, Е(г) is the median surface radius of the i-th shell, is the i-th shell thickness.

The asymptotic analysis carried out in [7] showed that the intensity of deformations in (1), (2) can be considered on the middle surface (z = 0) for longitudinal waves. Consequently, forces acting on the element of the shell middle surface are determined by the formulas

4°/2

N(J) =

jo* = ML fM _,, M

~ d = ' ° [ dx ,0 r(<)

-h^/2

1 — ,2 \ dx

)x

( 2 \ 2 ( / dU(i)\ 2 ( w(i) \

w*) ,1 Ц+[Wj

_ ml 2 \ 2 1 + ё' ^

h(P /2

nA =

Og* ^dZ =

dx

dU(i) W(i) dx R(i)

~hf/ 2

)X

( 2 \ 1 ( ( dU(i)\ 2 ( w(i) \

w*) ц1Ц+ (,

_ ml 2 \ 2

1+ Ё* ^

+ P2“

9^(i) M« R(i)

dU(i) M«

+

(3)

and the moment is defined as

h^/2

M« =

f (i) , Eh0)3 (d2W(i) w(i) \

J O*zdz = — ~toT + ,0Ж) ’

-h^/2

(4)

We write the dynamics equations for the i-th shell

dNf]

dx

P0h0° — #

dt2

d^MA d ( dW(i)

dx2

+ TZ~

(+ -^y N® = poh0° dx у dx x j Rd) g K0 0

d2W(i) dt2

— (—1 )г-Ап

R«

(5)

Here p0 is the г-th shell material density, qx , qn are the shear and normal fluid stresses; r, x are the cylindrical coordinates, t is the time.

Substituting (3), (4) into the shell dynamics equations, we obtain the equations in displacements

Eh0) d l dU(i)

1 — ,2 dx \ dx

W(i)

m

~E

^0 R(£) + E (^3) ^ [

dU(i) W(i)

dx

,0

R(i)

,1

( dU(i) \ у dx J

+

+

f W(i) \ ' \W )

du(i) w(i) + ,2 ~ЖЖ

P л«д2и(г) P0h0 dt2

J Ri)

2

1

2

4

2

]

Eh0 d2 / h°)2 f d2W(i) W(i)\ \

12 (1 — m°) dx2 \ 12 ^ 9ж2 + M° R(i)2 J I +

+

Eh° d / dW(i)

1 — m2 dx\ dx

('

dU(i)

W(i) m ( 2 \ 1 I rdU(i)

M°-

+

/ W(i) \

J

(A)

m , _

+~Ё 1 ~

M°-

dx r"° R(i) E

dU(i) W(i)

+

dU(i) W(i)

Ш

ж Ж [

w(i)

dx

Mo

m°-

Mi

dx R(i)

\ + ML J_/

/ 1 — m2 r"’\

) '+(D')

R(i)

dU(i) W(i) dx R(i)

( dU(i) V

у dx J

+

dUW

Mi|t g.x MT + M2feT

>

+

P°h0] — [( —1)1 1^n]R(i) . (6)

2

2. Analysis of equations for shells containing fluid by perturbation method

To analyze the wave process in the channel walls, we consider a dimensionless axisymmetric problem and choose small parameters of the problem. Namely, we assume

W(i) = wmu3),

U(i) = umyi\

x = —

, = 2,

r =

R(i)

wm — h°, Um —

X

~v

h°l

R(i)'

r

(7)

here c° = д/E/(p°(1 — m°)) is the sound velocity in the shell, l is the wavelength, um, wm are the characteristic values of elastic shells displacements. Let us make the following assumptions

h«

h°

R(i)

£ < 1,

R(d2

l2

О (e2)

^ = OX),

h

^ ^ = 0(1) ™ = 0(1)

l h(d U(1)} E U(1)}

(8)

here e is the small parameter.

The method of two-scale expansions is applied and the dependent variables as an asymptotic expansion are represented

,(0 _ Jd

(i)

u\ = u\° + £ 2 u\ 1 + . . . ,U:

(г) . 1 (i) ,

= u(3° + e2 и\1 + ....

1

Independent variables are introduced in the form of

^ = X* — \J 1 — m2^*, T = £ 2 t*.

(9)

(10)

Here t is the slow time.

Substituting (7)-(10) into (5) in the zeroth approximation by e gives

(i) 9UW

46 = h0

« я 2 v(i)

10 u u'10

dl ’ dl2

ho-

дизо

w

= (1 - m0)

d2ul

dl2

Thus uf0 is an arbitrary function. Then the system of equations in the next approximation with account (11) is we obtain in the form of

d2u^

dldx E

10 + 1 Ш 4V1 - ^0 (hi + h2ho + hiho)

1 . . if 2 d2ui;0 h0V1 —he 94«i*0)

d l2

+

dl4

2V1 - h0 £2Р0^0г)с0 _

(#) _ ^i

. (12)

. R(i)

-u

(i)

The obtained Eqs. (12) are the generalized Schamel equations for -щ0. If the fluid is excluded, we will have two homogeneous uncoupled Schamel equations. To determine fluid stresses in Eqs. (12) it is required to study the fluid motion in the channel.

2

(

1

3. Determination of stresses acting on the shell from the liquid

Two coaxial infinitely long shells forming an annular channel with a viscous fluid whose density is constant are considered. The width of the slot occupied by the liquid 8 = Ei — R2, where Ri is the inner surface radius of the outer shell and R2 is the outer surface radius of the inner shell. The fluid motion equations of creeping flow for the problem under consideration have the following form [15]

1 dp

p dr

1 dp

p dx

d2Vr 1 dVr d2 К

----- +-----r +------

Qf2 rp

V-

dx2 r2

(d2Vx 1 dVx d 2VX

l dr2

+ -

r dr

+

dx2

) ■

v

v

1 d_

r dr

(rVr) +

dVx

dx

0.

(13)

At the boundary of the shells and the liquid, the no-slip conditions of the liquid are satisfied [15]

Vx

dU(i) dt

V-

dW(i) dt

at r

Ri — W(i),

(14)

here r, x is the cylindrical coordinates, VX, Vr are fluid velocity projections on the coordinate axis, p is the fluid pressure; p is the fluid density, v is the kinematic coefficient of fluid viscosity.

The fluid stresses ) and qn are determined at r = R(г)

qn

■p + 2pv

dVr dr ’

#

pv

fdVx + dvA \ dr dx J

(15)

Dimensionless variables and parameters are introduced

Vr

h« - v

hQ ^ Ur,

у — h(t) — v

VX — hQ g vX

*

Г —

- r(2)

R(2)

1

e 2,

l —

h

W

i

e 2.

h

(0

h

E(i>

W

Q_

l

r

t

g

g

Г

Со, * ж

:т', ж — V р

5 g 3

£ 4 , Re

1 ’

pvcolh^ p g3 g Scq

1 ~ — £.

The variables (16) are substituted into Eqs. (13), (14), (15), and the following decompositions

P — PQ + e1P1 + ... vr

VQ + £ 2 v\ + . .

Vx

Vx + £ 2 Vx +

i

(17)

are taken into account. After that, for the first terms of expansion (17) are obtained dynamics equations of the thin fluid layer for creeping flow [7, 15]

dPQ 0 dPQ 0,

Qp*

dx*

d2vx dvx dvx о

dr*2 ’ dx* + dr*

(18)

with boundary conditions

vQ — --

du{ 1>

3 „,Q

dui2>

vux — 0 at r* — 1, vQ — -:-q3y, V° — 0 at r* — 0

dt*

(19)

as well as with the accuracy to ф, £ 2 from (15) are obtained, too

(1> hX^cx dvQ *

q9) — -pv at r

— 1 Xx) — -

g2 dr*

Solving the problem (18), (19) we obtain

pv

hQ2')cx dvQ* g2 dr*

at r* — 0, qn — —

pvCxlhQ) Q

g3

P Q.

(20)

PQ — 12

/[/ f#—¥)**

dx*, —£■ — 6 (r*2 — r*

’ dt* v

/(

9242) dr*

dt*2 dt*2

(21)

Bearing in mind (9), the new variables l, т (10) and with the accuracy to £ 2 we write (21) as

P Q

1<2\/1 — hQj (4Q> — 4Q>) dl, — 6^1 — hQ (2r* — 1) (4Q> — 4Q>) > (22)

and then

dP Q

~ж

1<2\[1 4 (u3Q u3Q^j,

9vx

Qp*

r* = 1

б^1 — hQ (4Q> — 4Q>) ,

9vx

Qp*

r=Q

dvx

Qp*

(23)

r* = 1

Given that according (11) w3o = ^odu^j^/9% submiting (22), (23) into (20) and assuming 1] = 2) = R, h01] = h02) = ho due to smallness ф, X the right-hand side of the first equation

(i = 1) of system (12) was obtained as

« 2 P^ -6ho

poho Rc0e2 V 8.

(R\Vduw _ duw \ W \ 9? d l )

(24)

and the right-hand side of the second equation (i = 2) of system (12) was obtained in the form of

« 2 P^ -6Ио

poho Rao£2 \ 8.

(R\3 f дЦц _ c]uw_\ W V d I )

(25)

4. Governing equations of deformation waves

Taking into account the found right-hand parts of Eqs. (24) and (25), the system of Eqs. (12) is rewritten as

d2 u1o] m 3

dldx E 4

+ f 4 Y - Ho ((b1 + h2ho + h1b2)

9u$j\2 92u(11o) , Иа V1 - ^o

dl2

+

914

2 pl v /R\3/9мЮ) ^Mio)

6Ho poho Ecoe ( 8 )

91

92u(2,) m 3

dldx E 4

+ f 4 V1 - ho ((И1 + b2ho + ИИ^) ^

2 92иЦ) . wV1 - ho

9l2

+

= -6w

2 pZ v /R\3/

poho Rco£ \ 8

(?)

9l

91

)

The following notations are introduced

du^/dl = сзф(1), du^/dl = сзф(2), n = ci l, f = C2T,

2 P V

C2 = 6p0„ _2

C3 =

po£2 8coe4

C2 E

C1 = (2c2^Ио^1 - ho

C1 m 3y/1 - ho(2Д/3)2 (h1 + h2ho + h1ho) 4

(27)

Considering (27) in (26) the system of governing equations for the study of longitudinal deformation waves in the walls of the annular channel is obtained

9ф(1) (1) 2 9ф(1) 93ф(1) /1) (2)

аТ + 6ф(1) + "^ + ф(1) - ф(2) = °,

9ф(2) (2)2 9ф(2) 93ф(2) /2) /1)

-фг+6ф(2) ’ -Ьт + -^Г +ф(2) - ф(1) = “■

(28)

v

v

2

2

2

4

6

System of Eqs. (28) has exact partial solution

ф(!) = ф(2) = 25к4 (i+chк (p — 4k2t)) 2 ,

but the general case requires a numerical solution to this system. By implementing a numerical solution to equations (28), the initial conditions at t = 0 in the form of solutions (29) can be used

Ф(1)(0, n) = ф(2)(0, n)

25

T

fc4(1+ch кц) 2

(30)

or

ф(1)(0, n)

25

T

fc4(i+ch кц) 2,

ф(2)(0, n) = 0.

(31)

5. Computational experiment results

The computational experiment was carried out similarly to [7], but taking into account the fractional nonlinearity. The desired difference scheme for the numerical solution of the system of equations (28) was obtained using the Grobner basis technique in the Maple computer algebra system. The resulting difference scheme is similar to the Crank-Nicholson scheme for the heat equation [16] and has the form

„и) »+■—„dr*+4 (u(i),'s—«.<1)>/2"+;)+(„<1)V2 -+,—u(i),'s; 1)

4h

+

+

(«.(1) —2«.(1) +2»(1) —u(1)^21) («(1)"+2—2»(1) ■+1+2«(1) ”. 1—««;_2)

--------------------7~Tn------------------ + Л-------------------777!-----------------~ +

4h3

4h3

+

(1) "+1+u(1): u(2) "+1+u(2)

„(2) »+■—„(2)» (u(2)>'s;++;—»(2),'s;+;) + («(2)i/s;+1—»(2)s/2 ”_ 1)

_Г + 4T_______________________L___b_____________________L +

3

т

4h

+

(u(2) "++21 — 2u(2) "++1 + 2u(2) "+1 — u(2) "+21) (m(2) i+2 — 2u(2) ™+! + 2u(2) ! — u(2) 2)

4h3

4h3

+

+

uy

(2) "+1 + U(2) ? U(1) "+1 + u(1)

0.

(32)

т

3

0

2

2

3

2

2

Within this scheme, terms with fractional nonlinearity for the next time step are linearized as

3/2 _ v3/2 k+1

V+1 - Vfc

3/2 3/2

3 + vk _

(V^21 - (Vfc+1 + v1+21Vl/2 + Vfc) +

3/2

1/2 1/2

(v1/2 l/2^ vfc+1 + vfc (v I v1/2 v1/2 I .Л I v3J2~. (v v l3 v1/2 2v3/2

^vfc+1 - vfc ) ~TJ2 1/2 vfc+1 + vfc+1vfc + vfcJ + vfc ~ (vfc+1 - vfc1 2Vfc - 2 k _

1/2 fk+1

v1J2 + v1J2 vfc+1 + vfc

3 1/2

-v

2

к

vfc+1 - 2v

1 3/2

(33)

к

к

Models (28), (30) and (28), (31) were numerically studied by using this difference scheme. We consider the initial condition (30) with к _ 0.2, the numerical simulation results of the wave process are shown in Fig. 1.

According to Fig. 1, it can be seen that the waves propagate to the right without changing the speed and amplitude (supersonic speed). The numerical solution coincides with the analytic solution (29). Then the initial conditions (31) are consider with к _ 0.2 and the calculation results are shown in Fig. 2.

According to Fig. 2, it can be seen that in the presence of a disturbance in the outer shell and its absence in the inner shell at the initial moment of time, the wave amplitude in the outer shell decreases with time, while in the opposing shell it increases. The wave amplitudes are equalized, which indicates the transfer of energy through the liquid layer between the shells.

Let us consider the case when at the initial moment a perturbation is given in the form of two waves (30) with different amplitudes and speeds assuming к _ 0.225 for the first wave and к _ 0.2 for the second wave. The numerical simulation results are presented in Fig. 3.

Fig. 1. Results of numerical solution of equations (28) with initial conditions (30) (color online)

ф(|)

0.0025 • 0.0020 ■ 0.0015 ■ 0.0010' 0.0005 ■ 0-

ф4-'

Л / = 0 0.0025 — 1 = 0

[\ — /—36.23 — / = 36.23

/ \ —/=72.45 0.0020 — / = 72.45

1 —/=108.68 — /=108.68

1 —/=144.90 0.0015 — /=144.90

Ж 0.0010 Ш

0.0005 11/ш

0

С—-

-80 -60 -40 -20 0 20 40 60

-80 -60 -40 -20 0 20 40 60

Fig. 2. Results of numerical solution of equations (28) with initial conditions (31) (color online)

Fig. 3. Results of numerical solution of equations (28) with two initial conditions (30) for fc=0.225 and k=0.2 (color online)

It follows from Fig. 3 that there is an elastic interaction of waves as particles. This means that deformation waves are solitons.

Summary and conclusion

The numerical simulation of nonlinear wave process in the walls of an annular channel indicates the need of accounting the presence of viscous fluid in the channel in the study of longitudinal deformation waves propagation. Excitation of the strain wave in the outer shell at the initial moment of time leads to the appearance of the strain wave in the opposite shell. In other words, energy transfer from one shell to the other occurs via the liquid. This process is accompanied by a decrease in the amplitude of the wave in the outer shell, which leads to a decrease in the rate of propagation of the deformation wave in this shell. At the same time, the amplitude of the wave in the opposite shell increases. Due to fluctuations in amplitudes and velocities, their amplitudes be come equal in cause of time. In addition, for the case where a solitary strain wave is excited in each shell at the initial moment of time, calculations have shown that these waves are solitons. The results obtained can be used for the development of non-destructive methods of control of pipelines with viscous liquids used in devices, machines and units, as well as the control of working processes.

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Могилевич Лев Ильич — родился в Саратове (1946). Окончил с отличием механикоматематический факультет Саратовского государственного университета по специальности «Механика» (1969) и аспирантуру по кафедре теоретической механики и аэрогидродинамики (1972). Защитил диссертацию на соискание учёной степени кандидата физикоматематических наук на тему «Некоторые задачи теории коротких волн» по специальностям «Механика жидкости газа и плазмы» (1972, СГУ). Защитил диссертацию на соискание учёной степени доктора технических наук по специальности «Приборы навигации» (1989, Киевский политехнический институт). Является действительным членом Международной академии информатизации (с 1996). С 2015 года работает на кафедре «Прикладная математика и системный анализ» Саратовского государственного технического университета им. Гагарина Ю. А. в должности профессора. Научные интересы — нелинейные колебания и волны в механике сплошных сред, асимптотические методы решения нелинейных задач. Опубликовал свыше 300 научных статей по указанным направлениям.

Россия, 410054 Саратов, Политехническая, 77

Саратовский государственный технический университет имени Гагарина Ю. А.

E-mail: mogilevichli@gmail.com ORCID: 0000-0003-2642-762X AuthorID (eLibrary.Ru): 2927

Попова Елизавета Викторовна — родилась в Саратове (1997). Окончила с отличием механико-математический факультет Саратовского государственного университета по специальности «Прикладная математика и информатика» (2019). В настоящее время обучается в аспирантуре СГТУ имени Гагарина Ю. А. Научные интересы — нелинейные колебания и волны в механике сплошных сред. Опубликовала более 40 научных статей.

Россия, 410054 Саратов, Политехническая, 77

Саратовский государственный технический университет имени Гагарина Ю. А.

E-mail: elizaveta.popova.97@bk.ru ORCID: 0000-0001-6374-7468 AuthorID (eLibrary.Ru): 847784