Difference methods for solving some classes of multidimensional loaded parabolic equations with boundary conditions of the first kind Текст научной статьи по специальности «Математика»

УДК 519.63

doi: 10.21685/2072-3040-2024-2-3

Difference methods for solving some classes of multidimensional loaded parabolic equations with boundary conditions of the first kind

Z.V. Beshtokova1, V.A. Vodakhova2, M.Z. Khudalov3

institute of Applied Mathematics and Automation KBSC RAS, Nalchik, Russia 2Kabardino-Balkarian State University named after H.M. Berbekov, Nalchik, Russia 3North Ossetian State University after K.L. Khetagurov, Vladikavkaz, Russia 1zarabaeva@yandex.ru, 2v.a.vod@yandex.ru, 3hmz@mail.ru

Abstract. Background. In the literature, loaded differential equations are usually called equations containing functions of the solution on manifolds of smaller dimension than the dimension of the domain of definition of the sought function. The purpose of the work is to study a difference scheme of second order accuracy in terms of mesh parameters for solving the first boundary value problem for loaded parabolic equations in a multidimensional domain with variable coefficients. Two different types of equations are considered. Problems of this type arise when studying the movement of groundwater, in problems of managing the quality of water resources, when a pollutant of a certain intensity enters a reservoir from n sources, when constructing a mathematical model of the transfer of dispersed pollutants in the atmospheric boundary layer when describing the mass distribution function of drops and ice particles, taking into account microphysical condensation processes, coagulation (combination of small drops into large aggregates), fragmentation and freezing of drops in convective clouds, as well as in the study of natural processes and phenomena that take into account the memory effect. Materials and methods. The finite difference method and the method of energy inequalities are used to obtain a priori estimates for the solution of difference schemes. Results. For each problem, a difference scheme with the order of approximation O h|2 +xm°) is constructed, where mo = 1, if o^ 0,5 and mo = 2, if

o = 0,5; an a priori estimate was obtained using the method of energy inequalities to solve the difference problem. From the obtained estimates it follows that the solution is unique and stable with respect to the right-hand side and the initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding initial differential problem with a speed O(|h|2 + xm°) at o = 0,5. Conclusions. New numerical

schemes of second order approximation have been developed to solve the problems posed.

Keywords: first initial-boundary value problem, loaded equation, a priori estimate, difference scheme, parabolic equation

For citation: Beshtokova Z.V., Vodakhova V.A., Khudalov M.Z. Difference methods for solving some classes of multidimensional loaded parabolic equations with boundary conditions of the first kind. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2024;(2):25-39. (In Russ.). doi: 10.21685/2072-3040-2024-2-3

Разностные методы решения некоторых классов многомерных нагруженных параболических уравнений с граничными условиями первого рода

© Beshtokova Z.V., Vodakhova V.A., Khudalov M.Z., 2024. Контент доступен по лицензии Creative Commons Attribution 4.0 License / This work is licensed under a Creative Commons Attribution 4.0 License.

З. В. Бештокова1, В. А. Водахова2, М. З. Худалов3

1Институт прикладной математики и автоматизации Кабардино-Балкарского научного центра РАН, Нальчик, Россия 2Кабардино-Балкарский государственный университет имени Х. М. Бербекова, Нальчик, Россия 3Северо-Осетинский государственный университет имени К. Л. Хетагурова, Владикавказ, Россия

1zarabaeva@yandex.ru, 2v.a.vod@yandex.ru, 3hmz@mail.ru

Аннотация. Актуальность и цели. Нагруженными дифференциальными уравнениями в литературе принято называть уравнения, содержащие функции от решения на многообразиях меньшей размерности, чем размерность области определения искомой функции. Главной целью работы является исследование разностной схемы второго порядка точности по параметрам сетки для решения первой краевой задачи для нагруженных параболических уравнений в многомерной области с переменными коэффициентами. Рассмотрены два разных вида уравнений. Задачи такого типа возникают при изучении движения подземных вод, в задачах управления качеством водных ресурсов, когда в водоем поступает из n источников загрязняющее вещество определенной интенсивности, при построении математической модели переноса дисперсных загрязнений в пограничном слое атмосферы при описании функции распределения по массам капель и ледяных частиц с учетом микрофизических процессов конденсации, коагуляции (объединение мелких капель в большие по размеру агрегаты), дробления и замерзания капель в конвективных облаках, а также при изучении процессов и явлений природы, учитывающих эффект памяти. Материалы и методы. Используется метод конечных разностей, метод энергетических неравенств для получения априорных оценок решения разностных схем. Результаты. Для каждой задачи построена разностная схема с порядком аппроксимации O h|2 + тт°), где

то = 1, если о Ф 0,5 и то = 2, если о = 0,5; методом энергетических неравенств для

решения разностной задачи получена априорная оценка. Из полученных оценок следуют единственность и устойчивость решения по правой части и начальным данным, а также сходимость решения разностной задачи к решению соответствующей исходной дифференциальной задачи со скоростью Oh|2 +т™° ) при о = 0,5. Выводы. Разработаны новые численные схемы второго порядка аппроксимации для решения поставленных задач.

Ключевые слова: первая начально-краевая задача, нагруженное уравнение, априорная оценка, разностная схема, параболическое уравнение

Для цитирования: Beshtokova Z. V., Vodakhova V. A., Khudalov M. Z. Difference methods for solving some classes of multidimensional loaded parabolic equations with boundary conditions of the first kind // Известия высших учебных заведений. Поволжский регион. Физико-математические науки. 2024. № 2. С. 25-39. doi: 10.21685/20723040-2024-2-3

Introduction

Loaded differential equations in the literature are usually called equations containing functions of the solution on manifolds of smaller dimension than the dimension of the domain of definition of the desired function [1]. Boundary value problems for loaded differential equations arise when studying the movement of groundwater, in problems of managing the quality of water resources, when a pollutant of a certain intensity enters a reservoir from n sources, and when

constructing a mathematical model for the transfer of dispersed pollution in the boundary layer of the atmosphere.

The purpose of this paper is to construct and study a second-order accuracy difference scheme for solving the first boundary value problem for loaded parabolic equations in a multidimensional domain with variable coefficients.

The scientific novelty of the work is the development of new numerical schemes of the second order of approximation for solving the problems. For each problem, a difference scheme is constructed with the order of approximation

O (|h|2 +1™°), where m° = 1 if c^0.5 and m° = 2, if c = 0.5; an a priori

estimate is obtained by the method of energy inequalities for solving the difference problem. The obtained estimates imply the uniqueness and stability of the solution with respect to the right-hand side and initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding original

differential problem at a rate of O (|h|2 +Tm°) for c = 0.5, where A, are the grid

parameters, a is the parameter that determines the approximation and stability of the difference scheme.

Problems of this kind arise when describing the mass distribution function of drops and ice particles, taking into account the microphysical processes of condensation, coagulation (association of small drops into large aggregates), crushing and freezing of drops in convective clouds [2-7], as well as when studying the processes and phenomena of nature, taking into account the effect of memory [8, 9].

Many papers of the authors' are devoted to the study of various boundary value problems for multidimensional partial differential equations with variable coefficients; we note the papers [10-13] among them.

1. Statement of the first initial boundary value problem

In the cylinder QT = G x[0 < t < T], whose base is a /»-dimensional rectangular parallelepiped G = {x = (xx , x2, • • •, xp ): 0 < xa < la, a = l2^-^ P} with border r, G = G u r, the problem is considered

^ = Lu + f (x,t), (x,t)e Qt, (1) at

u |r = 0, 0 < t < T, (2)

u(x,0) = uq (x), xe G, (3)

where

P

d

( du >

Lu = ^Ax^ Lau = d t)

a=1 dx«

V dxa

m

(x t)u(x1,--,xa-1,^a,xa+1,--,xp,t),

s=1

0<с0 <ka(x,t)<Ci,

ZM^t)

s=1

< C2, a = 1, p,

(4)

are fixed points of interval (0,la), 5 = 1,2,...,m,c0,q,C2 are positive constants, QT = Gx(0 < t <T].

In what follows, we assume that the solution of the differential problem under consideration exists, is unique, and has all the continuous derivatives necessary in the course of the presentation.

Denote by Mi (i = 1,2,...) positive constants that depend only on the input data of the original differential problem (1)-(3).

2. Difference scheme construction

Let us partition the p -dimensional space of variables xj,..., xp (hyperplane

of dimension p) (p -1) -dimensional hyperplanes

X a = 7 h

a = 1,2,...,p, where

ha = ——), ia are integers, into p -dimensional parallelepipeds [14]. The vertices of

Na

these parallelepipeds will be called grid nodes. The set of nodes belonging to the open domain G = G \r will be called internal nodes and denoted by

f

®h = i xi =

.(il)

X1 ,., xp

0 < la< Na

The set of nodes belonging to the boundary r will be called the boundary nodes Yh = {xi e r}. On the interval 0 < t < T we introduce the grid

MT=|tj = jT, j = 0,1,., m, mT = t}.

Thus, a uniform mesh ®hT is introduced in the closed domain Qt [14]:

®hT=«h xwT=|(xi,tj), xe«h, tewT},

®h = n ®ha, ®ha = { = iaha,ia = 0,l. • ^Na, Naha = ¿a} a=1 a a L J

®T=|tj = jT, j = 0,1,.,m, mT = T}.

On the grid Mhx, we assign to the differential problem (1)-(3) a difference scheme with weights of the order of approximation O(|h|2 + Tm° ):

(5)

(6) where

28

yt = Л(t )У(0)+Ф, (x,t)efflht, У |Yh = 0, У(x,0) = u0 (x),

A(f) = ±A0(f). A0(f)= (aoy^ _tdsa + y£Ua,

a=1 s=1

y<0>=cy + (1 _o)y, a(+1»)=aOJa+1, y = yj+\ y = yj,

yx = *Zz_L,, = y+izy,y, = zi+LV, m<j =J2- ^ = 0-5-

h ha yt T G 11, ifg^0.5,

= ka(x 05a,tj) dsa = 4sa(xi,tj) 9i = f(xi,tj) x(( +1)

x- =:ra_x + x(as)<ps < J^+1)

xi„ ~ i > i ~ i ' a ^a a

"a as "a

x"05a = xb..., Xa_1, Xa- 0.5ha, Xa+1,..., xp, t = (( + 0.5)t = tj + 0.5t = tj+o.5,

[ a >

= ax =

t a =t j + =

j+- P p

a j +-

v P /

T, | h |2 = h2 + h2 + • + hp, T,h are grid steps.

Note that the expression

(°) r(as +1)_Ps (°) £s _ r(as ) y\<°) xa_za + yy°> ^a xa

ia s ha ias +1 ha

that we use to approximate u( x1, •••, xa_ 1, ^a, xa+1, •, xp, t) was previously

introduced in [15] to approximate the loaded part.

3. Stability and convergence of the difference scheme

To obtain an a priori estimate for the solution of the difference problem (5), (6), we use the method of energy inequalities. We introduce the scalar product in the following form:

(u,v)= Z u(x)v(x)hjh2 ••• hp = xera h

N1 _1N2 _1 Np _1

= Z Z ••• Zu(iphp,•,iphp)v((iphp,•,iphp)h1h2••• hp; i1 =1 i2 =1 ip =1

Nj_1 N2 _1 Na_1 Np _1

(u,v)a = Z Z ••• Z ••• Z u(x)V(x)h1h2 "h =

i1=1 i2 =1 ia=1 ip =1

= z

lß*la

Z u(x)v(x)ha

V ia =1 /

H / ha,

Nj-1 К2 -1 Na NP -1

(u, v]a = E I "' Z "' Z U (X )V (X )h1h2 hP

г1=1 '2=1 га=1 ip=1

= Z

'ß^a

Jva

H /ha, H =Hha-a=1

form:

Zu(x)v(x)ho

,ia=1

In the function space, we define the norm and introduce it in the following

(u,u)=||u II2, (u,u]=||u]|2, (u,v] = Z(u,v]a, ||Yj]|2 = £ || Yja] |2 .

a=1 a=1

Assuming a = 0.5 , we multiply (5) scalarly by Y = y + y:

(t,Y) = 0.5(A(t )Y,Y) + (cp,Y). (7)

Let us transform the sums included in identity (7), taking into account the first Green's difference formula [14], we obtain the inequality

,1 Л (1, ^ )-(1' ) 2 (t, Y ) = |±(( - y), (j) + y)) = ^-^-'- = (1, y 2)r,

f p Л

(Л(t )Y,Y) = ¿Ла(Г)Y,Y = I^(F)Y,Y)a =

p I

| a=1

p

Z

) a=1

P 2 P f m f Л 2

-Z(«a,Y2 ]a — Z Z^^a fYaA. + \ +< J'Y

a=1 a=1| s=1

a

Using (8), (9) from (7) we find

(8)

(9)

(1,y2)t + 0,5Z(«a,YX \

a=1

P f m f Л ^

= -0-5Z Zdsa f Ya,X—s +Yas +< I^ + (<P,Y)•

a=11 s=1 )a

(10)

Let us estimate the sums in identity (10) using the Cauchy inequality with £ and Lemma 1 [16]:

(1, y2) t = (II y II2) t, Z( «a, Y?] a> со Za yX£] a= cq(1, yX ]= cq || Yx ]|2,

(11) (12)

=1

=1

p I m i \ I

y Yds | Y x- + Y +1x+ I ,Y2

a 1 1 as 1 as 'as l „ |

<

a=1 V s=1

P I

= -y( Ya

a=1

I

p y a=1 1 (y1 xl 2 V las

s as as 'at I a

V s=1

/ a I

/ a

s2 , I

s2 I

/a

<

< Mi (e || Yxa ]|2 +c (e)|| Y || (cp, Y )< 1 (|| Y y

2 + n,„ ii2

(13)

(14)

Taking into account estimates (11)-(14), after a series of transformations from (10) we obtain the inequality

(|| y y2)t + Co y Yx] |2< eM1 y Yxa ] |2 +M2 y Y ||2 +M3 y p] |2 . (15)

Co

Choosing e = ——, from (15) we find 2M1

(|| y y2 )t + y Yx] |2<M4 y Y y2 +M5 y p] |2 . (16)

Multiply both sides of (16) by T and sum over j' from 0 to j, then we

have:

j j I j ^

\yJ +1 y2 + y y Yx]|2 T<M4 y y YJ y2 t+M6 y y pJ y2 t+ yy0 ||2

j'=o j=0

j

V j

'=0

. (17)

Denoting by F(tj) = y y p' y2 T+ y y0 ||2, from (17) we obtain

j=0

yyJ +1 y2 + y y Yx]|2 T<M4 y ||YJ y2 t+M6F(tj). j=0 j'=0

We transform the first term on the right-hand side of (18), then we have

y y YJ y2 T= y (yJ+1 + yJ y2)t.

(18)

j=0

j'=0

(19)

Taking into account the inequality y zJ"+1 + zJ y2 < 21| z J+1 y2 +2 y zJ y2 : from (19) we have the estimate

Z II Yj' II2 T< 2II yj+1 II2 t + 2II y0 II2 T + 4Z II yj' II2 T. (20)

/=o j'=i

Due to (20), from (18) we find

Iy^+1 I2 + Z I Jx]|2 T<

j =0 j '

< 2M4 I yj+112 T+M7 Z Iyj I2 T+MgF(tj). (21)

j'=1

Choosing t<t0 = —1—, from (21) we get

0 4M4

Iyj+1 I2 + Z IYx]|2 T<M8Z Iyj I2 T+M9F(tj). (22)

j '=0 j=1

Applying to (22) an analogue of the Gronwall lemma for grid functions [17, Lemma 4, p. 171], we obtain an a priori estimate

\yj+1 II2 + £ II((+1 + yj')x]|2 T<M(T)f £ у 9j" II2 т+ IIy0 II2

j'=0 U=0

(23)

where M (T) is a positive constant, independent of |h| and T .

Thus, based on (23), we have proved the uniqueness and stability of the

solution of the difference problem (5), (6) with respect to the initial data and the

j ■' ■'

right-hand side in the sense of the norm I y I2 =II yj+1 IIQ2 + Z II (yj +1 + yj )x]|2 T

j =0

on the layer.

Theorem 1. Let conditions (4) be satisfied, then in the class of sufficiently smooth coefficients of the equation and boundary conditions for the solution of the difference problem (5), (6) for c = 0.5 and small t<Tq (, q, C2), the a priori

estimate (23) holds for each time layer in the sense of the I y I norm.

Let u(x,t) be a solution to problem (1)-(3), yj be a solution of the difference problem (5), (6), then denote by zj = yj _ uj the approximation error. Substituting yj = zj + uj into (5), (6) and considering u(xi,tj) as a given function, we get the problem for z :

zt = A (t )z(o)+y, (24)

z|Yh = 0, z (x,0 ) = 0, (25)

where yi = Z¥a = O(|h|2 + Tm°), m° = 1, if a* 0.5 and m° = 2, if c = 0.5 is

a=1

the approximation error on the solution of the original problem for each fixed t = T, due to the construction of the operator A.

Applying the a priori estimate (23) due to the linearity of the difference problem (5), (6) to the problem for the error (24), (25), we obtain the estimate

I zj+1 Ib2 + Z II(z/+1 + ^)x]|2 T<M Z II ¥j' I2 T . (26)

j '=0 j '=0

The a priori estimate (26) implies the convergence of scheme (24)-(25) for

,2 p ■

c = 0.5 at the rate O(| h |2 +Tm°), | h |2 = h12 + h2 +• + '-2

4. Statement of the first initial-boundary value problem for a multidimensional integro-differential equation

Instead of equation (1), we consider the equation of the following form

— + fp1 (x t,T)udt = Lu + f (x,t), (x,t)e QT, (27)

dt J

t

dt

where

p

Lu = ZLau, Lau =~x~ ka(x,t)d~u~ _qa(x,t)u _ jP2,a(xt a=1 a V a) 0

|qa(x,t)) |P1 (x,t,T)) |p2,a (x,t)|< c2. (28)

5. Construction of a difference scheme for the differential problem (2), (3), (27)

On the grid ®hT, we associate the differential problem (2), (3), (27) with

a difference scheme with weights of order of approximation O(| h |2 +Tm°):

yt = A (t) y(o)+9, (x, t )e«hT, (29)

y lyh = 0, y(x,0) = uq (x), (30)

where

A(t ) = ZAa(F)

a=1

Aa(F )y(0) = (aay(])Xa _ day(o) _ Z jy^ _ -1 t^T,

ia=0 p s=0

l

da= Яа(х>t ) Pi (x,t,t) = P- (x,t, t ), P2,a = P2,a (xt ) t =t. -•

J

6. Stability and convergence of the difference scheme

To obtain an a priori estimate for the solution of the difference problem (29), (30), we use the method of energy inequalities. Assuming c = 0.5, we multiply (29) scalarly by Y = y + y:

(y,Y) = 0.5(A(t )Y,Y) + (cp,Y). Repeating arguments (7)-(14), taking into account the estimates

-£(da,Y2)a< C2 ¿(1,Y2)a= ¿2 H Y y2; a=1 a=1

(31)

-Z

a=1

N„

Z Р2,Л ,y

V га=0

/а

< i2-y 2 i+z

а=1

f Na >

Z P2,aYh°

2 ]

V г'а 1

/а

< -2 У Y II2 +M- Z

a=1

1 Na

- Z

ia=1

f a ] J

Y. p

ia

V /

2 ]

<M2 y Y У2;

f

1 F J 1 F

-pz|Zpi>t.y| <V2p•Y2]+Z

" a=1 V 5=0 /a V ^ J a=1

Л

f

f

f

2 P

J

\

2]

Z PiVT

V 5=0

/a

< — У Y Ib2 +M3 Z У y Ib2 T,

2P

5=0

after a series of transformations from (31) we obtain the inequality

(il y У2 )t + У Yx] |2<M4 y Y y2 +M5 Z У y Ib2 T+Мб

5=0

(32)

Multiply both sides of (32) by T and sum over j' from 0 to j, then we get:

J J ,

lyJ+- У2 + Z У Yx ]|2 T< M4 Z IIYJ У2 T +

J =0

J =0

J J

+M5 ZZII У I02 TT+ M7

J=05=0

Z У & У2 T+ У y1

0 ii2

V J

J =0

(33)

Let us estimate the second term on the right-hand side of (33), then we

obtain

ZZiiyIb2 TT<TZ IyIb2T. (34)

j '=0s=0 /=0

Due to (20), (34) from (33) we find

j j '

II yJ +1 I2 + Z II Yx]|2 T< 2M5 IyJ+112 T+M8 Z IyJ I2 T+M9F(tj), (35)

j '=0 /=1

j '

where F(tj )= Z II 9J II2 t+ I y0 II2.

j =0

Choosing t<t0 = —1—, from (35) we get 0 4M5

I y^+1 I2 + Z I Yj I2 T<M10 Z I yj I2 T+M11F(tj). (36)

j '=0 /=1

Applying to (36) an analog of the Gronwall lemma for grid functions [17, Lemma 4, p. 171], we obtain an a priori estimate

j

j

lyj+112 + Z II (yj'+1 + yj)xll2 T<M(T) j =0

j

Z II I2 T+II y0 II2

V / =0

(37)

where M (T) is a positive constant, independent of |h| and T .

Thus, based on (37), we have proved the uniqueness and stability of the

solution of the difference problem (29), (30) with respect to the initial data and the

j

right-hand side in the sense of the || y if =|| yj+11|2 + Z II (yj +1 + yj )x]|2 t norm

j =0

on the layer, and also, due to the linearity of the problem under consideration, the

2 2

convergence of scheme (29), (30) for c = 0.5 at the rate O(| h | +t ),

| h |2 = h12 + hf + •..+ hp .

Theorem 2. Let conditions (4), (28) be satisfied, then in the class of sufficiently smooth coefficients of the equation and boundary conditions for the solution of the difference problem (29), (30) for c = 0.5 and small t<Tq ((, C1, C2), the a priori estimate (37) holds for each time layer in the sense

of the | y |1 norm.

Comment. To solve the resulting systems of difference equations (5), (6) and (29), (30), iterative solution methods can be used. Note that the iterative method makes sense if it converges, i.e., || yk _y 0, for k ^^ .

Each iterative method has its own limited area of applicability, since, firstly, the iteration process may be divergent for a given SLAE and, secondly, the convergence of the process may be so slow that it is practically impossible to achieve a satisfactory closeness to the exact solution [18, p. 204].

When solving the resulting algebraic problem Ay = f by the iterative method, where A is the coefficient matrix of the system of difference equations, it is necessary that the sufficient condition for the convergence of the iterative process is satisfied, i.e. the matrix has line diagonal dominance:

Max £

i j*

aij

£ j * n\

— „п.

= max—jp^-j—- < 1, i, j = 1,2,., N.

i

If the matrix A has row-by-row diagonal dominance, then the termination condition for the iterative process is the fulfillment of the following condition

|| Ayk - f ||<£ || Ayo - f ||, 0 <£< 1,

where £ is a relative error with which an approximate solution to the problem Ay = f can be found; k is the number of iterations providing the required accuracy of approximation to the exact solution.

Let's represent Ay = f as yk+i = Byk + F, where yo is an arbitrary initial approximation, detB ^ 0, then for the convergence of the iterative process, in particular, in the methods of successive approximations, simple iteration (or method Jacobi), Gauss - Seidel, it is enough that all eigenvalues of the matrix B are less than unity in absolute value, and this is possible if

|| B ||= max< q < 1, i = 1,.,N, ' j

then

|| Ayk - f ||< qk || Ayo - f 0.

It can be seen from the latter that the number of iterations k , needed to achieve the required approximation accuracy £, can increase, as q tends to 1, i.e., it depends on the specific form of the constructed matrix B .

Conclusion

In this paper, loaded parabolic equations in a multidimensional domain with boundary conditions of the first kind are studied. Such problems arise in the study of groundwater movement, in the problems of water resources quality management when a water body receives a pollutant of a certain intensity from n sources, in the construction of a mathematical model of dispersed pollution transport in the atmospheric boundary layer, in the description of the mass distribution function of droplets and ice particles, taking into account the microphysical processes of condensation, coagulation (combining small droplets into large-sized aggregates), crushing and freezing of droplets in convective clouds. For each problem, a difference scheme with approximation order O (|h f + t ) if c^0.5 and O (|h\2 + t2 ) if

° = 0.5 is constructed. An a priori estimate is obtained to solve the corresponding difference problem by the method of energy inequalities. From the obtained estimates in difference form the uniqueness and stability of the solution on the right part and initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding initial differential problem with the

rate of O(|h|2 + Tm°) for c = 0.5.

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Информация об авторах / Information about the authors

Зарьяна Владимировна Бештокова младший научный сотрудник, Институт прикладной математики и автоматизации Кабардино-Балкарского научного центра РАН (Россия, г. Нальчик, ул. Шортанова, 89а)

E-mail: zarabaeva@yandex.ru

Валентина Аркадьевна Водахова

кандидат физико-математических наук, доцент кафедры алгебры и дифференциальных уравнений, Институт физики и математики, Кабардино-Балкарский государственный университет имени Х. М. Бербекова (Россия, г. Нальчик, ул. Чернышевского, 173)

E-mail: v.a.vod@yandex.ru

Марат Захарович Худалов кандидат физико-математических наук, доцент кафедры прикладной математики и информатики, Северо-Осетинский государственный университет имени К. Л. Хетагуров (Россия, г. Владикавказ, ул. Ватутина, 44-46)

E-mail: hmz@mail.ru

Zaryana V. Beshtokova Junior researcher, Institute of Applied Mathematics and Automation of Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences (89a Shortanova street, Nalchik, Russia)

Valentina A. Vodakhova Candidate of physical and mathematical sciences, associate professor of the subdepartment of algebra and differential equations, Institute of Physics and Mathematics, Kabardino-Balkarian State University named after H.M. Berbekov (173 Chernyshevskogo street, Nalchik, Russia)

Marat Z. Khudalov

Candidate of physical and mathematical sciences, associate professor of the subdepartment of applied mathematics and computer science, North Ossetian State University after K.L. Khetagurov (44-46 Vatutina street, Vladikavkaz, Russia)

Авторы заявляют об отсутствии конфликта интересов / The authors declare no conflicts of interests.

Поступила в редакцию / Received 22.01.2024

Поступила после рецензирования и доработки / Revised 24.02.2024 Принята к публикации / Accepted 14.04.2024