HOLMGREN PROBLEM FOR ELLIPTIC EQUATION WITH SINGULAR COEFFICIENTS Текст научной статьи по специальности «Математика»

Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 114-126. ISSN 2079-6641

MSC 35A08, 35J25, 35J70, 35J75 Research Article

Holmgren problem for elliptic equation with singular coefficients

T. G. Ergashev1'3, A. Hasanov1'2

1 V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, M. Ulugbek str. 81, 100125, Tashkent, Uzbekistan

2 Department of Mathematics, Analysis, Logic and Discrete Mathematics of Ghent University, B 9000, Gent, Krijgslaan 281, Belgium

3 Institute of Irrigation and Agricultural Mechanization Engineers, 100100, Tashkent, Kari-Niyazi st., 39, Uzbekistan

E-mail: [email protected]

In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella's hypergeometric function in many variables. Then using an "abc" method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green's function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella's hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem.

Keywords: Holmgren problem, multidimensional elliptic equations with several singular coefficients, decomposition formula, summation formula, Lauricella hypergeometric function in many variables, Green's function

DOI: 10.26117/2079-6641-2020-32-3-114-126

Original article submitted: 04.07.2020 Revision submitted: 07.10.2020

For citation. Ergashev T. G., Hasanov A. Holmgren problem for elliptic equation with singular coefficients. Vestnik KRAUNC. Fiz.-mat. nauki. 2020,32: 3,114-126. DOI: 10.26117/2079-66412020-32-3-114-126

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

Ergashev T. G., Hasanov A., 2020

Introduction

It is well-known that due to the many applications in aerodynamics [1] and irrigation problems [2], the theory of boundary value problems for singular partial differential equations (PDEs) has become a rapidly developing area in the general theory of PDEs. In addition, generalized axisymmetric potentials have been studied using various methods [3, 4, 5].

Funding. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors

Relatively few works are devoted to the study of boundary value problems for multidimensional (more two-dimensional) elliptic equations with singular coefficients. Let Rm+ be denote a 2nth part of the m-dimensional Euclidean space:

Rm+ := {(xi,...,xm) : x1 > 0,...,xn > 0, 1 < n < m, m > 2}.

In the present work we study the Holmgren problem for multidimensional elliptic equation with several singular coefficients

Ha , )(u) = L TT + = 0 (1)

i=1 d x2 = xk d xk

in the domain lying in Rm+, where a := (a1,...,an); ak are constants with 0 < 2ak < 1,k e K, K = {1,...,n}; n is a number of the singular coefficients of equation (1).

Holmgren problem is that in part of the boundary of the considered domain where the solution is sought, the values of the desired function are known, and in the remaining part of the boundary, the values of the derivative of the same function are given. In 1926 Holmgren [6] posed a new problem for equation

a. ^201 _0

ux1x1 + ux2x2 + ux1 — 0 x1

in a finite domain lying in a half-plane x1 > 0.

In the three-dimensional case [7] the solution of the Holmgren problem for equation

^201 , 202 i 203 -0

ux1x1 + ux2x2 + ux3x3 + ux1 + ux2 + ux3 — 0

x1 x2 x3

in the first octant of the ball is written out through the Lauricella hypergeometric

(3)

function in three variables F ) (for definition, see a formula (4)).

The present paper is organized as follows: First, we give some preliminary information, which will be used in what follows. Second, we formulate the problem and prove a uniqueness theorem. In the rest of the paper handling the method of Green's function we find an explicit solution of the problem with the Holmgren conditions for equation (1). Finally, we state our main result as a theorem.

Preliminaries

Below we give some formulas for Euler gamma-function, Gauss hypergeometric function, multiple Lauricella hypergeometric function (that is, Lauricella hypergeometric function in several variables), which will be used in the next sections.

Let be C set of complex numbers and N set of the natural numbers : N = {1,2,3,...}. It is known that the Euler gamma-function r(a) has property [8, Ch.1, 1.2(2)]

r(a + p) = r(a)(a)p, a e C, p e NU{0}.

Here (a)p is a Pochhammer symbol, for which the equality (a)p+q = (a)p(a + p)q is true. A function

F (a, b; c;y) = F

a, b;

c;

~ (a)p (b)p

I VrMV, |y| < !, a,b,c,y G C, to (c)pP!

y

is known as the Gauss hypergeometric function and an equality

F (a,b; c; 1) = r (c)r (c - a ~ [c = 0, _i, _2, .... R (c - a - b) > 0] (2) r (c _ a) r (c _ b)

holds [8, Ch.2, 2.1(14)]. Moreover, the following autotransformer formula [8, Ch.2, 2.1(22)]

F (a, b; c;y) = (1 — y) b f( c — a, b; c; y

y — 1

(3)

is valid.

The Lauricella hypergeometric function in n g N variables has a form [9, Ch.1, 1.4(1)]

FA ) (a,bi,...,bn;ci,...,cn;zi,...,Zn)

= F (n)

A

z1, ... , zn

a, bi,..., bn;

c1, ..., cn;

(a)mi+...+mB (b1)mi ... (bn)mB Zm Z

mB n

m1,...mn=0 (ci)mi ... (cn)m

m1f mn!

[ck = 0,-1,-2,...; |zi| + ... + |z„| < 1; a,bk,Ck,zk G C]. We give some elementary relations for necessary in this study:

(4)

d d Zk

3^FAn) (a,bi,...,bn;ci,...,cn;zi,.

., Zn)

abkF (n) c/A

a + 1, bi,..., bk—i, bk + 1, bk+i,. ci,..., ck—i, ck + 1, ck+i,..., cn;

• ., bn;

zi,..., Zn

, k g k,

(5)

k=i ck

a + 1, bi,..., bk—i, bk + 1, bk+i,..., bn; ci,..., ck—i, ck + 1, ck+i,..., cn;

=F

(n)

A

a + 1, b1,..., bn;

c1, ..., cn;

Zi, ..., Zn

F

(n)

A

Zi,...,Zn

a, bi,..., bn;

c1 , ..., cn;

Zi,..., Zn

(6)

Relations (5) and (6) can be proved in two ways: by comparing coefficients of equal powers of zi, ..., zn on both sides or mathematical induction.

For a given multiple hypergeometric function, it is useful to fund a decomposition formula which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. Burchnall and Chaundy [10, 11] systematically presented a number of expansions and decomposition formulas for some double hypergeometric functions in series of simpler hypergeometric functions. Using the Burchnall-Chaundy method, Hasanov and Srivastava [12, 13] found decomposition formulas for a whole class of hypergeometric functions in several (three and more) variables. For example, the hypergeometric Lauricelli function F^, defined

n

by formula (4) has the Hasanov-Srivastava's decomposition formula [12] FA(n) (a,bi,...,bn;ci,...,cn;zi,...,zn)

(a),

..+mn (bz)m2 ••• (bn)mK m2+...+mn m2 mn ,„„..1 ™ t^^ ^ A Zi Z2 ...^n

m2,...,mn=0 m2'...mn! (ci)m2+...+mn (c2)m2 ••• (c n mn

x F (a + m2 + ••• + mn,bi + m2 + ••• + mn; ci + m2 + ••• + mn;zi)

(7)

x fa

(n-l)

a + m2 + ••• + mn, b2 + m2 ,•••, bn + mn ; C2 + m2, •••, cn + mn;

•••,zn

,n G

However, due to the recurrence of formula (7), additional difficulties may arise in the applications of this expansion. Further study of the hypergeometric Lauricelli function showed that formula (7) can be reduced to a more convenient form. Lemma 1 [14]. The following decomposition formula holds true at n g N\{1}

FA(n) (a, bi, ••••, bn; ci,--, Cn ; zi,-, zn)

E(a)A(n,n) -pr

m-■! П

m;, j=0 ! k=i

(2<i< j<n)

,n) B(fc,n)p/ a+A(k,n), bk + B(k,n);

Zk I Ck + B(k, n);

(ck )B(k,n)

zk

(8)

where

k+i n k n

A(k,n) = EEm;,j, B(k, n) = £ m;,k+ £ mk+i, ;=2 j=; ;=2 ;=k+1

(9)

Lemma 2. Let a,b1,...,bn e C be with a = 0, -1, -2,... and Re (a — b1 —... — bn) > 0. Then the following summation formula holds true at n e N\{1}

E

^j=

(2<;< j<n)

(a)A( n,n) A (bk)B(k,n) (a - bk)A(k

П

M a - E bk

k=i / т-n-1

m;ij=0 m;j ! k=i

(a)

'A(k,n)

Гп-1 (a), (10)

П Г (a - bk) k=i

Formulae (8) and (10) are proved by the method mathematical induction [14]. Obviously, in the case when n = 2 the formula (10) coincide with the summation formula (2).

The fundamental solutions of equation (1) in the domain Rm+ were found in [14], one of which looks like:

where

q(x;%) = Yr 1(aF<^) (Xc,ai,-,an;2ai,-,2«n;o), x := (xi,•••,Xm),% := (€i,-,%m), o := (oi,-,on) ;

(11)

„9x m Г (a)-A Г (ak) _ m - 2 Л лл „

Y = 22 -mnv2tП r&t), * = -2" + E Xk, 0 < 2xk < i;

r2 =

E (x; - %;)2, r2 = (xk + %k)2 + E (x; - %;)2, Ok = i - r|, k G K ;=i ;=i,;=k

(12) (13)

;

It is easy to verify that the fundamental solution q (x; £) has the property

2akd q(x;£ Y

d Xk

= 0, k g K.

(14)

Xk=0

Formulation of the problem and the uniqueness of the solution

Let Q be a finite domain in R+, bounded by the Lyapunov surface S [15] and the hyperplanes xi = 0, ..., xn = 0 . The boundary of the domain Q on the hyperplane xk = 0 is denoted by Sk.

We introduce the following notations:

x(2a) .= x2ai r2aB r(2a) .= r2ai Y2«k-i Y2«k+i r2aB.

1 ...xn , Xk

.x,

k—1 ^k+i

xk — (xb xk-1,xk+1,...,xm) ; xk0 — (xb xk-1, 0,xk+b xm) ;

dx — dx1...dxm; dxk — dx1...dxk-1dxk+1...dxm, k G K.

Holmgren problem. To find a function u (x) g C (Ù) n C2 (Q), satisfying equation (1) in Q and conditions

2ak

d u d Xk

= Vk (Xk), Xk G Sk, k g K,

Xk=0

u|S = Ç (x) , X G S,

(15)

(16)

where vk (xk) and ^ (x) are given functions, and, moreover, vk (xk) can reduce to an infinity of the order less than m _ 1 _ 2ak on the boundaries of Sk.

Uniqueness of the solution of the Holmgren problem. One can readily check the validity of the following relation

,(2« )

uHrV)—wHr,n)(u)

m d i=i c x;

,(2« )

d w d u \

d Xi d Xi /

, n < m.

(17)

Integrating both sides of the identity (17) in a domain Q and using the Gauss-Ostrogradsky formula, we obtain

r(2« )

Iß

uffr; (w) — wHim,n) (u)] dx = ^x(2a^u|w — wdf) dS,

( m, n)

(18)

where S' is the boundary of Q, N is the outer normal to S' and

= r ^ cos(Nx-) dN r dx,- C0S(N,xJ

is the normal derivative with respect to x.

Assuming that w = 1 and replacing u by u2 in (18), we obtain

iß

i=i

d u

.(2a )

d u

/(2a)E ( dX ) dx = L /, xf a)Tk (Xk) Vk (Xk) dSk — £x(2aV(x)dudS

k=1 J Sk

d N

where u is a solution of equation (1) and Tk (Xk) := u| =0, k g K.

k

k

2

To prove the uniqueness of the solution, as usual, we suppose that the problem has two v and w solutions. Denoting u = v — w we have that satisfies homogeneous Holmgren problem (vk = 0, ^ = 0). Further we have to prove that the homogeneous problem has only trivial solution. In this case from (19) one can easily get

/, x(2a)| (|)* = 0.

Hence, it follows that uxi =... = uxm = 0, which implies that u is a constant function. Considering homogeneous condition (16), we conclude that u(x) = 0 in Q.

The existence of the solution

We prove the existence of the solution in a special case of the domain Q in order to get the solution in an explicit form. Let Q and S be a 2nth part of the multidimensional ball and sphere with the radius a and center in the origin respectively:

Q = {x :x2 + ... + x;m < a2, x1 > 0,...,xn > 0},

S = {x : x2 + ... + x;; = a2, x1 > 0,...,xn > 0}.

We find a solution of considered problem using method Green's functions [16]. Therefore, first we give a definition of Green's function for the formulated problem.

Definition. We call the function G(x; £) as Green's function of the Holmgren problem, if it satisfies the following conditions:

this function is a regular solution of equation (1) in the domain Q, expect at the point £, which is any fixed point of Q; it satisfies boundary conditions

2ak d G (x;£)

d Xk

= 0, k G K; G(x;£)|S = 0;

Xk=0

it can be represented as

G (x; £ )= q (x; £)+ q* (x; £), (20)

where q(x;£) is the fundamental solution found earlier (see a formula (11)), a function

/ a \ 2a

q* (x; £) = — (q (x;£)

is a regular solution of equation (1) in the domain Q. Here

t

" ■"'1 1 ••• 1 "mi " "31 1 ••• 1

It is easy to see that a regular solution q* (x; £) also has the property (14). Excise a small ball with its center at £ and with radius p > 0 from the domain Q. Designate the sphere of the excised ball as Cp and by Qp denote the remaining part of Q.

£:= (£?,...,£m), £« = ^£«, a2 = x? +... + x^, R2 = £i2 +... + £m

Applying formula (18), we obtain

X

(2a)

/cp

d G (x; % ) d u (x)

u (x)—— G (x; %)

d N

d N

dCp

— tjSkXka G (Xk0; %) Vk (Xk) dSk — jfx(2a)ç (x) dS

where

(21)

Sk = < x : u x2 = a2, x1 > 0,...,xk—1 > 0,xk+1 > 0,...,xn > 0 >,k G K. i=i,i=k

First, we consider an integral

/cpx-12"'u (x) ^

Taking (20) into account we rewrite it as follows

dCp.

1X<2a'«(X)IGdC, - L X<2a'»(X)^dCp + L X(2a'-(X)^

d Cp

= Ii(1, p )+ /2(1, P ).

Using the formula of differentiation (5) and the adjacent relation (6), we calculate

dq (x; % ) m dq (x; % )

d N

i=1 dxi

cos (N, xi).

(22)

-v / , C \

Below we get detailed evaluations for —^—, when 1 < - < n. Indeed, using the

formula of differentiation (5), we get = _ 2a Y (x, _ 1,) r_2a _2F,(n)

d Xi

d Xi

A

(X, a1,..., an; 2ai,..., 2 an;

— 2aY (Xi — %i) r—2a—2 £ 2a-OkF?"

k=i 2ak

1 + (X, ai,..., ak—i, 1 + ak, ak+i,..., an;

— 2 a Y%i r——2Fa

2aX —2 (n)

A

2ai,..., 2ak—i, 1 + 2ak, 2ak+i,..., 2an;

1 + a, a1,..., ai—1, 1 + ai, ai+1,..., an; 2ai,..., 2af—i, 1 + 2ai, 2ai+i,..., 2an ;

Considering adjacent relation (6) we obtain

^ = — 2aY (Xi — %i) ^F^

d Xi

1 + a , a1,..., an;

— 2a Y%i r—2a—2 FAn)

2a1,..., 2an;

1 + a, a1,..., ai—1,1 + a,-, ai+1,..., an; 2a1,..., 2ai—1,1 + 2aj, 2ai+1,..., 2 an;

, 1 < i < n.

Similarly, we calculate

d q (x, % )

d Xi

—2a y (Xi — %i) r—20—2FAn)

1 + a , a1,..., an; 2a1,..., 2an;

a

, n + 1 < i < m.

(23)

(24)

Taking (22), (23) and (24) into account we calculate

dq (x,£ )= - a y r-2a

d N

1 + a, a?,..., an; 2ai,..., 2a„;

^ rin r2]

d Nl j

- 2aYr-2a-2 £ £kF k=i

(n)

1 + (x, a?,..., ak-i, 1 + ak, ak+i,..., an; 2ai,..., 2 ak-i, i + 2ak, 2ak+i,..., 2a„;

cos (N ; xk ).

Now consider the integral

/c x(2a}u(x) dqd|£)dCp = In (£, p) + /12(£, p),

where

iii(£,p)= - ay u(x)r-2aF

20^77 (n)

Ii2(£,p)= - 2ay/ w(x)r

A

,-2-2a

i + a, ai,..., an; 2ai,..., 2an;

d

— [ln r2]dCp,

/Cp

X

n

£ £kFA(n)

A

k=i

i + a;,ai,...,ak-i, i + ak,ak+i,...,an;

where

2a1,..., 2ak—1,1 + 2ak, 2ak+1,..., 2an;

We use the following generalization spherical system of coordinates:

x,- = £, + p$¿(9), i = 1,...,m,

9 = (91,..., 9m—1) ; ^1(9) = cos 91, (9) = sin 91 sin 92 ■ ■ ■ sin 9,—1 cos 9,, i = 2,..., m — 1, ^m(9) = sin 91 sin 92 ■ ■ ■ sin 9m—2 sin 9m—1 (0 ^ p ^ r, 0 ^ 91 ^ n, ..., 0 ^ 9m—2 ^ n, 0 ^ 9m—1 ^ 2n). Then we have

cos (N ; xk )dCp.

2n

n

Iii(£, p ) = 2a Yp

-2ai-...-2an

J d^m-iy sin9m-2d9m-2

00 n

X / sin2 9m-sd 9m-3- / W (£i + p $i(9 ),..., £m + p $m(9 ))

n

X n [£k+p ^ (9 )]2akFAn)

k=i

i + ax, o?,..., an; , N

2a, J...!^; a(p,9)

sinm 2 9id9i,

where

a(p, 9) = ( i - pp,..., i - pp ) , r2p = 4£2 + 4p£k^k(9) + p2

p2

p2

First we evaluate a Lauricella's hypergeometric function in (25):

Ff (p, 9) := Ff

i + a, ai,..., a„; , , 2ai ,..., 2an; a(p,9)

(25)

(26)

p

n

Using the decomposition formula (8) and then auto-transformation formula (3), we rewrite a formula (26) as follows

, , n _ _

Ff (p, 9) = p2ai+...+2an n r-2* ■ P(p, 9), (27)

k=i

where

B(k,n)

P(P, 9 ) =

^ (i + <5; )a( n,n) ,n) I P 1 1

¿0 ! n (2«k)ß(k,n^rfp- y

m~ 0 m,j! k=1 (2ak)B(M Vrfe

(2<;< j<n) (28)

Af/ 2ak _ (a _ 1 + B(k, n) _ A(k, n), ak + B(k, n); _p2

X WH 2«k + n); 1 _ rfp

where A(k,n) and B(k,n) are expressions defined in (9).

It is easy to see that when p ^ 0 the function P becomes an expression that does not depend on 9. Applying the summation formula (2) to each hypergeometric function F(a,b;c;1) in the sum (28), we get

limP(p, „) =—1-- ¡I r(2(k)r(( + a _ )

p^0 rn (1 + a) k=1 r(ak)

^ (1 + a)a(»,») A (ak)B(fe,n) (1 + a _ ak)A(k

X m^ ^ 1=1 (1 + a)A(k,n) '

(2<i< j<n)

Taking into account the identity (10), we obtain

lim P(P, 9) = ft ^' (29)

p^o (r (1 + a) f={ r (ak) V '

If we take into account (25), (27), (29) and (12), then we will have

lim/„(1,p)= u(1)' (30)

p ^o

By similar evaluations one can get that

lim /12(1, p ) = lim /2(1, p ) = 0' (31)

p ^0 p ^0

If we consider an integral

d u (x)

/cp x(2a * (x; 1 )1Nf dCp,

using above given algorithm for evaluations (in this case calculations will be more simple), we can prove that

lim f x(2a)G (x; 1) dCp = 0. (32)

p Cp d N

Now, by virtue of (30), (31) and (32), from (21) one can easily get a solution of the Holmgren problem in the following form:

where

« « ) =-Y E Î42"1

k=:r Sk

Y-2a F (n-i) Yk fa

f-2ö F(n-i) 4 fa

(X, ai,..., , ak+i,..., an ; 2ai,..., 2ak-i, 2ak+i,..., 2an;

OfcQ

<7k0

+ 2aY x(2a)FA(n)

(X, ai,..., ak-i, afe+i,..., an; 2ai,..., 2ak-i, 2ak+i,..., 2an ;

" R2 — a2

Vk (Xt ) dxt

i + a, ai,..., an; 2ai ,..., 2an;

Rr2+2a

a Ф (x) dS,

CTto:= (о^.., of-^ o■jQ+l,..., <nQ

<Jto:= (<JiQ ок-^ <J/Q+l,..., <<nQ

(33)

<Q ^ <XQ = _

X2 , s

a2 4xs R2 Y?

, s = i,...,n, s = к;

a2 = Ex2, R2 = E£2, X2 = £2 + E (& -Xi)2, i=i i=i i'=i,i'=t

Y2 = E (a -i=i ,i=k

a

i

+ "2 EE - (m - 2) a2, k G K. a

i=i ,i=kj=i, j=i

Here constants a, 7 and variables o :=(oi,...,on), r2 are defined in (12) and (13), respectively.

The formula (33), and with it all the proof, requires that m > 2. However, the formula (33) is also valid for m = 2.

Hence, the main result of the paper is formulated as the following theorem:

Теорема. If vk (xk) g C2 (Sk), к g K and 9 (x) g C2 (S) are given functions, then the Holmgren problem with conditions (15) and (16) for equation (1) has unique solution represented by formula (33).

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

S

2

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Вестник КРАУНЦ. Физ.-Мат. Науки. 2020. Т. 32. №. 3. С. 114-126. ISSN 2079-6641

УДК 519.644 Научная статья

Задача Холмгрена для эллиптического уравнения с сингулярными коэффициентами

Т. Г. Эргашев1,3, А. Хасанов1,2

1 Институт Математики имени В. И. Романовского Академии наук Узбекистана, г. Ташкент, ул. Мирзо Улугбека 85, 100170, Республика Узбекистан

2 Кафедра математики, анализа, логики и дискретной математики Гентского университета, г. Гент, Бельгия

3 Ташкентский институт инженеров ирригации и механизации сельского хозяйства, 100100, г. Ташкент, ул. Кари-Ниязи, 39, Республика Узбекистан

E-mail: [email protected]

В данной работе мы исследуем задачу Холмгрена для многомерного эллиптического уравнения с несколькими сингулярными коэффициентами. Мы используем фундаментальное решение уравнения, содержащее гипергеометрическую функцию Лауричеллы от многих переменных. Затем методом «abc» доказывается единственность решения проблемы Холмгрена. Применяя метод функции Грина, мы можем найти решение задачи в явном виде. Более того, формулы разложения и суммирования, формулы дифференцирования и некоторые смежные соотношения для гипергеометрических функций Ла-уричеллы от многих переменных были использованы для нахождения явного решения поставленной задачи.

Ключевые слова: задача Холмгрена, многомерные эллиптические уравнения с несколькими сингулярными коэффициентами, формула разложения, формула суммирования, гипергеометрическая функция Лауричеллы от многих переменных, функция Грина.

DOI: 10.26117/2079-6641-2020-32-3-114-126

Поступила в редакцию: 04.07.2020 В окончательном варианте: 10.10.2020

Для цитирования. Ergashev T. G., Hasanov A. Holmgren problem for elliptic equation with singular coefficients // Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 114-126. DOI: 10.26117/2079-6641-2020-32-3-114-126

Конкурирующие интересы. Авторы заявляют, что конфликтов интересов в отношении авторства и публикации нет.

Авторский вклад и ответсвенность. Все авторы участвовали в написании статьи и полностью несут ответственность за предоставление окончательной версии статьи в печать. Окончательная версия рукописи была одобрена всеми авторами.

Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru)

Ergashev T. G., Hasanov A., 2020

Финансирование. Исследование выполнялось без финансирования