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Vladikavkaz Mathematical Journal 2022, Volume 24, Issue 4, P. 30-47
YAK 517.968
DOI 10.46698/i8323-0212-4407-h
INVERSE PROBLEM FOR VISCOELASTIC SYSTEM IN A VERTICALLY LAYERED MEDIUM#
A. A. Boltaev12 and D. K. Durdiev13
1 Bukhara Branch of the Institute of Mathematics at the AS of Uzbekistan, 11 M. Ikbal St., Bukhara 200117, Uzbekistan; 2 North Caucasus Center for Mathematical Research VSC RAS, 1 Williams St., village of Mikhailovskoye 363110, Russia; 3 Bukhara State University, 11 Muhammad Ikbal St., Bukhara 200117, Uzbekistan E-mail: [email protected], [email protected], [email protected]
Abstract. In this paper, we consider a three-dimensional system of first-order viscoelasticity equations written with respect to displacement and stress tensor. This system contains convolution integrals of relaxation kernels with the solution of the direct problem. The direct problem is an initial-boundary value problem for the given system of integro-differential equations. In the inverse problem, it is required to determine the relaxation kernels if some components of the Fourier transform with respect to the variables x1 and x2 of the solution of the direct problem on the lateral boundaries of the region under consideration are given. At the beginning, the method of reduction to integral equations and the subsequent application of the method of successive approximations are used to study the properties of the solution of the direct problem. To ensure a continuous solution, conditions for smoothness and consistency of initial and boundary data at the corner points of the domain are obtained. To solve the inverse problem by the method of characteristics, it is reduced to an equivalent closed system of integral equations of the Volterra type of the second kind with respect to the Fourier transform in the first two spatial variables x1, x2, for solution to direct problem and the unknowns of inverse problem. Further, to this system, written in the form of an operator equation, the method of contraction mappings in the space of continuous functions with a weighted exponential norm is applied. It is shown that with an appropriate choice of the parameter in the exponent, this operator is contractive in some ball, which is a subset of the class of continuous functions. Thus, we prove the global existence and uniqueness theorem for the solution of the stated problem.
Key words: viscoelasticity, resolvent, inverse problem, hyperbolic system, Fourier transform. AMS Subject Classification: 35F61, 35L50, 42A38.
For citation: Boltaev, A. A. and Durdiev, D. K. Inverse Problem for Viscoelastic System in a Vertically Layered Medium, Vladikavkaz Math. J., 2022, vol. 24, no. 4, pp. 30-47. DOI: 10.46698/i8323-0212-4407-h.
Introduction
A perfectly elastic material does not exist in nature; in fact, inelasticity is always present. This inelasticity results in energy dissipation or damping. Therefore, for a wide class of materials, it is not enough to use an elastic model to study their mechanical behavior. Therefore, viscoelastic foundational models have often been used to model the behavior of polymeric materials with respect to time variable.
#The research is supported by the Ministry of Education and Science of Russia, agreement № 075-022022-896.
© 2022 Boltaev, A. A. and Durdiev, D. K.
Let be x = (^1,^2,^3) € M3. Let us denote by <7^ the projection onto the Xi axis of the stress acting on the area with the normal parallel to the xj axis, and Ui are the projection onto the xj axis of the vector particle displacement. According to Hooke's law for viscoelastic media, stresses and deformations are related by the formulas [1, pp. 449-455], [2, ch. 3]:
(cfu' cFH'
+ di' +^AdivI1
t
+ Kjj (t - t)
(cfu' cFH'
0
(x,t) (It, i,j = 1,2,3, (1)
here ^ = ^(x3), A = A(x3) are Lame coefficients, Sj is Kronecker symbol, Kj(t) are functions responsible for the viscosity of the medium and Kj = Kjj, i, j = 1,2,3.
The equations of motion of a viscoelastic body particles in the absence of external forces have the form
j=1 J
where p = p(x3) is medium density, u(x,t) = (ui(x,t),u2(x,t),us(x,t)) is displacement vector. Throughout this work, A, p are considered to be given functions.
Note that (1) can be considered as integral Volterra equations of the second kind with
/ p.— Q-y . \
respect to the expression /x ( ^ + g^ J + 5ijXdivu, i,j = 1,2,3. For each fixed pair (i,j) solving these equations, we get
t
' ' \ f -Q^- + -Q^- j + A div u+ / Vij{t - T)(Tij (x,t) (It, i,j = 1,2,3, (3)
j j 0
where rj are the resolvents of the kernels Kj and they are related by the following integral relations [3, 4]:
t
rjj(t) = -Kjj(t) - J Kjj(t - t)rjj(t) dT, i,j = 1, 2, 3. (4)
0
From the condition Kjj = Kjj implies the rjj = rjj.
Differentiating (3) with respect to t and introducing the notation m = -j^Ui, we get
t
t du* du'\ I = II. I -- -I--3- I -I- A Hivl/. -I- r/.iOW/. il t) +
dt
(Qu ■ du' \ f
~dx + ~dx j + + nj(0)aij(x, t)+ r^t - r)aij (x, r) dr. (5)
j j 0
Then the system of equations (1) and (2) for the velocity uj and strain oj (oj = ajj) in view of (3)-(5) can be written as a system of first-order integro-differential equations.
t
/ d d d d \ f
{Am + BWt + °W2 +DW,+F) « = JR(t-T) dT- <«>
where U = (ui,u2,u3, an, ai2, ai3, a22, a23, a33)*, * is the transposition sign,
A =
(PJ)3x3 (O)3x3
(0)3x3
(0)3x3
(1)3x3 (0)3x3
(0)3x3 (0)3x3 (')3x3
-(A + 2^) Bi = I 0
0
0 0
0 0 /
(0)3x3 (Ci)3x3 (C2)3x3
(C3)3x3 (0)3x3 (0)3x3
(C4)3x3 (0)3x3 (°)3x3
(0)3x3 (-1 )3x3 (0)3x3
(Bi)3x3 (0)3x3 (0)3x3
V (B2)3x3 (0)3x3 (0)3x3
( ~ A00 1
B2 = | 0 0 0 ,
\ " A00 1
C2 =
D2 =
0 0 0 I 0 -A 0
-1 0 0 ) , C3 = I - -p 0 0
0 -1 0 V 0 0 0
(0)3x3 (Di)3x3 (D2)3x3 '
D I =I (D3)3x3 (0)3x3 (0)3x3
V (D4)3x3 (°)3x3 (0)3x3
0 0 0 4 I 0 0 -A
0 -1 0 ) , D3 = I 000
0 0 -1 V- 0 0
Ci =
C4 =
0 -10 000 000
0 -(A + 2^) 0
0
-A
D
i=
D4 =
0 0 -1 0 0
00 00
0 0 -A 0 0 0 0 -(A + 2^)
F =
(°)3x3 (0)3x6
(0)6x3 diag (rii(0), r22 (0), r33 (0), ri2 (0), ri3 (0), r23(0))
R(t) =
(0)3x3 (0)3x6
(O) 6x3 diag (rii, r225 r33> r12> r13, r23)
The system (6) can be reduced to a symmetric hyperbolic system [5]. We reduce the system (6) to canonical form with respect to the variables t and x3. To do this, multiply (6) on the left by A-1 and compose the equation
A-iD - vII =0,
(7)
where I is the identity matrix of dimension 9. The last equation with respect to v has following solutions:
vi
V9
- Vp = -4
'A + 2^
P
V2,3
V7,8
-Vs = -,
V4,5,6 = 0,
(8)
here vs and vp define velocities of the transverse and longitudinal seismic wave, respectively. Now we choose a nondegenerate matrix Y(x3,t) so that the equality
T-iA-iDT = A
(9)
is hold, where A is a diagonal matrix, the diagonal of which contains the eigenvalues (for each fixed x3) (8) of the matrix A-1D that is A = diag (—vp, —vs, — vs, 0,0,0, vs, vs, vp).
From the formula (9) implies the equality
A-i DT
TA,
0
which means that the column with the number i of the matrix Y is an eigenvector of the matrix A-1DY, corresponding to the eigenvalue Aj. Direct calculations show that the matrix Y, satisfying the above conditions, can be chosen as (not uniquely)
\
Y(xs) =
0 0 1 0 0 0 0 1 0
0 1 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0 1
a 0 0 1 0 1 0 0 _ a
Up Up
0 0 0 0 1 0 0 0 0
0 0 pvs 0 0 0 0 —pvs 0
X_ 0 0 0 0 1 0 0
Up Up
0 pvs 0 0 0 0 —pvs 0 0
K pvp 0 0 0 0 0 0 0 —Pvp
We introduce the vector function $ by the equality
U = Y$.
Making this change in the equation (6) and then multiplying it on the left by Y-1A-1,
we get
then
( d d d d \ f
(ia + V + + Cl aS + Fl) = J m ~ T' ^ T)
(10)
where
Bi(xa) = Y-1A-1BY = (bjj), Ci(xs) = Y-1A-1CY = (cj)
dY
Fi(x3) = T~1A~1D-— + T-U-^T = fa),
dX3
R1(x3 ,t) = Y-1A-1RY = (fjj)
/ Ik 2
0 0
vv 0
vp 0 0
r33
V —t
0 0
ik 2
0 0 0
0
ri 2
0
0 0 0
r'n 0 0 0 0 0
0 0 0 0
r12 0
0 0 0
0 0 0
r'n — r:
22
0
r22 0 0 0
0 0
ll 2
0 0 0 0
lia 2
0
r33 \ --2~
0 0
X(r'22-r'll)
v-p 0
Hr'33-r22)
0
0
r33 2 /
(11)
The purpose of this article is to study the direct and inverse problems for the system (11). Moreover, the direct problem is an initial-boundary value problem for this system in domain D = {(x1, x2, x3, t) : (x1,x2) € R2, x3 € (0, H), t > 0}, H = const, and in the inverse problem, the elements of the matrix R are assumed to be unknown, which are included in the definition of the matrix R1 (12).
The is organized as follows. Section 1 presents the formulations of the direct and inverse problems and investigates the direct problem. In Section 2, the inverse problem is reduced
t
0
r
V
p
r
r
2
to solving of an equivalent closed system of integral equations. In Section 3, we present the formulation and proof of the main result, which consists in the unique global solvability of the inverse problem. At the end there is a list of literatures used in the article.
1. Statement of the Direct and Inverse Problems
Consider the system of equations (10) in the domain D with a bounded r = To U r1 U r2:
T0 = {(x,t) : (xi,x2) eR2,0^x3^H,t = 0},
Ti = {(x,t) : (xi,x2) € M2, x3 = 0, t > 0} , r2 = {(x,t) : (xi,x2) € M2, x3 = H,t> 0} .
For this system the direct problem we pose as follows: determine the solution of the system of equations (10) at the following initial and boundary conditions:
^\t=0 = ^(x), i = l,...,9, (12)
$i\X3=H = gi(x1,X2,t), i = 1, 2, 3, $i\x3=0 = gi(x1,X2,t), i = 7, 8, 9. (13)
Here <fi(x), gi(xi,X2,t) are given functions. It is known that [5, 6] the problem (10), (12), (13) is posed well.
The inverse problem is to determine the nonzero components of the matrix kernel R, that is Tij(t), i, j = 1,2,3 (R is included in R1 according to the formula (12)) in (10) if the following conditions are known:
$j|x3=0 = hi (x1,X2,t), i = 1,..., 6, (14)
where hi(x1, x2, t), i = 1,..., 6, are the given functions.
In the inverse problem, the numbers rij(0), i,j = 1,2,3, are also considered to be given.
Currently, the problems of determining kernels from one hyperbolic integro-differential equations of the second order [7-22] have been widely studied. One- and multidimensional inverse problems are investigated and unique solvability theorems are obtained. Typically, second-order equations are derived from systems of first-order partial differential equations under some additional assumptions.
The inverse problem of determining the convolution kernels of integral terms from a system of first-order integro-differential equations of general form with two independent variables was studied in [23]. The theorem of local existence and global uniqueness is obtained. In the work of [24], the method for studying the work of [23] was applied to the investigating of the inverse problem of determining the diagonal relaxation matrix from the system of Maxwell's integro-differential equations.
It seems completely natural to study inverse problems on the determination of the kernels of integral terms of a system of integro-differential equations directly in terms of the system itself. This article is a natural continuation of this circle of problems and to a certain extent generalizes the results of [23] to the case of a three-dimensional system of viscoelasticity equations (1), (2).
Let functions <p(x), gi(x\,X2,t) included in the right-hand side of (10) and the data (12), (13) are compact support in x1, x2 for each fixed x3, t. From the existence for the system (10) of a compact support domain of dependence and compact support with respect to x1, x2 of the right-hand side (10) and data (12), (13) implies the compact support in x1, x2 solutions to the problem (10)-(13).
Let us study the property of solution to this problem. More precisely, we restrict ourselves to studying the Fourier transform in the variables xi, x2 of the solution. In what follows, for convenience, we put x3 = z and introduce the notation
$ (ni,n2,z,t) = J 0(xi,x2,z,t)ei[nixi +n2X2] dxi dx2,
R2
where ni, n2 are transformation parameters. We fix ni, n2 and for convenience, we introduce the notation i?(ni,n2,z,t) = $(z,t).
In terms of the function $ we write the equations (10) as
^ ' fc=1 n fc=i
(15)
j = 1,..., 9,
where pj = -inibjfc - i^Cjfc - j.
We will use a similar notations for the Fourier images of functions included in the initial, boundary and additional conditions (12)-(14):
tf,;
z=H
tf4=0 = 'tfi(z), i = 1,..., 9, = tf,(t), i = 1, 2, 3, = tf,(t), i = 7, 8, 9
z=0
= h;(t), i = 1,..., 6.
(16)
(17)
(18)
Where <^(z), i = 1,..., 9, <7i(t), i = 1,2,3, 7,8,9, hj(t), i = 1,..., 6, are the Fourier images of the corresponding functions from (12)-(14) for n1 = 0, n2 = 0. We also denote by DH the projection of D onto the plane In what follows, we will consider the system of equations (15) in the domain DH U r under the conditions (16) and (17). Where ro ={(z,t) : 0 < z < H, t = 0}, ri = {z,t) : z = 0, t > 0}, T2 = {(z,t) : z = H, t > 0}, r = To U ri U r2. ^
For the purpose of further research let us introduce the vector function w{z,t) = z,t). To obtain a problem for a function w(z,t) similar to (15)-(18) differentiate the equations (15) and the boundary conditions (17) with respect to the variable t, and the condition for t = 0 is found using the equations (15) and the initial conditions (16). In this case, we get
t
/ d d \ 9 9 (dt +Ujdz) UJj(^Z,t"> = ^Vjk{z)ujk{z,t) + ^rjk{z,t)$k{z) ^ ' fc=i fc=i
t 9
+ / X] fjfc(z,r)wfc(z,t - t) dr, j = 1,..., 9, (19) 0 k=i
^|t=0 = -^r^- + YjPMvAz) =: $i(z), i = 1,... ,9, (20)
j=i
Ui\z=H = Jt9i$), ¿ = 1,2,3, = i = 7,8,9. (21)
For functions wi additional conditions (18) gets
\ d -
w4=o = ^(i), i = !,•••, 6. (22)
Let us pass from the equalities (19)-(22) to the integral relations for the components of the vector $ with integration flux along the corresponding characteristics of the equations of the system (19). Recall that the characteristics corresponding to vp and have a positive slope, and the characteristics corresponding to —vp and —have a negative slope. We denote
z
lM(z)= f ~777Tj ¿ = 1,2,3,7,8,9, tn(z) = 0, ¿ = 4,5,6.
J MP) 0
Inverse functions to ^(z) will be denoted by z = Using the introduced functions, the
equations of characteristics passing through the points (z, t) on the plane of variables t can be written in the form
t = t + ^ (£) — ^i(z), i = 1,..., 9. (23)
Consider an arbitrary point (z, t) € DH on the plane of variables t and draw through it the characteristic of the i th of the system (15) equation tell to intersection in the domain t ^ t with boundary r. The intersection point is denoted by (z0, t0). Integrating the equations of the system (15) along the corresponding characteristics from the point (z0, t0) to the point (z, t) we find
4 9
w»(z,t) = w»(z° ,t°) + |£ PifcWfc({,r)
40
dr
4 - 9 T 9 1 (24)
+
0
(^,r)&(0+ / (^,r - ®)Wfc(£,a) da
Lfc=l 0 k=l
i = 1,..., 9.
dr,
We define in (24) t°. It depends on the coordinates of the point (z,t). It is not difficult to see that t°(z,t) has the form
t°(zt) = it - ^¿(z) + t ^ Mz) - ), i = 123 \o, 0 < t < ^i(z) - №(H), ' ' '
t°(z,t) = 0, i = 4, 5, 6, t°(z,t) = it - ^ t ^ i = 7, 8, 9.
Then, from the condition that the pair (z°, t°) satisfies the equation (23) it follows
zs(z,t)H H 1 ' ^ ^ - ^), i = 1, 2, 3,
V-1 (^i(z) - t), o < t < ^j(z) - ^i(H),
0, t ^ ^j(z),
^u-1 Gui(z) - t), 0 <t<^j(z),
zi(z,t)= z, i = 4,5,6, zi(z,t) = { „ \ ^ i = 7,8,9.
The free terms of the integral equations (24) are defined through the initial and boundary conditions (20) and (21) as follows:
wi(z0 ,to) —
Wt9i{t~ Hi{z) + Hi(H)), t ^ Hi{z) - Hi(H),
$i (^r1 (№(z) - t))
0 < t < ^¿(z) - (H),
i — 1,2,3,
Wi(z0, t0) — $i(z), i — 4, 5,6,
,, (V fi \ _
I 1 -t))
t ^ ^i(z), 0 < t < ^¿(z),
i — 7, 8, 9.
Let the following conditions hold dp (t)
(i (H ) — pi(0) and
(i(0) — pi (0) and
dt
-Vj
dpi (z)
t=o
dz
z=H
+ ^pij(H)(• (H), i — 1, 2, 3, (25)
dpi (t)
dt
— -Vj
d(i(z)
t=o
dz
9
+ > Jj(0)(j(0), i — 7, 8, 9. (26)
z=0
j=1
It is easy to see that the conditions for matching the initial and boundary data (16), (17) (20), (21) in corner points of the domain coincide with the relations (25) and (26). Hence it is clear that at the fulfillment of the same equalities (25) and (26) equations (24) will have unique continuous solutions Wj(z, t), or the same
Suppose that all given functions included in (24) are continuous functions of their arguments in . Then this system of equations is a closed system of integral equations of the Volterra type of the second kind with continuous kernels and free terms. As usual, such a system has a unique solution in the bounded subdomain = {(z,t) : 0 < z < H, 0 < t < T}, T > 0 are some fixed number.
Theorem 1. Assume functions ^>(x), g(x^x2,t) have compact support in xi, x2 for each fixed z, t. Let be p(z),^(z), A(z),<p(z) € C 1[0,H], p(t) € C1 [0,T], p(z) > 0, A(z) > 0, ^(z) > 0, rj(t) € C [0,T], i, j = 1,2,3 and conditions (25), (26) are satisfied. Then there is a unique solution to the problem (19)-(21) in the domain .
The problem (15)-(17) in the domain is equivalent to a linear integral equation of the second kind of Volterra type with respect to p As follows from the theory of linear integral equations, it has a unique solutions [3]. So we drop it.
2. Reduction of the Inverse Problem
In this section, the inverse problem is reduced to solving of an equivalent closed system of integral equations. Consider an arbitrary point (z, 0) € To and draw through it the characteristics (23) for i = 1,2,3, up to the intersection with the boundary of the domain . Integrating the first six components of the equation (19), we obtain
Ui(z, 0) = jh%(t\) -
99
Lj=1 j=1
dr
?=Mi 1 [t+№(z)l
1 T 9
/ / Ê fija)wjT - a) da
0 0 j=1
where 4 — -^i(z), i — 1,2,3, 4 — t, i — 4, 5,6.
dr, i — 1,..., 6, (27)
t
1
For the purpose of further research we introduce the following notation for the unknowns:
ui(z,t)= Wi(z,i), i = 1,...,9, u2(t) = rii(t), u2(t) = ri2(t), u2(t) = ri3(t), (28)
d
^(i) = r£2(i), u§(i) = r£3(i), v26(t) = r'33(t), vf(z,t) = —Ui(z,t), i = 4,5,6, (29)
d
r3 3(t0 )
dy d
t>3(z,i) = —Ui(z,t) - (^(4) - & —ij, i = 1,9,
dz d
t>3(z,i) = - - z = 2,7,
^ - 9' * r'13№,) Oh(4) ~ < = 3,8.
Vi(z,t) =—Ui(z,t) - 2
dz dz'
(30)
(31)
(32)
Taking into account these notations and the explicit forms of the functions rj (z,t) in terms of rj (t) by the formula (11), we rewrite the equations (24) in the form
t r o
/9 2( \
dT
t T
u6 (a) / 1 „1
(u{ — (£, t — a) da
ti 0
l0
dT, i = 1,9, (33)
1 [t-t+W(z)]
t r a
/9 2( \
Epyco^&r) - - (o
dT
1 [t-t+W(z)]
t T
ti 0
t0
dT, i = 2,7, (34)
1 [t-t+W(z)]
t
u^(z,t) = u0i(z,t) + J
(^)u)(e,T)
Lj=i
u2(T)
(<$3 — (£)
dT
1 [t-t+W(z)]
t T
ti 0
t0
dT, i = 3,8, (35)
1 [t-t+W(z)]
t T
0 j=i
t T
u|(z, t) = f £p4j(z)uj(z,T) dT + J Ju^a)-](z,T — a) dadT
00
— — Vg) (z,r — a) + v\(z,t — dadr
-2 —
+
K — u|) (T)
($i — <$9) (z) + <$6(z)^ + -2(t)<$4(z)
dT, (36)
1
u
0
2
1
u
0
2
0
t
t
A
V
p
t T
u^Zji) = J (z)uj(z,r) dr + J Jt>2(a)^1 (z,r — a) dadr + Ju^t)^(z) dr, (37) 0 j=1 0 0 0
t T
u1(z,i) = JJ
— (f| - v2) (a) (vI — v\) (z, r - a) + f2(a)f| (z, r - a)
da dr
00
t 9 t
+ /Êp6J(z)ul(z,r) dT + J 0 j=1 0
(38)
(u| — t>2) (t) ((pi — <p9) (z) + u2(r)<pe(z)
dr,
where ui01(z,t) — Wi(z0,t0), i — 1,2,3, 7,8,9.
Consider (27) the initial conditions (20), we differentiate (27) with respect to z for i — 1,2,3 and for t for i — 4, 5,6. After simple calculations, taking into account (28)-(32), we pass to integral equations
A f id d \ id
vi(t) =vf{t)--Ml vl(T)l—h1-—ggUt-T)dT-M1 vi(T)-h4(t-T)dT
— M1
dd
K " vj) (r) ( -h - -g9 ) (i - r) - (ü2 - f2) (T)—h6(t - r)
dt
dr
t t 9
d
0 j=1
t T
+ AMW / u2(a) [(u? — u?3) (£, r — a) — u2(t) (( — (9) (0] da
— M1
Ç=M-1[t-T ]
dr
dr (39)
00 t
Ç=M-1[t-T ]
i M(T) - "»<T)> (Is- - (i -T) + -T)
dr,
^l(i) = ^2(i) -M2f vl{r)jh,{t - t) dr,
(40)
^32(i) = - M4 I v£(t) [ ps - ) (t - r) dr
t T
+ M? J J u2(a) [(u3 — u3) (e, r — a) — u2(t) ((? — (s) (C)) da 00
t t 9 -Ms If2(r)^ (p3 — <p8) (£) dr + 2M3 f^-^Psj^v^r)
J dz [t-T] J dz „•=1
Ç=M2-1[t-T ]
dr (41)
dr,
Ç=M2-1[t-T ]
t
t
A
t
t
t
A
t
t
A f id d
vi(t) = vf(t) - -Ms / vi(r) -M - -gg ) (t - T) dT+
dt i dt
t T
+ AM5
^ — -3) (e, T — a) — u^t) (ft — £9) (e)] da
0 0
dT
?=^r1[t-T ]
t t 9 +A M5[vi(r)-?-(0 dr+ 2AM5
J dz ?=M-1[t-T] J dz =
(42)
d
0 j=i
dT
?=^r1[t-T ]
— M5
W " W " r) " " r)) + t^r)^ - r)
dT,
t>2(i) = t>°2(i) + M6 J (ft - ft) (£)
dT
?=M2-1[t-T ]
t T
+ M6
^ — u73) (e, T — a) — u2(t) (ft — fr) (e)] da
00
?=^21[t-T ]
dT (43)
Mr J vUr) (Jfo -(t - r) dr + 2M6 j JL r)
0 0 j=i
t
Ǥ(*) = +M8J vi(r)^-z (ft - ft) (0
?=^21[t-T ]
dT,
?=^21[t-T ]
t T
+ Ms J J -2(a) [(-3 — -3) (e, T — a) — u2(t) (ft — ft) (e)] da 00
t t dd
?=m21[t-T ]
dT (44)
// d d \ f d vl{r) [jhr - -gg] (t -T)dT + 2 M8 I — r)
0 0 j=i
?=m21[t-T ]
dT,
where
u02 (t) = MiQi(t), u°2(t) = M2Q2(t), u02 (t) = — 2M3P3(t),
u402 (t) = M5Q4(t), u502 (t) = —2M6 P2 (t), u302(t) = —2Ms Pi (t),
d2 _ d 1 9 i>i(t) = Wi) - - —y£^(0H(0,t),
i( ) j=i
and
d2 9 d2 9 Qi = - £^(0)^(0, i) - Ms " J>6j(0H(0,i) - 2MgPi(z)
j=i \ j=i /
j=
9
d2 9 d2 9 Q2 = ¿pMt) - £p5j(0)vj(0,t), Ql = he(t) - £p6j(0)uj(0,t) - 2M8P1(z), j=i j=i
t
p
t
t
t
t
M2 =
M1 =
A (ft(0) - £9(0)) + vpft(0) + vpft(0)
, M3 =
V2
£3(0) - ft(0)'
M4 =
£3(0) - ft(0):
Me =
V2
£2(0) - ft (0)
, M5 =
A (ft (0) - ft(0)) + vpft(0):
M7 =
1
ft(0) - ft(0):
M8 =
Vl
ft(0) - ft(0):
M9 =
1
ft(0) - ft (0)'
The equation (39)-(44) contains unknown functions j = 1,... ,9. For them we will receive integral equations from (24) by differentiating them with respect to the variable z. Using the notation (28)-(32), we obtain the integral equations for them
t
u3 (z,t) = u03(z,i) + J
d_
dz
EPij (^ )u) )
Lj=1
u2(t )
(ft -(0
dT
1 [T-t+W(z)]
+ I (v\-vl)(zi,4-t)dr
(45)
t T
u6(a) d / ,1 „,1
2 dz
(u1 - vD (£, t - a) da
ti 0
l0
dT, i = 1,9,
1 [t-t+W(z)]
1
V
P
1
V
P
2
0
Kz,t) = vf(z,t) + Jj-z - ^ (ft-ft) (0
« Lj=1
dT
1 [T-t+W(z)]
+ ¿¿0 I (vt-v$)(4,tb-T)dT
(46)
t T
v5 (T) d / ,1 „1
2 dz
(u1; - v1) (£, t - a) da
ti 0
t0
dT, i = 2, 7,
1 [t-t+W(z)]
t
u
0
t
u3(z,t) = u03(z,i) + J
d_
dz
9 u2(T) E^K^-^^r (o
Lj=1
dT
1 [t-t+W(z)]
t T
j j^lH-^r-a) ön
ti 0
t0
1 [t-t+W(z)]
dT, i = 3, 8,
0
0
t g t r d
v\(z,t) = j ^^^\Pij{z)v)(z,T)\dT + J J v\(a)—v\(z,t - a) dadr
o j=1 o o
0 j=1 t T
+ [ [ (<*(«)-<*(«)) |
— {v\ — fg) (z,t — a) + vl(z,r — a)
da dr
oo t
+ i
(v\{t) - vj(r)) (A (^1 - (*) + ) +
dz V V
dz'
dr, (48)
t g t T t
f5 = — (^5^(2;, r)) dr + J J v\{a)—v\{z,T— a) da dr + J vl{r)—(pi{z)dt, (49) 0 j=1 00 0
t 9 d t T d vl(z,t) = J [P6j(z)v}(z,T)] dr + J J v\(a)—vl(z,t - a) dadr
0 j=1 0 0
t T
+
IJ (uf (a) - u62(a))
00
+
uf - u6) (T
A
dz
A
dz
(u1 - u91) (z,T - a)
— (<Pl(z) -<P9(z))
da dT
d
+ vi(r)-ipe(z)
dT, (50)
where
d d vf{z,t) = -ul{zlQ--tl
(z0) w (z0 ,t0)
Lj=1
i = 1, 2, 3, 7, 8, 9.
A [p1(0) - <pg(0)] + vppe(0)+ Vp&(0) = 0, £5(0) = 0, <ps(0) - <ps(0) = 0, (51) A [p1(0) - <pg(0)] + Vpip6(0) = 0, p(0) - ^7(0) =0, fr(0) - <pg(0) = 0. (52) We require the fulfillment of the matching conditions
-Vj
dpi(z)
dz
d
z=0
+ = —h
j=1
i = 1,..., 6.
t=0
(53)
A
Vp
3. Main Result
The main result of this work is the following theorem:
Theorem 2. Let the conditions of Theorem 1 are satisfied, besides function h(x1; x6; t) have compact support in x1, x6 for each fixed t, <pj(z) € C6 [0, H], i = 1,..., 9, pj(t) € C6 [0, H], i = 1,2,3, 7,8,9, pi(t) € C6 [0, H], i = 1,..., 6, equalities (51), (52) and matching
conditions (25), (26), (53) hold. Then for any H > 0 on the segment [0, H] there is a unique solution to the inverse problems (15)—(18) from class rj(t) € C [0, H], i, j = 1,2,3.
< Equations (33)-(50) supplemented by the initial and boundary conditions from the equalities (19) forms a closed system of equations for the unknowns w(z,t), i = 1,... ,9, rljit), i,j = 1,2,3, ^u)i(z,t), i = 1,...,9. Consider now a square Do := {(z,t) : 0 < z < H, 0 < t < H} .
Then, these equations show that the values of the functions Wj(z,t), i = 1,..., 9, rj(t), i, j = 1, 2, 3, j i), i 1,..., 9 at (-Z, i) (E -Do expressed through integrals of some
combinations of the same functions over segments lying in D0.
The system of equations (33)-(50) we rewrite in the operator form
u = Au,
(54)
where the operator A = (A|,A2, A3), i = 1,..., 9, j = 1,..., 6, in accordance with the right-hand sides follow equations (33)-(50).
Let Cs (D0), (s ^ 0) be the Banach space of continuous functions induced by the family at weighted norms ||-|L ,
u||s = max < max sup |ui1(z,i)e st I , max sup |u2(i)e st I, max sup |u3(z,t)e
-,-st I
1^i^9(z,t)eDo
1<i<6
te[0,H ]
1<i^9(z,t)eflo
Obviously, Cs with s = 0 is the usual space of continuous functions with the ordinary norm, denoted by ||-|| in what follows. Because e-sH||u|| ^ ||u||s ^ ||u||, the norms ||u||s and ||u|| are equivalent for any H e (0, to). We choose that number s later.
Next, consider the set of functions S(u0,r) C Cs(D0), satisfying the inequality
u - u0^s ^ r,
(55)
where r is a known number, the vector function
u0(z,t) = (uf^t), i = 1,...,9, u^(t), i = 1,...,6, uf^t), i = 1,...,9) ,
defined by the free terms of the operator equation (54). It is easy to see that for u e S(u0, r) the estimate ||u||s ^ ||u0||s + r ^ ||u01| + r := r0. Thus, r0 is a known number.
Let us introduce the following notation:
£0 := max \ ||£i ||C2[0H] f , g0 := max
¿=1 9 l" [0,H]J J i=1 ,2 , 3 ,7 , 8 ,9
He2[0,H]} , h0 := .m ax6|
C 2[0,H ]
M10 = si llMi(z)llc[0,H]} , M20 = t max. 9{ ^ (z)^C[0,H]}
M30 = max 3 ¿=1,2,3,7,8,9
A(z)
VP(z)
C[0,H ]
, 11A(z) l
|dt1
C1[0,H]' II ^ llc[o
H] f , M0 = ma^M0, M20, M30}
Note that the operator A maps the space Cs(D0) into itself. Let us show that for a suitable choice of s (recall that H > 0 — is an arbitrary fixed number) it is on the set S(u0, r) a contraction operator. First, let us make sure that the operator A takes the set S(u0, r) into itself, that is, it follows from the condition u(z,t) e S(u0, r) that Au e S(u0, r), if s satisfies
some constraints. In fact, for any (z,t) € Do and u € S(u0,r) the following inequalities hold:
IKu - <||s = sup |(A}u - uO1) e-st| s (z,t)eDo
<
/ + (ft - ft) (0 i Lj = 1
3-s(t-T)
dr
1[t-T(z)]
t
+ I e
fOi
-s(i-r) fvUa)e_^_
(u1 - ug1) (£,r - a)e-s(T-a) da
dr
1 [t-T+Mi(z)]
< (9M°||f ||s + <A)|M|s + IMUIMU) f e-'V-^dT^^idMO + vo + ro) :=
t0
Using similar calculations for the remaining equations. Finally, we get
||Av - < j max |j.maxg {7y} , .max^ {72i} , ,maxg {73j} j :=
where
Yij := 9M0 + po + ro, j = 2, 3, 7, 8, 9, 714 := 13M0 + 4M0ro + 3ropo + 3po, 715 := 9M0 + po + ro, 716 := 9M0 + 9Moro + 4M0po + Po, 721 := 2ro + 2poro + 5 (M0)2 (go + ho) + 4ho +9 (M0)3 , 722 := Moho, 724 := (M0)2 (3go + 3ho + 2po + 18) , Y2j := 2M0 (ro(1 + Po) + Po + ho + 9) , j = 3, 5, 6, 73j := 11M0 + po + ro, j = 1, 2, 3, 7, 8, 9, 734 := 13M0 + 4M0ro + 3po(ro + 1), 735 := 9M0 + ro + po, 736 := 9M0 + 9M0ro + po(4M0 + 1).
Choosing s > (1/r)7°, we get that the operator A maps the set S (u°, r) into itself. Now, let u and U be two arbitrary elements in S(u0,r). Using the obvious inequality
|ukui - UkU |e-st ^ |uk - Uk||ui |e-st + |Uk ||u- - u] |e-st ^ 2ro||u - U||s, (z,t) € Do,
after some easy estimations, we find that for (z,t) € Do,
„-st^ ||u - U||s m* ,0 1
||A1u - A1U|s = sup |(A1u - A1U) e-st| < s (z,t)eDo
[9M° + + 4r0] := -741 ||f - v
and hence we have
||Au - AU||s =
- U|s
max < max {74,}, max {75,}, max {767} s 1 j=1,...,9 j=1.....^ j j=1,...,^ j
:= -71 ||f — v |
where
Y4j := 9M0 + po + 4ro, j = 2, 3, 7, 8, 9, 744 := 13M0 + 4Moro + 12ropo + 3po,
745 := 9M0 + po + 4ro, 746 := 9M0 + 9Moro + 4Mopo + 4po,
t
2
t
s
s
Y51 := 2r0 + 4^0r0 + 10 (M4)2 (g4 + M + 10h) + 9 (M4)3 , 752 := M%, 754 := (M0)2 (12g0 + 3h0 + 10^0 + 18) , Y5j := M0 (^(1 + <^4) + <^4 + 4h) +9), j = 3, 5, 6, 76j := 11M0 + ^0 + r0, j = 1, 2, 3, 7, 8, 9, 754 := 13M0 + 4M4r4 + 3^4 + 1), 765 := 9M4 + r4 + ^4, 766 := 9M4 + 9M% + ^M4 + 1).
Choosing now s > 71, we get, that the operator A compresses the distance between the elements u, U to S (u4, r) .
As follows from the performed estimates, if the number s is chosen from conditions s > s* := max{7°, 71}, then the operator A is contracting on S (u4, r) . In this case, according to the principle Banach the equation [25, pp. 87-97] (54) has the only solution in S (u4,r) for any fixed H > 0. Theorem 2 is proved. >
By the found functions rj(t), i, j = 1,2,3, the functions rj(t), i, j = 1,2,3, are found by the formulas
t
rij(t) = rij(0) + y rjj(t) dr, i, j = 1, 2, 3.
0
Note that by the functions rij(t), i, j = 1,2,3, the functions Kj(t), i, j = 1,2,3, are defined as solutions of integral equations (4).
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Received October 8, 2021 asliddin a. boltaev
Bukhara Branch of the Institute of Mathematics at the AS of Uzbekistan, 11 M. Ikbal St., Bukhara 200117, Uzbekistan, PhD Student;
North-Caucasus Center for Mathematical Research VSC RAS,
I Williams St., village of Mikhailovskoye 363110, Russia,
Researcher
E-mail: [email protected]
https://orcid.org/0000-0003-0850-6404
durdimurod k. durdiev
Bukhara Branch of the Institute of Mathematics at the AS of Uzbekistan,
II M. Ikbal St., Bukhara 200117, Uzbekistan, Head of Branch;
Bukhara State University,
11 Muhammad Ikbal St., Bukhara 200117, Uzbekistan, Professor
E-mail: [email protected], [email protected]
https://orcid.org/0000-0002-6054-2827
Vladikavkaz Mathematical Journal 2022, Volume 24, Issue 4, P. 30-47
ОБРАТНАЯ ЗАДАЧА ДЛЯ ВЯЗКОУПРУГОЙ СИСТЕМЫ В ВЕРТИКАЛЬНО-СЛОИСТОЙ СРЕДЕ
Болтаев А. А.1,2, Дурдиев Д. К.1,3
1 Бухарское отделение института Математики АН РУз, Узбекистан, 200117, Бухара, ул. М. Икбола, 11; 2 Северо-Кавказский центр математических исследований ВНЦ РАН, Россия, 363110, с. Михайловское, ул. Вильямса, 1; 3 Бухарский государственный университет, Узбекистан, 200117, Бухара, ул. М. Икбола, 11 E-mail: [email protected], [email protected], [email protected]
Аннотация. В данной работе рассматривается трехмерная система уравнений вязкоупругости первого порядка, написанная относительно перемещение и тензора напряжения. Эта система содержит свёрточные интегралы ядер релаксации с решением прямой задачи. Прямая задача есть начально-краевая задача для данной системы интегродифференциальных уравнений. В обратной задаче требуется определить ядра релаксации по заданным для некоторых компонент Фурье преобразования по переменным x1 и x2 решения прямой задачи на боковых границах рассматриваемой области. В начале методом сведения к интегральным уравнениям и последующим применением метода последовательных приближений изучаются свойства решения прямой задачи. Для обеспечения непрерывного решения получены условия гладкости и согласования начальных и граничных данных в угловых точках области. Чтобы решить обратную задачу методом характеристик она сводится к эквивалентной замкнутой системе интегральных уравнений вольтерровского типа второго рода относительно преобразования Фурье по первым двум пространственным переменным x1, x2, для решения прямой задачи и неизвестных обратной задачи. Далее к этой системе, написанной в виде операторного уравнения применяется метод сжимающих отображений в пространстве непрерывных функций с весовой экспоненциальной нормой. Показывается, что при подходящем выборе параметра в показателе экспоненты, этот оператор являются сжимающим в некотором шаре, который является подмножеством класса непрерывных функций. Таким образом, доказывается глобальная теорема существования и единственности решения поставленной задачи.
Ключевые слова: вязкоупругость, резольвента, обратная задача, гиперболическая система, преобразование Фурье.
AMS Subject Classification: 35F61, 35L50, 42A38.
Образец цитирования: Boltaev, A. A. and Durdiev, D. K. Inverse Problem for Viscoelastic System in a Vertically Layered Medium // Владикавк. мат. журн.—2022.—Т. 24, № 4.—C. 30-47 (in English). DOI: 10.46698/i8323-0212-4407-h.
