Inverse problem for viscoelastic system in a vertically layered medium

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.

Let be x = (х 1 ,х 2 ,х з ) € R ³ - Let us denote by O ij the projection onto the x i axis of the stress acting on the area with the normal parallel to the x j axis, and u i are the projection onto the x i 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]:

∂u i    ∂u j

^ij(x,t) = ^dj + dx-J + ^ij A div u

t

+ j Kij(t - т) |a (dX + dX) + 5ij A div uj (х,т) dT, i,j = 1, 2, 3, (1)

0       ji here ^ = ^(хз), A = А(хз) are Lame coefficients, 6ij is Kronecker symbol, Kij(t) are functions responsible for the viscosity of the medium and Kij = Kji, i, j = 1, 2, 3.

The equations of motion of a viscoelastic body particles in the absence of external forces have the form

∂ ² u i ^ρ ∂t ²

3 ∂σ

= У ,, i = 1,2,3, ∂xj j=1

where p = р(х з ) is medium density, U(x,t) = (U i (x,t),U 2 (x,t),U 3 (x,t)) is displacement vector. Throughout this work, µ , λ , ρ are considered to be given functions.

Note that (1) can be considered as integral Volterra equations of the second kind with respect to the expression ^ (ddl i + d j) + ^ ij A div U, i, j = 1, 2, 3. For each fixed pair (i, j) solving these equations, we get

t

∂ui   ∂uj oij (x,t)= ^(dX" + aX )+ 5ij A div u +   rij (t - T )ст. (х,т) dT, i,j = 1, 2, 3,   (3)

ji      ₀

where r ij are the resolvents of the kernels K ij and they are related by the following integral relations [3, 4]:

t rij(t) = -Kij(t) - j Kij(t - т)rij(t) dT,   i,j = 1, 2, 3.

From the condition K ij = K ji implies the r ij = r ji .

Differentiating (3) with respect to t and introducing the notation u i = dt U i , we get

t

∂

∂t ^{σ ij}

∂u i   ∂u j

(x,t) = ^(dX" + dX/+ ⁵ ij A div u + r ij (°)^a ij (x,t)+    rij (t - т )ст. (х,т) dT. (5)

ji

Then the system of equations (1) and (2) for the velocity u i and strain ct . ( ct . = j ) in view of (3) — (5) can be written as a system of first-order integro-differential equations.

t

(A-X + B-X— + C"X— + D-X— + f) U(X,t)= [ R(t - т)U(х,т) dT,

∂t ∂x1     ∂x2

where U = (u i ,U 2 ,U 3 , ^ 11 , ^ 12 , ^ 13 , ^ 22 , ^ 23 , ^ 33 ) * , * is the transposition sign,

C2 =

D2 =

A =

(pI) 3 x 3	( O ) 3 x 3	( O ) 3 x 3
( O ) 3 x 3	(I) 3 x 3	( O ) 3 x 3
( O ) 3 x 3	( O ) 3 x 3	(I) 3 x 3 .
	/ — (A + 2ц)   0
B 1 =	0	- µ
	0	0
	( O ) 3 x 3	(C 1 ) 3 x 3
C = (	(C 3 ) 3 x 3	( O ) 3 x 3
	(C 4 ) 3 x 3	(O) 3 x 3

,

⁽C 2 ) 3 x 3

( O )

( O )

- µ

-

,

C3 =

- µ

B=

3 x 3

3 x 3

-

λ

,

( O ) _{3 x 3}

⁽B 1 ) 3 x 3

⁽B ² ) 3 x 3

B2 =

,

,

C1 =

-

D=

^(O) 3 x 3

(D 3 ) _{3 x 3}

(D 4 ) _{3 x 3}

-

,

-

F =

(O)

(O)

3 x 3

6 x 3

^{(D 1 )} 3 x 3

( O ) 3 x 3

^(O) 3 x 3

D3 =

^(D 2 ) 3 x 3

( O ) 3 x 3

( O ) _{3 x 3}

- µ

,

-

λ

,

^{( —} I) 3 x 3

(O)

(O)

3 x 3

3 x 3

- λ

- λ

C4 =

D1 =

D4 =

( O ) _{3 x 3}

( O ) 3 x 3

( O ) _{3 x 3}

,

,

-

,

-

(A + 2ц)

-

-

- µ

,

λ

- µ

-

,

-

λ

(A + 2ц)

,

( O ) _{3 x 6}

diag (rn(0), Г22(0), Г33(0), Г12(0), Г13(0), Г23

(0))    ,

p(+\ = ( ^(O) 3 x 3               ^(O) 3 x 6              A

(t)               dbrnY'

\ ^{( O)} 6 x 3 ^{diag (r} 11 , ^r 22 , ^r 33 , ^r 12 , ^r 13 , ^r 23) /

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 X 3 . To do this, multiply (6) on the left by A ^{- 1} and compose the equation

| A ^{- 1} D - vI I =0,

where I is the identity matrix of dimension 9. The last equation with respect to ν has following solutions:

ν 1

-V9 = -Vp =

IA + 2ц ρ

^v 2,3 = ^{— v} 7,8 = ^—v s

-

^v 4,5,6 = ⁰,

here v _s and v _p define velocities of the transverse and longitudinal seismic wave, respectively. Now we choose a nondegenerate matrix Y(x 3 ,t) so that the equality

Y-1A-1DY = Л

is hold, where Л is a diagonal matrix, the diagonal of which contains the eigenvalues (for each fixed X 3 ) (8) of the matrix A ^{- 1} D that is Л = diag ( — v _p , —v _s , —v _s , 0, 0, 0, v _s , v _s , v _p ).

From the formula (9) implies the equality

A-1 DY = YЛ, which means that the column with the number i of the matrix Υ is an eigenvector of the matrix A-1DΥ, corresponding to the eigenvalue λi. Direct calculations show that the matrix Υ, satisfying the above conditions, can be chosen as (not uniquely)

		0 0	0 1	1 0	0 0	0 0	0 0	0 1	1 0	0 0
		1	0	0	0	0	0	0	0	1
		λ	0	0	1	0	1	0	0	λ
		ν p								ν p
Y(X 3 ) =		0	0	0	0	1	0	0	0	0
		0	0	ρν s	0	0	0	0	-PV s	0
		λ	0	0	0	0	1	0	0	λ
		ν p								v p
		0	ρν s	0	0	0	0	-PV s	0	0
		ρν p	0	0	0	0	0	0	0	-Pvp /

We introduce the vector function ϑ by the equality

U = Υϑ.

Making this change in the equation (6) and then multiplying it on the left by Υ ^{- 1} A ^{- 1} , then we get

t

( 4 +Л-^- + Bvl— + C1^— + F1) ^(x,t)= [ Ri(t - т,хз)^(Х,т) dr,

∂t    ∂x 3      ∂x 1      ∂x 2

where

	B 1 (x 3 )	= Υ - 1 A - 1 BΥ = (b ij ), F 1 (x 3 ) = Υ - 1 A - 1 D ∂Υ ∂x 3 R 1 (x 3 , t) = Υ - 1 A				C 1 (x 3 ) = Υ - 1 A - 1 CΥ			= (c ij ) ,
						+ Υ - 1 A - 1 F Υ =		(p ij ) ,
						- 1 RΥ =	^ ij )
/	r 3′ 3 2	0	0	0	0	0	0	0	r 3′3 2	\
	0	r ′23 2	0	0	0	0	r 2′ 3 2	0	0
	0	0	r 1′3 2	0	0	0	0	r 1′3 2	0
	λ(r 1 ′ 1 - r 2 ′ 2 ) ν p	0	0	r 1 ′ 1	0r	′′ 11 r 22	0	0	λ(r 2 ′ 2 - r 1 ′ 1 ) ν p		(11)
	0	0	0	0	r 1 ′ 2	0	0	0	0		.
	λ(r 2 ′ 2 - r 3 ′ 3 ) ν p	0	0	0	0	r 2 ′ 2	0	0	λ(r 3 ′ 3 - r 2 ′ 2 ) ν p
	0	r 2′3 2	0	0	0	0	r 2′3 2	0	0
	0	0	r 1′3 2	0	0	0	0	r 1′3 2	0
1	r 3′3 2	0	0	0	0	0	0	0	r 3′ 3 2

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 = { (x 1 , x ₂ , X 3 , t) : (x 1 ,x ₂ ) E R ² , X 3 E (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 R 1 (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 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 Г = Г о U Г U Г 2 :

Г 0 = { (x,t) : (x _i ,x ₂ ) G R ² , 0 <  x ₃ ^ H, t = 0 } ,

Г ₁ = { (x, t) : (x 1 ,x ₂ ) G R ² , x ₃ = 0, t > 0 } ,

Г 2 = { (x, t) : (x _i ,x ₂ ) G R ² , x ₃ = 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:

^ i \ _t =₀ = ^ i ^(x), i = ¹,..., 9,                                   ⁽¹²⁾

^ i \ _x 3 = _H = g i ^(x i ^,x 2 ^,t), i = ¹ 2, ³ ^ i ^| _x 3 =0 = g i ^(x i ^,x 2 ^,t), i = 7 8, 9.         ⁽¹³⁾

Here ^ i (x), g i (x i ,x 2 ,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 r j (t) , i,j = 1, 2, 3 ( R is included in R i according to the formula (12)) in (10) if the following conditions are known:

^ iL     = h i (x i ,^x 2 ,t), i = 1,..., 6,                             ⁽¹⁴⁾

x3= where hi (xi , x2, t), i = 1, . . . , 6, are the given functions.

In the inverse problem, the numbers r ij (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 ^(x), g i (x i ,x 2 , t) included in the right-hand side of (10) and the data (12) , (13) are compact support in x i , x ₂ for each fixed x ₃ , t . From the existence for the system (10) of a compact support domain of dependence and compact support with respect to x 1 , x 2 of the right-hand side (10) and data (12) , (13) implies the compact support in x i , x 2 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 x i , X 2 of the solution. In what follows, for convenience, we put X 3 = z and introduce the notation

----

^ (n i ,n 2 ,z,

t)=/

R ²

^(x 1 , x ₂ , z, t)e ^i[n 1 ^x 1 ⁺ⁿ 2 ^x 2 1 dx 1 dx ₂ ,

where n i , П 2 are transformation parameters. We fix n i , П 2 and for convenience, we introduce the notation ^(n i , П 2 , z, t) = ^(z, t) •

In terms of the function ϑ we write the equations (10) as

/ d d \         x"-      -        /A"-

— т) dr,

I dt + vjdzpj(z,t) = Z^P'jk(z^k(z,t) +   ZVi^-k)^k(z,t z             k=1                 0 k=1

j = 1,..., 9, where pjk = —inibjk — in2Cjk - Pjk-

We will use a similar notations for the Fourier images of functions included in the initial, boundary and additional conditions (12)–(14):

^ilt=0 =

A|z=H = 9г(t), i = 1, 2, 3, ^i|z=0 = /(t), i = 7, 8, 9,

^i|z=0 = hi(t), i = 1,..., 6.

Where A(z), i = 1, • • •, 9, g_i(t), i = 1, 2, 3, 7, 8, 9, hi(t), i = 1,..., 6, are the Fourier images of the corresponding functions from (12)-(14) for ni = 0, n2 = 0. We also denote by DH the projection of D onto the plane z, t. In what follows, we will consider the system of equations (15) in the domain Dh U Г under the conditions (16) and (17). Where Г_о= {(z,t) : 0 < z < H, t = 0}, Г₁= {z,t) : z = 0, t > 0}, Г₂= {(z,t) : z = H, t > 0}, Г = Го u ri U Г2. ^^

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 ^νj∂z

""

j Uj (z,t) = ^pj'k (z)wk (z,t) + ^Tjk (z,t)$k (z)

k=i

k=i

t"

+ / ^/jk(z,T)шк(z,t — т) dr, j = 1,..., 9, (19)

0 k=i

U, t ,, = —vi d^d(z) + ^ Pij (z)iPj (z) =: фi(z), i = 1,..., 9, j = i

Ui|z=H = -dg/i(t),  i = 1, 2, 3,   и,|z=0 = -dgi(t),  i = 7, 8, 9.

dtdt

For functions ωi additional conditions (18) gets

।             d x

Ш I =    h (t), ilz=0     dt iv ,

i = 1,..., 6.

Let us pass from the equalities (19)–(22) to the integral relations for the components of the vector ϑ with integration flux along the corresponding characteristics of the equations of the system (19). Recall that the characteristics corresponding to ν_pand ν_shave a positive slope, and the characteristics corresponding to -ν_pand -ν_shave a negative slope. We denote

z

№i(z) = I           i = 1, 2, 3, 7, 8, 9,

J Vi(e)

№(z) = 0, i = 4, 5, 6.

Inverse functions to №i(z) will be denoted by z = щ^-10. Using the introduced functions, the equations of characteristics passing through the points (z, t) on the plane of variables £, т can be written in the form

т = t + щ (£) - ^i(z), i = 1,..., 9.

Consider an arbitrary point (z, t) G Dh on the plane of variables £, т and draw through it the characteristic of the i th of the system (15) equation tell to intersection in the domain т ^ t with boundary Г. 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

t 9

Wi(z,t) = Wi(z0,t0)+ / ^pikШк(^,т) t0 k=1

dτ

Е=-1\т -t+»i(z)]

Я9                ^T9                     "I

^ Тгк,⁽£^,т)$г⁽€⁾⁺ / ^ Tk⁽£^{,т - а)ш}к ⁽£^{,а) da} ₀k=1                  ₀ k=1

(24) dτ,

Е=-1\т -t+»i(z)]

i = 1,..., 9.

We define in (24) tg. It depends on the coordinates of the point (z,t). It is not difficult to see that tⁱ₀(z, t) has the form

tⁱ₀(z, t) =

t - №i(z) + Щг(Н), t > ^i(z) - щг(Н),

0,                     0 < t < ^i(z) - щг(Н),

i = 1, 2, 3,

t0^(z,t) = ^0, i = ^4,5,⁶, t0^(z,t) = ^

t - ^i(z), t > №i(z),

0,

0

i = 7, 8, 9.

Then, from the condition that the pair (z0i , ti0) satisfies the equation (23) it follows x f H,               t > щ(z) - №(H),

z0(z,t) = < ’                                 P V h i = 1,2,3,

(№i 1 (^i(z) - t), 0 < t < №i(z) - щг(Н), zi(z, t) = z, i = 4, 5, 6,    zi(z, t) = < , ,                                ^’          i = 7, 8, 9.

⁰        ,              ⁰       V-¹Mz) -1) , 0 №(z),       , ,

The free terms of the integral equations (24) are defined through the initial and boundary conditions (20) and (21) as follows:

^i^(z0,to) =

(55 Hi⁽t - №^(z) + m⁽H)), I ^фi (r, ' ⁽№^{(z) -}t)),

t > ^(z) - m(H),

0 < t < ^i(z) - ^i (H),

i = 1, 2, 3,

^i(z0,t0) = Фi(z),

^i^(z0,to) =

( d Иг^(t- №^(z)) ,

( ^фi (г, ' ⁽№^{(z) -}t)) ,

i = 4, 5, 6, t ^ ^i(z), 0  №(z),

i = 7,8,9.

Let the following conditions hold

Hi (H )= Hi (0) and

ддг (t)     = - d<3i (z)

^dtt=0     j^{3   dz}

+ £Hij (h )Hj (h ), ^z=^H  j=i

i = 1, 2, 3, (25)

+ Vp. (0)<Hj (0), i = 7, 8, 9. (26) z=⁰ j=i

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 Dh 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 Wi(z,t), or the same 57 M^t), i = 1,... , ⁹

Suppose that all given functions included in (24) are continuous functions of their arguments in DH . 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 Dht = {(z,t) : 0 < z < H, 0 < t < T}, T > 0 are some fixed number.

Theorem 1. Assume functions y(x), g(xi,X2,t) have compact support in xi, X2 for each fixed z, t. Let be p(z), ^(z), X(z),Cp(z) G C ¹[0, H], g(t) G C¹[0,T] , p(z) > 0, A(z) > 0, ^(z) > 0, rj (t) G 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 Dht■

The problem (15)–(17) in the domain DHT is equivalent to a linear integral equation of the second kind of Volterra type with respect to ϑ. As follows from the theory of linear integral equations, it has a unique solutions [3]. So we drop it.

j    ^dgi^(t)

^i(0) = gi (0) and dt

_      d<3i(z)

j ∂z

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) G Г and draw through it the characteristics (23) for i = 1, 2, 3, up to the intersection with the boundary of the domain Dht. Integrating the first six components of the equation (19), we obtain

^wi(z, 0) = dt ^Hi (ti)

i

Я9               9            "I

^ Pij⁽£)^3 ⁽£^,T) ⁺^ Tij⁽£^,T)Hj⁽£)

j=i                  j=i

i t1 t 9

-j ] Z1 rij⁽£^,а)ш3 ⁽£^{,T - a) da}

where ti = —^i(z), i = 1, 2, 3, ti = t, i = 4, 5, 6.

dτ е=м-1 [t+^i(z)]

dT, i = 1,..., 6, (27)

?=^i ^1[t+^i^(z)]

For the purpose of further research we introduce the following notation for the unknowns:

u1(z,t)= Wi(z,t), i	= 1,..., 9,	"2(t) = r11(t),	"2(t) = r12(t),	"3(t) = r‘3 (t)-	(28)
"4 (t) = r2 2(t),   "5(t)	= r2 з(t),	"6(t) = r3 3(t),	∂ "i (Z,t) = Sy^i	Cz-t)-    i = 4,5-6-	(29)
ui3(z, t) =	∂ dz^i(z,t)	- r332r- (^0)	- m=0 )) d^to-	i = 1,9,	(30)
"i3(z, t) =	∂ dz '^z t	- r23^ (ft(z0)	- ?7(z0)) dzfo,	i = 2, 7,	(31)
"i3(z, t) =	∂ дzШi(z,t)	- r13^ (^CzO)	- n8(=0)) дZt0-	i = 3, 8.	(32)

Taking into account these notations and the explicit forms of the functions rij(z,t) in terms of rij (t) by the formula (11), we rewrite the equations (24) in the form

"H^t)

t₉

= ^U01(z,t)+ / ^Pij⁽⁴⁾"j^(4,T) _ti₀    j =1

-

(n - ПО)]

dτ

ξ=µ_i^-1[τ -t+µi(z)]

- / j ^U2₂a⁾ ("1 - "9) ⁽4'T " “) ^Д“ _ti₀  0

dT, i = 1, 9, ξ=µ_i^-1[τ -t+µi(z)]

^Ui(z,t)

t₉

= ^u01(z,t)+ /  ^Pij^(4)u1^(4,T)

_ti₀    j =1

-

"2(т)

(n - ^7) (4)]

dτ

ξ=µ_i^-1[τ-t+µi(z)]

-

j j ^ ("2 - "7) (4,t - a) da tⁱ0 0

t

"H^t)

01 υ_i

^(z,t)+ / ^Pij⁽O^uj⁽C,^T)

i    j=1

-

i t₀

-

dT, i = 2, 7,

ξ=µ_i^-1[τ -t+µi(z)]

^ (й - ft) (4)

j j ^ (u2 - u2) (4,t - a) da ti0  0

dτ

ξ=µ_i^-1[τ-t+µi(z)]

dT, i = 3, 8,

ξ=µ_i^-1[τ-t+µi(z)]

t₉                   tτ

^Ui(z,t)

= j ^^ p4j(z')uj(z,T) dT + У У u2(a)u4 (z,T - a) da dT

₀j=1                       _{0 0}

tτ

+ У j (u2 - U2) (a) VJ( (u1 - U9) (z, T - a) + u6(z, t - a)^ da dT t +/ 0

υ1²

-

"4)^(т) (^ ⁽$i - $9)^{(z) +}^(z))

+ "2(т)П4(z) dT,

t9                     tτt u1(z,t) = j ^ P5j (z)uj (z,T) dT + j j u2(a)u1 (z,t — a) da dT + ju2(t)hi (z) dT, 0 j=1                        0 00

tτ u1(z,t) = j j ^7 ^u2 - Ue) (a) (u1 — u1) (z,t — a) + u2(a)u6(z,T — a)] dadT 00 p t9

+ / ^P6j(z^Uj(z,T)dT + j [V- (U4 — Ue) (t) (hi — $9)(z) + U2(T)

where u⁰ⁱ(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 tt u2 (t) = U02(t) — —Mi [ U2(T) (^hi — d-дЛ (t — t) dT — Mi [ u.(t)-d^(t — t) dT νp              dt dt                             dt

• ^—Mi / Г7" (^U2^{— U}4) ^(t) f ;d^hi^—;d^h9)^{(t — T})^—(^U2^{— U}4)^(T) ;d^he^{(t — T})1^dT ν_p                dt dt                            dt

t

+AMi j ^{u2 (t} ) dz ^(hi^—<$9) ⁽€)

∂⁹

d^{T + 2}AMi — $Г.Р1з⁽£ )Wj ⁽^^,t )

ξ=µ1^-1[t-τ]                  ^∂z j=1

tτ

^+AMi /1^u2(a) [(^U3^{— U}9)⁽^

t — a)

^{— U}6^{(t) (<h}i ^—^M^] ^da

dτ ξ=µ₁^-1[t-τ]

dτ ξ=µ₁^-1[t-τ]

^—Mi / [7" (^u2(t) ^{— u}6^(t)) f7+^hi ^—779*^^{(t — T) + u}4^(t)7+^h6^{(t — T})1

ν_p                   dt dt                    dt

dτ,

u2 (t) = .. (t) — M2 t U3²(T) ddh₅(t — T) dT, 0

-

t

^U3^(t) = ^U3^{2(t) —}M4 У ^u2(t) (ddt^h3

-

d dt^g8

(t — T) dT

tτ

⁺M31 /^U3^(a)f(^U3^{— U}8) ⁽^^{,t — a) — U}3^{(t) (}3^{— )(}^))^da 00

ξ=µ₂^-1[t-τ]

dτ

t

M3 jU.1T)dZ ($3 — _й)(5)

∂⁹

dT + 2M3 — V.P33 (£ )uj(7T)

ξ=µ2^-1[t-τ]               ^∂z j=1

dτ, ξ=µ₂^-1[t-τ]

t u2(t) = u42(t) - —M5 [ и2(т) (-dhi - d-gA (t - т) dT+ νp              dt dt tτ

^{+ X}M5 / 1 ^U2^(a) ^3 - ^ ⁽^

^{T - a) - u2(t) (—}. ^-£9) (£)] ^da

dτ ξ=µ₁^-1[t-τ]

+XM,Ju²(T)dZ (—-*)(£)

ξ=µ₁^-1[t-τ]

^t∂⁹

d^T +2AM5 / — У pij⁽£^)u1(C,^T)

₀^z j=1

dτ ξ=µ₁^-1[t-τ]

-^M5 / [vp ^(u4^{- 6 (T}) (sh'^{(t - T})^-    9 t ^-T))

+ U2(T) dd h6 (t - t)j dT,

^u2(t) = ^и0 ^{(t) +} M6 У ^{u2 (t} ) dZ^(—2 ^-£7) (£)

dτ ξ=µ₂^-1[t-τ]

tτ

⁺M61 /^u2(a) [(^u3^{- u}7^⁽^

^{T - a) - u}2 ^{(t) (}<2^-7) ⁽€)] ^da

dτ ξ=µ₂^-1[t-τ]

tt

- M7 [ u2(t) (hh- - У g7\ (t - T) dT + 2M6 [ ^yj p2j(p)uj (^,T) dt dt∂z

0                                               0

^u2^(t) = ^u62^{(t) +}M8 j^u6 ^(t) dZ ⁽ⁱ<1 - ⁽?9^{) (}£)

dτ ξ=µ₁^-1[t-τ]

dτ, ξ=µ₂^-1[t-τ]

-

tτ

⁺M8 1 / ^U2(a) [(^U3

t

^{- u}3) ⁽^^{,T - a) - u}6^{(t) (i}?1^-<9⁾⁽£)] ^da

ξ=

dτ µ₁^-1[t-τ ]

M9 / ^u6 (^T) (dthi 0

-

^d_й dt^g9

∂⁹

j ⁽t-^т)d^{T + 2}M8 j y^Lpij⁽^)^и(^^,T)

0 j=1

dτ, ξ=µ₁^-1[t-τ]

u0²(t) = MiQ1(t), u0²(t) = M2Q2(t), U3⁰²(t) = -2M3P3(t), u4²(t) = M5Q4(t),   u0²(t) = -2M6P2 (t),   u3²(t) = -2M8 Pi (t),

Pi (t) = 1Г2 ^i (t1) - d^Z фi(0) - ViB У ^ (°)^j (°’t)’ and

_Q1₁

d²

dt2 ^h4^(t)

.—й

-

У P4j (°)^j (°,t) - M5[ j=1

d²

dt2 h^{6 (t)}

.——

-

У P63(°b(°,t) - 2M8Pi(zA , j=1

29                                 29

Q2 = ^2Mt) - E—5j(°)^j(°,t),  Q4 = ^2Mt) - E—6j(°)^j(°,t) - 2M8Pi(z), dt                                      dt j=1

j=1

M²   ^5(0)’  M¹   A (71(0) - £9(0))+^£б(0) + v_p$4(0)’

³   £3(0) - w(0)’

M⁴   V3(0) - 78(0) ’  M⁶

V2(0) - .(0)’      ⁵

νp

A (vM0) — V9(0)) + vpV6(0) ’

£2(0) - £7(0) ’   M⁸   71(0) - V9(0) ’  M⁹   n(0) - .9(0) ■

The equation (39)-(44) contains unknown functions ^Z, 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’^t)+ /ddZ ^ p'ij ^{(4)U1 (}^’^T) ,i      Lj=1

-

^ (71 - 79) (4)

i t₀

tⁱ

+^ /

- 0 ’t0^{- T}) d^T

tτ

^- / / ^U62^a)dZ ^u -^u9) ⁽⁴’^{T - a) da} to ⁰

u3(z’t) = u^Cz’t)+ / d|; [ ^ Pij(4 )u1 (4’t ) -+i      Lj=i t0

dτ,

?=^i ^1[t-t+Mi(z)]

ЕэИ

(7 - Vr)«)j

i t0

⁺ IZt0 / ^^{u1 - u}7 ^ ^(z0 ’ t0^{- T})^dT

tτ

^- / / ^и52^т) dz ^^U1^{- u}1) ⁽⁴’^{T - a) da ti}0 ⁰

dτ,

§=^-^1[t-t+^i^(z)]

t

^ui^3(z’^t) = ^u03

(Z’t)+ fib ^ 7ij (4)uj «’T)-ei ₊i       Lj-=1

ЕзИ

(73 - 78) (4)

i t₀

tⁱ

+ddZt0 J

ii

₀’ t₀

- T) dT

dτ

§=^-1 [t-t+_W(z)]

i = 1’ 9’

dτ

e=^i¹[t-t+^i(z)]

i = 2’ 7’

dτ

§=Ц-1[т-t+^i(z)]

tτ

- ti0  0

из(а)

Э

2 dz

(^u3^{- u}8) ⁽⁴’^{t - a) da}

§=^i ^1[t-t⁺^i^(z)]

dT’ i = 3’ 8’

t9  ∂                    tτ      ∂ u3(z,t) = ^ Ё d- [p4j (-)uj1 (z,T)] dT + j j U2(a) —u4(-,T - a) dadT

0 j=1                             0 0

tτ

+

j j (u>2.(a) - ^u4(^a)) d-

00 t

d- [ip M - ^u9 ’ ^(-T

- a) + u6(z,T — a) dadT

+

/[(^u2^(t) - ^u2(T)) d-^ ⁽й 0

^-<9^{)(-) +}<6^(-)) ^{+ u}2^(T)д|-<4^(-) d^T, ⁽⁴⁸⁾

t9  ∂ u3 = / Ё d- ^T^ dT+

0 j=1

tτ

t

j j u1(a) d-u¹(-,t - a) dadT + j и2(т) ^-<1 (-) dT, (49)

t9  ∂                    tτ      ∂ u6(-,t)= j^ d— [p6j (z)uj (-,t)] dT + j y‘u2(a) —u6(-,T - a) dadT

0 j=1                             0 0

+

tτ j j (u4(a) - u6(a)) d- vA (ui - u9) (z

00                p

T-

a)

dα dτ

t

+ / (u2 - u2) (t)— №(-) - (?9(-)) + u2(T)77" <6(-) dT, (50) ∂z νp                       ∂z where

∂

^U03(-,t) = ^i^(-° ’t^o)

-

d-to ^ pij^{(-9) ш}з^(-o ,to)j,

i = 1, 2, 3, 7, 8, 9.

A [<6i(0) - <69(0)]+ v_p<66(0) + v_p<64(0) = 0,   <65(0) =0,   <6з(0) - <68(0) = 0,       (51)

A [<i(0) - <9(0)] + Vp<6(0) = 0,   <2(0) - <7(0) = 0,  <1 (0) - <9(0) = 0.     (52)

We require the fulfillment of the matching conditions d<6i(-) j ∂z

+ Ё P.j (0)

^, t=0

i = 1,..., 6.

• 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(xi; X2; t) have compact support in xi, X2 for each fixed t, <6i(-) € C²[0, H] , i = 1,..., 9, gt(t) 6 C²[0, H] , i = 1, 2, 3, 7, 8, 9, hi(t) € C²[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 rij(t) € C [0,H] ,i,j = 1, 2, 3.

<1 Equations (33)-(50) supplemented by the initial and boundary conditions from the equalities (19) forms a closed system of equations for the unknowns wi(z,t), i = 1,..., 9, ri’j (t), i,j = 1, 2, 3, dZwi (z,t), i = 1,..., 9. Consider now a square Dq := {(z, t) : 0 C z C H, 0 C t C H} .

Then, these equations show that the values of the functions Wi(z,t), i = 1,..., 9, r’j^(t), i,j = 1, 2, 3, ddZWi(z,t), i = 1,..., 9 at (z,t) € Dq are expressed through integrals of some combinations of the same functions over segments lying in Dq.

The system of equations (33)–(50) we rewrite in the operator form и = Au,

where the operator A = (Ai^A?, A³), i = 1,..., 9, j = 1,..., 6, in accordance with the right-hand sides follow equations (33)–(50).

Let Cs(Dq), (s ^ 0) be the Banach space of continuous functions induced by the family at weighted norms HJIs ,

||u^_s= max < max sup ^|u¹(z,t)e (^1Ci^C9(z,tWo

st I , max ^sup H^{(t)e 1C}i^C6t_E[Q,H]

st¹, max sup ^|u³(z,t)e st¹>. ' Ki<9(z,t)_GDo' ’*          J

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| C |u|_sC |u|, the norms |u|_sand |u| are equivalent for any H € (0, to). We choose that number s later.

Next, consider the set of functions S(u^Q,r) C C_s(Dq), satisfying the inequality

^||u— ^и0^ ^Cr,

where r is a known number, the vector function uQ(z,t) = (uQ1(z,t), i = 1,...,9, uQ(t), i = 1,...,6, uQ3Cz,t), i = 1,...,9), defined by the free terms of the operator equation (54). It is easy to see that for u € S(uQ, r) the estimate |u|s C |uQ|s + r C |uQ|| + r := rQ. Thus, rQ is a known number.

Let us introduce the following notation:

^{hQ :}= iss Oh L -юн}¹

^Q ^:= i^mlaxg {^ ^lc2tQ’H 1} , go ^:= i=i^ma/7,8,9 {^ ^2[Q,H 1} ,

M^Q= ^max{ll^Mi^(z)llc[Q,H]} , M⁰ = • .max ^WpijИ^н]} , 1,...,9x.                                                         i,i 1«...«9

i

3S    , l^A(z)lc 1[Q^P l dl U’M^Q=max {^30} ^.

Vp(Z) c[q H]                ^{L J}       ^z     ^{L J} I

Note that the operator A maps the space Cs(Dq) 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(uQ, r) a contraction operator. First, let us make sure that the operator A takes the set S(uQ, r) into itself, that is, it follows from the condition u(z,t) € S(uQ, r) that Au € S(uQ, r), if s satisfies some constraints. In fact, for any (z,t) E Do and и E S(u0,r) the following inequalities hold:

||AiU - u01|_s= sup | (Aiи — u⁰¹) e ^st| (z,t)∈D0

t

V

[ ^ Pij(^j (^,T)

/    7 = 1

i^j

e ^ST+

EeHe

i t₀

t

τ

sτ

- (V1

^9)(€) e

s(t τ)

dτ

ξ=µ_i^-1[t-τ+µi(z)]

+

V^]!

sα

— ^(u1

— u9) (£,r — a)e ^{s(t a)}da

i t₀

dτ

ξ=µ_i^-1[t-τ +µi(z)]

< (9M0M. + «,||* + M.M.) /■)dT < rs (9M⁰+ vo + ro) := 211 ro.

t¹₀

Using similar calculations for the remaining equations. Finally, we get

||Au - u0||s ^ 1 max J max {Yij} , max {Y2j} , max {Y3j} f := r070, s           j =1,...,9            j=1,...,6            j=1,...,9                   s where

71j := 9M⁰+ vo + ro, j = 2, 3, 7, 8, 9, 714 := 13M⁰+ 4M⁰ro + 3_Wo + 3^0,

715 := 9M⁰+ vo + ro, 716 := 9M⁰+ 9M^oro + 4M⁰vo + vo,

721 := 2ro + 2voro + 5 (M0)2 (go + ho) + 4ho + 9 (M0)3 ,  722 := M%,

724 ^:= (^M0) ^(3go ^{+ 3}ho ^{+ 2}Vo ^{+ 18)}, Y2j ^:= ^{2M0 (r}o^{(1 +} Vo) ⁺ Vo ⁺ ho ^{+ 9)}, j = ³,⁵,⁶, 73j := 11M⁰+ vo + ro, j = 1, 2, 3, 7, 8, 9, 734 := 13M⁰+ 4M⁰ro + 3vo(ro + 1),

735 := 9M⁰+ ro + vo, 736 := 9M⁰+ 9M⁰ro + Vo(4M⁰+ 1).

Choosing s > (1/r)Y0, we get that the operator A maps the set S (u⁰, r) into itself.

Now, let u and v be two arbitrary elements in S(u⁰,r). Using the obvious inequality

|ukui — uku| |e-st ^ |uk — Vk||ul|e-st + |vk| |u| — V||e-st ^ 2ro^u — v^s, (z,t) E Do, after some easy estimations, we find that for (z,t) E Do,

^A1u — A1v||s = sup |(A1u — A1v) e-st| < ——— [9M0 + vo + 4ro] := 1741 ||u — v^s s    (z,t)∈D0                                     s                                     s and hence we have

|Au — Au|_s =

|u — VL г _r . _r . _r a

17¹B« — SUs ,

-------s max < max {74 Л , max {75 Л , max {76 Л > := s                  j=1,...,9     j j=1,...,6     j j=1,...,9     j where

74j := 9M⁰+ vo + 4ro, j = 2, 3, 7, 8, 9, 744 := 13M⁰+ 4M^oro + 12_Wo + 3vo,

745 := 9M⁰+ vo + 4ro, 746 := 9M⁰+ 9M% + 4M0vo + 4vo,

Y51 := 2ro + 4^oro + 10 (M0)2 (go + he) + 10h + 9 (M0)3 , Y52 := MX,

Y54 : = (M0) (12g0 + 3h0 + 10^0 + 18) , Y5j : = M0 (r0(1 + ^0) + ^0 + 4h0 + 9) , j = 3 5, 6

Y6j := 11M⁰+ ^0 + ro, j = 1, 2, 3, 7, 8, 9, Y64 := 13M⁰+ 4M⁰r, + 3^o(ro + 1),

Y65 := 9M0 + Г0 + ^0, Y66 := 9M0 + 9M% + ^(4M0 + 1).

Choosing now s > γ¹, we get, that the operator A compresses the distance between the elements u,U to S (u⁰,r) .

As follows from the performed estimates, if the number s is chosen from conditions s > s* := max{Y⁰, Y1}, then the operator A is contracting on S (u⁰, r) . In this case, according to the principle Banach the equation [25, pp. 87–97] (54) has the only solution in S υ⁰, r for any fixed H > 0. Theorem 2 is proved. >

By the found functions r’j(t), i, j = 1, 2, 3, the functions rij(t), i, j = 1, 2, 3, are found by the formulas

t

rij (t) = rij

(0)+

rij (t ) dT,

i,j = 1, 2, 3.

Note that by the functions rij(t), i,j = 1, 2, 3, the functions Kij(t), i,j = 1, 2, 3, are defined as solutions of integral equations (4).