An Inverse Two-Dimensional Problem for Determining Two Unknowns in Equation of Memory Type for a Weakly Horizontally Inhomogeneous Medium Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2024, Том 26, Выпуск 3, С. 112-134

УДК 517.958

DOI 10.46698/e7124-3874-1146-k

AN INVERSE TWO-DIMENSIONAL PROBLEM FOR DETERMINING TWO UNKNOWNS IN EQUATION OF MEMORY TYPE FOR A WEAKLY HORIZONTALLY

INHOMOGENEOUS MEDIUM#

M. R. Tomaev1 and Zh. D. Totieva1

1 North Ossetian State University, 44-46 Vatutina St., Vladikavkaz 362025, Russia E-mail: [email protected], [email protected]

Аннотация. A two-dimensional inverse coefficient problem of determining two unknowns — the coefficient and the kernel of the integral convolution operator in the elasticity equation with memory in a three-dimensional half-space, is presented. The coefficient, which depends on two spatial variables, represents the velocity of wave propagation in a weakly horizontally inhomogeneous medium. The kernel of the integral convolution operator depends on a time and spatial variable. The direct initial boundary value problem is the problem of determining the displacement function for zero initial data and the Neumann boundary condition of a special kind. The source of perturbation of elastic waves is a point instantaneous source, which is a product of Dirac delta functions. As additional information, the Fourier image of the displacement function of the points of the medium at the boundary of the half-space is given. It is assumed that the unknowns of the inverse problem and the displacement function decompose into asymptotic series by degrees of a small parameter. In this paper, a method is constructed for finding the coefficient and the kernel, depending on two variables, with an accuracy of correction having the order of O(e2). It is shown that the inverse problem is equivalent to a closed system of Volterra integral equations of the second kind. The theorems of global unique solvability and stability of the solution of the inverse problem are proved.

Keywords: inverse problem, delta function, Fourier transform, kernel, coefficient, stability. AMS Subject Classification: 35L20, 35R30, 35Q99.

For citation: Tomaev, M. R. and Totieva, Zh. D. An Inverse Two-Dimensional Problem for Determining Two Unknowns in Equation of Memory Type for a Weakly Horizontally Inhomogeneous Medium, Vladikavkaz Math. J., 2024, vol. 26, no. 3, pp. 112-134. DOI: 10.46698/e7124-3874-1146-k.

1. Problem Statement

Consider for x = (x\,x2,x3) € R3, t € R, x3 > 0, the integro-differential equation

I? = È ¿7 *3) J|) + / - T) E ¿7 -S3) J|) (*> r) dr, (1.1)

under the following initial and boundary conditions

u |t<0= 0, (1.2)

#The study was performed with the support from the Russian Science Foundation, project no. 23-27-00264, https://rscf.ru/en/project/23-27-00264/.

© 2024 Tomaev, M. R. and Totieva, Zh. D.

a(x2,0)

du dx3

t

(x,t) + / k(x\, t — r)——(x,t)c1t J dx3

= -5(xi )5(x2)5'(i), (1.3)

X3=+0

u(x,t) is the displacement function, a(x2,x3) is the velocity of propagation of transverse elastic waves, fc(xi , t) is the memory function showing the viscous properties of the medium; 5(-) is the Dirac delta function, 5'(■) is the derivative of 5(-).

The direct problem is to find the function u(x,t) from equation (1.1) under the initial and boundary conditions (1.2), (1.3).

The inverse problem: to determine the function u(x,t) coefficient a(x2,x3) and the memory kernel k(x1 , t), t > 0, if additional information is known

FX1 tX2 M(x3,t,v,A)|X3=+o = g(t, v, A), t> 0, v, A € R, (1.4)

where g(t, v, A) is the measurement data and

oo

Fx1;x2 [u](x3 ,t,v,A)= J u(x,t)e-i(vX1 +Axa) dxi dx2

-o

is the Fourier transform of the function u(x,t) by variables x1, x2 (next, i is imaginary unit).

Definition. A pair of functions a(x2,x3) € (R x [0, to)), k(x1,t) € (R x [0, to)) is called the solution of the inverse problem (1.1)-(1.3) if the solution of the direct problem (1.1)-(1.3) u(x, t) from the class of generalized functions D'(R+ x R) satisfies (1.4) for g(t, v, A), belonging to the class D'([0, to)) for a fixed nonzero (v, A).

The problems of determining the kernel of the integral convolution operator is a trend in the theory of inverse problems that arose at the end of the last century [1-8]. A more detailed analysis of the sources is presented in the monograph [9], which is one of the latest fundamental works in the theory of inverse problems for equations of memory type. It presents the results of a study of the well posedness of one-dimensional and multidimensional inverse problems for hyperbolic integro-differential equations of memory type. Theorems on the unique solvability of the inverse problems are proved, and stability estimates are obtained. Among the first results on inverse problems of linear viscoelasticity (close to this) can be noted [5, 10, 11]. In [5], the local solvability and global uniqueness in the one-dimensional inverse problem of determining the kernel of the integral convolution operator of the viscoelasticity equation with constant coefficients are obtained. In this paper, the direct problem is the Cauchy problem with continuous data. The inverse problem is replaced by a system of Volterra integral equations of the second kind. In [10, 11], the method of separation of variables is used to solve inverse problems in a limited domain, by which the problems are reduced to a system of integral equations of the Voltaire type with respect to unknown functions depending on a time.

The further results of research, in particular, over the past ten years is shown, for example, in [12-29]. In [12-17] there are inverse problems on the determination of kernels having a special structure. The goal is to reduce the initial problem to solving problems of integral geometry using a singular source (delta function) of wave disturbance. The unknowns to the inverse problems are the coefficients of the equation and the spatial parts of the kernel. In articles [18, 19], the main results are the global unique solvability of one-dimensional inverse problems using spaces of continuous functions with a weighted norm. In recent years, there has been an increasing of the number of publications on numerical calculations of the integral operator kernels [20-24].

Of most interest are multidimensional kernel determination problems when the unknowns depends on two or more variables. The multidimensional inverse problem for (1.1)-(1.3) and additional information (1.6) have been investigated in [25]. In this work, based on a combination of the method of scales of Banach spaces and the method of weight norms, a global unique solvability of the problem of determining the kernel of k(x, t) in the class of functions analytic in the variable x and smooth in the variable t was obtained.

In [26], the problem of determining the two-dimensional kernel of an integro-differential equation in a medium with weakly horizontal inhomogeneity is considered, in which method from [27] is developed.

Among the works devoted to coefficient inverse problems for viscoelastic media, which also determine the kernels of integral operators, one can note the works [28, 29]. For example, in [29] the one-dimensional problem of simultaneous determination of the wave propagation velocity and the kernel of the integral operator was studied. It is shown that both unknowns are uniquely determined by setting the Fourier image for the spatial variable of solving a direct problem on the boundary of a half-space. A conditional assessment of the stability of the solution of the problem is established.

The fundamental difference from the above results and at the same time the significant novelty of this work is the fact that it presents a multidimensional inverse problem of simultaneously determining the coefficient of the viscoelasticity equation and the kernel of the integral operator describing the properties of a viscoelastic medium for a half-space.

It should be noted that simultaneous recovery of several parameters for media with aftereffect is undoubtedly an actual problem from the point of view of applications, since it becomes possible to analyse the influence of the memory of the medium, for example, on the velocity of wave propagation in space. For practical applications, it is more interesting when the characteristics of the environment depend on two or more variables. For example, for geophysics, one of the main problem is the quantitative assessment of horizontal inhomogeneities in the velocities of seismic waves [30].

In this paper, which is a continuation of the study presented in [31], a new approach to the simultaneous determination of parameters depending on two variables in the viscoelasticity equation for a half-space is proposed. The novelty of the approach lies in the assumption that k(x1,t), a(x2,x3) weakly depend on the horizontal variables x1, x2 as follows:

a(x 2,x3) = ao + ex2a1(x3) + O(e2),

0 (1.5)

k(x1,t) = ko(t) + ex^t) + O(e2),

where e is small parameter.

In the equations (1.5) a0 is a given positive constant.

The main purpose of this work is to construct a method for finding k0(t) and a1(x3), k1(t) with an accuracy of O(e2). To do this, as we will see later, it is enough to set the g(t, v, A) for two different non-zero sets (vj, Aj), j = 1,2.

The necessary and sufficient conditions for the global unique solvability of the inverse problem (1.1)-(1.4) and its stability estimate represent the theoretical significance of the work.

The theoretical results are useful for applications in solving seismic problems and the numerical implementation of this study. It has been shown [32] that with an increase in the strength of an earthquake, the soil behaves not as an elastic, but as a viscoelastic body. Soils are medium with memory, that is, the state of such medium at the current time depends on the entire background of the process. This is indicated, for example, in [33], which provides

a detailed review of studies to clarify the nature of absorption of seismic waves in soils and examines the main patterns of absorption of stress waves in dispersed and semi-bedrock. As shown in [34], failure to take into account the absorbing properties of the medium leads to significant distortions in the restoration of the velocity model of the medium. The author is going to make a numerical analysis of the effect of the memory function on the wave propagation in half-space later. The algorithms given in the monograph [35] are also the basis for numerical analysis.

We seek the solution to (1.1)-(1.3) in form of the series in powers of e

u(x,t) = ^ £jUj (x,t).

(1.6)

j=0

Using (1.4) and (1.6), we have

Fxi!x2 [u](x3 A)

X3=+0

=: U(x^x2,t) = ^ejUj(xi, x2,t).

j=0

It is not difficult to verify that Uj (hence Uj) are even in xi, x2 for even j and odd for odd j. Thus, according to the well-known function U(x1,x2,t), Uo(x1,x2,i) and U1(x1,x2,t) can be found up to O(e2) [27]:

TT U(X1 ,X2,t) + U(-X1, -X2,t) f/0(xi,x2,i) =-

Ui(Xi,X2 ,t) =

2

U (xi, x2, t) — U (—xi, — x2, t) 2 '

Since the method presumes determining a1(x3), k0(t), k1(t) with accuracy O(e2), by inserting (1.8) and (1.7) in (1.1), we obtain two inverse one-dimensional problems of the successive definition of k0 (t) and a1(x3), k1(t).

(i) The problem of determining k0(t) and u0(x,t) from the equalities

1 d 2 u0 a0 dt2

d2uo d2uo

dxi dx2

+

d (duo 9x3 \dx3

t

+ J Ä0(i - T)

d 2 u0 d2u0 dxi dx2

_d_ i<9-u0\ 9x3 \dx3J

(x, t) dT,

fl0

U0 |t<0 = 0,

t

f ko(t — r)^^(x,r) dr 0x3 J 0x3

= -5(xi)5(x2 )5'(i),

X3=+0

Fxi,X2[U0](x3,t,v, A)|x3=+0 = Fx1;x2[U0](t, v, A) =: g0(t, v, A), t > 0.

(1.7)

(1.8) (1.9)

(1.10)

(ii) The problem of determining a1(x3),k1(t) and u1(x,t) from the equalities d2u1

dt2

L

d / dun duA d ( duo\

fco, t— ^201(^3)7^--h a0T— + T— Xïaiix-i)——

dx3 \ dx3 dx3 / dx2 \ dx2 /

+

dx2 dx2

+ X2ai (X3)

cruo dxl

t

+ J xi ki(r )

ao

d2u0 d2u0

+

dxl dx2 j

+ — (a —^

V 0 dx3 J

(x, r) dr,

(1.11)

ui|t<0 = 0,

L

i /,Mdu0 dui

dx3 dx3

t

/, du0

Ki(i — rj——(a;, r) (it dx3

0,

X3=+0

[ui](x3,t, V, A)| X3=+0 — FX1,X2 [Ui](t, V, A) —: gi(t, v, A), t >

(1.12)

(1.13)

(1.14)

2. The Problem of Determining k0(t) and u0(x,t)

Introduce the variable z by the formula

x3

Let

w(z, t, v, A) :—

2 := —=, c0 := a/öö.

V^o

v(z,t, v, A) :— Fx1 ,x2 [u0](c0z, t, v, A)

t

v(z,t, v, A) + y k0(t — r)v(z,r, v, A) dr

exp(-k0(0)t/2) .

Then

t

(z, t, v, A) — exp (Ä0 (0)t/2)w(z,t,v, A) + J m(t — r)exp (k0(0)r/2)w(z,r, v, A) dr, (2.1)

t

r0(t) — —k0(t) — J k>(t — r)r0(r) dr.

We obtain the following equations for the functions w(z, t, v, A) and r0(t):

t

d2w d2w

dt2 dz2

+ H(v, A)w — J h(t — r)w(z, r, v, A) dr, z > 0, t € R,

(2.2)

(2.3)

dw dz

z=+0

w|t<0 = 0,

(2.4)

(2.5)

v

t

w|z=+0 = go(t,v, - T)go(t, v, A) dT, (2.6)

o

h(t) := r0'(t) exp (ro(0)t/2) , go(t, v, A) := FXl>X2 [go](t, v, A) exp(ro(0)t/2), fco(t) := ko(t) exp (ro(0)t/2) .

Here, for example, ro, ro mean the operations of one-time and double differentiation. The derivative of the transformation parameter will be denoted, for example, (t, v, A). We used the equality ko(0) = —ro(0) in (2.5) which results from (2.2). By the theory of hyperbolic equations, the function w(z,t,v, A), as a solution to (2.3)-(2.5), possesses the property w = 0, t < z, z > 0, and has the following structure in the neighbourhood of the characteristic line t = z:

w(z, t, V, A) = —5(t -z)+ w(z, t, V, \)6{t - z), (2.7)

Co

where W(z, t, v, A) is a regular function. Then

9o(t,v,X) := -5(t) +go{t,v,\)6{t), gQ(t,v, A) := goo(t, v, A) exp(ro(0)i/2), Co

here goo(t, v, A) is the regular part of go(t, v, A).

Inserting (2.7) in (2.3)-(2.6) and using the method of separation of singularities, we find that W(z, t, v, A) satisfies the following equations for t > z > 0 (w = W for t > z):

d2w d2w dt2 dz2

t

+ H{v, \)w - —h(t - z) - I h{t- t)w{z, t, v, A) dr, (2.8)

Cq J

Ht=,+o = -^M0)-tf(*/,A)z) :=ß(z,v,X), (2.9)

2cq

dw

0, (2.10)

dw dz

z=+Q

t

w\z=+0 = go(t, v, A) + I ko(t - T)go(T, v, A) dr + —(2.11) j cq

Q

Thus, the inverse problem of determining kQ(t) and uQ(x,t) from (1.7)-(1.10) reduces to the problem of finding kQ(t) and w(z,t, v, A) from (2.8)-(2.11). Next we will find unknown quantities rQ(0), rQ(0).

We will require continuity of functions w(z,t,v, A), (^j) {z,t,v, A) for z = t = 0 and from (2.9), (2.11) we find:

ro(0) = 2CQkQ(0, v, A), (2.12)

rQ(0) = —q(0) + (v2 + A2)cQ - c2kQ(0, V, A) - 2cqkQ(0, v, A). (2.13)

For the last equalities, we used the relations

t

k'(t) = -r'(t) - r(0)k(t) - J r'(t - r)k(r) dr,

fc'(O) = -r'(0) + r2(0), k'(0) = ^^ - r'(0).

Next, note the r(0), r'(0) are already known. The following equalities v, A have fixed nonzero reals.

Lemma 2.1. Suppose that g00(t,v, A) € C3[0,T], for a non-zero real v, A, where T > 0 is fixed. Then the inverse problem (2.8)-(2.11) for (z,t) € DT, DT = {(z,t)| 0 ^ z ^ t ^ T — z} is equivalent to the problem of finding a vector-function w(z,t,v, A), (If) {z,t,v, A), (g^) (z,t,v,\),h(t),h\t),lto(t),l&(t),%¿(t),k¡¡'(t) from the following non-linear system of integral equations:

t

/dw

(z, r, z/, A) dr,

(2.14)

dw 1 1

— (z, Í, 1/, A) = A) + - (&(* - V, A) - ro(0)5o(i - v, A))

t-z

1 1 1 /" ~2c^ + 2 J hott - z ~T)ÏÏo(T,iy,\)dT

0

(z+t)/2

+ -

H{v, \)w{£, t + z-£,v, A)--+ z - 2£)

c0

t+z-2Ç

J h(r)w(£,t + z - £ - r,v, A)dr

d£.

t-z

i J H{u,\)w{i,t-z + i,u,\)- j h{T)w{i,t-z+i-T,u,\)dT

d£

h(t) = -2co

=: Gi [w, h, ", "],

d2w d r " "

A) = — Gi

1

i/, A) - ro(0)^o(i, f, A) + roi£o(i, f, A) -~H(y, \)ß[-,v,\

2

t

2

(2.15)

(2.16)

-2",'(t) - 2c0 / "0'(t - r)"0(r, v, A) dr - C0 / h(r)ß

t — r

v,AI dr

! r Î-2Ç

+2co I J h(T)^,t-Z-T,v,\)dT

=: G2

0

dw

d£

h,"0'

(2.17)

z

t

t

h' (t) = (ö2

— h k"

(2.18)

fco(t) = -ro(0) + roit + / (t - T)*#(t) dT,

(2.19)

t

A?0 (t) = roi + J ko (t) dT,

(2.20)

t

ko'(t) = —h(t) + rooko(t) - J h(t - T)ko (t) dT.

(2.21)

t

ko''(t) = —h'(t) + rooko(t) — h(0)ko(t) — J h'(t — T)ko(T) dT, (2.22)

where

roo =

'°™-r>o(0), rol = ^-r>o(0).

ro2(0)

4

2

< Note that the following are valid:

Taking this into account, integrate (2.8) along the corresponding characteristics of differential operators of the first order for (z,t) € . Integrate along the characteristic of the operator Jj — Jj from (z,t) to ((z + t)/2, (z + t)/2) in the plane of variables (£, r). Using the equality G! + rn) w((z + (z + ^ A) = A)> resulting from (2.9) after differentiation

with respect to z, we have

d d \ 1 - + - u<z,t,,,X) = -H(,, A)

(z+t)/2

+

H{v, \)w{£, t + z-£,v,\)--+ z-2£)

Co

(2.23)

i+z-2Ç

J h(T)w(£,t + z — £ — t,v, A)dT

d£.

Integrate along the characteristic of Jj + qz from (0, t z) to (z,t). Using (2.10), (2.11),

we get

d d

dt dz

w(z, t, v, A) = ko(t — z, v, A) — ro(0)ko(t — z, v, A)

t-z

1 1 k /" k

--— z)z H--k0(t — z) + / k0(t — z — t^o^jU, X) dr

co co J

o

t-z

\Î H(V, A)W(^,t — z + V, A) — / h(Tt — z + ^ — T, V, A) dT

d£.

(2.24)

t

z

From (2.23) and (2.24) we can easily obtain (2.15). Putting z = 0 in (2.23), and using (2.10), (2.11), we obtain

g'oit, v, A) - ro(0)go(t, v, A) + -k'Q(t) + [ k'0(t - r)g0(t, v, A) dr

c0 J

0

i/2- i-2£

Differentiate this equality by t and arrive at (2.17) after simple computations.

The remaining equations of the system are obvious and are used to close the system of integral equations. The h(0), k''(0) are knowns if we solve for t = 0 a system of two equations (2.21) and (2.17). The Lemma 2.1 is proven. >

Theorem 2.1. Suppose that the conditions of Lemma 2.1 hold. Then there is a unique solution ko(t) € C3[0,T] to (1.7)-(1.10) for every fixed T > 0.

Let r(K0) be the set of functions k0(t) € C3[0, T], satisfying ||fe0(t)||c3[0,T] ^ K0 for t € [0, T] with a positive constant K0.

Theorem 2.2. Let A^t), fc02) (t) € r(K)) be solutions to (1.7)-(1.10) with the set of data j g0'0)(t,v, A) j for j = 1,2 respectively. Then there exists a positive constant C =

C(Ko,ho(v, A),co, T), ho(v, A) = max j ||g0o)(i, v, A) ||C3[0)Tp j = 12 estimate holds:

(1) (2)" < C

k(1) _ k(2) k0 k0

C3[0,T ]

g00)

(2) goo

C3[0,T ]

such that the stability (2.25)

< Proof of Theorem 2.1. The main idea of the proof consists in application of the Contraction Mapping Principle to the non-linear system of the integral Volterra equations of the second kind (2.14)-(2.22). Write the system of equations as an operator equation

with = ], j = 1, 2,..., 9 :

<^i(z, t, v, A) := w(z, t, v, A),

(2.26)

, dw , 1 w \ 1 -r-,, .

<p2{z, t, v, A) := -rrr{z, t, v, A) + —h{t - z)z - —k0{t - z),

dt 2C0 2C0

<£3(z,t,v, A) :=

d2w

1

1

„ t, i/, A) H--h'(t — z)z— . .

dt2 2cq ' 2co(0)

P4(t) := h(t) + 2A?0(t), <^(t) := h'(t) + 2A0"(t) + C0h(t)£(0, v),

^a(t) := k0(t), ^z(t) := k0(t), Ps(t) := Aft(t) + h(t) - nAt),

^g(t) := A#'(t) + h'(t) - r00^0(t) + h(0)A?0(t).

The operator A is determined on the set of vector-functions ^ € C[DT] and, by (2.14)-(2.22), has the form A = (Ab A2,..., A9) :

Ai^ = ^01+j

<p2{z, t, v, X)-J-z (2(f8(t - z) - ip4(t - z) + 2roo^6(r - z))+-^—<p7{t-z) 2C0 2C0

dr,

i

z

t-z

A2(f = <a)2 + ^ J it - Z - t)#o(t, V, \)dr + 7; J

H(v, A)^i(C,t - z + C,v, A)

t-z

- J (2^8(t) - (t) +2roo^6(t))^1 (C,t - z + C - r,v)dr

1 + 2

"2"

H{y, \)tpi(£,t + z-Z,v)--(2<ps(t + z - 2£) - <pA(t + z - 2£) + 2roo<p6(t + z - 2£))

c0

t+z-2£

dC,

J (2^8(t) - ^4(t) +2roo^6(t))^1 (C,t + z - C - T,v, A)dT o

t-z

= <£03 + ^ J (<Ai (i - z-t) - Lps(t - z-t) - rooveit - z - r))go(t, v, A) dr

4

t-z

/dw

(2^8 (r) - ^4(r) + 2r00^6(r)) —(C, i - z + v)dr

dC

~~2~

+

2

dw 1

# ("> i - ^ + e, A) - + z - 20

dt co

-(2^(t + z - 2C) - ^4(i + z - 2C) + 2roo^6(i + z - 2C))ß(C, v)

t+z-2£

/dw

(2 <p8(r) -<P4(r) +2r00^6(r)) — (€,t + y-£-T, v, A )dr

dC,

-co J (2^8(t) - ^4(t) + 2roo^6(T)) ß

= ^o4 - 2co y fco'(t - T)go(t, v) dT o

t/2

i ( ) dr + 2cq

t-2£

dC,

/dw

(2^8 (r) - ^4(r) + 2roo^6(r)) A )dr

o

t

= ^o5 - co J (^4(t - T) - ^8(t - T) - roo^6(t - T)) (o(t, v, A) dT

-y J(2<ps(r) - '-Pa(t) + 2roo^6(r)/? ' ^ ^ dr

z

z

1

t

t

t

+c„j

o

t/2

d2w

H("> A) ^ — í->v) — (2^8 (i - 20 - <4(¿ - 20)

t—2Ç

d2w

dw /" d2w

+2roo^6(i-20 —- j (2щ(т) - щ(т) + 2г00<б(т)) t - Ç - т, v)dr

o

t

= <06 + У (t - т) (<4(t - т) - ^8(t - т) - roo <6(t - т)) ¿т, o

t

= <07 + J (^4(t - т) - ^8(t - т) - roo^6(t - т)) ¿т, o

t

= <08 - J (2^8(т) - <4(т) + 2Г00^6(т)) <6(т) ¿т, o

t

Ag< = <09 - J h'(t - т)^6(т) ¿т, o

with <o = [<01, <02, • • •, <09] :

<ol(z,v, Л) := e(z,v, Л),

<02(z,i,ï/, A) := - z, г/, Л) - ro(0)go(t - z, v, А)) +-^-Я(г/, Л),

2 4 Co

¿С,

í, г/, Л) := i ДО - у, и, А) - ro(0)$,(í - у, и, А)) + i

<04(t, v, Л) := -2Co

ш и, А) - г0(ОШ*, ^ А) + roi5b(í, А) - ^Я (г/, Л) ß Vj Л

<05(t, v, Л) := <£04(t, v, Л) <06(t) := -r(0) + rolt, <07(t) := Г01, <08(t) := 0, <09(t) := 0^ In (2.26) we have

h(t) = 2<8(t) - <4(t) + 2roo<6(t), h'(t) = 2<g(t) + 2roo<7(t) - <5(t)

+Co (2<8(t) - <4(t)) + 2 (rooCoe(0, v) - h(0)) <6(t), (2^27)

k0' (t) = <4 (t) - <8 (t) - roo <6 (t),

^(z, i, v, A) = <2(2, í, V, A) - ^(2<s(i - 2) - m(t - z)

+2r00<6(i - + 77-- z), 2Co

z

d2w z 1 л 1 /*

-^(z, t, V, A) = <3(z, t, V, A) - — h'{t -z) + —k'¿(t -z)- -h(t -z) J /?(£, и, A)

o

k0''(t) = <g(t) - h'(t) - roo<7(t) + h(0)<6(t)•

In the last two equalities, instead of h(t), h'(t), k0'(t) on the right-hand side, we take their expressions via the components of ^ (2.26).

Introduce the Banach space of continuous functions C0, generated by the family of weighted norms

= max < sup |^(z,t,v, A)e-ot| = 1,2,3, sup |^j(t)e-oi | , j = 4,..., gl ,a ^ 0. [(z,i)eDT te[o,T] J

For a = 0, this space is the space of continuous functions with the usual norm ||^>||. By the inequality

e-oT||p|| < |M|a < ||p|| (2.28)

the norms ||^>||0 and ||^>|| are equivalent for every fixed T € (0, to). The positive real a will be chosen later. Let (<^0, ||^>0||) =: | ||^> — ^>0||0 ^ ||^0||} be the ball of radius ||^>0|| with center at ^>0 from some weighted space C0 (a ^ 0). For ^ € (^>0, ||^>0||), the following estimate holds: ||^||CT ^ ||^0110 + ||^01| ^ 2||<^01|.

Let <^(z, t, v, A) € (^>0, ||^>01|). Next, we will show that for an appropriate choice of a > 0 the operator A takes into . Give, as an example, the estimating technique for the second nonlinear equation of (2.26); the estimates are obtained similarly for other equations [18]. For (z, t) € DT, we have

IIA - ^02 IIa = sup |(A - ^02 )e

(z,t)6DT

t-z

sup

(z,t)eDr

z

+-J

0

\ J mt - z - t)go(r,^a)e~a{t~z~t)e~a{z+t) dt 0

H(v, A)^i(e, t - z + e, V, A)e-CT(t-z+?)e-CT(z-?)

t-z

- y (2^s(t) - <^(t) + 2roo^e(r))e-aTpi(e, t - z + e - t, v, A)e-a(t-z+?-T)e-ff(z-e)dr

de

"2

1

z

-a(2^g(t + z - 2e) - ^4(t + z - 2e) + 2roo^(t + z - 2e))e-a(t+z-2?)e-a(2?-z)

t+z-2£

J (2^8(t) - ^4(t) + 2roo^e(T))e-aTpi(e, t + z - e - t, v, A)e-a(t+z-?-r)e-a(?-z) dT 0

(l - e"^ + A(2M<7 + II^H, + 2r00|M|.)i(e-CTZ - e"^) 2

de

< 2

^G + JFfo+(3 + 2|roo|)(^ + T||<o||

a

:=2||vq ||x2 (cq,g,hq ,roo,t,

Ho := maXze[o)T/2] |H(v, A)|, G := maxie[o;T] |£q(î, v, A)| . Thus, for all equations of (2.26), we have

||AjV - Voj H* < 2||VoHXj

1

a

j = 1, 2,..., 9

(Xj are the constants depending on the same values as x2).

Choosing a ^ a0 := 2max1^j^g {xj} , we find, that A takes the ball Qa(<0, ||<o||) into

Qa(<0, |<0|)-

Let <1, be two arbitrary elements from Qa(<0, ||<01|). Using the auxiliary inequalities of the form

I, „1,„1 ,„2 „21 at ^ I, „11 I, „1 , „2 I at i I, „21 I, „1 ,„2| at ^ aH,~ II IL „1 ,„2M

l^i^j- ^j Ie ^ I - ^j Ie + | - Ie ^ 4|Ny ||< - < ||a,

we have \\Atp1 — Atp2II ^ ^ lit/?1 — <2|| , where <too is determined the same way as oq (the

only difference between a00 and a0 is that the constant ||<0|| in the coefficients Xj is doubled [18]).

If a is chosen from the condition a > a* := max{a0,a00}, then the operator A is contracting on Qa(<0, ||<0||). Then, by the Banach Contraction Mapping Principle, equation (2.26) has a unique solution in Qa(<0, ||<0||) for any fixed T > 0.

Since fc0(t) := exp(r0(0)t/2)k0(t), by the obtained fc0(t) the function k0(t) is found by the formula

ko(t) = exp [ - ro(0)t/2 fco(t). (2.29)

Theorem 2.1 is proven. >

< Proof of Theorem 2.2. Since the conditions of Theorem 2.1 are valid, a solution to (2.26) belongs to Qa(<o, ||<o||) and||<i||a < 2||<o||, i = 1,2,..., 9. Thus,

max |ko(t)| < 2 te[0,T ]'

exp(|ro(0)|T) =: Kq.

Let v(j), j = 1,2 be the vector-functions that solve (2.26) with the set of data {gQ0(t, v, A)} respectively. From the arguments in the proof of Theorem 2.1, we obtain the estimate for

a > a*

- <(2)

< Cl

(1) _ (2) gQQ gQQ

a*

c3[o,T] a

V

(1)

V

(2)

(2.30)

where C1 depends on the same arguments as C in Theorem 2.2. The estimate

£(1) - fr(2) k0 k0

<

aC1

a a*

(1) _ (2) gQQ gQQ

C3[0,T ]

follows from (2.28) and (2.30). Then, considering equation (2.29) for {fc01),£?02)}, {k^,^02)} and using (2.29), we obtain (2.25).

3. The Problem of Determining a1(xs), k1(t) and u1(x,t) Next we will use the bilinear integral operator

t

L [k0(t),u(x, t)] = u(x,i) + y k0(t — r)u(x,T) dr.

1

a

a

Pass from the functions ui(x,t) and u0(x,t) to the Fourier images Uj(x3,t, v, A) := Fxi,x2[uj](x3,t,v, A), j = 0,1. Then inverse problem (1.11)-(1.14) can be rewritten in terms of U1 as follows:

crüi

L

+L

(3.1)

+i y aoki(t - t) o

d2U

ov

- (2vuo + (A2 + v2)uov)

(x3, t, v, A) dT,

L

fco,iai(+0)—--ha0T—

dx3 dx3

ui |t<o = 0,

— Mo [ k\(t — (xs,t, v, X)dT

J dx3

o

= 0,

(3.2)

(3.3)

X3=+o

Ui(0,t,v,A)= Fri ,x2 [Ui](t, v, A) := gi (t,v,A), t> 0 (3.4)

(in (3.1) and (3.3) the subscript v (A) denotes differentiation with respect to v (A)). Let

V(z,t,v, A) = L[&0,Ui(^-1(z),t,v, A) exp (r0(0)t/2).

Then (3.1)-(3.4) take the following forms for z > 0, t € R :

d2V _ d2V dt2 dz2

+ H (v, A)V -J h(t - t )V (z, t, v, A)dT - iAci (z)w

-i (A2 + z/2) CI(Z)wa + --h ~2Cl(Z)-

n dz Cn

dz2

(3.5)

t

+i exp(ro(0)t/2^y ki (t - t)

dz2

- (A2 + v2)covv - 2vc2v(z, t, v)

dT,

V |t<o= 0,

m1(+0)^ + ^+ieXp(ro(0)i/2) J k^t-r^dr

z=+o

V |

z=+o =

L

ko,ki (t,v, A)

(3.6)

(3.7)

(3.8)

where

ci (z) := ai(coz), ki (t, v, A) = gi(t, v, A) exp(ro(0)t/2).

By (2.1) and (2.7)

v = exp(ko(0)t/2)

co

5(t - z) + w(z, t, v, A)0(t - z)

t

o

0

1

t

+ y ro(t - T) exp(ko (0)t/2)

co

5(t — z) + w(z, t, v, A)0(t — z)

dT

(henceforth we omit the tilde over w),

t

Vv = exp(ko(0)t/2) [wv(z, t, v, A)0(t — z)] + J ro(t — t(z, t, v, A) exp(ko(0)T/2) dT.

Note that the initial-boundary value problem obtained by differentiation of (2.8)-(2.11) with respect to v is valid for wv:

d2wv d2wv

dt2 dz2

t

+ H(v, A)wv + Hv(v, A)w — J[h(t — t)wv(z, t, v, A)] dT,

(3.9)

Wv|t=z+0 = ^v(z,v, A), Stoiy _

Wv |z=+0 = L A?0, #0v(t, v, A) Summarizing the above, we have

(3.10)

(3.11)

(3.12)

t

exp(ro(0)i/2) J hit-r^dr

= (t — t )

dw f dw

A) + / ?o(t -ï])-^(z,ï],u, X)dï]

dT, (3.13)

where fci(t) := ki(t) exp(ro(0)t/2), fo(t) := ro(t) exp(ro(0)t/2),

/" d2V exp(ro(0)í/2) h (í-r)-^dr

= fci (t — t )

d2w /* d2w

+ / fo(r - V, v, A) drj

dT, (3.14)

exp (ro(0)t/2) y ki(t — t) (q(z) — (A2 + v2)co)v vdT

= fci(t — t )( q(z) — (A2 + v2) co)

w v(z, t, v, A)

T

+ J Fo(t — n)w v(z, n, v, A)dn

dT, (3.15)

1

t

t

t

t

t

t

exp(ro(0)t/2)y ki(t - t)2vcov(z,T, v, A) dT

= 2vc

ki (t - z) + / ki(t - t)ko(t - z)dT

+ / ki(t - t) w(z,T, v, A) + / ko(T - n)w(z, n, v, A)dn dT

(3.16)

Observing (3.11) and (3.13)-(3.16), we can rewrite (3.5)-(3.8) for z > 0, t € R as follows:

d2V d2V

dt2 dz2

+ H(v, A) V - h(t - t) V(z, t, v, A)dT + vcki(t - z)

-¿Aci(z) ^¿(i - -z) + ^(i - z)j ~ ¿(A2 +

t

i i d2w f ■___■

h----1" "^cli-2) n 9 + / v-, - r) (it,

dz ' Co

dz2

(3.17)

where

V |t<o= 0,

dz z=+o

V |t=z = 0,

V |z=o= L ko,ki(t, v, A)

(3.18)

(3.19)

(3.20)

(3.21)

c = 2ico, p(z,t, v, A) = cko(t - z) - iLo

d2w

ro, - (A2 + v2)c%wv - 2vc0w

(the difference between L0 in the definition of p(z, t, v, A) and L is that the subscript of the integral in the operator is changed for z).

Thus, the inverse problem of determining ai(x3), ki(t) from (1.13)-(1.16) reduces to the problem of determining ci(z), ki(t) from (3.17)-(3.21). By means of the d'Alembert formula, we obtain

V(z, t, v, A) = - (L koj'giit — z,v, X) +L koj'giit + z,v, X)

t + z

2 Z t+Z-S,

I {

0 t-z+£ v

2

vcki(T - e) + H(v, A) V(e, T, v, A)

(3.22)

h(T - n)V(e, n, v, A) + ki(t - n)p(e, n, v, A) dn / dTde := F V, ci, ci, ki

t

t

T

t

o

where

N(£, r, v, A) := i Aw + (A2 + v2)wA —

i d2W* c0 dz2

(£,r,v)

Passing to the limit in (3.22) as t ^ z + 0 with V|t=z = 0, we derive

L

fco,g1(2z,v, A)

¿A

Co

C1(C )d£

z 2z-£,

+

0 ?

vch(t -0+ H{y, \)V{i, r,!/, A) - Cl{i)N{i, r,!/, A) +

(3.23)

h(r — n)V(e, n, v, A) + A;1(r — n)p(e, n, v, A) dn > drd{.

From (3.23) it follows that <^1(0,v, A) = 0. Replacing 2z by t and differentiating (3.23) with respect to t, we get

L

iA

t/2,

ko,g[(t, v, A)J =__Cl(i/2) + J ivck1(t-2£)+H(v,\)V(Z,t-Z,v,\)

o ^

-cim& a) + A)

t-2£

dz

h(r)v(e, t—e — n, v, A)+*1(rMe, t—e—r, v, A) dr de.

(3.24)

It's obviously that ci(0) = 0, z/, A). Differentiating (3.24) with respect to t, then

substituting the values A1, A2 sequentially and making up the difference of the equalities for a fixed v, we can obtain the equation for c1(z)(z = t/2) :

c1(z) =

+

M (z)

A* < L

^o,^1A(2z,v, A)

2Ax{N(z,z,u, A)} M(z)

c1(z)

NVa (z,v,A)

M(z)

z i dV dN

J A\ H{V, a)—(e, t - e, i/, A) - ci (o-^tf, t - e,A)

o v

2z-2£

c0 dtdz

2z — £ — t,v,\) + h (r)^(C, 2 z-i-t,u, A) dr ¡> d£,

(3.25)

where A*{-} is the difference of the values {■} for A = A1 and A = A2. In particular, A*{N(z, z, v, A)} := N(z, z, v, A1)—N(z, z, v, A2). Next, by A v{■} we will denote the difference of values for v1, v2.

z

T

0

1

1

Note that if Ai = A2, then we have

M(z) := i(Ai - A2)

4co

= 0.

Differentiating equation (3.24) by t (after replacing the variable in the first integral t — 2{ = t), and then using the parameters v^v2 (vi = v2), we can obtain the equation for fci(t) (t/2 = z):

h(t) = -

2

+

c(vi - V2) Av {2N (z,z,v,A)} , x 2

A J L

(t,v, A)

c(vi - V2)

Ci(z) -

c(vi - V2)

NV (z,v,A)

o

dN ^ . i , 2wa

o

-ci (6 ^r (e, i - e, A) + ^ c; (o —^ (e, r, A)

dt

dtdz

i-2g

^(r)^(e, t - e - r,!/, A)+t - e - r, A) dr \

(3.26)

Next, the obvious equalities are used:

z

ci (z) = ci(0) + J ci(0 d£,

(3.27)

dV d , - -

— {z,t,v,\) = —F F,ci,Cl,A:i dt dt

(3.28)

Equations (3.22), (3.25)-(3.28) are equivalent to equalities (3.17)-(3.20) and form a closed linear system of Volterra integral equations of the second kind in the domain DT with respect to V{z,t,v, A), A), ci(z), c'^z).

Next, we need that the functions N(z, t, v, A),p(z, t, v, A) € C 1[DT]. Therefore, it must be shown that WA,WV € C3[DT].

Indeed, using the d'Alembert formula for the problem (3.13), (3.15), (3.16) we obtain a linear integral equation of the Voltaire type with a continuous free term and a continuous kernel in the domain Dy:

1

Wv

L

-0,-0v(t - z, V)

+ L

-0 ,-0v (t + z,V)

z i+z-g, .

J J (e,v)wv+Hv(e,v)w(e,r,v)-Jh(r-nK(e,n,v))dnUrd£. (3.29)

0

+

0 i-z+g

It follows from the theory of integral equations that equation (3.29) has a unique continuous solution in DT. The smoothness of the solution is determined by differentiating equation (3.29) a sufficient number of times. It is easily checked that the right part of the differentiated equation will be continuous, and therefore the left part will also be continuous [30]. Thus, wv € C3[DT].

0

2

1

2

Similarly, it can be proved that w* € C3[DT].

The following theorems of unique global solvability and stability of the inverse problem of determining a1 (y), k1(t) are the main results of this section.

Theorem 3.1. Under the conditions of Theorem 2.1, let g1 (t,v, A) € C2[0, T] for fixed non-zero (v, A), and gi(0,v,\) = 0, g[(0, v, A) = Then there is a unique solution of

inverse problem (1.11)-(1.14) d(z) € C 1[0,T/2], fc^t) € C[0,T] for every fixed T > 0.

Theorem 3.2. Let c(11)(z), c12)(z) € C 1[0,T/2], k(1)(t), k(2)(t) € C[0,T] be solutions to (1.11)—(1.14) with

|g(j)(t, v, A), k0j)(t), 4j)(x3,t,v,A)}

for j = 1,2 respectively. Since the conditions of Theorem 2.2 are valid, there exists a positive number C = C(C,h(v, A)),

h1(v,A)=maxj ||g(j)(t,v,A)||C2[0;Tp ||N(j)(z,t,v,A)||Cl(DTr Mp(j)(z,t,v,A)|cl(DT), j = 1, 2 ^

such that the stability estimate holds:

(1) „(2)1

c — c

IC1[0,T/2]

+ 11^ -fcS2)|

ic[0,t ]

^ C

I ~(1) _~(2) ||

lg1 g1 llc2[0,T]

+iik0i) - a

C[0,T ]

. (3.31)

< Proof of Theorem 3.1. System (3.22), (3.25)-(3.28) is a closed system of the linear integral Volterra equations of the second kind with continuous free terms and kernels in . The idea of proving existence of the unique solution to the given system consists in application of the generalized contraction mapping principle. Write the system (3.22), (3.25)-(3.28) as the operator equation

^ = B^, (3.32)

^ : =

1

t + z

2

V(z,t,v,X) + — / ci(£)d£, c;(z) + iVA(z,z/,A)ci(z),

2 Co J "-v-'

t-z 2

02

■01

fci(2z) - Nv(z,v, X)ci(z), ci(z),

04

^ 1

c1

03

t + Z

- c1

t — z

+ —- z)cz

05

Then V(z,t,v, A), c^z), fci(2z), ^(z,t, v, A) can be defined through the components of ^

z±£

2

V{z,t,v, A) = Vi(£,t,z/) H--/

2c0 J

1

2

ci (z) = -02(z, V, A) - Na(z, V, A)"04 (z), -i (2z) = 03(z, V, A) + N(z, V, A)04(z),

dV. ^ , , 1

— (z, t, V, A) = ip5(z, t, V, A) - —

( ^T^ ) - t Z

22 03((t - z)/2, V, A) + Nv(z/2, V, A)04(z/2)

cz.

The operator B = (B1,B2,B3, B4,B5) is determined on the ^ € C(DT) for fixed v, A. Similarly, as it was done in [31], it can be shown that some degree of n (n is natural number) of the linear map B-0 is compression. Let

H^ll = max < max (z,t, v, A)|, j = 1,..., 5> .

[(z,i)eDT J

Let ^(1), ^(2) be continuous vector-functions in DT satisfying a linear system of integral equations (3.32). Let

A(z,t) = {(£,t): 0 < £ < z, t — z + £ < t < t + z — £| ,

£(z,t,0 = {t : (C,t) € A(z, t)} .

Then, by virtue of the linearity of (3.32) for (z, t) € DT according to the equations (3.22), (3.25)-(3.28) we have (the parameters v, A will be omitted from the argument list)

Bj"(i) - Bj"(2) (z, t) < ^z "(i) - "(2)

j = 1,... 5,

where are constants depending on the parameters of C (Theorem 3.2). If M := max{^1,^2,^3,^4,^5|, then we have

max 1<K5

Bj"(i) - Bj"(2) (z, t) < Mz "(i) - "(2) , (z, t) € Dt

Next, the following estimate are hold [31]

and,

max

KK5

max

KK5

2!

(z, t, V)

(z, t) € Dt

Bn"(i) - Bn"(2)

n!

T A™

^ M

n V 2 ,

n!

"(i) - "(2)

For every fixed T we can choose the number n so large that

Mnili_ =: a< L

n!

Then Bn is a contraction. By the generalization of the Contraction Mapping Principles the equation B-0 = ^ has one and only one solution belonging to C(DT). This solution can be found by successive approximations. >

< Proof of Theorem 3.2. Let be a vector of functions which are solutions to (3.32) with {gj)(i), (t), w(j)(z,i)}, j = 1,2, respectively. Obviously, the function 1/M(z) can be estimated:

1

M (z)

<

1

|Ai - A2I'

Further, from the arguments of Theorem 3.1, we obtain

^ + a

(3.33)

where

Y :=

(1) (2) g1 - g1

C2[0,T ]

+

k(1) _ k(2) k0 k0

C [0,T ]

and 70 depends on the parameters of C. It follows from equality (3.33) that

with 7 = 70/(1 — a).

Considering the equation k1(t) = exp[fc0(0)t/2]A;1 (t) for {fc(1), fc(1)}, {fc(2),fc(2)} and using (3.33), we obtain (3.31). Theorem 3.2 is proven. >

References

1. Lorenzi, A. and Sinestrari, E. An Inverse Problem in the Theory of Materials with Memory I, Nonlinear Analysis: Theory, Methods and Applications, 1988, vol. 12, no. 12, pp. 1317-1335. DOI: 10.1016/0362-546X(88)90080-6.

2. Durdiev, D. K. The Inverse Problem for a Three-Dimensional Wave Equation in a Memory Environmentu, Matematicheskij analiz i diskretnaya matematika, Novosibirsk, Izd-vo Novosibirskogo Universiteta, 1989, pp. 19-27 (in Russian).

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Received March 28, 2024

Marat R. Tomaev

North Ossetian State University,

44-46 Vatutina St., Vladikavkaz 362025, Russia,

Researcher

E-mail: [email protected]

Zhanna D. Totieva

North Ossetian State University,

44-46 Vatutina St., Vladikavkaz 362025, Russia,

Docent

E-mail: j annatuaeva@inbox. ru

https://orcid.org/0000-0002-0089-074X

Владикавказский математический журнал 2024, Том 26, Выпуск 3, С. 112-134

ОБРАТНАЯ ДВУМЕРНАЯ КОЭФФИЦИЕНТНАЯ ЗАДАЧА ДЛЯ ОПРЕДЕЛЕНИЯ ДВУХ НЕИЗВЕСТНЫХ В УРАВНЕНИИ С ПАМЯТЬЮ ДЛЯ СЛАБО ГОРИЗОНТАЛЬНО НЕОДНОРОДНОЙ СРЕДЫ

Томаев М. Р.1, Тотиева Ж. Д.1

1 Северо-Осетинский государственный университет им. К. Л. Хетагурова, Россия, 362025, Владикавказ, ул. Ватутина, 44-46 E-mail: [email protected], [email protected]

Аннотация. Представлена двумерная обратная коэффициентная задача определения двух неизвестных, которые являются коэффициентом и ядром интегрального оператора свертки в уравнении упругости с памятью в трехмерном полупространстве. Коэффициент, зависящий от двух пространственных переменных, представляет собой скорость распространения волн в слабо горизонтально-неоднородной среде. Ядро интегрального оператора свертки зависит от временной и пространственной переменной. Прямая начально-краевая задача представляет собой задачу определения функции смещения при нулевых начальных данных и граничное условие Неймана специального вида. Источником возмущения упругих волн является точечный мгновенный источник, представляющий собой произведение дельта-функций Дирака. В качестве дополнительной информации задается образ Фурье функции смещения точек среды на границе полупространства. Предполагается, что искомые величины обратной задачи и функция смещения разлагаются в асимптотические ряды по степеням малого параметра. В работе построен метод нахождения коэффициента и ядра, зависящих от двух переменных, с точностью до поправки, имеющей порядок O(e2). Показано, что обратная задача эквивалентна замкнутой системой интегральных уравнений Вольтерра второго рода. Доказаны теоремы глобальной однозначной разрешимости и устойчивости решения обратной задачи.

Ключевые слова: обратная задача, дельта-функция, преобразование Фурье, ядро, коэффициент, устойчивость.

AMS Subject Classification: 35L20, 35R30, 35Q99.

Образец цитирования: Tomaev M. R. and Totieva Zh. D. An Inverse Two-Dimensional Problem for Determining Two Unknowns in Equation of Memory Type for a Weakly Horizontally Inhomogeneous Medium // Владикавк. мат. журн.—2024.—Т. 26, вып. 3.—С. 112-134 (in English). DOI: 10.46698/e7124-3874-1146-k.