CC BY
2DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 3, 2021 Electronic Journal, reg. N §077-39410 at 15.04.2010 ISSN 1817-2172
http://diffjournal.spbu.ru/ e-mail: [email protected]
Ordinary differential equations
Inverse problem for incomplete Sobolev type equation of higher order
Zamyshlyaeva A. A1, Lut A. V2 1,2South Ural State University, Russia, Chelyabinsk E-mail: 1 [email protected], [email protected]
Abstract. The article is devoted to the study of the inverse problem for a high-order Sobolev type equation. Mathematical models based on such equations describe various problems of hydrodynamics and elasticity theory. The main result of the article is to find sufficient conditions for the existence and uniqueness of a solution to the original problem. At first using the theory of relatively bounded operators, the original problem is reduced to the equivalent system of two problems, which are usually called regular and singular. Thus, the solution to the original problem is represented as the sum of the solutions of these two problems. Further, the regular problem is reduced to a first-order equation. Then, by the method of successive approximations, the required smoothness for the function q is achieved by obtaining sufficient conditions for the existence and uniqueness of a solution to the regular problem. To find a solution of the singular problem, the phase space method and the results of the study of the regular problem are used. The results obtained can be applied to the study of various mathematical models, such as the model of sound waves in smectics or the model of oscillation of a rotating viscous fluid taking into account the viscosity coefficient.
Keywords: Sobolev type equation, inverse problem, method of successive approximations, relative boundedness of operator.
1 Introduction
Let U, F, Y be Banach spaces, operators L, M e L(U; F), C £ L(U; Y), ker L = {0}, given functions x : [0,T] ^ L(Y; F), f : [0,T] ^ F, ^ : [0,T] ^ Y. Consider the following problem
where (1) is an incomplete, inhomogeneous high-order Sobolev type equation, (2) is the Cauchy condition, and (3) is the overdetermination condition. The problem of finding a pair of functions v £ Cn([0,T]; U) and q £ C:([0,T]; Y) from relations (1) - (3) is called the inverse problem.
A large number of works is an devoted to the study of Sobolev type equations [1, 3, 4, 5, 6, 7, 11, 12]. The article [4] gives sufficient conditions for the existence of positive solutions to both the Showalter - Sidorov problem and the Cauchy problem for an abstract linear equation of the first order (n = 1). The Cauchy problem for a nonlinear Sobolev type differential equation in the space of continuous functions was studied in [11]. The paper [12] is devoted to initial-boundary value problems for a Sobolev type equation with a Gerasimov-Caputo fractional derivative with a memory effect.
Sobolev type equations find their application in mathematical modeling [3, 5, 6]. In [6], the sufficient conditions for the solvability of a high-order semilinear Sobolev type mathematical model were obtained. Consideration of optimal equation problems for linear Sobolev type models with the initial Cauchy condition is presented in [5].
Recently, a large number of works [3, 7, 8, 9, 10] are devoted to the study of inverse problems in various aspects. The question of well-posedness of the inverse problem of determining the source function for a second order quasilinear parabolic system in Sobolev spaces was considered in [9]. The article [8] is devoted to establishing the uniqueness of the solution to the coefficient inverse problem of wave tomography in the non-overdetermined setting. The solvability of the inverse problem for first-order partial differential equations was studied in [10]. In [7] the solvability of several inverse problems belonging to the same class of Sobolev type equations is studied. Work [3] establishes the unique solvability of the inverse problem for a complete second-order Sobolev type equation.
Lv<">(t) = Mv(t) + x(t)q(t) + f (t), t e [0,T],
V(0)= V0, ..., v<"-1»(0) = V„_1, Cv(t) = *(t),
(1)
(2) (3)
2 Preliminary Information
To find a pair of functions v(t) and q(t) we use the results obtained in the research of higher order Sobolev type equations [1].
Definition 1 The sets
and aL(M) = C\pL(M) will be called the L-resolvent set and the L-spectrum of the operator M, respectively.
Definition 2 The operator-function (^L — M)-1 with domain pL(M) will be called the L-resolvent of the operator M.
Definition 3 The operator M is called (L,a)-bounded if
Lemma 1 Let the operator M be (L, a)-bounded. Then the operators
are projectors. Here y = £ C : = r > a}.
Put U0 = ker P, F0 = ker Q, U1 = im P, F1 = im Q. From the previous Lemma it follows that U = U0 © U1, F = F0 © F1. Let Lk(Mk) denote the restriction of the operator L(M) onto Uk, k = 0,1.
Theorem 1 [1] Let the operator M be (L, a)-bounded. Then the actions of the operators split:
(i) Lk £ L(Uk;Fk), k = 0,1;
(ii) Mk £ L(Uk;Fk), k = 0,1;
(iii) there exists an operator (L1)—1 £ L(F1; U1);
(iv) there exists an operator (M0)—1 £ L(F°;U0).
Denote H = M0—1% £ L(U0), S = L—lMx £ L(U1).
PL(M) = [ß £ C : (ßL - M)-1 e L(F;U)}
3a £ R+ Vß £ C (|ß| > a) ^ ((ßL - M)-1 £ L(F;U)) .
P
Y
Definition 4 The point to is called
(i) a removable singular point of the L-resolvent of operator M, if H = O;
(ii) a pole of order p e N of the L-resolvent of operator M, if Hp = O, but Hp+1 = O;
(iii) an essentially singular point of the L-resolvent of operator M, if Hk = O for any k e N.
Remark 1 If the operator M is (L, a)-bounded, and to is a pole of order p £ {0} U N of the L-resolvent of the operator M, then the operator M is called (L,p)-bounded.
3 Reduction of the Initial Inverse Problem
Let the operator M be (L,p)-bounded, then v(t) can be represented as v(t) = Pv(t) + (I - P)v(t). Denote Pv(t) = u(t), (I - P)v(t) = u(t). Suppose that U0 C ker C. Then, by virtue of Theorem 1 and Lemma 1, problem (1) - (3) is equivalent to the problem of finding the functions u e Cn([0, T]; U1),
u e Cn([0,T];U0), q e C1([0,T]; Y) from the relations
u(n)(t) = Su(t) + (L1)-1Qx(t)q(t) + (L1)-1Qf (t), (4)
u(0) = U0, ..., u(n-1)(0) = Un-1, (5)
Cu(t) = #(t) = Cv(t), (6)
Hu(n)(t) = u(t) + (M0)-1(1 - Q)x(t)q(t) + (M0)-1(1 - Q)f (t), (7)
u(0) = U0, ..., u(n-1)(0) = un-1, (8)
where u0 = Pv0, ..., un- = Pvn-1, U0 = (I - P)v0, ..., un_i = (I - P)vn-1, t e [0,T]. The inverse problem (4) - (6) is called regular, and problem (7), (8) is called singular.
4 Solution of the Regular Inverse Problem
Rewrite problem (4) - (6) in the notation [2]. Let X = U1, operators S e L(X), C e L(X, Y), operator-function $ : [0,T] ^ L(Y; X), functions h : [0,T] ^ X,
*:[0,T] ^Y
u(n)(t) = Su(t)+$(t)q(t) + h(t), t e [0,T], (9)
u(0) = U0, ..., M(n-1)(0) = un-1, Cu(t) = #(t).
(10) (11)
Theorem 2 Let the operator M be (L,p)-bounded, moreover, C £ L(X; Y), $ £ C 1([0,T]; L(Y; X)), h £ C 1([0,T]; X), # £ Cn+1([0,T]; Y), for any t £ [0,T] the operator C$(t) be invertible and (C$)—1 £ C1([0,T]; L(Y)). If the compatibility condition Cun—1 = 1)(0) is satisfied, then there exists a unique solution q £ C1 ([0,T]; Y), u £ Cn([0,T]; X) to the inverse problem, (9) - (11).
Proof. Reduce problem (9) - (11) to the problem for the first order equation:
z'(t) = Az(t) + Q(t)q(t) + H(t), t £ [0,T], (12)
z(0) = Z0, (13)
Bz (t) = vj>(t), (14)
( u(t) \ /0 I ... 0\ /0^
where z(t) =
H(t) =
u(n-2)(t) \ u(n-1)(t) ,
0
A =
V
00 S 0
I 0
/
Q(t) =
V
0 *(t)
V
0
h(t)
z (0) =
/
u(0)
u
(n-2)(0)
B = ( 0 ... 0 C^, #(t) =
\
0
u(n-1)(0)
/
Z0 =
< u0 \
un-2 y un—1 J
0
\
^(n—1)(t)
/
Put R(t) = — (C$(t)) Therefore, all the conditions of Theorem 6.2.3 from [2], are fulfilled, and the function q(t) satisfies the integral equation
q(t) = q0(t) + R(t)CS V1 n(t — s)$(s)q(s)ds,
(15)
where
q0(t) = — R(t) (Vn)(t) — CSV1 1(t)u0 — CSV1 2(t)u1 — ...—
t
-CSV n(t)un- - CS J V n(t - s)h(s)ds - Ch(t)^.
0
Thus, there exists a unique solution q G C 1([0,T]; Y), z G C 1([0,T]; Xn) to the inverse problem (12) - (14). And we obtain that the solution to the regular inverse problem (9) - (11) exists and is unique, with q G C1 ([0,T]; Y), u G Cn([0, T]; X). □
In order to obtain a solution to the singular problem, we need a greater smoothness of the function q from the solution of the regular problem, than class C 1([0,T]; Y). Next, we need the following Lemma from [3].
Lemma 2 Let l G N, V G C1-1([0,T]; L(X)), g G C1 ([0,T]; X). Then
/ ¡-t \ (0 1-1 ¡-t
/ V (t - s)g(s)ds = £ V (1-k-1)(t)g(k)(0)+ / V (t - s)g(1)(s)ds. \J0 J k=0 Jo
The following theorem provides sufficient conditions for the existence of solution q G Cn(p+1)([0, T], Y) of the regular problem.
Theorem 3 Let the operator M be (L,p)-bounded, p G N0, moreover, C G L(X; Y), $ G Cn(p+1)([0, T]; L(Y; X)), h G Cn(p+1)([0, T]; X), ^ G cn(p+2)([0, T]; Y), for any t G [0,T] operator C$(t) be invert-ible, with (C$)-1 G Cn(p+1)([0, T]; L(Y)) and the compatibility condition Cun-1 = ^(n-1)(0) be satisfied for some un-1 G U1. Then there exists a unique solution of (9) - (11). Moreover q G Cn(p+1)([0, T]; Y).
Proof. Write the propagators of the homogeneous equation (9) in a matrix, defining the resolving group of homogeneous equation (12)
V (t) =
/
Vi i(t) V i(t)
Vi 2(t) V2 2(t)
V
Vn-i i(t) Vn-i 2(t) Vn i(t) Vn 2(t)
Vi n-i(t) Vi n(t) ^
V2 n-i(t) V2 n(t)
Vn i n i (t) Vn-i n(t)
Vn n-i(t) Vn n(t) J
2ni
(ßnL - M)
1
M
ßn-1L .
( ßn-1L ßn-2L ... ßL I ^
ß2L ßI
ßn-1L ßn-2I
M ßn-1I y
ßn-3M ßn-4M . y ßn-2M ßn-3M .
e^dß,
where I is the identity operator.
Earlier, in the proof of Theorem 2 it was established that the function q(t) satisfies integral equation (15). Take the natural number l < n(p+1). Assuming that q £ Cl([0,T}; Y) by Lemma 2 we obtain the equality
l1
Z-fc-1
9<0(i) = q<Z)(t) + £CfcR(fc>(t)CS £ V(Z-t-m-1)(i) ($„)<•"» (0)+
fc=0
Z Z-fc
m=0
+ Cfc,mR(k\t)CS f V1 n(t - s)$(m)(s)q(Z-fc-m)(s)ds,
fc=0 m=0 0
where Cfc = Hj^ , cfc,m = jim^
!m!( i—fc—m>!
and
q0Z) (t) = - ^ CZfcR(fc)(t) ( ^(Z-fc+n)(t) - CSV(Z-fc)(t)uo-
r(Z-fc)
fc=0
t
-CSV(n2-fc)(t)u1 - ... - CSV(nn-fc)(t)un-1 - CS V1 n(t - s)h(Z-fc)(s)ds-
Z1
Un-1 CS I r 1 n \ 0
Zfc1
-Ch(l—k)(t)J CkR(k)(t)CSj2 V(l—k—m—1}(t)h(m)(0)
' k=0 m=0
exists due to the conditions of the theorem for l = 0,1, ...,n(p + 1).
Show that q £ Cn(p+1)([0,T], Y), for this purpose denote r0 = q0(0), and for l = 1, 2, ...,n(p + 1) determine the following values
l 1 l k 1 m
n = q0Z)(0) + £CfcR(fc>(0)CS £ V1(';fc-m"1)(0^Cm$(j>(0K,-j.
fc=0
m=0
j =0
Consider the system of integral equations
t
qb(t) = q0(t) + R(t)CS J V1 n(t — s)$(s)qb(s)ds,
0
1
t
l—1 l—k—1 m
«(t) = q<(l)(t) + ECkR(k)(t)cs E Vi"—k—m—1)(i) Ecm$(j)(0)rm—j+
k=0 m=0 j =0
l l—k t + EE Cf'mR(k)(t)cW Vi n(t — s)$(m)(s)«l—k—m(s)ds,
k=0 m=0 0
l = 1, 2,..., n(p + 1). (16)
Reduce (16) to the Volterra equation of the second kind
t
g(t) = go(t) + J K(t, s)g(s)ds 0
on the space (C ([0, T]; Y ))n(p+1)+1 with a matrix operator function K (t, s) given on the triangle A = {(t,s) G R2 : 0 < t < T, 0 < s < t}. By virtue of the continuity of all data of system (16), it has a unique solution
(q~o,qV..,q~n(p+i)) G (C([0,T]; Y))n(p+1)+1.
This solution will be the limit of the sequence of approximations
t
io,i(t) = qo(t) + R(t)C^y V1 n(t - s)$(s)5o,i-1(s)ds,
o
l—1 l—k—1 m
«,.№ = «¿"(t) + £ CkR(k)(t)CS £ V<',—k—m—1)(t) £ cm,$(j)(0)rm—j +
k=o m=o j =o
i i—k t + £ £ Cf'mR(k)(t)cW V1 n(t — s)$(m)(s)q—k—m,i—1(s)ds, k=o m=o o
l = 1, 2,..., n(p + 1), i G N, (17)
which for i ^ œ on the interval [0,T] converge uniformly to the functions q, l = 0,1, ...,n(p + 1). Set the initial approximation q>o = 0, l = 0,1, ...,n(p + 1), then ®+1>o = qo, l = 0,1, ...,n(p + 1) — 1. In addition, from (17) it follows that
q>i(0) = ri, l = 0,1,...,n(p +1), i G N. (18)
Assume that for all t = 1,2,...,i the equalities q;l+1,T (t) = q[T (t), l = 0,1,...,n(p +1) — 1 are true. Then, using Lemma 2 and equalities (18) we obtain
EE CfcmR(fc)(t)CS J V1 n(t - s)$(m>(s)ql-fc-m,i(s)ds
Z Z-fc
(m)
R (t)CSJ -1 nl'~......
fc=0 m=0 0
Z Z-fc t
EE Cfcm R(fc+1)(t)C^y V1 n(t - s)$(m>(S)q_fc_m,i(s)dS+
fc=0 m=0 0
Z Z-fc
fc
fc=0 m=0
+ EE CZfc,mR(fc) (t)CSV1 n(t)$(m>(0)q_fc_m,i(0)+
Z Z-fc t
+ EE CZfc,mR(fc) (t)CS / V1 n(t - s)$(m+1>(S)q~Z_fc_m,i(s)dS+
fc=0 m=0
0
t
+ EE Cfc,mR(fc)(t)CS [ V1 n(t - s)$(m>(S)q_fc_m+1,i(s)dS =
fc=0 m=0 0
Z+1 Z-fc+1 *
E E C?-1,mR(fc)(t)CS J V1 n(t - s)$(m>(S)q~Z_fc_m+1,i(s)dS+
0
Z Z-fc
+ ^ CfcR(fc)(t)CSV1 n(t)E C-fc^(m)(0)rZ-fc-m+
Z R '(t)CSV1 n(t) CZ-fc^ (0)nZ
fc m+
fc=0 m=0
t
Z Z-fc+1 „
+ EE CZfc,m-1R(fc)(t)CS / V1 n(t - s)$(m>(S)q~Z-fc-m+1,i(s)dS+
fc=0 m=1 ^
0
Z Z—fc t
+ EECZfc,mR(fc)(t)CS / V1 n(t - S)$(m)(S)q~Z-fc-m+1,(s)dS. (19)
fc=0 m=0 0
Denote
afc,m = V1 n(t - s)$(m>(s)gZ-fc-m+1,i(s)ds, l = 2, 3, ...,n(p +1).
t
Taking into account the equalities
nk I rik-1 _ s~ik s~ik,m . ^k-1,m . ^k,m-1 _ ^k,m
C1 + C1 = C1+1, C1 + C1 + C1 = C1+1 ,
we get
1 1-k 1+1 1-k+1 1 1-k+1
E E C1 'mflk,m + E E C1 1,mak,m + E E C1 'm 1ak,m
k=0 m=0 k=1 m=0 k=0 m=1
l l-k l l \ / l l-k E E C '^m + E 'Vo + E Cf^m I + ( E E C i'
k=i m=i k=i m=0 / \ k=i m=i
1 k-10 ' A-1Z-A+1 /0 \
k=i k=i
l l-k l+i l + ( E E C/ 'm iak,m + E C°'m ia0,m + E C/ ' ak,1-k+i
k=i m=i m=i k=i
= E E Ck+mak,m + E Cf-+01ak,0 + E C0+7a0,m + E CZ+01ak,Z-k+1 + k=1 m=1 k=1 m=1 k=1
1+1 1-k+1
+C°'°a0,0 + C/0' a0,1+1 + C '0® 1+1,0 = E E C1-+mak,m. (20)
k=0 m=0
For l = 0,1 fulfilment of (20) can be checked directly. From (19) and (20) it follows that
Jt( EE Ck,mR(k)(t)CS i V1 n(t - s)$(m) (s)®-k-m,i(s)d^ =
V k=0 m=0 0 J
1+1 1-k+1 t
E E Clk+mR(k)(t)C^ Vi n(t - s)$(m)(s)q~n-k-m+i'i(s)ds +
k=0 m=0 0
-k
+ E CkR(k)(t)CSV1 n(t) E Cm-k$(m)(0)r-k-m. (21)
k=0 m=0
Changing the summation indices and re-grading the sums we get
-1 -k-1 m
EckR<k)(t)CS e ^k^-1^)E cm$<j)(0)r„_^ =
k=0 m=0 j =0
Z-1 Z-fc-1 m
E CfcR(fc)(t)CS e v'1-m>(t) E Cm$(j')(0)rm—j+
fc=0 m=0 j =0
Z-1 Z-fc- 1
m
+ £CfcR(fc+1>(i)C^ V1(Z-Cj$(j>(0)r^j =
fc=0 m=0 j=0
Z-1 Z-fc-1 m
= e CfcR(fc)(t)CS E v,^-m>(t) E Cm$(j)(0)rm-j+
fc=0 m=0 j =0
l l-fc m
¿-1 Z?(fc) (+\nQ V^ T/(Z-fc-m)
+ £ Cfc-1R(fc>(i)C^ V( „ ">(t)£ Cm, $(j>(0)r„—j =
fc=1 m=0 j =0
l-1 l-fc-1 m
E C R(fc>(t)CS e v/1-"■>№ E Cm $(j >(0)rm-j+
fc=1 m=0 j=0
l-1 m
+C0R(t)CS E v(Z-m)(t) E Cm $(j >(0)rm-J+
m=0 j=0
-1 l-fc-1 m
+ ( £ Cfc-1R(fc>(t)CS E V1(Z-fc-">(i) E Cm,S(j>(0)rm-j +
fc=1 m=0 j =0
+ E Cfc-1R(fc>(i)CSV1 B(i)E j$«(0)^fc-j+
fc=1 j=0
+CZZ-1R(Z)(t)CSV1 n(t)C00$(0)r0
l l-fc m
E Cfc+1R(fc> (t)CS E v1(Z-fc-™>(t) E Cm, $(j>(0)
fc=0 m=0 j=0
Z Z-fc
rm-j
^ CZfcR(fc)(t)CSV1 n(t^ Cz5-fc$(m)(0)rZ-fc-m. (22)
fc=0 m=0
Differentiating (17), and also using (21) and (22), we obtain the equalities q,i+1 = tfl+M+b l = 0,1,...,n(P + 1) — 1
Thus, the sequence f/o,^ converges as i ^ to to the function q0 uniformly on the interval [0,T], and the sequence q0 i = q'1,i converges as i ^ to to the function ^ uniformly on the segment [0,T]. Therefore, the function
q0 is continuously differentiable and q0 = qr1 - The equalities of q = g1+1, l = 1,2,...,n(p +1) - 1, are proved in the same way, which implies that ^0 = q G Cn(p+1)([0, T]; Y) and, therefore, q(1) = qh l = 1, 2,...,n(p + 1). □
5 Solvability of the Original Inverse Problem
Now we can formulate the solvability conditions for the problem (1) - (3).
Theorem 4 Let the operator M be (L,p)-bounded, p G No, operator C G L(U; Y), Uo C ker C, x G Cn(p+1)([0, T]; L(Y; F)), f G Cn(p+1)([0,T]; F), ^ G Cn(p+2)([0, T]; Y), for any t G [0,T] operator C(L1 )—1Qx be invertible, with (C(L1)—1Qx)—1 G Cn(p+1)([0, T]; L(Y)), the condition Cun—1 = ^(n—1)(0) be satisfied at some initial value un—1 G U1, and the initial values wk = (I — P)vk G Uo satisfy
p
dnj+k
k = 0,1,...,n — 1.
wk = ^ (Mo)—1 ^ [(/ — Q)(x(0)q(0) + f (0)) j=o
Then there exists a unique solution (v,q) of inverse problem (1) - (3), where q G Cn(p+1)([0, T]; Y), v = u + w, whence u G Cn([0,T];U1 ) is the solution of (4) - (6), and the function w G Cn([0,T]; Uo) is a solution of (7), (8) given by
w(t) = — £ (Mo)—1 ddlj [(I — Q)(x(t)q(t) + f (t))]. (23)
j=o
Proof. The conditions of Theorems 2 and 3 are satisfied, and therefore there exists a unique solution (q,u) of problem (4) - (6), where q G Cn(p+1)([0,T]; Y), u G Cn([0,T];U1).
Using the result of [1] and the required smoothness of the function q, we obtain that there exists a unique solution (7), (8) w G Cn([0,T];Uo), given by (23). □
Acknowledgements. The reported study was funded by RFBR, project number 19-31-90137.
References
[1] Zamyshlyaeva A.A. and Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher order, Bul-
letin of the South Ural State University. Series: Mathematics. Mechanics. Physics 8(4) (2016), pp. 5-16.
[2] Prilepko A.I., Orlovsky D.G. and Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics, New York: Marcel Dekker (2000), 744 p.
[3] Zamyshlyaeva A.A. and Lut A.V. Inverse Problem for Sobolev Type Mathematical Models, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software 12(2) (2019), pp. 25-36.
[4] Banasiak J., Manakova N.A. and Sviridyuk G.A. Positive Solutions to Sobolev Type Equations with Relatively p-Sectorial Operators, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software 13(2) (2020), pp. 17-32.
[5] Zamyshlyaeva A.A., Manakova N.A. and Tsyplenkova O.N. Optimal Control in Linear Sobolev Type Mathematical Models, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software 13(1) (2020), pp. 5-27.
[6] Zamyshlyaeva A.A. and Bychkov E.V. Research of One Semilinear Mathematical Model of the Sobolev Type of Higher Order, Mathematical Methods in Technics and Technologies - MMTT 8 (2020), pp. 3-6. (in Russian)
[7] Kozhanov A.I. and Namsaraeva G.V. Linear Inverse Problems for a Class of Equations of Sobolev Type, Chelyabinsk Physical and Mathematical Journal 3(2) (2018), pp. 153-171. (in Russian)
[8] Kokurin M.Y. Completeness of asymmetric products of solutions of a second order elliptic equation and uniqueness of a solution to an inverse problem for a wave equation, Differential Equations 57(2) (2021), pp. 255-264. (in Russian)
[9] Pyatkov S.G., Uvarova M.V. and Pronkina T.V. Inverse Problems for a Quasilinear Parabolic System with Integral Overdetermination Conditions, Mathematical Notes of NEFU 27(4) (2020), pp. 43-59. (in Russian)
[10] Sulaimanov B.E., Myrzapayazova Z.K. and Toktogulova A.S. Inverse Problem for Differential Equations in Individual Derivatives, Bulletin of the Kyrgyz State Technical University named after Razzakov 2(54) (2020), pp. 95-102. (in Kyrgyz)
[11] Umarov KH. G. The Cauchy Problem for The Nonlinear Long Longitudinal Waves Equation in a Viscoelastic Rod, Siberian Mathematical Journal 62(1) (2021), pp. 198-209. (in Russian)
[12] Beshtokov M.Kh. Numerical study of initial-boundary value problems for a Sobolev type equation with a time-fractional derivative, Computational Mathematics and Mathematical Physics 59(2) (2019), pp. 185-202. (in Russian)
