The Inverse Problem of Determining the Coefficients Elliptic Equation
BASTI ALIYEVA
Faculty of Economics of Turkish World,
Department of Economics and Business Administration,
Azerbaijan State University of Economics (UNEC),
Baku, Istiglaliyat str. 6, AZ1001,
https://orcid.org/0000-0002-3274-5301
AZERBAIJAN
Abstract: - The paper considers the inverse problem in determining unknown coefficients in a linear elliptic
equation. Theorems of existence, uniqueness and stability of the solution of inverse problems for a linear
equation of elliptic type are proved. Using the method of sequential measurements, a regularizing algorithm is
constructed to determine several coefficients.
Key-Words: - Inverse problem, elliptic equation, quasilinear elliptic equation, regularizing algorithm, correct
problem, unknown coefficient.
Received: August 29, 2022. Revised: March 27, 2023. Accepted: April 22, 2023. Published: May 9, 2023.
1 Introduction
Inverse problems include the tasks of determining
some physical properties of objects, such as density,
thermal conductivity, elastic moduli depending on
the coordinates or as functions of other parameters.
The solution of inverse problems is carried out,
as a rule, within the framework of some
mathematical model of the object under study. It
consists in determining either the coefficients of
differential equations, or the domain in which the
operator acts, or the initial conditions.
Inverse problems have a number of features that are
unpleasant from a mathematical point of view. First,
they are usually non-linear, that is, an unknown
function or an unknown parameter enters an
operator or functional equation in a non-linear
manner. Secondly, the solutions of inverse problems
are usually non-unique. Thirdly, inverse problems
are not well-posed.
Inverse problems of mathematical physics are
currently a rapidly developing part of 1 modern
mathematics. An increasing part of mathematical
models is becoming harmonious and reliable
precisely due to the achievements of the theory of
inverse problems.Mathematical models of many
established processes of various physical natures
lead to elliptic differential equations.It is enough to
specify stationary problems of thermal conductivity
and diffusion, the problem of determining the
current in a conductive medium, and problems of
electrostatics.The problems of identification of these
models are investigated as inverse problems of
mathematical physics.To date, the number of studies
of inverse problems for an elliptic equation, ranging
from theoretical to specific applied problems, has
increased significantly.
The papers [1], [2], [3], [4] present methods
for solving various inverse problems with boundary
conditions. In [5], the classical solution of a
nonlinear inverse boundary value problem is
studied.To solve the problem under consideration, a
transition is made from the original inverse problem
to some auxiliary inverse problem. Monograph [6]
is devoted to the theory of inverse problems of
mathematical physics and applications of such
problems.Modern results on the problem of
uniqueness in integral geometry and on inverse
problems for partial differential equations [7] are
presented very broadly. In [8], the inverse problem
for a second-order quasilinear elliptic equation with
an unknown coefficient was considered. In the class
of continuously differentiable functions, inverse
problems of determining the source and coefficient
of an elliptic equation in a rectangle are studied [9].
In [10], two inverse problems are considered. A
numerical method for solving these inverse
problems is proposed.
In this paper, we investigate the correctness of
one class of inverse problems of determining the
coefficients of an elliptic equation. A regular
algorithm for determining the coefficients is
constructed.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.35
Basti Aliyeva
E-ISSN: 2224-2880
292
Volume 22, 2023
2 Materials and Methods
Let
1
,DD
be bounded regions
)1( n
dimensional and
n
dimensional Euclidean spaces,
respectively, and an arbitrary point
1
D
of the region
can be represented in the form
),( yx
),,,...,,( 121 yxxx n
where
.),...,,( 121 Dxxxx n
By
Г
denote the
boundary of the area
1
D
, which is assumed to be
sufficiently smooth. Let
21,
be the parts of the
boundary
Г
such that their projections on the
dimensional
)1( n
subspace coincide with the
domain
D
and there are sufficiently smooth
functions
)(xFi
such
,),(:),( DxxFyyx ii
.2,1i
that it is obvious
2,1, i
i
that it can be,
for example, a piece of the boundary
Г
such that
their projections on the
)1( n
dimensional space
cover the domain
.D
Let
,,1,1,...,221 nInI
.
10 Ii
Consider the problem of determining
),(
0xai
),(xc
),( yxu
from the following
conditions:
1
1
),()()()(
n
iyyn
i
x
i
xi yxhuxcuxauxa
,
,),( 1
Dyx
(1)
),,(),(
fyxu Ã
,),( Ã
(2)
),,(),()( 1
10
gyxuxai
1
),(
(3)
),,(),()(),()( 2
20
gyxxcyxuxai
,),( 2
)4(
here
),()(0 DCxai
0 0 1
2
1,2,..., 1, 1,..., , ( , ) ( ),
( , ) ( ),
i i i n h x y C D
fCГ
),(
i
g
),(i
Ñ
2,1i
set functions,
0
,1
direction of the internal normal of
the surface
,
i
i
yxu
),(
,),( i
yx
v
u
.2,1i
Definition.Functions
),(),(
0xcxai
),( yxu
are
called the solution of the problem
)1(
),4(
if there
0
21
1
0 ( ), ( ) ( ), ( , ) ( ) ( )
i
a x c x C D u x y C D C D
are limits of functions
),,( yxu i
x
,,...,2,1 ni
yxn
at
2,1,),(),( iyx i
and the correlations
are satisfied
).4()1(
Below, everywhere constant numbers that do not
depend on the estimated values are denoted by
.,...,2,1, niNi
It is not difficult to verify that if a solution to the
problem
)4()1(
exists, then under accepted
assumptions about the smoothness of the problem
data,
0( ), ( ) ( ),
i
a x c x C D
21
( , ) ( ).u x y C D
Indeed, under accepted assumptions from the
general theory of elliptic equations, it follows,
)(),( 1
2DWyxu p
)( 1
1DC
that
pn
therefore, from the additional
)3(
and
)4(
conditions
).()(),(
0DCxcxai
It follows
that therefore
),( yxu
)( 1
2DC
Task (1) can also be written in the following form
0
000 0
1
11
( ) ( ) ( )
( ) ( , )
in
i x x i x x i x x
i i i
i i i i i i
a x u a x u a x u
c x u h x y
,
1
),( Dyx
.
Suppose, in addition to the task
)4()1(
, there is
also a task
),4()1(
where all the functions that
)1(
),4(
are input to are replaced by the
corresponding functions with a dash.
Let 's put
),(),(),( yxuyxuyx
,
)()()( 000 xaxax iii
,
)()(
1xax ii
),(xai
,,...,1,1,...,2,1 00 niii
),()()( xcxcx
),(),(
1yxhyx
),,( yxh
),,(),(),(
2
ff
),,(),(),(
3
.2,1),,(),(),(
3
igg iii
The uniqueness of the solution of the inverse
problem
)4()1(
under the assumption of its
existence is proved by Theorem 1.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.35
Basti Aliyeva
E-ISSN: 2224-2880
293
Volume 22, 2023
Theorem 1.
Let,
0),(
1
g
,
0),(
,
1
1NmesD
,then
the solution of the problem
)4()1(
is unique and
the following estimate is true:
)(
2
)()( 1
00 )()()()( D
p
WDCDC
ii uuxcxcxaxa
1
0
1
0
1()
1
()
( ) ( )
( ) ( )
i
iCD
i
n
iCD
ii
i
i
N a x a x
a x a x
)( 1
),(),( D
p
L
yxhyxh
)(
2
),(),( Г
p
W
ff
)( 2
),(),(
C
+
2
()
1
( , ) ( , ) i
i i C
i
gg
,
,pn
(5)
1
,NN
positive constants, depending on the data of
the solution set.
Proof. From
)4()1(
, respectively, we
subtract
)4()1(
.Then we get
0
0
000
1
11
1 1 1
1
( ) ( ) ( )
( , ) ( ) ( ) ( ) ,
ii
i
nn
i x x i x x i x x
i i i i
i x x
i i i i i i
ii
a x x u x u
x y c x x u x u
,),( 1
Dyx
)6(
),(),( 2
Ã
yx
,
Г),(
(7)
,),(),()(),(),()( 1010
11
4
yxyxuxayxux ii
),(
,
1
)8(
02
02
1
1
( ) ( ) ( , ) ( , )
( , ) ( ) ( , )
i
i
x a x x y
u x y x
),(),(),()( 1
53
2
xc
,
2
),(
)9(
For a function
),( yx
satisfying the equation
(6) and the condition (7), the estimate is true (10):
1
000
0
01
2 2 1
1
`
11
1
22
( ) ( )
( , ) ( , ) ( )
( ) ( , ) ( ) ( )
i
i x x
W D W i
n
i x x i x x
ii
Гii
pp
i i i i
x y N x u
x u x y c x x u
1
()
( ) ,
LD
p
xu
(10)
Let's put it
)()(),( 0
,xхmaxхmayxxта x
i
xyx
.
Then from the embedding theorems, under the
conditions assumed above, we obtain
111
0
0
3
()
1
4 1 1
( ) ( )
11
2()
( , ) ( , )
( ) ( )
C D W D
in
ii
C D C D
i i i
p
x y N x y
N x x
1
1 2 1
2
( ) ( )
( , ) ( , )
DWLГ
pp
x y mes D
(11)
If we take
)11(
account the estimate in the right
part
)8(
and
)9(
then under the assumptions of the
theorem we get:
)(
))(
0DC
ix
,
15 mesDN
15
)(
)( mesDNx DC
)12(
where
is the value contained in curly brackets on
the right side
)5(
. It
)11(
follows from the obtained
)12(
estimate that
1
mesDN
.Thus,
under the condition
1
1NmesD
that the evaluation
is performed
.
N
From the evaluation
)11(
and
)12(
the validity of the evaluation follows
)5(
. The uniqueness of the solution of the problem
)4()1(
directly follows from the evaluation
).5(
Theorem proved.
The existence of a solution to the problem
)4()1(
is proved under additional assumptions.
Lemma. Let
,0),(,0),(
fyxh
to solve the
problem
,0)(
1
n
ii
x
i
xi xa
),(),(
fyx Г
the
estimates
,0),(,0),( 21
yxyx
nixai,...,2,1),(
)(xc
and the strictly positive
continuous functions given are correct.Then , to
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.35
Basti Aliyeva
E-ISSN: 2224-2880
294
Volume 22, 2023
solve the problem of determining
),( yxu
from the
conditions
)2()1(
, the estimates are correct:
11
1
( ) ( ) ()
, ( , )
( , ) , 1,2.
C D C ГCD i
i
u f c h u x y
x y i
(13)
Proof. The evaluation of a function
),( yxu
in a
uniform metric follows from the maximum
principle. Now let's prove the validity of the
estimate
,),(),( ii yxyxu
.2,1i
Let's
.
u
Assume that the function
),( yx
is a
solution to the Dirichlet problem, so it follows from
the maximum principle
0),(
f
, provided that
the
.0),( yx
Function
),( yx
satisfies the
equation:
1
( ) ( ) ( , )
( ) ( , )
n
i x x
iii
a x c x h x y
c x x y
(14)
and a homogeneous boundary condition.The right
side of the equation
)11(
is negative, therefore
0),( yx
. Hence,
0),(
i
yx
and
,),(),( ii yxyxu
.2,1i
The Lemma is
also proved.
Theorem 2. Let where
1
2
( , ) 0, ( , ) 0, ( , ) 0,
( , ) 0, ( , ) 0,
h x y f g
g
,1 1NmesD
here
N
be a positive number
determined by the data of the problem
),4()1(
for solving
,0)(
1
n
ii
x
i
xi xa
),(),(
fyx Г
the problem the estimates are correct
,0),( 1
yx
then the problem
)4()1(
has at
least one solution.
Proof. The proof is carried out by the method of
successive measurements. Let the functions
),(),(),()()()(
0yxuxcxa sss
i
have already been found
and
0
),(
)(
0xa s
i
)(
)( xc s
),(DC
),(
)( yxu s
)( 1
2DC
. Let's consider
the problem of determining
),(
)1( yxu s
from the
conditions:
0
000 0
1( 1) ( ) ( 1) ( 1)
11
( ) ( 1)
( ) ( ) ( )
( ) ( , ),
in
s s s s
i x x i x x i x x
i i i
ss
i i i i i i
a x u a x u a x u
c x u h x y
1
),( Dyx
, (15)
),,(),(
)1(
fyxu Ã
s
,),( 1
Г
(16)
This problem has a single solution belonging to
)(
2DC
. Function
),(
)1( yxu s
in the ratio
),,(),()( 1
)1()1(
10
gyxuxa s
v
s
i
,),( 1
(17)
02
( 1) ( 1) ( 1)
2
( ) ( , ) ( ) ( , )
( , ),
s s s
i
a x u x y c x x y
g
,),( 2
(18)
are used to determine ,
)(
)1(
0xa s
i
,.
).(
)1( xc s
It
follows from the statement of the lemma that the
sequence
),(
)( yxu s
is uniformly bounded,
.),(),( 11
)1(
yxyxu s
And therefore, from
the condition
)18()17(
under the assumptions
assumed above, we obtain that:
01
1
( 1) 1
0 ( ) max ( , ) ( , )
, 1,2,...,
s
i
a x g x y
As
(19)
2
( 1)
1
0 ( )
max ( , ) ( , )
, 1,2,....
s
cx
A x y
Bs
(20)
Now let 's evaluate
)(
)(
1
),( DC
syxu
. The
function
),(),(),( )1()1( yxyxuyx ss
satisfies a homogeneous boundary condition and the
equation:
0
00 0
00
1( 1) ( 1) ( 1)
( ) ( )
11
( ) ( )
( ) ( )
in
s s s
ii
x x x x x x
ss
i i i
ii
i i i i i i
a x a x
a x a x
.)(),()( )1()(
1
)(
0
sss
iuxcyxhxa
Therefore, the estimate is correct for the function
),(
)1( yx
s
:
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.35
Basti Aliyeva
E-ISSN: 2224-2880
295
Volume 22, 2023
0
11
1
( 1) ( ) ( 1)
2() ()
( ) ( , ) ( )
s s s
i
WD LD
pp
N a x h x y c x u
Hence , taking into account the uniform limitation
of sequences
,)(
)(
0xa s
i
),(
)( yxu s
, we obtain:
1
11
1
( 1) ( 1)
1
()
11
2
22
()
( ) ( )
2() ,
ss
WD
W D W D
s
WD
p
pp
p
u
N mesD
where
1
N
is determined by the data of the problem
and
.0)( )()(
0 ss
ixa
Then from the embedding
theorems we have the following estimate:
11
( 1)
()
( , )
s
CD
u x y
)(
2
)1(
21
D
p
W
s
uN
1
)(
13 1
s
mesDN
,
where
0
3N
is determined by the data of the
problem
).4()1(
Let
,0
0130
1
3
)0( amesDNgN
where
.),(
1
),(
0min
gg Г
Prove that under the
conditions
,...,.2,1,
0
)( sa
s
of the theorem,
the proof is carried out by induction.Let us
,
0
)( a
k
.,...,2,1 sk
, check
.
0
)1( a
k
, that
it follows from the relation
)19(),17(
that:
)1(s
1
1
()
0 3 3 1
1
1
2
0 3 3 1 0 3 1 0,
s
g N N mesD
g N N mesD g N mesD a
Therefore,
)(
)(
1
1
),( DC
syxu
).1( 1
013
amesDN
Given this estimate in
),17(
we get
0
()
0
0 ( ) .
s
i
a a x A
Then it follows from the general theory of
elliptic equations that, under the conditions of the
theorem, the sequence
),(
)( yxu s
is uniformly
bounded by the norm
),(1
2DWp
.np
, therefore
),(
)( yxu s
compact in
).(1
1DC
, while from the
condition
)17(
,
)18(
, it follows that the sequence
,)(
)(
0xa s
i
)(
)( xc s
will be compact in
)(DC
.
Hence and from follows
)14()13(
compactness in
the system
)18()15(
moving to the limit
s
, we obtain that there is a function
satisfying the conditions
),(),(),(
0yxuxcxai
of
the problem . The theorem is proved.
Let
.
20 Ii
The Problem (1) also be written in
the following form
000
00
0
1
2
1
( ) ( ) ( )
( ) ( , ) ( , )
nn
ii
n
n x x i x x i x x
i
ii i i i
a x u a x u a x u
c x u h x y x y D
,
Theorems 1,2 are proved anologically.
3 Results
Thus, the inverse problem in determining unknown
coefficients in a linear elliptic equation was
considered. Existence, uniqueness, and stability
theorems for the solution of inverse problems for a
linear equation of elliptic type are proved. Using the
method of successive approximations, a regularizing
algorithm is constructed to determine the
coefficients.
References:
[1] Aliev. R.A.: On the definition of unknown
coefficients at the senior derivatives in the
LEE. Vest.Sam.Gos.tec.uz. ser.fiz.-mat. nauk,
(2014) № 3 (36), 31-43.
[2] Aliyev.R.A.: Abdullayev.A.Kh.: On the
determination of unknown coefficients at
higher derivatives in a linear elliptic equation.
Proceedings of IAM, (2015) V.4, N.2, 183-
194.
[3] Baev. A.V., : Solution of Inverse Problems
for Wave Equation with a Nonlinear
Coefficient Pleiades Publishing, Ltd (Road
Town, United Kingdom), (2021) volume 61,
№ 9 ,1511-1520 DOI
[4] Baev.A.V., :On the Solution of One Inverse
Problem for Shallow WaterEquations in a
Pool of Variable Depth. Mathematical Models
and Computer Simulations, (2021)volume 13,
№ 4, 543-551 DOI
[5] Farajov.A.S.: On a solvability of the non-
linear inverse boundary value problem for the
boussinesq equation. Advanced Mathematical
Models & Applications 7(2), (2022) 241-248.
[6] Inverse problems of mathematical physics
and methods of their solutions, (2015)
Supervisors of R&D: Denisov A.M.,
Dmitriev.V.I., Division: Direction of
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.35
Basti Aliyeva
E-ISSN: 2224-2880
296
Volume 22, 2023
technological breakthrough of Russia:
Strategic information technologies
[7] Lavrent'ev.M.M..Romanov,S.P.
Shishatsky.M.M.Romanov,S.P. Shishatsky,
M.P.:Uncorrected problems of mathemati-cal
physics and analysis. Russia, Moscow:
Nauka(1980)
[8] Lyubanova,A.S.,Velisevich,A.V.:An Inver-se
Problem for a Quasilinear Elliptic Equation. J
Math Sci 270, 591–599 (2023).
[9] Solov'ev.V.V. :Inverse problems for
determining a source and coefficient of an
elliptic the equation in a rectangular equation.
J. of computational mathematics and
mathematical physics, (2007) vol. 47, №8
1365- 1377.
[10] Tuikina,..S.R. A numerical method for the
solution of two inverse problems in the
mathematical model of redox sorption 2020,
Computational Mathematics and odeling,
Consultants Bureau (United States), vol. 31,
№ 1, 96-103 DOIshab625.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research did not receive any financial support.
Conflict of Interest
The author has no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.35
Basti Aliyeva
E-ISSN: 2224-2880
297
Volume 22, 2023
