Inverse Problem of Determining the Unknown Coefficients in an

Elliptic Equation

BASTİ ALİYEVA

Faculty of Economics of Turkish World,

Department of Economics and Business Administration,

Azerbaijan State University of Economics (UNEC),

Baku, Istiglaliyat str. 6, AZ1001,

AZERBAIJAN

https://orcid.org/0000-0002-3274-5301

Abstract: - The inverse problem of determining the coefficients of an elliptic equation under different boundary

conditions in a given rectangle is considered. These problems lead to the necessity of approximate solution of

inverse problems of mathematical physics, which are incorrect in the classical sense. The existence, uniqueness,

and stability theorems for the solution of the set inverse problem are proved and a regularizing algorithm for

determining the coefficient is constructed.

Key-Words: - Inverse problem, elliptic equation, regularization algorithm, the Green's function, successive

approximation method, existence, and uniqueness of the solution.

Received: August 15, 2023. Revised: March 24, 2024. Accepted: April 19, 2024. Published: May 15, 2024.

1 Introduction

The monographs [1], [2] about inverse problems for

differential equations tell the history of this area of

mathematical physics, its problems, areas of

application, existing solution methods, etc. are

widely given. When studying direct problems, the

solution of a given differential equation or system

of equations is carried out through additional

conditions, whereas in inverse problems the

equation itself is unknown. Both the definition of

the basic equation and its solution require the

imposition of additional conditions, rather than

problems directly related to them.

Newton's problem of discovering the forces that

set the planets in motion according to Kepler's laws

was one of the first inverse problems in the

dynamics of mechanical systems. It covers the

subject of similar problems, including a fairly

complete and systematic theory of inverse

problems. By developing an approach to the

existence, uniqueness and stability of solutions, this

work represents a systematic development of the

theory of inverse problems for all main types of

partial differential equations. Here we discuss

modern methods of linear and nonlinear analysis,

the theory of differential equations in Banach

spaces, applications [1].

The book, [2], offers in-depth coverage of inverse

problems for second-order equations and for

hyperbolic systems of first-order equations,

including the kinematic problem of seismology, the

Lamb dynamic problem for equations of the theory

of elasticity, and the problem of electrodynamics.

The third edition, [3] is intended for ordinary

graduate students of physical sciences who do not

have extensive mathematical training. The book is

complemented by a companion website that

includes MATLAB codes corresponding to

examples, illustrated with simple, easy-to-

understand problems that highlight the details of

specific numerical methods. Updates in the new

edition include more discussion of Laplace

smoothing, expansion of exercises with basis

functions, addition of stochastic descent, improved

presentation of Fourier methods and exercises, and

much more.

The main classes of inverse problems for

equations of mathematical physics and their

numerical solution methods are considered in this

book which is intended for graduate students and

experts in applied mathematics, computational

mathematics, and mathematical modelling, [4].

This book, [5], explores methods for specifically

solving inverse problems. The inverse problem

arises when it is necessary to determine the reasons

that caused a particular effect, or when trying to

indirectly estimate the parameters of a physical

system. The author uses practical examples to

illustrate inverse problems in the physical sciences.

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Developing an approach to the question of

existence, uniqueness and stability of solutions, this

work presents a systematic elaboration of the theory

of inverse problems for all principal types of partial

differential equations. It covers up-to-date methods

of linear and nonlinear analysis, the theory of

differential equations in Banach spaces,

applications of functional analysis, and semigroup

theory, [6].

The paper, [7] 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.

A huge number of mathematical models are

called Boussinesq-type equations. The classical

solution of one nonlinear inverse boundary value

problem for the linearized sixth-order Boussinesq

equation with an additional integral condition is

considered. The first method is based on the

application of the Fourier method. The second

method is based on the application of the

compressive method, which consists in the fact that

it is required to determine together with the solution

the unknown coefficient depending on the variable t

at the unknown function. The problem is

considered in the rectangular domain. When

solving the original inverse boundary value

problem, a transition from the original inverse

problem to some auxiliary inverse problem is

carried out. With the help of compressed mappings

the existence and uniqueness of the solution of the

auxiliary problem are proved. Then the transition to

the original inverse problem is made again, and as a

result the conclusion about the solvability of the

original inverse problem is made. The proposed

methods of finding solutions to the inverse problem

can be used in the study of solvability for various

problems of mathematical physics, [8].

The identification of an unknown coefficient in

the lower term of elliptic second-order differential

equation M u + ku = f with the mixed boundary

conditions of the third type is considered. The

identification of constant based is based on an

integral boundary data. The local existence and

uniqueness of strong solution for the inverse

problem is proved, [9].

For a mathematical model with external-

diffusion kinetics, we consider an inverse problem

of determining the inverse isotherm and a kinetic

coefficient from two dynamic output curves

observed at two points in a single experiment. A

gradient-type iterative method utilizing the adjoint

problem technique is proposed for this inverse

problem, and numerical results are reported, [10].

The purpose of this paper is to prove the

uniqueness and existence of solutions of the inverse

boundary value problem for the second order

elliptic equation.

In the face of higher-order derivatives, the

coefficients coincide with the given problem for the

rectangular region. In the case under review, similar

issues with different border conditions are

considered.

2 Problem Formulation

Let

 

,2,1I

12

1

,,

0

10 



 i

i

iIei

,



,1,0, qk

),2()1( kqqktw t

qkt 



qkt





)1(1  kk





qt t

)1(

,

.1,0t

Through

,, tt db

1,0t

denote the constants, which

are defined as follows:

,,:0,0 0000 tttt dwbqk





:1,0  qk

,, 1010 tttt dwb





:0,1  qk

,, 0101 tttt dwb





:1,1qk

,,

1111 tttt dwb





.1,0t

Let us consider the problem with fixed

parameters

qkei ,,,

0

and



),(2

0xai



),( 21 xxu

the following conditions:

),,()()()( 2122221 2211 xxhuxcuxauxa xxxx 

,),( 21 Dxx 

(1)

),(),0( 212 xxu





),(),()2(),()1( 222121

1xxluexlue x





,

22

0lx 

(2)

),()0,()0,( 111110 2xxubxub x





,0 11 lx 

(3)

),(),(),( 12211210 2xlxudlxud x





,0 11 lx 

(4)

),(),0()( 222 00 1xgxuxa ixi 

,0 22 lx 

(5)

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whereby

),0()()(),0()0()0( 221121011110 22



 ldldbb xx

2

02 (0)

x

b





)0(

21



b

).()()(),(1

)1(

22212201

)1(

12lldldl e

x

e 



Here

},0,0|),{( 221121 lxlxxxD 

),(),,( 221 xxxh i



2,1),(),(21 0ixgx ii



 given

functions

),(),(),,( 2121 11 DCxxhxxh xx





],,0[)( 2

2

21 lCx









],,0[)( 2

3

22 lCx e









)( 11 x



],,0[ 1

11lC b





)( 12 x



],,0[ 1

11lC d





)( 2

0xgi

],,0[ 2

lC



.10 



Definition. The functions





),(),(212

0xxuxai

are called a solution of

problem (1)(5), if

)()(),(],,0[)(0 2

2122

0DCDCxxulCxai

and satisfy the relation (1) (5).

It is easy to check th if solutions (1) - (5) exist, then

under the assumed assumptions on the smoothness

of problem da

).(),(],,0[)( 2

2122

0DCxxulCxai







Indeed,

with the assumptions accepted, it follows from the

general theory of elliptic equions th

)(),( 2

21 DWxxu p



)(

1DC







2p

.

Therefore, from the additional condition (5) it

follows th

].,0[)( 22

0lCxai





Therefore,

),( 21 xxu

).(

2DC





Equion (1) can also be written in the following

for:

11

1

000

2 2 2

1 2 1 2

( ) ( ) ( )

( , ),( , ) .

i

i i i i

x x x x

i

a x u a x u c x u

h x x x x D

   



3 Uniqueness and Stability of the

Solution

Let us now consider the uniqueness and stability of

the solution. Suppose, besides problem (1) - (5),

there is a problem where all functions included in

),5()1( 

are replaced by the corresponding

functions with a line. Put:

0

00

1 2 1 2 1 2 2

22

( , ) ( , ) ( , ), ( )

( ) ( ),

i

ii

Z x x u x x u x x x

a x a x



  



),()()( 2221 11 xaxax ii 



),()()( 2222 xcxcx 



),()()( 2222 xxx iii







),()()( 1114 xxx iii







,2,1i

 ),(),( 21217 xxhxx



),,( 21 xxh

).()()( 2228 00 xgxgx ii 



We denote

),,(

~

21 xx

e



the functions on the

boundary each



Ieqk  ,1,0,

coinciding,

respectively, with

),(22 x

i



2,1),(14 

ix

i



and

belonging to

)(

2DC





. Denote,

ei

kj

eijei

l

lgiek )2(

1)

2

()(),1)(1( 1



,

1)1(

1





ej

je

dej

,

j

i

j

qkj

jjiqk l

xli

xxL

qkj )1)(1(

)

2

1

()1()2(

)(

1

)1)(1(





 



.

The function

),(

~

21 xx

e



is defined as follows:





 







)()(),(),(

~

1

)1(

2

1,

2121 j

eij

j

jiji

ji ijeexxPlllxx m



 

)(),( 121 xPlln iije







)( 2

xPj

e



1

)(

4)1( li

ei

k

j

t





.

Here

jeijeijjipeij tllnmxPll ),,(,),(),(21

are

defined as follows:

at

:0,0  qk

,1),(),(,),()(),(),( 112100121  jeieijejeijjijieijeij tlgllndmxLxPlglll

at

:1,0  qk

1 2 1 10

1 2 1 1

( , ) ( ), ( ) ( ), ,

( , ) ( ), 1,

eij eij i j i j eij ej

eij ei j

l l l g l P x L x m d

n l l g l t

  



at

:0,1  qk

( 1) (2 )( 1)

1 2 2 1 1

01

( , ) ( 1) ( ), ( )

( ),

j i i j

eij ei i j

ij

l l l l g l P x

Lx

  

  



,2 )1)(1( ej

ji

eij dm 



,2),()1(),( 1

11

2

221

  j

jei

jj

eij tlgllln

at

:1,1  qk

),()(),(),( 111

)1)(1(

221 jijieij

ji

eij xLxPlgllll  

,2 )1)(2( ej

jii

eij dm 



.2),(),( 1

11

1

221

  j

jei

j

eij tlgllln

Lemma 1. Let the solutions to problem (1) - (5)

exist. Then the following estimates are true

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max,( 21 xxu



,

)(

),(

max

1

21

xc

xxh

D

,)(max,)(max)2(,)(max 1112221 1

22

xbxex xxx











)(max 121 1

xd x



,

2

11

12

2

12 2

( , )

( , max ,

()

( 1) max ( ) ,

xx

xx D

x

h x x

u x x ma x cx

ex









(6)

,)(

max

max)

2( 21

2

xe i

x

i







.)(

max

,)(

max 121111 11

1

11

1

xdxb xx

x

xx

x



Here

22

()

2 2 2 2 2 2 1 1 2

12

1

( ) ( ) ( ) ( ) ( ) ( , )

()

ei

k

ie ix x i

x a x x c x x h il l x

ax

  



    



1,2i

Proof: The first inequality for problem (1) - (3) at

each is

Ie

obtained from the maximum

principle. By differentiating equation (1) twice and

using the maximum principle we obtain the second

estimate. Analogously at

2e

we obtain the

evaluation (6). Lemma 1 is proved.

The uniqueness of the solution of the inverse

problem (1)-(5) under the assumption of its

existence is establishesis.

Theorem 1. Then let

.1,0)( 212

0 lNlxgi

the solution of the problem (1)-(5) be singular and

the following evaluation be true:

2

00

22

[0, ]

( ) ( )

ii

Cl

a x a x

2

11

1 2 2

() [0, ]

22

[0, ]

( ) ( )

ii

CD Cl

Cl

u u N a x a x

c x c x



    







111

1 2 1 2 1 1 1 1 [0, ]

()

2 1 2 1 [0, ]

( , ) ( , ) ( ) ( )

( ) ( )

b

d

Cl

CD

Cl

h x x h x x x x

xx





    

  

 ],0[

2222

],0[

2121 2

3

2

2)()()()( lClC e

xxxx









],0[

22 2

)()( 00 lC

ii xgxg

(7)

1

,NN

 positive constants that depend on the task

data.

Proof. From

),5()1( 

respectively, subtract

(1)(5) and put

),( 211 xxZ

).,(

~

),( 2121 xxxxZ e





Then we get

1 1 2 2

1 2 1 2 2 1

1 1 2 2 1 2

( ) ( )

( , ) ( ) ( , )

x x x x

ee

a x Z a x Z

x x c x x x



  

  

),(),()( 221

2

1

12 xxxZxc i

ii









(8)

0),0( 21 xZ

,

,0),()1(),()2( 211211 1 xlZexlZe x

(9)

,0),()0,( 2111110 2 lxZbxZb x

(10)

,0),(),( 21112110 2 lxZdlxZd x

(11)

),,0()()()( 2111222 10 xZxxx xei





(12)

Here

,),( 00

21

0i

x

i

xi uxx 



,)],0()[()( 1

222 10



 xuxax xi



11

1 1 2 1 2 1 2 2 2

2

7 1 2 2 1 2

1

( , ) ( , ) ( ) ( )

( , ) ( ) ( , ),

e

i ex x

i

ii

xx

x x u x x x x u

x x a x x x

  





  











1

0

1

2 2 2 2 8 2 2

( ) ( ) ( ) ( ) (0, ) .

e i ex

x x a x x x

   





  



Using the Green's function [9] from (8)-(11),

we define the function

),( 211 xxZ

through the right

side of equality and substitute this expression into

the condition (12). Then we obtain:

),( 21 xxZ

1 2 1 2 1 2

( , ) ( , , , )

e

D

x x G x x

  







),( 211



e

2 1 2

( ) ( , )cZ

  





00

1 2 2

( , ) ( )

ii

    

21



dd

,





D

xei xGxxx ),,,0()()()( 21221222 10





),( 211



e

),()( 212



Zc



0

01 2 2

( , ) ( )

ii

    



.

21



dd

(13)

The following estimates are valid for the

Green's [9] function:

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,

)()(

1

ln),,,( 2

22

2

11

12121





xx

MxxG

 

,)()(),,,( 2/1

2

12

2

1122121

1







xxMxxGx

.2,1,0  iMi

Now in the system (13) let's put





21 ,

max

xx

)(max),( 221 0

2

xxxZ i

x





.

From system (13) we obtain







 

)(

21

)(

217

2

1],0[

22 2

2),(

~

),()( DC

e

DC

ilC

ixxxxxN





.)()( 2/1

213

],0[

28 2llNx lC





3,2, iNi

are some positive numbers. Hence,

given the condition of the theorem, we obtain that

the evaluation of stability (7) is correct at

),( 21 xx

D

. The uniqueness of the solution to the problem

follows from evaluation (7) and the theorem is

proved.

4 Method of Successive

Approximations

The method of successive approximations for

solving problem (1)- (5) is applied according to the

scheme:

11

1

000

( ) ( 1) ( 1)

22

( ) ( )

s s s

i

i i i i

x x x x

i

a x u a x u



  

),,()( 21

)1(

2xxhuxc s 

,),( 21 Dxx 

(14)

),(),0( 212

)1( xxu s







),(),()1(),()2( 2221

)1(

21

)1(

1xxluexlue s

x

s



 

,0 22 lx 

(15)

),()0,()0,( 111

)1(

11

)1(

02xxubxub ss

x



 

11

0lx 

, (16)

),(),(),( 1121

)1(

121

)1(

02xlxudlxud ss

x



 

11

0lx 

, (17)

),(),0()( 22

)1(

2

)1(

00 1xgxuxa i

s

x

s

i



22

0lx 

, (18)

According to the scheme (14) - (18) successive

iterations are carried out as follows: first some

0)( 2

)0(

0xai

belongings are chosen

],0[ 2

lC



and

substituted into equation (14). Then problem (14) -

(17) is solved and

),( 21

)1( xxu

. The function

),,0( 2

)1(

1xux

from the conditions (18) is found

)( 2

)1(

0xai

and this function is used for the next

iteration step.

Theorem 2. Let the solution of problem (1)-(5)

exist and for all

,...,1,0s

),( 21

)( xxu s

 

1

0

2 ( )

22

()

1 2 2 1 2

( ), ( ) 0, ,

( ) (0, ) 0, 1

s

i

s

x

C D a x C l

g x u x Nl l







and the derivatives of the function:

),( 21

)( xxu s

up to second order are uniformly

bounded.

Then the functions



),(2

)(

0xa s

i



),( 21

)( xxu

s

obtained by the method of successive

approximations (14)-(18) at

s

uniformly

converge to the solution of problem (1)-(5) at the

rate of geometric progression.

N

-positive

constant, depending on the given tasks.

Proof. Assume

0

00

( ) ( ) ( )

1 2 1 2 1 2 2

()

22

( , ) ( , ) ( , ), ( )

( ) ( ).

i

s s s

i

s

ii

Z x x u x x u x x x

a x a x



  



From (1) - (5), respectively, subtracting (14) - (18)

we get:

  )1(

1

)1(

1

)1(

1)()()( 22

1

00

0

ss

ii

i

s

ii ZxcZxaZxa xxxx

i

),(),( 2

)(

21

)(

0

0xxx s

i

s

i





,),( 21 Dxx 

(19)

( 1) ( 1)

2 1 2

( 1)

12

(0, ) 0,(2 ) ( , )

( 1) , ) 0,

ss

s

x

Z x e Z l x

e Z Z x





  

  

(20)

,0)0,()0,( 1

)1(

11

)1(

02  xZbxZb ss

x

(21)

,0),(),( 21

)1(

121

)1(

02  lxZdlxZd ss

x

(22)

)( 2

)1(

0x

s

i



),,0()( 2

)1(

2

)(

1xZx s

x

s





(23)

Where

,),( )1(

21

)(

00

0



s

ii

s

ixx

uxx



.)],0()[()( 1

2

)1(

12

)(

1

0



 xuxax s

x

i

s



Using the Green's function from (19)- (22) we

define

),( 21

)1( xxZ s

through the right side of

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equality (19) and substitute this expression in the

condition (23).

Then we obtain:



1

0

00

()

2 1 2 2 1 2

( ) ( )

1 2 2 1 2

( 1) ( ) ( ) (0, , , )

( , ) ( ) .

s

ix

D

ss

ii

sx x G x

dd

   

      









(24)

Put

.)(max 2

)()(

0

2

x

s

i

x

s





In the former way,

as a proof of Theorem 1, it follows from system

(24) that



)1(s



.)( 2/1

213

)( llN

s



Thus the

theorem is proved.

5 Existence of a Solution

Let us first prove one lemma. We will take

.1)( 2xc

Lemma 2. Let the problem

uuxauxa xxxx  22

2221 )()( 11

),,( 21 xxh

,),( 21 Dxx 

(25)

),(),0( 212 xxu





),(),()1(),()2( 222121 1xxluexlue x





,0 22 lx 

)26(

),()0,()0,( 111110 2xxubxub x





,0 11 lx 

(27)

),(),(),( 12211210 2xlxudlxud x





(28)

,0 11 lx 

satisfying conditions:

22

0 1 1 1 1 0 1 2 1 1 2

2

(0) (0) (0), ( ) ( )

(0),

xx

b b d l d l

    



   



 )0()0( 2120 2



bb x

2

( 1) ( 1)

1 1 0 2 2 1 2 2 2 1

( ), ( ) ( ) ( )

ee

x

l d l d l l

   



  

at a

given

),(21 xa

0)( 022 



xa

has a solution

belonging to

)(

2DC

)(DC

and

0,),( 211  xxhMl

,0)()1( 22

1 x

e



,0)0()( 101 



bb

,0)0(

2



,)0(,)0( 021011 11 ddbb xx 



,0)( 21 22 x

xx



  



1

1 0 1 1 0 1 1

1 1 1 1

(0) '(0)) ( ) (0)

( ) ,

b m b m l x b b

x b Mx





   



 

1

0 2 0 2 1 1 2

2 1 1 1

'( ) ( )) (0)

( ) .

d m l d m l l x

x d Mx





  



Then

1

1 2 0 1 2

1

21

( )(2 ) (0, )

( ) ,

x

M x l u x

m x l





   



(29)

Where,



1

2

1

1 1 2 2 2 1 1 1

max max [ ( ) ( )], max ( ) ,

x

xx

M l x x b x

  







,)(max 121 1

1

xd x

x



2

2 2 2 0 1

0 2 1 2

( ) [0, ], ( ) 0, '(0) 0,

'( ) 0, "( ) 0

m x C l m x b m b

d m l d m x

   

  

Proof. Suppose

1

1 2 1 2 2 1 1 1 2 1 2

1 2 1 2

( , ) ( , ) ( ) ( ), ( , )

( , ) ( )

x x u x x m x x l x V x x

u x x x







   

  

).()2)((111

1

0211 xlxxMx  



It is not difficult to check that satisfies the

conditions of the problem

1 1 2 2

1 2 2 2 1 2

11

2 1 1 2 2 2 1 1 1 2

( ) ( ) ( )

( ) ( ) "( ) ( , ),

x x x x

a x a x x

m x l x a x m x x l h x x

   



     

  

,0),0( 2x



 

1

1 2 1 2 2 1

2 1 2 2 2

( 1) ( , ) (2 ) ( , ) ( 1) ( )

(2 ) ( ) ( ) ( ),

x

e l x e l x e m x l

e m x x x







     

   

 

2

0 1 1 1 1 1

/ 1 1

0 1 1 1 1 1 1 1

( ,0) ( ,0) ( )

(0) (0) (0) (0) ,

x

b x b x x

b m l x b m l x

  





  



     





2

0 1 2 1 1 2 2 1

/1

0 1 2 2 1 1

( , ) ( , ) ( )

( ) ( )

x

d x l d x l x

d l m l l x

  





  



   







1

12211 )()( xllmld 





.

Given the conditions of the lemma, we obtain

that the largest positive value of the function

),( 21 xx



is reached at

0

1x

.

Then

0),0( 2

1x

x



, in other words

1

122 )()(0,

1



 lxmxux

(30)

Similarly, after substituting

),( 21 xxV

in (25) -

(28) and considering the conditions of the lemma,

we obtain that the largest positive value of the

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function

),( 21 xxV

is achieved at

0

1x

.

Therefore

0),0( 2

1xVx

, or

)(0,)2)((2x1

1

021 1xulxM  



(31)

Combining estimates (30) and (31), we obtain

estimate (29). Lemma 2 is proved.

Theorem 3. Empty

0,),( 211  xxhMl

,0)()1( 22

1 x

e



,0)0()( 101 



bb

,0)0(

2



,)0(,)0( 021011 11 ddbb xx 



,0)( 21 22 x

xx



 

1

1 2 1 2 2 2

12

0 1 1 2 1

( ) (2 ) ( ) ( )

(1 ) (2 ) ( ) ,

e

l m x e x x

e l x Ml









   

  

 

1

1 0 1 1 0 1

1 1 1 1 1

(0) '(0)) ( )

(0) ( ) ,

b m b m l x b b

x b Mx





  



 

1

0 2 0 2 1 1 2

2 1 1 1

'( ) ( )) (0)

( ) ,

d m l d m l l x

x d Mx





  



,0)( 21 xg



)( 210 xgg

,)(

2

1

121 lx



)( 2

xm

 is such a non-negative function that it is

1

221 )]()[(

xmxg

bounded,

0

g

a positive number.

Then problem (1) - (5) has at least one solution.

Proof. The proof is done by the method of

successive approximations. It follows from the

statement of the lemma that:

,)()0,(])2)((1[ 1

122

)1(

1

021 1



 



 lxmxulgxM s

x



22

0lx 

,

then

1

2212

)1(1

0})]()[({max)(

2

0lxmxgxaMg x

s

i



Thus, for all approximations, the function

)( 2

)(

0xa s

i

is strictly positive, continuous, and

uniformly bounded. Then it follows from the

general theory of elliptic equations that, under the

conditions of the theorem, the sequence

 

),( 21

)( xxu s

is uniformly bounded by the norm

.2,

2pWp

Therefore

 

),( 21

)( xxu s

, it is compact

in

).(

1DC

It follows from condition (18) that the

sequence

)( 2

)1(

0xa s

i

will be compact in

].,0[ 2

lC

Hence, and from (14) - (17) the compactness

 

),( 21

)( xxu s

in

)(

2DC

. In the system (14) - (18)

passing to the limit at

s

we obtain that there

exists a pair of functions



),(2

0xai



),( 21 xxu

satisfying conditions (1) - (5). The theorem is

proved.

At

0,0  qk

we get the following expression

for the function

),(

~

21 xx

e



:





)(

2

)()(),(

~

24

1

23

1

11

21 x

l

x

l

xl

xx e

e











)0()()( 5

1

11

15

2

22



e

l

xl

x

l

xl

 

( 1)

12

5 1 6 1

1

12

( ) ( )

2

ee

e

xx

lx

ll





  

11

6

1

( ) (0)

e

lx

l









)(

21

)1(

6

1

1l

l

xe

e

e





.

At

1,0,0  eqk

to solve problem (1) -

(5) we can take the following functions:

2

5

)

2

1

2

(),( 11

2

21

22 



 xx

x

xxu

,

.1)(,1)(),

2

5

(

2

1

)( 2222221  xcxxaxxa

In this case

)( 2

xm

is defined as follows:

1

2

22

1

)( l

xx

xm 



.

The conditions of Theorem 3 are satisfied for

this function.

6 Results

Thus, the inverse problem of finding the

coefficients of a linear elliptic equation under

various boundary conditions in a given rectangle

was studied. To solve the inverse problem,

theorems on existence, uniqueness, and stability

were proven. Using the method of successive

approximations, a regularization algorithm was

constructed to determine several coefficients. The

inverse problem of finding the coefficients and

solving a linear elliptic equation in a given

rectangle is studied. A theorem of existence,

uniqueness, and stability of the solution to the

posed inverse problem is proven.

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[1] Prilepko A.I., Orlovsky D.G. Aleksey I., Igor

A. Vasin. Methods for Solving Inverse

Problems in Mathematical Physics, 732

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[3] Aster, Richard C.,

Borchers, Brian, Thurber, Clifford H.

Parameter Estimation and Inverse

Problems, Elsevier, 3rd ed., 2019.

[4] Samarskii A. A., Numerical methods for

solving inverse problems of mathematical

physics, Walter de Gruyter, 2007.

[5] Kern, Michel, Numerical methods for

inverse problems, Wiley-ISTE, 2016.

[6] Prilepko A.I., Kosten A.B.,Solovev V.V.

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spaces. Journal of Mathematical Sciences,

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[7] Aliyeva. B.M: The Inverse Problem of

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https://doi.org/10.37394/23206.2023.22.35.

[8] Farajov. A.S.: On a solvability of the

nonlinear inverse boundary value problem for

the boussinesq equation. Advanced

Mathematical Models & Applications, 7(2),

(2022) 241-248.

[9] Lyubanova, A.S., Velisevich, A.V. An

Inverse Problm for a Quasilinear Elliptic

Equation, JMath Sci 270, 591–599 (2023).

[10] Tuikina S.R. A numerical method for the

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(2020), Computational Mathematics and ode

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https://doi.org/10.1007/s10598-020-09478-8.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally 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

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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