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IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

 95

ON THE CORRECTNESS OF A NONLOCAL PROBLEM 
FOR THE SECOND ORDER MIXED TYPE EQUATION OF 

THE SECOND KIND IN A RECTANGLE 

SIROZHIDDIN ZUHRIDDINOVICH DJAMOLOV* 
The Institute of Mathematics at the National University of Uzbekistan,  

100125, Tashkent, Academgorodok, Durmon yuli street, 29. 

*corresponding author: siroj@mail.ru  

 (Received: 15th Apr. 2016; Accepted: 1st Jul. 2016; Published on-line: 30th Nov. 2016)  

ABSTRACT: In this work, the correctness of a nonlocal problem from the Sobolev 
spaces is proven under some restrictions to the coefficients of the considered second 
order mixed type equation of the second kind. The proof is accomplished with multiple 
methods namely, "ε-regularization", priory estimates, and the Galerkin method.  

ABSTRAK: Dalam karya ini, ketepatan satu masalah bukan setempat dari ruang Sobolev 
telah dibuktikan di dalam beberapa sekatan kepada koefisien persamaan jenis bercampur 
perintah kedua daripada jenis yang kedua yang dipertimbangkan. Buktinya dicapai 
dengan pelbagai kaedah iaitu, "ε-regularization", anggaran priori dan kaedah Galerkin. 

KEYWORDS: second order mixed type equation of the second kind; Sobolev space; Galerkin 
and ε-regularization methods   

1.   INTRODUCTION  
The basic concepts of the theory of differential equations in partial derivatives were 

formed in the study of classical problems of mathematical physics and are currently well 
understood. However, the current problems of natural sciences lead to the necessity of 
setting and qualitatively studying new tasks, which is a striking example of a class of 
nonlocal problems. 

As nonlocal problems we call such problems that determine relationships between the 
value of the solution or its derivatives at the boundary and interior points of a considered 
domain. In recent decades, nonlocal problems for partial differential equations are being 
actively studied by many mathematicians. 

Among the first studies of non-local problems, we note a work by Bitsadze and 
Samarskii [4]. In this work, spatial and non-local problems for a certain class of elliptic 
equations were formulated and studied, which led to the study of non-self-adjoint spectral 
problems. Subsequently, the problem formulated in [4], has been named a Bitsadze-
Samarskii problem. 

The investigation of nonlocal problems is caused by both theoretical interest and 
practical necessity. This is due to the fact that the mathematical models of various 
physical, chemical, biological, and ecological processes are often problems in which, 
instead of the classical boundary conditions, definite connection values of the unknown 
function (or its derivatives) on and within the boundary are given. Problems of this type 
may arise in the study of phenomena related to plasma physics, the spread of heat, the 



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

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process of moisture transfer in capillary porous media issues, demographics, mathematical 
biology, and some technological processes. Nonlocal problems also have practical value in 
solving the problems of solid mechanics. They allow to control the stress-strain state and 
these are similar to the control tasks [4, 8-12, 16, 17, 19]. 

For the first time, non-local problems in certain weight and negative spaces for 
equations of mixed type were investigated by functional methods in the works [1, 6, 8, 10, 
11, 17]. 

2.   FORMULATION OF A PROBLEM 
In a rectangle (0,1) (0, )Q T   we consider a second order differential equation 

( , ) ( , ) ( , ) ( , ) (1) .tt xx tLu K x t u u x t u c x t u f x t      

Problem: To find a generalized solution of Eq. (1) from the Sobolev space 2 ( )
lW Q , ( 2 l  

is integer), satisfying boundary conditions 
 ( , 0) ( , ), (2 )u x u x T   

 (0, ) (1, ) 0, ( 3)u t u t   
here 0const   . 

In the previous works [6, 7, 11, 17] the correctness of problem Eqs. (1)-(3) in the case 
where ( , 0) ( , ) 0K x K x T   was proven under comparatively strong conditions to the 
coefficients of Eq. (1). In the present work, we consider the case where, 

( , 0) 0 ( , )K x K x T  . 

Note that Eq. (1) is the second kind of mixed type equation, since there is no restriction 
inside the domain Q  to the sign of the variable t  of the function ( , )K x t  [5]. 
First, we consider the case where 2l  . Assuming the coefficients of Eq. (1) are smooth 
enough and ( , ), ( , )x t c x t  are periodic in function with respect to variable t . 
 
3.   UNIQUENESS FOR PROBLEM 
 
Theorem 1:  
Let aforementioned conditions to the coefficients of Eq. (1) are fulfilled, moreover let 

1 22 0; 0t tK K c c           , where 
2

ln
T

  , such that (1, )   . Then for 

any function  2( , ) ( )f x t L Q  a generalized solution of problem Eqs. (1)-(3) from the 
space 22 ( )W Q  is unique and the following inequality holds true 

1 0
.u m f  

Here (, )l  and  l  are respectively regular scalar product and norm of the Sobolev space 

2 ( )
lW Q , 2 l  is an integer number and it is known that at 0l  , 02 2( ) ( )W Q L Q   [2, 5, 

13-15]. Here, and further, by m  we designate positive constants, the exact values of which 
are not of interest. 
 
 
 
 



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

 97

Proof:  
For any function 22 ( )u W Q , one can easily get the following equality by using integration 
by parts: 

 

2 2 2

2 2 2
0 1 0 0

2 exp( ) exp( )(2 ) ( )

(4)
exp( ) 2

t t t x t
Q Q

t x t x
Q

Lu t u dxdt t K K u u c c u dxdt

t Ku u u u cu ds

     

    


           

    

 


 

where 0 1( cos( , ); cos( , ))t x        is a unit vector of the inner normal to the Q . 
Conditions of Theorem 1 guarantee non-negativity of the integral over the domain Q . Let  

2
2 ( )u W Q  satisfy boundary conditions Eqs. (2)-(3), on condition that Theorem 1, with 

respect to function ( , )K x t , must meet the condition of ( , 0) 0 ( , )K x K x T  . With 
respect to function ( , )c x t , the periodicity of variables t  and 2 Te   satisfy conditions 
Eqs. (2)-(3), thereby leading to positivity of the following boundary integral. 
 

   2 2 2 2 2 20 1 0 0 0exp( ) 2 exp( )t x t x t x
Q Q

J t Ku u u u c u ds t Ku u cu ds      
 

            
1 1

2 2 2 2
1

0 0

2 [ ( , ) ( , 0)] ( , 0) [ 1] ( , 0)T Tx t t x
Q

u u ds K x T e K x u x dx e u x dx    


         
1

2 2

0 0

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

T
x t x tc x T e c x u x dx u t u t u t u t dt

      …
1 1

2 2

0 0

[ ( , ) ( , 0)] ( , 0) [ ( , ) ( , 0)] ( , 0) 0  .tK x T K x u x dx c x T c x u x dx   … …  
 Omitting non-positive items from Eq. (4) we get 
 

  2 2 22 exp( ) exp( ) (2 ) ( ) } . (5)t t t x t
Q Q

Lu t u dxdt t K K u u c c u dxdt                 

 
Applying Young's inequality to Eq. (5) [5, 13, 14] the following estimate is obtained 

 
1 0

,u m f„  

from which the uniqueness of the solution of problem Eqs. (1)-(3) follows. 

 
4.   EXISTENCE FOR PROBLEM  

In order to prove the existence of the solution of problem in Eqs. (1)-(3) in  22 ( )W Q  
the "ε-regularization" method is used with the Galerkin method [7, 13-15]. Consider a 
nonlocal problem for a composite type equation 

 ( , ) (6)L u u Lu f x t
t   




    


 

 
0

; 0,1, 2 (7) q qt tt t TD u D u q      

 (0, ) ( ) 8) 1, 0 (u t u t    



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

 98

Here ,
p

p
t p

u
D u

t





 0 , 0,1, 2,tD u u p   
2 2

2 2
u u

u
t x

 
  

 
 is the Laplace operator in a plane, 

  is a small positive number, and 0const    such that (1, )   . 
Eq. (6) acts as an  -regularization of Eq. (1) [5, 7, 13, 14]. The class of functions

2
2( , ) ( )u x t W Q  , 2 ( )u L Qt 


 


 are designated through V , satisfying conditions in Eqs. 

(7), (8). 

Definition:  

As the regular solution of problem in Eqs. (6)-(8), function  ( , )u x t V   satisfies Eq. (6). 
 

Theorem 2:  

Let the following conditions holds true in Q : 
 1 22 0; 0,t tK K c c            

where 
2

ln
T

   such that (1, )   .  Then for any function 2, ( )tf f L Q , 

( , 0) ( , )f x f x T    there exists a unique regular solution of problem in Eqs. (6)-(8) and 
the following inequalities are true: 
 

 
2 22 2

2 2
2 1 0

0 0

) ( ) ;I u u u m f
t t x  


 

  
  

„  
 

 
2

2 2 2

2 0 0
0

) .tII u u m f ft  


       
„  

 
Proof:    
The proof of inequality I) will be done similarly to the proof of Theorem 1, from which 
the uniqueness of the regular solution of problem in Eqs. (6)-(8) follows. Now, the second 
prior estimate is proven. 
 

Let ( , )j x t  are eigenfunctions of the following problem: 

 
2 2

2
2 2 , 9) (

j j
j j jt x

 
  

 
   

 
 

 
0

; 0,1, ( 10)p pt j t jt t TD D p     

 (0, ) (1, ) 0. (11)j jt t    

Due to the general theory of linear self-adjoint elliptic operators [2, 13, 15], it is known 
that all eigenfunctions of problem in Eqs. (9)-(11) form a fundamental system in ( )C Q , 
which is orthogonal in 2 ( )L Q . Solving problems in Eqs. (9)-(11), is 

( , ) cos sinj j jx t t x    , where 
2 2 22 ; ; ; 0,1, 2,...j j j j j

j
j j

T


           

Using these sequences of functions, a solution of the auxiliary problem is constructed 



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

 99

 2 , (12)
t

j
j je t

 
 

 
 


  

 ( , 0) ( , ). (13)j jx x T   

Obviously, problem in Eqs. (12)-(13) is uniquely solvable and its solution has the form 
 

1 2 2

0 0 0 0

2

2
2

1 1
exp( ) exp( ) sin cos cos

2 1 2 1

sin sin cos 1

4

t T t T t

j j j j j j j

t

j j j j

j

t
d dt x e tdt e tdt

e
x t t

 



 
       

 

   




       
 

     


   

It is clear that the functions ( , ) ( )j x t C Q
   are linearly independent.  Really, if 

1
0

m

j j
j

c 


  for some set of sequences of functions 1 2, ,..., m   , then acting on this sum 
by the operator   results in 

 
1 1

0,
m m

j j j j
j j

c c 
 

    
 
From which follows that 0jc   for any 1,j m . Note, from the construction of function 

( , )j x t  the following conditions to the functions ( , ) ( )j x t C Q
  follow: 

 
0

, 0,1, 2 ( 4) 1q qt j t jt t qD D q      

 (0, ) (1, ) 0 (15)j jt t    

Now an approximate solution of Eqs. (6)-(8) is searched for in the form 
1

N
N

j j
j

w u c 


   , 

where coefficients jc ; 1,j m   are defined as solutions of the algebraic system. 

 exp( ) exp( ) . (1
2

 6)
2

N
j j

Q Q

t t
L u dxdt f dxdt 

 
      

The unique solvability of the algebraic system in Eq. (16) is proven. Multiplying every 
equation of (16) by 2 jc  and summarizing with respect to j  from 1 to N , considering 
problem in Eqs. (9)-(13) results in 

 exp( ) exp( ) . (17) t t
Q Q

L w t w dxdt f t w dxdt       

From which, by virtue of Theorem 2, by integrating the identity in Eq. (17) an 
approximate solution of problem in Eqs. (6)-(8) is obtained that estimates I) i.e. 

 
2 22 2

2 2
2 1 0

0 0

.u u u m f
t t x  


  
    
    

 

This implies the solvability of Eq. (16) [6, 8, 13, 14]. In particular, from the estimate I) we 
obtain a weak solution of problem in Eqs. (6)-(8). 



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

 100

Now the second priory estimate II) must be proven. Thanks to Eqs. (9)-(13), from the 
identity in Eq. (16) comes 
 

 2 2
1 1

exp( ) exp( ) . (18)
2 2j jj jQ Q

t t
L w dxdt f dxdt

 
 

 
          

 

Multiplying each equation of (18) by 22 j jc  and summing  j  from 1 to N , taking Eqs. 
(13),(14) into account from Eq. (18), the following is derived 
 

 
2 2

2 2 (19)
4 4

 t ttt t tt t
Q Q

w w
L we w w dxdt fe w w dxdt

t t
 


 

  
    

             
   

 
Integrating Eq. (19) according to the conditions of Theorem 2 and the boundary conditions 
in Eqs. (14)-(15), we obtain the following inequality 
 

  
2

2 2 2 2
0 0

0

(2 ) (2 )tt t tt t tx xx
Q

w
m f f e K K w K K w w dxdt

t
     

          
     

    
3

2 2 2
0 1

1
( ) 2 (20) t ttt tx xx tt tx t tx xt tt i

iQ Q

e K w w w ds e Kw w w w w w ds J    
 

          
 

where 1J  is integral along the domain and iJ , 2, 3i   are integral along the boundary. 
Considering the condition of Theorem 2, using Young's inequality, results in: 

 
2

2 2 2
1

0

exp( ) . (2 1)tt tx xx
Q

w
J m t w w w dxdt const

t
 


          

 
Based on boundary conditions in Eqs. (14),(15) and the condition of Theorem 2, 

 0, 1, 2. (22)iJ i   

c const  does not depend on N , hence from Eqs. (17)-(22) the second estimate for the 
approximate solution of Eqs. (6)-(8)  follows. An estimate of Eq. (17) together with Eq. 
(21) allows to pass to limit at N    and conclude that a subsequence  kNu  converges 
in view of the uniqueness (Theorem 1) in 2 ( )L Q  together with a first-and second-order to 
the desired regular solution ( , )u x t  of Eqs. (6)-(8) possessing the properties stated in 
Theorem 2 [6, 8, 10, 13, 14]. 
For ( , )u x t  by virtue of Eq. (21), the inequality holds true 
 

 
2

2 2 2

2 0 0
0

(23)tu u m f ft  


     
 

 

Theorem 2 is thus proven. 
Now using the method of "ε-regularization" the solvability of Eqs. (1)-(3) is proven. 

Theorem 3:   
Let all conditions of Theorem 2 be fulfilled. Then the generalized solution of the problem 
from 22 ( )W Q  exists and is unique. 



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

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Proof: 

The uniqueness of the solution of problem in Eqs. (1)-(3) from 22 ( )W Q   is proven in 
Theorem 1. Now the existence of a generalized solution of Eqs. (1)-(3) from 22 ( )W Q  is 
proven. For this, consider Eq. (6) in domain Q  with boundary conditions in Eqs. (7), (8) at 

0  . Since all conditions of Theorem 2 are fulfilled, then there exists a unique regular 
solution of problem Eqs. (6)-(8) at 0   and for this solution estimates I), II) are true. 

From here follows, under the known theorem of weak compactness, that from a set of 
functions  , 0u    it is possible to take a poorly converging sub-sequence of 
functions in V such that  

i
u u    at 0i  . It will be shown that limiting function 

( , )u x t satisfies the equation Lu f (1).  In reality, as the sequence  
i

u  converges 

poorly in 22 ( )W Q and the sequence  
3

3 , ( 0)
i

u
t
 





 is in regular intervals limited in

2 ( )L Q , and the operator L - linear it is had      

( ) (24)i i
i ii i

u u
Lu f Lu Lu L u u

t t
 

  
   

      
 

 

From here, the following inequality will be performed 

0 0 2
0 0

( ) (25)i i
i ii i

u u
Lu f L u u m u u

t t
 

  
   

     
 

„  

From (25) approaching a limit at 0i   we get the unique solution of Eqs. (1)-(3). Thus, 
theorem 3 is proven. 

5.   SMOOTHNESS OF THE SOLUTION 
Now, the more general case of 3l… must be proven. Further, the coefficients of Eq. 

(1) are assumed to be infinitely differentiable in a closed domain Q . 

Theorem 4:  

Let the conditions of Theorem 3 be fulfilled and let 2( ) 0t tmK K K       . Then, 

for any function ( , )f x t  such that 12 2( ), ( )
m m

tf W Q D f L Q
  ,   

0

m m
t tt t T

D f D f
 
 ,  

there exists a unique generalized solution to problem (1)-(3) from the space 22 ( )
mW Q , 

where 0,1, 2, 3,...m   
 
Proof: 
From the smoothness of the solution to problem in Eqs. (9)-(13), the following conditions 
for the approximate solution of Eqs. (6)-(8) follow: 
 

 0
( ); ; 0,1, 2, 3, 4,...,

(0, ) (1, ) 0.

N q q
t tt t T

w u C Q D w D w q

w t w t
 


 

   

 
 

 



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

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Considering the conditions of Theorem 2 at 0  , nonlocal conditions at 0,t t T   and 
from the equality 

 2 2 2 2
0 0 0

( , )
t T t T t T

t t t t

t t t

e L u e u e Lu e f x t
t

   

   
  

  


  

     
         

     
 

then 
 

0
( , 0) ( , ) .ttt tttu x u x T const      

Hence, it follows that function ( , ) ( , )tv x t u x t   belongs to V  and satisfies the following 
equation 
 (26 )  

tt t t t t
P v L v K v f u c u F              

From Theorem 2, it follows that the set of functions   F  is uniformly bounded in the 
space 2 ( )L Q , i.e. 

 
2 2

0 0 0t
F m f f      

Further, from the conditions of Theorem 3, one can easily see that operators ,( 0)P    
satisfy the conditions of Theorem 4. From here, based on estimates of I) and II) for a 
function  v , the following analogical estimates are obtained 

  
2 22 2

2 2 2
2 1 0 0

0 0

(27) tv v v m f ft t x  

  
    
    

 

 
2

2 2 2

2 1 0
0

(28)ttv v m f ft  


      
 

Function  u  satisfies a parabolic equation with conditions in Eqs. (2),(3) 

 ( 1) ,  (29)t xx tt tu u u f u Ku u cut       
 


          


 

here 2 ( )L Q  . The set of functions    is uniformly bounded in  12 ( )W Q  i.e. 

 2 2 2 2
1 1 0 2

(3 0)ttm f f m f        

Based on priory estimates for parabolic equations [4,6] and inequality in Eq. (30) , then 

 2 2
3 2

.u m f   

Similarly, one can prove that 
 2 2

2
,

m m
u m f    

where  2, 3,...m   

 
 
 



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 103

6.   CONCLUSION 
Solvability of the problem considered can be formulated in the terms of smoothness 

of K (x, t) in the case where, ( , 0) 0 ( , )K x K x T  . In this the second kind of mixed type 
equation studied, since there is no restriction inside the domain. Under the assumption that 
the coefficients of the equation are smooth enough and ( , ), ( , )x t c x t  are periodic with 
respect to variable t . 

 
REFERENCES  
[1] Alimov NA. (1983) On a nonlocal boundary value problem for a non-classical equation.   

The theory and methods for solving ill-posed problems and their applications. Novosibirsk, 
237-239. 

[2] Berezinsky YM. (1965) Expansion in eigenfunctions of selfadjoint operators. Kyev. 
[3] Berdyshev AS. (1993) Nonlocal boundary value problems for equations of mixed type in a 

deviation from the characteristics. Differensial.Uravn., 29(12):2118-2125. 
[4] Bitsadze AV, Samarskii AA. (1969) On some simple generalizations of linear elliptic 

boundary value problems. DAN SSSR, 185(4):793-740. 
[5] Vragov VN. (1983) Boundary problems for non-classical equations of mathematical physics. 

Novosibirsk, 216pp.  
[6] Glazatov SN. (1985) Nonlocal boundary problems for mixed type equations in a rectangle. 

Siberian Math. J., 26(6):162-164. 
[7] Djamalov SZ.  (1989) On correctness of nonlocal boundary problems for many-dimensional 

mixed type equation. Application of method of functional analysis to the non-classical 
equations of mathematical physics. Novosibirsk, 63-70. 

[8] Yogorov IE. (1995) On smoothness of a solution to a nonlocal boundary value problem for 
an operator-differential equation with variable time direction. Mat. Zametki YaGU, 2(1):98-
104. 

[9] Il'in VA, Moiseev YI. (2000) On the uniqueness of solution of mixed problem for wave 
equation with nonlocal boundary conditions. Differens. Uravn., Ç6(5):656-661. 

[10] Karatkoprakliev GD. (1987) Nonlocal boundary problems for mixed type equations. 
Differens. Uravn., 23(1)78-84. 

[11] Karatopraklieva MG. (1991) A nonlocal boundary-value problem for an equation of mixed 
type. Differens. Uravn., 27(1):68-79. 

[12] Kalmenov TS. (1990) On the Dirichlet problem and nonlocal boundary problems for wave 
equation. Differens. Uravn., 26(1):60-65. 

[13] Kozhanov AI. (1990) Boundary problems for equations of mathematical physics of odd 
order. Novosibirsk, 132 pp. 

[14] Kuzmin AG. (1990) Non-classical mixed type equations and their applications to the gas 
dynamics. Leningrad 271pp. 

[15] Ladyjenskaya OA. (1973) Boundary problems of mathematical physics. Moscow 470 pp. 
[16] Moiseev YI.  (2001) On a solvability of a nonlocal boundary problem. Differens. Uravn., 

37(11):1565-1567. 
[17] Terekhov AN. (1985) Nonlocal boundary problems for equations of variable type. Non-

classical equations of mathematical physics. Novosibirsk, 148-158. 
[18] Nakhushev AM. (1982) On an approximate method for solving boundary value problems for 

differential equations and its application to the dynamics of soil moisture and groundwater. 
Differens. Uravn., 18(1):72-84. 

[19] Beilin SA. (2006) On a mixed nonlocal problem for a wave equations. Electronic J. Diff. 
Equations, 103:1-10. 

[20] Berdyshev AS, Karimov ET. (2006) Some-non-local problems for the parabolic-hyperbolic 
type equation with non-characteristic line of changing type. Central European J. Math. 
(CEJM), 4(2):183-193. 



IIUM Engineering Journal, Vol. 17, No. 2, 2016 Djamolov 

 104

[21] Rassias JM, Karimov ET. (2010) Boundary-value problems with initial non-local condition 
for parabolic equations with parameter. Eur. J. Pure Appl. Math. (EJPAM), 3(6):948-957.