Contact-Boundary Value Problem in The
Non-classical Treatment for One Pseudoparabolic
Equation
Ilgar Mamedov
A.I.Huseynov Institute of Cybernetics of NAS of Azerbaijan. Az 1141,
Azerbaijan, Baku st. B. Vahabzade, 9
E-mail: [email protected]
Abstract — In the paper, the contact - boundary value problem
with non-classical conditions not requiring agreement conditions
is considered for a pseudoparabolic equation. The equivalence of
these conditions is substantiated in the case if the solution of the
solution of the stated problem is sought in S.L.Sobolev isotropic
space. (Abstract)
Keywords— Contact - boundary value problem, pseudoparabolic
equation, discontinuous coefficients equation. (key words)
I.
Consider equation


V u x   a xD
i1 0 i2 0

Wp4, 4  G   ux  : D1i1 D2i2 ux   L p G , i1  0,4, i2  0,4 
4
4, 4
Therewith, the important principal moment is that the
equation under consideration has discontinuous coefficients
that satisfy only some p-integrability and boundedness
conditions, i.e. the considered pseudoparabolic differential
operator has no traditional conjugation operator.
Under these conditions, we'll look for the solution u x  of
equation (1) in S.L.Sobolev isotropic space
PROBLEM STATEMENT (HEADING 1)
4
ai1 , 4 x   Lxp1,,x2 G , i1  0,3 
i1 ,i2
i1
1
D2i2 ux   Z 4, 4 x   L p G  , 
where a4,4 x  1 .
where 1  p   .
For equation (1) we can give the contact-boundary
conditions of the classic form as follows: (see fig.1 ):
Here ux   ux1 , x2  is a desired function determined on
G; ai1 ,i2 x  are the given measurable functions on G  G1  G2 ,
where Gk  0, hk , k  1,2 . Z 4, 4 x  is a given measurable
function on G; Dk   / xk is a generalized differentiation
operator in S.L.Sobolev
transformation operator.
sense,
Dk0
is
an
identity
Equation (1) is a hyperbolic equation that has two real
characteristics x1  const, x2  const , the first and second one
of which is four-fold. Therefore, we can consider equation (1)
in some sense as a pseudoparabolic equation [1]. This equation
is a generalization of the equation of thin spherical shell
bending (2, p. 218) [1].
In the paper, we consider equation (1) in the general case
when the coefficients ai1 , i2 x  are non smooth functions
satisfying the following conditions:
ai1,i2 x   L p G , i1  0,3 i2  0,3 ;
a4,i2 x   Lx1,, xp2 G , i2  0,3 ;
Fig. 1. Geometric interpretation of classic contact-boundary conditions

 u x1 , x2  x h  1 x2 ; u x1 , x2  x 0   1 x1 ;
1 1
2

 u x1 , x2 
u x1 , x2 
  2 x2 ;
  2 x1 ;

x2
 x1
x1  h1
x2 0
 2
 2
  u x1 , x2 
 2u x1 , x2 
  3 x2 ;
  3 x1 ;

2
2
x2
x1  h1
x2 0
 x1
 3
 3u x1 , x2 
  u x1 , x2 
  4 x2 ;
  4 x1 ;
3
 x1
x23
x

h
x

0
1 1
2

If the function u Wp4, 2  G  is a solution of the classical
form contact boundary value problem (1), (2), then it is also a
solution of problem (1), (4) for Z i1 ,i2 , determined by the
following equalities:
Z 0,0  1 0   1 h1 ;  Z 0,1  1 0    2 h1 ;
Z1, 0   2 0    1 h1 ;  Z1,1   2 0   2 h1 ;
Z 2, 0  3 0    1h1 ;  Z 2,1  3 0    2h1 ;
Z 3, 0   4 0    1h1 ;  Z 3,1   4 0    2h1 ;
Z 0, 2  10    3 h1 ; Z 0,3  10    4 h1 ;
where k x2 ,  k x1 , k  1,4 are the given measurable
functions on G . It is obvious that in the case of conditions (2),
in addition to the conditions
Z1, 2   20    3 h1 ;  Z1,3   20    4 h1 ;
Z 2, 2  30   3h1 ;  Z 2,3  30    4h1 ;
Z 3, 2   40   3h1 ;  Z 3,3   40   4h1 ;
x1 ;  Z 4,2 x1    3IV  x1 ;
 IV 
 IV 
Z 4,1 x1    2 x1 ;  Z 4,3 x1    4 x1 ;
 IV 
 IV 
Z 0, 4 x2   1 x2 ;  Z 2, 4 x2   3 x2 ;
 IV 
 IV 
Z1, 4 x2   2 x2 ;  Z 3, 4 x2   4 x2  
Z 4,0 x1    1
 k x2   W p( 4) G2   k x1  Wp( 4) G1  
( IV )
the given functions should satisfy also the following
agreement conditions:
1 0   1 h1 ;  2 0   1 h1 ;
 
1 (0)   2 h1 ;  2 (0)   2 h1 ;
1(0)   3 h1 ;  20   3 h1 ;

1 (0)   4 h1 ;  20   4 h1 ;

3 0   1h1 ;  4 0   1 h1 ;
3 0   2h1 ;  4 0   2 h1 ;

30   3h1 ;  40   3h1 ;
 0   h ;  0   h .
4 1
4
4 1
 3


3
It is easily proved that the inverse is also true. In other
words, if the function u Wp4, 4  G  is a solution of problem
(1), (4), it is also a solution of problem (1), (2) for the
following functions:
1 x2   Z 0, 0  x2 Z 0,1 

Obviously, conditions (2) are close to the boundary
conditions of the Goursat problem from [3-6].
Consider the following non-classical boundary conditions:
Vi1 ,i2 u  D1i1 D2i2 u h1 ,0  Z i1 ,i2  R, ik  0,3, k  1,2;
V u x   D D ux ,0  Z x   L G , i  0,3; 
V u x   D D uh , x   Z x   L G , i  0,3 .
4,i2
i1 , 4
1
2
4
1
i1
1
i2
2
4
2
1
1
2
4,i2
1
p
i1 , 4
2
p
1
2
2
2
x    Z  d ;
x3
 2 Z 0, 3   2
0, 4
3!
3!
0
x
3
 2 x2   Z1,0  x2 Z1,1 

x22
Z 0, 2 
2!
x22
Z1, 2 
2!
2
x    Z  d ;
x3
 2 Z1,3   2
1, 4
3!
3!
0
x
3

5

6

7

8
1
If the function ux   W p4, 4  G  is a solution of the classical
form contact - boundary value problem (1), (2), then it is also a
solution of problem (1) (4) for Z i1 ,i2 , determined by the
following equalities:
 3 x2   Z 2,0  x2 Z 2,1 

x3
 2 Z 2,3 
3!
x22
Z 2, 2 
2!
x2   3 Z  d ;
2, 4

x2
0
3!
Vi1 ,i2 u  D1i1 D2i2 u h1 ,0  Z i1 ,i2  R, ik  0,3, k  1,2;

V u x   D D ux ,0  Z x  L G , i  0,3; 4
V u x   D D uh , x   Z x  L G , i  0,3 .
4,i2
i1 , 4
1
2
4
1
i1
1
i2
2
4
2
1
1
2
4,i2
1
i1 , 4
2
p
p
1
2
 4 x2   Z 3,0  x2 Z 3,1 
2
1

x22
Z 3, 2 
2!
2
x    Z  d ;
x3
 2 Z 3, 3   2
3, 4
3!
3!
0
x
3
 1 x1   Z 0,0  x1  h1 Z1,0 


x1  h1 3 Z
3!
x1
3, 0

3!

x1  h1 
3!
2!
2 ,1
x1
Z 3, 2  

9

10

x1   3 Z  d ;

3,1
4,1

3!
h1
3
x1  h1 2 Z
x1
 3 x1   Z 0, 2  x1  h1 Z1, 2 

2, 0
agreement type. Note that some boundary value problems in
non-classic treatments were considered in the author’s papers
[7-9].
3!
h1
x  h 3
 1 1 Z
2!

x1   3 Z  d;
4, 0
 2 x1   Z 0,1  x1  h1 Z1,1 

x1  h1 2 Z
x1  h1 2 Z
2!
2, 2

x1    Z  d ;
4, 2
h1
Fig.2. Geometric interpretation of contact-boundary value
conditions in non-classical treatment.

3
11
REFERENCES
3!
[1]
 4 x1   Z 0,3  x1  h1 Z1,3 

x  h 3
 1 1 Z
3!
x1  h1 2 Z
2!
2,3
x1   3 Z  d ;
3, 3  
4,3
x1
h1

[2]

12
3!
Note that the functions (5)-(12) possess an important
property, more exactly, the agreement conditions for all Z i1 ,i2 ,
possessing the above-mentioned properties are fulfilled for
them. Therefore, equalities (5)-(12) may be considered as a
general form of all the functions k x2 , k x1 , k  1,4
satisfying agreement conditions (3).
So, the classic form contact-boundary value problem (1),
(2) and in non-classic treatment (1), (4) (see. Fig.2) are
equivalent in the general case. However, the contact-boundary
value problem in non-classic treatment (1), (4) is more natural
by the statement than problem (1), (2). This is connected with
the fact that in the statement of problem (1), (4), the right sides
of boundary conditions don’t require additional conditions of
[3]
[4]
[5]
[6]
[7]
[8]
[9]
A.P.Soldatov, M.Kh.Shkanukov, “Boundary value problems with A.A.
Samarsky general non-local conditions for higher order equations,”
Dokl. AN SSSR, 1987, т.297, No 3. pp.547-552.
I.N.Vekua, New solution methods of elliptic equations M.:L.Gostekhizdat, 1948, 296 p.
E.A.Utkina, “To the general case of Goursat problem, ” Izv. Vuzov .
Mathematica, 2005, No 8 (519), pp.57-62.
E.A.Utkina, “Increase of the order of normal derivatives in boundary
conditions of Goursat problem,” Izv. Vuzov. Mathematica, 2007, No 4
(539), pp. 79-83.
E.A.Utkina, “On a boundary value problem with displacements in four dimensional space,” Izv. Vuzov. Mathematica, 2009, No4, pp.50-55.
S.S.Akhiyev, “Fundamental solutions of some local and non local
boundary value problems and their representation,” DAN SSSR, 1983,
т.271, No 2. pp.265-269.
I.G.Mamedov, “One Goursat problem in a Sobolev space,” Izv. Vuzov.
Mathematica, 2011, No 2, pp. 54-64.
I.G.Mamedov, “Fundamental solution of initial-boundary value problem
for a fourth order pseudoparabolic equation with nonsmooth
coefficients,” Vladikavkazskiy mathematichskiy jurnal, 2010, vol.12,
No1, pp. 17-32.
I.G.Mamedov, “Contact-boundary value problem for a hyperbolic
equation with multiple characteristics for a hyperbolic equation with
multiple characteristics Functional analysis and its applications,”
Proceedings of the International Conference devoted to the centenary of
academician Z.I.Khalilov. Baku, 2011, pp.230-232.