Dirichlet Problem in the Non-classical Treatment
for One Pseudoparabolic Equation of Fourth
Order
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 Dirichlet problem with non-classical
conditions not requiring agreement conditions is considered for a
fourth order pseudoparabolic equation with non-classical
coefficients. The equivalence of these conditions with the classic
boundary conditions is substantiated in the case if the solution of
the stated problem is sought in S.L.Sobolev isotropic space.
(Abstract)
Keywords—
Dirichlet
problem,
non-smooth
pseudoparabolic equations (key words)
I.
coefficients
PROBLEM STATEMENT (HEADING 1)
The first boundary value problem or the Dirichlet problem
(i.e. a problem in which a closed contour is an input medium)
known well for elliptic type differential equations is one of the
basic boundary value problems of mathematical physics [1-2].
From this point of view this paper is devoted to urgent
problems of mathematical physics.
Consider the equation

2
1
2
2
2 ,1
2
1
2
1, 2
1
2
2
 a2, 0 x D12u x   a0, 2 x D22u x    ai1 ,i2 x D1i1 D1i2 u x  
1
1
Fig. 1. Geometrical interpretation of Dirichlet classic conditions.
In the paper we consider equation (1) in the general case
when the coefficients ai1 ,i2 x  are non-smooth functions
V u x   D D ux   a x D D ux   a x D D ux  
2, 2
generalization of the Boussenesq-Liav equation [4] describing
longitudinal waves in a thin elastic bar with regard to lateral
inertia effects.


i1 0 i2 0
 Z 2, 2 x   L p G .
satisfying only the following conditions:
a2,i x  Lx,, px G , i2  0,1 ; ai , 2 x   Lxp ,,x G , i1  0,1 ;
1
2
1
2
2
1
ai ,i x   Lp G , i1  0,1, i2  0,1 .
1 2
Here ux   ux1 , x2  is a desired function determined on
G; ai ,i x  are the given measurable functions on G  G1  G2 ,
1 2
where G j  (0, h j ), j  1,2; Z 2, 2 ( x) is a given measurable
function on G; D j   / x j is a generalized differentiation
operator in S.L.Sobolev sense, j  1,2 .
Equation (1) is a hyperbolic equation that possesses two
real characteristics x1  const, x2  const , the first and the
second one of which are two-fold. The equation of type (1) in
the paper of A.P.Soldatov and M. Kh.Shkhanukov [3] are
called pseudoparabolic ones. Different special cases of
equation (1) arise by modeling various processes of applied
character (generalized equation of moisture transfer, telegraph
equation, string vibration equation, heat conductivity equation,
Aller’s equations and etc.). Furthermore, this equation is a
Therewith the important principal moment is that the
equation under consideration possesses non-smooth
coefficients that satisfy only some p -integrability and
boundedness conditions, i.e. the considered pseudoparabolic
differential operator V2, 2 has no traditional conjugation
operator. In other words, the Green function - the source
function for such an equation can’t be investigated by the
classic method of characteristics.
Under these conditions, we’ll look for the solution u x  of
equation (1) in S.L.Sobolev space


Wp2, 2  G   ux  : D1i D2i ux   Lp G ; i j  0,2, j  0,1 ,
1
2
where 1  p   . We’ll define the norm in space Wp2, 2  G  by
the equality:
2
2
ux  W 2, 2  G    D1i1 D2i2 ux 
p
i1 0 i2 0
L p G 
V u  u 0,0   Z  R;
0, 0
 0, 0
V1, 0u  D1u 0,0   Z1, 0  R;

V0,1u  D2u 0,0   Z 0,1  R;

2
V2, 0u x1   D1 u x1 ,0   Z 2, 0 x1   L p G1 ;
V u x   D 2u 0, x   Z x   L G ;
2
2
2
0, 2
2
p
2
 0, 2
 h1 
h1 
V0, 0 u  u h1 ,0   Z 0, 0  R;
 h1 
h1 
V0,1 u  D2u h1 ,0   Z 0,1  R;
 h2 
h2 
V0, 0 u  u 0, h2   Z 0, 0  R;
V h2 u  D u 0, h   Z h2   R;
1
2
1, 0
 1, 0
 V h2 u x   D 2u x , h   Z h2  x   L G ;
1
1
1
2
2, 0
1
p
1
 2, 0
 V0,h21 u x2   D22u h1 , x2   Z 0h, 12 x2   L p G2 ;

.
For equation (1) we can give the classic form Dirichlet
condition [5] as follows
u (0, x2 )  1 x2 , u ( x1 ,0)   1 ( x1 ),

u (h1 , x2 )   2 ( x2 ), u ( x1 , h2 )   2 ( x1 ),
(2)
where  j x2  and  j x1 , j  1,2 are the given measurable
functions on G . Obviously, in the case of conditions (2), in
addition to conditions


~
~
G    x  : D  x  L G , i  0,2, 1  p  , j  1,2 :
 j x2  Wp2  G2   ~x2  : D2i ~x2   L p G2 , i2  0,2 , 1  p  , j  1,2 :
2
 j x1  W p2 
1
1
i1
1
1
p
1


1
the given functions should also satisfy the following agreement
conditions:
1 0   1 0,  2 h2    2 h1  ,

1 h2    2 0,  2 0   1 h1  .
(3)
(4)


If the function u Wp2, 2  G  is a solution of the classic
form Dirichlet problem (1), (2), then it is a solution of problem
h 
(1), (4) for Z i ,i and Z i1 ,ij2 ,defined by the following equalities:
1 2
Z0,0  1 (0)   1 0; Z1,0   1 (0) ; Z0,1  10;
Z 2, 0 x1    1( x1 ) ; Z 0, 2 x2   1x2 ;
Z0h,01   2 (0)   1 (h1 ) ; Z0h,11   2 0;
Z0h,02    2 0  1 h2 ; Z1,h02    2 0;
Z 2h,02  x1    2x1 ; Z 0h,21  x2    2 x2  .
It is easy to prove that inverse one is also true In other
words, if the function u Wp2, 2  G  is a solution of problem
(1),(4) then it is also a solution of problem (1), (2) for the
following functions:
x2
1 x2   Z 0, 0  x2 Z 0,1   x2   Z 0, 2  d ;
(5)
0
x2
 2 x2   Z 0h, 0   x2 Z 0h,1    x2   Z 0h, 2   d ;
1
1
1
(6)
0
x1
 1 x1   Z 0, 0  x1Z1, 0   x1   Z 2, 0  d ;
Fig. 2. Geometrical interpretation of Dirichlet non-classical conditions.
(7)
0
x1
 2 x1   Z 0h, 0   x1Z1,h0    x1   Z 2h, 0   d ;
Consider the following non-classical boundary conditions:
2
2
2
(8)
0
Note that the functions (5), (8) possess an important
property, more exactly, agreement conditions (3) for all Z i ,i
h j 
1 2
and Z i1 ,i2 , possessing the above-mentioned properties are
fulfilled for them automatically. Therefore, we can consider
equalities (5)-(8) as a general form of all the functions
 j x2 ,  j x1 , j  1,2 , satisfying the agreement conditions
(3).
So, the classic form Dirichlet problems (1), (2) and of the
form (1), (4) are equivalent in the general case. However, the
Dirichlet problem (1), (4) in nonclassic treatment is more
natural by statement than Dirichlet problem (1), (2). This is
connected with the fact that in the statement of Dirichlet
problem (1), (4), the right sides of boundary conditions have
no additional conditions of agreement type.
It should be especially noted that in the papers [6-7] the
author suggested a method for investigating boundary value
problems in non-classical treatment for pseudoparabolic
equations with non-smooth coefficients of higher order.
Note that the Dirichlet non-classical problem in treatment
(1), (4) is investigated by means of integral representations of
special type for the functions ux  Wp2, 2  G  :
u x   u 0,0   x1  D1u 0,0  x2 D2u 0,0   x1 x2 D1 D2u 0,0  

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