Final-boundary Value Problem in the NonClassical Treatment for a Sixth Order
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 a rectangular domain we consider a final - boundary value problem for the equation
equation, heat conductivity equation, telegraph equation, string
vibration equation and etc.).
V u x, y   D D ux, y   a x, y D D ux, y  
In the present paper we consider equation (1) in the general
sense when the coefficients ai, j x, y  are non - smooth
4
x
4, 2
2
y
3
x
3, 2
 a4,1 x, y Dx4 D y u x, y  
4
2
y
2
  a x, y D D ux, y   Z x, y 
i, j
i
x
j
y
4, 2
i  0 j 0
i  j 5
For this equation we consider a final-boundary value problem
with non-classical conditions not requiring agreement conditions.
Equivalence of these conditions with the classic boundary
condition is substantiated in the case if the solution of the stated
problem is sought in S.L.Sobolev anisotropic space. (Abstract)
Keywords— final boundary value problem, pseudoparabolic
equation, discontinuous coefficient equations. (key words)
I.
PROBLEM STATEMENT (HEADING 1)
Consider the equation
V u x, y   D
4, 2

4
x
D y2 u x, y   a3, 2 x, y D x3 D y2 u x, y  
4
functions satisfying only the following conditions:
ai, j x, y   L p G , i  0,3 j  0,1;
a4, j x, y   Lx,,yp G , j  0,1;
ai , 2 x, y   Lxp,,y G , i  0,3.
Under these conditions the solution u x, y  of equation (1))
will be sought in S.L.Sobolev space


Wp4, 2  G   ux, y  : Dxi Dyj ux, y   L p G , i  0,4, j  0,2 
where 1  p   . For equation (1) we can give the final
boundary conditions of classic form as (see. Fig.1.)
2
 a 4 ,1 x, y D x4 D y u x, y     ai , j x, y D xi D yj u x, y    
i 0 j 0
i  j 5
 Z 4 , 2 x, y   L p G  .
Here u x, y  is a desired function determined on G ;
ai , j x, y  are the given measurable functions on G  G1  G2 ,
where Gk  0, hk , k  1,2; Z 4, 2 x, y  is a given measurable
function on G ; Dtn   n / t n is a generalized differentiation
operator in S.L.Sobolev sense and Dt0 is an identity
transformation operator.
Equation (1) is a hyperbolic equation possessing two real
characteristics x  const, y  const , the first of which is fourfold, the second is two-fold. Therefore, in some sense, we can
consider equation (1) as a pseudoparabolic equation [1]. This
equation is a generalization of many model equations of some
processes (for example, Boussinesq-Liav equation, Manjeron’s
Fig.1. Geometrical interpretation of classic final-boundary conditions
 u x, y  x  h  1  y ; u x, y  y  h   1 x ;

 u x, y 
u x, y 
  2  y ;
  2 x ;


x
y y  h

xh


  2 u  x, y 
 3 u  x, y 
  3  y ;
  4  y ;
 x 2
x 3 x  h
xh



1

y

2
1
1
2
1  y   Z 0, 0   y  h2 Z 0,1    y   Z 0, 2  d ; 
5
h2
2
y

2  y   Z1, 0   y  h2 Z1,1    y   Z1, 2  d ; 
6
h2
1
where k  y , k  1,4 и  1 x , 2 x  are the given measurable
functions on G . Obviously, in the case of conditions (2), in
addition to the conditions
 k  y   W p2  G2 , k  1,4   1 x  W p4  G1 , 2 x   Wp4  G1  
y

3  y   Z 2, 0   y  h2 Z 2,1    y   Z 2, 2  d; 
7
h2
y

 4  y   Z 3, 0   y  h2 Z 3,1    y   Z 3, 2  d ; 
8
h2
the given functions should satisfy also the following agreement
conditions:

 1 h1   1 h2 ;
 
 1 h1    2 h2 ;

 1h1   3 h2 ;
 1h1    4 h2 ;
 2 h1   1h2 ;
 2 h1    2 h2 ;
 2h1   3 h2 ;
 2h1    4 h2 .
 1 x   Z 0, 0  x  h1 Z1, 0 


3

If the function u Wp4, 2  G  is a solution of classic type
final boundary value problem (1), (2), then it is also a solution
of problem (1), (4) for Z i, j , i  0,4, j  0,2 , determined by the
following equalities:
1
3!
x
Z 3, 0  
h1

x  h 
2

1
3!
x
Z 3,1  
h1
2
1
2!
Z 2, 0 
x    Z  d ;
3
x  h 
9

10
2
1
2!
Z 2 ,1 
x    Z  d ;
3
3!

4,0
3!
 2 x   Z 0,1  x  h1 Z 1,1 
Consider the following non - classical boundary conditions:
Vi , j u  Dxi D yj u h1 , h2   Z i , j  R, i  0,3, j  0,1;

4
j
V4, j u x   Dx D y u x, h2   Z 4, j x   L p G1 , j  0,1  4

i
2
Vi ,, 2 u  y   Dx D y u h1 , y   Z i , 2  y  L p G2 , i  0,3
x  h 
3

x  h 
4 ,1
Note has the functions (5)-(10) possess one important
property, more exactly, agreement conditions (3) for all Z i, j ,
having the above-mentioned properties are fulfilled for them
automatically. Therefore, we can consider equalities (5) - (10)
as
a
general
form
of
all
the
functions
k  y , k  1,4, 1x, 2 x , satisfying agreement conditions
(3).
Z 0, 0  1 h2    1 h1 ;  Z 0,1  1h2    2 h1 ;
Z1,0   2 h2    1 h1 ;  Z1,1   2 h2    2 h1 ;
Z 2, 0  3 h2    1h1 ;  Z 2,1   3 h2    2h1 ;
Z 3, 0   4 h2    1h1 ;  Z 3,1   4 h2    2h1 ; 
 IV 
( IV )
Z 4,0 x    1 x ;  Z 4,1 x    2 x ;  Z 0, 2  y   1 y ; 
Z1, 2  y    2 y   Z 2, 2  y    3 y   Z 3, 2  y    4 y  
It is easy to prove that the inverse one is also true. In other
words, if the function u Wp4, 2  G  is a solution of problem
(1), (4), then it is also a solution of problem (1), (2) for the
following functions:
Fig.2. Geometric interpretation of final - boundary conditions in non classical
treatment.
So, the classic form final boundary problem (1), (2) and in
non-classical treatment (1), (4) (see fig.2) are equivalent in the
general case. However, the final-boundary value problem in
non-classical 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
agreement type. Note that such boundary value problems in
non-classical treatment were considered in the author’s papers
[2-4].
REFERENCES
[1]
[2]
[3]
[4]
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