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Second order Fuchs type systems of partial differential equations with a finite number of
singular lines.
1
Tungatarov A.B., Altynbek S.A., 2 Abdymanapov S.A.3
Abstract.The solution for the equation of Fuchs type in angular domain is searched for in
an obvious way. Under certain boundary conditions the uniqueness of the solution is shown.
Key words: Second order PDE of Fuchs type, angular domain, Dirichlet boundary value
problem, integral representations.
Mathematics subject classifications: 30G 20; 35 J 70
1.Introduction.
The systems of first order partial differential equations are well studied,as they have early
arisen in problems of mathematical physics [1 3] .The systems of partial differential equations of
2-nd order part have not been studied regularly yet.In the work a variety of continuous solutions of
partial differential second order systems are constructed and Dirichlet boundary problem are solved.
2.General solution.
Let 0 < 0 2 and G = {z = re i : 0 r < ,0 0 } .
Consider the equation
4 z
2
2V
z z
4 z z
2V
z z
4z 2
2V
V b( )V =
zz
f ( )r
m
( y k
j
x)
j
(1)
j =1
in G , where a( ), b( ), f ( ) C[0, 0 ] and 0 < < 1,
> 0,
, , , are real
m
parameters, = j ,0 < j < 1, j = tan j , 0 < j < 0 , ( j = 1, m)
j =1
Here
1
ei 1
= (
i ) =
(
),
2 r r
z 2 x y
1
ei 1
= (
i ) =
(
),
z 2 x y
2 r r
2
e 2i 2 2i 2i 2
1 1 2
=
(
)
4 r 2 r 2 r r r r r 2 2
z z
1
Al-Farabi Kazakh National University, Al-Farabi ave., 71, Almaty, 050038, Kazakhstan.
E mail : tun mat@list .ru
2
Al-Farabi Kazakh National University, Al-Farabi ave., 71, Almaty, 050038,
Kazakhstan. E mail : Serik [email protected]
3
Kazakh University of economy, finance and internotional trade, Zhubanov str.,7, Astana,
010008,Kazakhstan.
2
1 2 1 1 2
= ( 2
),
r r r 2 2
z z 4 r
(2)
2
e 2i 2 2i 2i 2
1 1 2
=
(
)
zz
4 r 2 r 2 r r r r r 2 2
Particular kinds of equation (1) are student in ([1],
.
[2]). Concerning the parameters
the solution of equation (1) are searched for in the S.L.Sobolev class(see[3])
Wp2 (G), 1 < p < 2
(3)
From the Sobolev imbedding theorem it follows Wp2 (G) C(G),1 < p < 2 .
Using formulas (2), equation (1) in polar coordinates is written in the form
2V
V
2
V
( )r
( )2i
( )2ir
( )r
2
r
r
r
2
( )
2V
V b( )V =
2
f ( )r
m
(cos k
j
j
(4)
sin )
j =1
For solving equation (4) method of separation of variables is applied.
Let
V (r , ) = r ( ),
where ( ) is a new unknown function from C 2 [0, 0 ] , satisfying the equation
( ) 2i( )(1 ) (( ) 2( )) =
(5)
= f1 ( ) b( ) ,
where
f1 ( ) =
f ( )
m
(cos k
j
j
sin )
j =1
Substituting
( ) = ea P( ),
where
(6)
a=
i( )( 1)
, q =
q
and P( ) is the new unknown function from C 2 [0, 1 ] , the last equation becomes
P P = b1 ( ) P f 2 ( ),
(7)
where
qa 2 2i( )(1 )a ( ) 2( )
=
,
q
b1 ( ) =
b( ) exp (2a )
,
q
f 2 ( ) =
f1 ( ) exp (a )
q
Solving equation (7) by applying the method of variation of constatant, we have
0
0
P( ) = b( , ) P( )d f ( , )d c1I ,0 ( ),
(8)
where
b1 ( )
sh ( ( )),
b ( )
b( , ) = 1
sin ( ( )),
b1 ( )( ),
if > 0,
if < 0,
if = 0,
f 2 ( )
sh ( ( )),
if > 0,
f ( )
f ( , ) = 2
sin ( ( )), if < 0,
f 2 ( )( ),
if = 0,
e ,
e ,
if > 0,
if > 0,
I ,0 ( ) = cos( ), if < 0,, J ,0 ( ) = sin ( ), if < 0,
,
if = 0,
1,
if = 0,
As 0 < < 1 the integral in f ( , ) is convergent.
For the construction of solutions of equation (8) the following functions and operators are
used:
I ,k ( ) = b( , ) I ,k 1 ( )d ,
J ,k ( ) = b( , ) J ,k 1 ( )d , (k = 1, ),
0
0
( Bf )( ) = b( , ) f ( )d ,
0
f 3 ( ) = f ( , )d ,
0
( B f )( ) = ( B( B
k
k 1
f )( ))( ),
k = (2, ), ( B1 f )( ) = ( Bf )( )
For these functions it is easy check that
( B(cI ,k ( )))( ) = cI ,k 1 ( ), ( B(cJ ,k ( )))( ) = cJ ,k 1 ( )
(9)
sh ( ) | b | | f |
1
0
0
(
)k
, if > 0,
k!
|q|
| b |0 k | f |0
| ( B k f )( ) | (
)
,
if < 0,
k
!
|
q
|
| b |0 2 k | f |0
(
)
,
if = 0,
(2 k )!
|q|
sh( 1 ) | b |0 k | f |0
)
, if > 0,
(
(
k
1)!
|
q
|
| f |0
| b |0 k 1
k
| ( B f 2 )( ) | (
)
, if < 0,
|
b
|
(
k
1)!
|
q
|
0
| b |0k | f |0 2 k 2
,
if = 0,
| q |k 1 (2 k 2)!
sh (
(
| I ,k ( ) |
1 ) | b |0 k e 1
)
, if > 0,
k!
|q|
| b |0 k 1
(
)
,
if < 0,
| q | k!
| b |0k 2 k 1
,
if = 0,
k
| q | (2 k 1)!
sh( 1 ) | b |0 k 1
)
, if > 0,
(
k!
|q|
| b |0 k 1
| J ,k ( ) | (
)
,
if < 0,
| q | k!
| b |0k 2 k
,
if = 0,
| q |k (2k )!
k = (0, ),
(10)
where c is any complex number, | f |0 =|| f ||C[ o, ] .
1
Using the specified notations, equation (8) is written in the form
P( ) = ( BP )( ) f 3 ( ) c1 I ,0 ( ) c2 J ,0 ( )
(11)
If we apply of operator B to both sides of equation (11) we have in view of (9)
( BP )( ) = ( B 2 P)( ) ( Bf 3 )( ) c1I ,1 ( ) c2 J ,1 ( )
(12)
From (11), (12) it follows
P( ) = ( B 2 P)( ) ( Bf 3 )( ) f 3 ( ) c1I ,1 ( ) c2 J ,1 ( )
(13)
c1 I ,0 ( ) c2 J ,0 ( )
If we again apply the operator B to both sides of equation (13) we have in view of (9)
( BP )( ) = B3 P)( ) ( B2 f3 )( ) ( Bf 3 )( ) c1I ,2 ( ) c2 J ,2 ( )
(14)
c1I ,1 ( ) c2 J ,1 ( )
From (11) and (14) it follows
P( ) = ( B 3 P)( ) ( B 2 f 3 )( ) ( Bf 3 )( ) f 3 ( ) c1 ( I ,2 ( ) I ,0 ( )) .
c2 ( J ,2 ( ) J ,0 ( )) c1I ,1 ( ) c2 J ,1 ( )
Continuing this process 2n and 2n-1 time, recpectively we receive the following
representations for solutions of equation (8)
2 n 1
n 1
n 1
k =0
k =0
k =0
P( ) = ( B 2 n P)( ) ( B k f 2 )( ) c1 I ,2 k ( ) c2 J ,2 k ( )
n 1
n 1
k =0
k =1
c1 I ,2 k 1 ( ) c2 J ,2 k 1
(15)
and
2n
n
n
k =0
k =0
k =0
P( ) = ( B 2 n 1 P)( ) ( B k f 3 )( ) c1 I ,2 k ( ) c2 J ,2 k ( )
n
n
k =1
k =1
c1 I ,2 k 1 ( ) c2 J ,2 k 1 ( )
If we pass to the limit as n in the representations (15), by virtue of (9) we receive
P( ) = ( BF )( ) c1P ,2 ( ) c2Q ,2 ( ) c1P ,1 ( ) c2Q ,1 ( ),
(16)
where
k =0
k =0
k =0
( BF )( ) = ( B k f 3 )( ), P ,2 ( ) = I ,2 k ( ), P ,1 ( ) = I ,2 k 1 ( ),
k =0
k =0
Q ,2 ( ) = J ,2 k ( ), Q ,1 ( ) = J ,2 k 1 ( )
Using the inequalities (10), we receive the following estimates
| f |
| b | sh( 1 )
0 (exp ( 0
) 1),
|q|
| b |0
| f |0
| b |0
| ( BF )( ) |
(exp (
) 1),
|
b
|
|
q
|
0
| f |0
| b |0
(exp (
) 1)
b0
|q|
e
| P ,2 ( ) |
e
| P ,1 ( ) |
1
1
| b |0 sh ( 0 )
),
|q|
| b |0 0
ch(
),
| q |
| b |0
|q|
sh ( 0
),
| b |0
|q|
ch(
| b |0 sh ( 0 )
),
|q|
| b |0 0
sh (
),
| q |
| b |0
|q|
sh( 0
),
| b |0
|q|
ch(
e0 ch( | b |0 sh ( 1 ) ),
|q|
| b |0 0
| Q ,2 ( ) |
ch(
),
| q |
| b |0
ch( 0
),
|q|
if > 0,
if < 0,
if = 0,
if > 0,
if < 0,
if = 0,
if > 0,
if < 0,
if = 0,
if > 0,
if < 0,
if = 0,
e0 sh ( | b |0 sh ( 1 ) ), if > 0,
|q|
| b |0 0
| Q ,1 ( ) |
sh (
),
if < 0,
|
q
|
| b |0
ch( 0
),
if = 0
|q|
By means of these estimates it is easy to show, that the function P( ) , given by formula (16), is a
solution of equation (7) from the class C 2 [0, 1 ] .
From (16), (6) and (5) we find
V (r , ) = r e [( BF )( ) c1 ( P ,2 ( ) c2Q ,2 ( ) c1P ,1 ( )
(17)
c2Q ,1 ( )]
Thus, the following result holds.
Theorem 1. When , > 0 the equation (1) is solvable in the class (3). The
general solution of equation (1) from the class (3) is given by formula (17).
3. Dirichlet problem.
Let us consider the Dirichlet problem for equation (1).
Problem D . It is required to find the solution of equation (1) from the class (3), satisfying
the conditions
| V (r , ) |= O(r ), r ,
(18)
V (r ,0) = b1r , V (r , 0 ) = b2 r ,
(19)
where b1 ,b2 are given complex numbers, > 0
We us formula (17) for solving problem D . Then automatically (18) takes place. From the
forms of the functions ( BF )( ), P ,1 ( ), P ,2 ( ), Q ,1 ( ), Q ,2 ( ) it follows
( BF )(0) = 0,
P ,1 (0) = 0,
Q ,1 (0) = 0
,
1, if 0
1, if 0
P ,2 (0) =
, Q ,2 =
,
0, if = 0
0, if < 0
Having substituted (17) in the first condition of (19), in view of (20) we receive
V (r,0) = (c1 ,2 c2 ,1 )r ,
(20)
where
1, if 0
1, if 0
, ,1 =
0, if = 0
0, if < 0
Therefore various algebraic systems of equations follow from the boundary conditions (19)
depending on the sign of rather that on c1 and c2 :
,2 =
1) when = 0
where 1 = b2 e
0
c2 = b1 ,
P ,2 (0 )c1 P ,1 (0 )c1 = 1 ,
( BF )( 0 ) b1Q ,2 ( 0 ) b1Q ,1 ( 0 ) ,
2) when < 0
c1 = b1 ,
Q ,2 (0 )c2 Q ,1 (0 )c2 = 2 ,
where 2 = b2 e
0
( BF )( 0 ) b1 P ,2 ( 0 ) b1 P ,1 (1 ) ,
3) when > 0
c1 c2 = b1 ,
(Q ,2 (0 ) P ,2 (0 ))c2 (Q ,2 (0 ) P ,1 (0 ))c2 = 2 ,
Solving the specified systems, we receive
1) When = 0, | P ,2 (0 ) || P ,1 (0 ) | ,
c2 = b1 ,
c1 =
1 P ,2 (0 ) 1 P ,1 (0 )
(21)
| P ,2 (0 ) |2 | P ,1 (0 ) |2
2) When < 0, | Q ,2 ( 0 ) || Q ,2 ( 0 ) | ,
c1 = b1 ,
c2 =
2 Q ,2 (0 ) 2Q ,1 (0 )
(22)
| Q ,2 (0 ) |2 | Q ,1 (0 ) |2
3) When > 0, | Q ,2 (0 ) P ,2 (0 |2 | Q ,1 (0 ) P ,1 (0 ) |2 ,
c2 =
2 (Q ,2 (0 ) P ,2 (0 )) 2 (Q ,1 (0 ) P ,1 (0 ))
| Q ,2 (0 ) P ,2 (0 ) |2 | Q ,1 (0 ) P ,1 (0 ) |2
,
(23)
c1 = b1 c2 .
Hence, the following result holds
Theorem 2. Promlem D1 has a unique solution if one of the conditions
1) = 0, | P ,2 (0 ) |2 | P ,1 (0 ) |2
2) < 0, | Q ,2 (0 ) |2 | Q ,1 (0 ) |2
3) > 0, | Q ,2 (0 ) | | P ,2 (0 ) |2 | Q ,1 (0 ) | | P ,1 (0 ) |2
holds. The unique solution then is given by formulas (17), (21)-(23).
References
[1] Abdymanapov S.A., Tungatarov A.B. Some classes of elliptic systems in the plane
with singular coefficient. Almaty: ’Galym’ 2005.
[2] Bitsadze A.B. Some classes of partial differential equations. Moscow: Nauka,1981
(Russion). Gordon a.Breach, New York, 1988.
[3] Vekua I.N. Generalized analitic function. Moscow: Fizmatgiz, 1959. Pergamon,
Oxford, 1962.
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