ABOUT THE ACCORDANCE BETWEEN THE VEKUA DIFFERENTIAL
EQUATION AND THE GENERALIZED LINEAR DIFFERENTIAL
EQUATION.
S.Brsakoska
University «Sts. Cyril and Methodius», Faculty for natural sciences and mathematics, Skopje,
Republic of Macedonia
In the paper two equations, the Vekua differential equation and the generalized linear differential equation, are
considered. The main result is the theorem with the condition which gives the accordance between this two
equations. Also the form of the solution is given for some equations.
Introduction
The equation
d̂W
 AW  BW  F
(1)
dz
where A  A  z  , B  B  z  and F  F  z  are given complex functions from a complex
variable z  D  is the well known Vekua equation [1] according to the unknown function
W  W  z   u  iv . The derivative on the left side of this equation has been introduced by
G.V.Kolosov in 1909 [2]. During his work on a problem from the theory of elasticity, he
introduced the expressions
ˆ
1  u v  v u   dW
(2)


i





2  x y  x y   dz
and
ˆ
1  u v  v u   dW
(3)


i





2  x y  x y   dz
known as operator derivatives of a complex function W  W  z   u  x, y   iv  x, y  from a
complex variable z  x  iy and z  x  iy corresponding. The operating rules for this
derivatives are completely given in the monograph of Г.Н.Положиǔ [3] (page18-31). In the

mentioned monograph are defined so cold operator integrals


f ( z )dz и
 f ( z )dz
from
z  x  iy and z  x  iy corresponding (page 32-41). As for the complex integration in the
same monograph is emphasized that it is assumed that all operator integrals can be solved in
the area D.
In the Vekua equation (1) the unknown function W  W ( z ) is under the sign of a complex
conjugation which is equivalent to the fact that B  B  z  is not identically equaled to zero in
D. That is why for (1) the quadratures that we have for the equations where the unknown
function W  W ( z ) is not under the sign of a complex conjugation, stop existing.
This equation is important not only for the fact that it came from a practical problem, but also
because depending on the coefficients A, B and F the equation (1) defines different classes of
generalized analytic functions. For example, for F  F  z   0 in D the equation (1) i.e.
d̂W
 AW  BW defines so cold generalized analytic functions from fourth class; and for
dz
d̂W
 BW defines so cold
dz
generalized analytic functions from third class or the (r+is)-analytic functions [3], [4].
Those are the cases when B  0 . But if we put B  0 , we get the following special cases. In
the case A  0 , B  0 and F  0 in the working area D 
the equation (1) takes the
ˆ
dW
 0 and this equation, in the class of the functions
following expression
dz
W  u  x, y   iv  x, y  whose real and imaginary parts have unbroken partial derivatives
A  0 and F  0 in D, the equation (1) i.e. the equation
ux , uy , vx and vy in D, is a complex writing of the Cauchy - Riemann conditions. In other
words it defines the analytic functions in the sense of the classic theory of the analytic
d̂W
 AW  F is the so cold areolar linear differential
functions. In the case B  0 in D i.e.
dz
equation [3] (page 39-40) and it can be solved with quadratures by the formula:




  A z  dz
 A z dz 
W e
z   F ze
dz  .
(4)




Here   ( z ) is an arbitrary analytic function in the role of an integral constant.
Main results
In the paper [5], the following lemma is proved.
Lemma : The equations
ˆ
dW
 f ( z,W )
(5)
dz
ˆ
dW
 g ( z,W )
and
(6)
dz
ˆ
df
where
 0 , have common solutions if and only if
dW
ˆ
ˆ
ˆ
ˆ
ˆ
df
df
dg
dg
dg
(7)

g

f
g.
dz dW
dz dW
dW
It is assumed that the operator derivatives in (7) exist and that they are continuous functions
in the working area D from the complex plane.
In this paper we are examining the accordance between the Vekua equation (1), on one side
and the generalized linear differential equation
d̂W
 W 
(8)
dz
on the other, where     z  and     z  are given complex functions from a complex
variable z  D  . The Vekua equation (1) is an equation of type (6) where
g ( z ,W )  AW  BW  F
and the generalized linear differential equation (8) is an equation of type (5), where
f ( z,W )  W  .
(9)
(10)
Here, the function f is an analytic function according to W , which means that
ˆ
df
 0 . That
dW
is the only condition to be accomplished, so we can use the mentioned lemma.
If we calculate all the derivatives in (7), we get that
ˆ
ˆ
df
dˆ
dˆ
df

W
,
 ,
dz dz
dz
dW
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
dg
dA
dB
dF
dg
dg

W
W
,
 A,
 B.
dz dz
dz
dz
dW
dW
And if we put them in (7) we get that
ˆ
ˆ
ˆ
dˆ
dˆ
dA
dB
dF
W
  AW  BW  F 
W
W
 A W    B AW  BW  F .
dz
dz
dz
dz
dz
Now we write the last equation in the following form
ˆ
ˆ
ˆ
 dˆ dA


 dˆ
dB
dF
W

 BB   W   B 
 B A 
F 
 A  BF  0.
 dz dz


 dz
dz
dz




This linear combination is true only if the following system of equation is satisfied
ˆ
 dˆ dA

 BB  0

dz
dz

ˆ

dB
 BA  0
.
(11)
 B 
dz

ˆ
 dˆ
dF
F 
 A  BF  0

dz
 dz
The second and the third equation of the system (11) gives us the relation between the
functions  and  of the equation (8) on one side and the functions A, B and F from the
equation (1) on the other side. From the second equation we get
ˆ
1 dB

 A.
(12)
B dz
The third equation is an areolar linear equation by  and it can be solved with quadratures
by the formula (4), i.e.



ˆ
ˆ    Adz 
 Adz 
 F dB
dF
 e
z  
 F A  BF 
dz  .
(13)
 e
 B dz


dz




Here   ( z ) is an arbitrary analytic function in the role of an integral constant.
Now if we find the areolar derivative of the function  given with (12), we get
ˆ dB
ˆ  dˆ A
dˆ
1  dˆ 2 B
dB
 2
B
.

dz B  dzdz
dz dz  dz
This derivative has the same left side as the left side of the first equation of the system (11)
and if we equal the right sides we get the following exspression
ˆ  dA
ˆ
ˆ dB
ˆ 
 dA
1  dˆ 2 B
dB
(14)


BB

B

 

2 
dz
dz
B
dzdz
dz
dz


 


which is the condition between the coefficients in the Vekua equation (1) in order to has
common solutions with the equation (8).




So, we have proved the following
Theorem 1:
The Vekua equation (1) and the generalized linear equation (8) have common solutions if and
only if the condition (14) is fulfilled and the relation between the coefficients of the two
equations are given with (12) and (13).
It is usual that instead of the Vekua equation (1), the homogeneous Vekua equation
d̂W
 AW  BW
(15)
dz
to be taken into consideration. Here, F  0 . It is interesting that the relation (12) and the
condition (14) are the same, but the relation (13) is much more concise and easier to
calculate, i.e.

   ze
 Adz
(16)
where   ( z ) is an arbitrary analytic function in the role of an integral constant.
So we can state the theorem 2 as a special case to the theorem 1.
Theorem 2:
The homogeneous Vekua equation (15) and the generalized linear equation (8) have common
solutions if and only if the condition (14) is fulfilled and the relation between the coefficients
of the two equations are given with (12) and (16).
Note: The generalized linear equation (8) is an equation which can be solved and its solution
is given with the formula




    z  dz
   z dz 
W e
  z     z  e
dz 
(17)




so if we put the expressions (12) and (13) we get the general solution of the Vekua equation
(1) which fulfils the condition (14) and if we put the expressions (12) and (16) we get the
general solution of the homogeneous Vekua equation (15) which fulfils the condition (14).
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