Nihan Aliyev, Yelena Mustafayeva – Some Special Conditions of ...

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Dec 1, 2013 (3 years and 10 months ago)

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SOME SPECIAL CONDITIONS OF INVESTIGATION mixed problem

Aliyev, NA, Mustafa
y
eva

Y.Y.

The faculty of Applied Mathematics and Cybernetics

Baku State University

nihan@aliev.info
,
helenmust@rambler.ru

We consider the mixed problem for the vibrating string with special linear
boundary conditions. These boundary conditions are such that, after the separation
of variables, or after the Laplace transform of the resulting spectral boundary
problem
either

has

no eigenvalues
or the whole complex plane is the spectrum of
this problem.

As is well known from
the study of the Cauchy problem for hyperbolic
equations

the
initial data are arbitrary (independent) functions.

From our
investigation we obtained some con
trary facts.

C
onsider the mixed problem:







,
0
,
1
,
0
,
,
,
2
2
2
2







t
x
x
t
x
u
t
t
x
u

(1)









,
0
;
2
,
1
,
0
,
,
1
,
,
0
1
1
0
0
1
0












t
i
x
t
x
u
t
u
x
t
x
u
t
u
x
i
i
x
i
i





(2)







,
1
,
0
;
1
,
0
,
,
0






x
k
x
t
t
x
u
k
t
k
k


(3)

If we apply the method of separation of variables or the Laplace transform
we arrive at the following auxiliary homogeneous boundary value problem with a
parameter:







,
1
,
0
,
0
2





x
x
X
x
X


(4)









,
2
,
1
,
0
1
1
0
0
1
0
1
0







i
X
X
X
X
i
i
i
i





(5)

where



C

is a complex parameter
.

It
has been

proved that if conditions (5) take the form









,
0
1
,
0
1
X
X
X
X






or









0
1
,
0
1
X
X
X
X





,

(6
)


t
hen

holds true the following

Theorem
1
.

If the entire complex plane is the spectrum for the auxiliary
problem (4), (
6
) with a complex parameter, then for the existence of the solution of
the mixed problem (1)
-
(3)) the initial data are not independent (arbitra
rily given)
functions. Both


x
0


and


x
1


should be given on one of the intervals
]
2
1
,
0
[

or
]
1
,
2
1
[

and be continued as odd
or even
functions.

If the boundary conditions (5) take the
form
















,
0
'
1
,
0
1
X
X
X
X





(
7
)

or
,
correspondingly
, the boundary conditions (2) take the form

x
t
u
x
t
u
t
u
t
u







)
,
0
(
)
,
1
(
),
,
1
(
)
,
0
(


, (
8
)

where


is an arbitrary constant,
1



, then holds the following

Theorem
2
.

If the auxiliary (spectral) problem (4), (5) does not have the
spectrum, i.e., hold true boundary condition (7) (or (8)), for the existence of
solutions of the mixed problem (1)
-
(3) the initial data are not arbitrary functions on
(0,1), i.e. they are depe
ndent, so that it is sufficient to give arbitrarily


x
0


and


x
1


on the interval

]
2
1
,
0
[

and define them on

]
1
,
2
1
[

by means of the relationships







,
1
1
2
1
1
1
2
1
0
1
1
t
t
t






























],
1
,
2
1
[

t
,







,
1
1
2
1
1
1
2
1
0
1
0
t
t
t































]
1
,
2
1
[

t
.

The resulting pathology
on

the study of solutions of a mixed problem for a
hyperbolic equation leads to the fact that the mixed problem may exist only in the
case when the initial data are not arbitrary,
and there is some connection

between

them
. This connection is established using the method of characteristics.