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# $ ! " #
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+ '( ( $ % * , ! #
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5 !* ) $% " ' $ ' --#
" % . %"* ) ? = {(t, x) | ? ? <
t < ?, 0 < x < 1} ' % 04
U ? U tt ? U ttxx + (sgn t)U xx = 0,
? ? ; $( ,"( *"( <* ) ? -%& U (t, x) % &!%& % U (t, x) ? C ? ? C 1 ? ? {x = 0} ? {x = 1} ? C 2 (? + ? ? ? ),
U (t, x) ? 0,
U (t, 0) = U (t, 1),
?
?? t ?,
0
U (t, x) dt = ?(x),
0
084
(t, x) ? (? + ? ? ? ),
U x (t, 0) = U x (t, 1),
U (t, x) dt = ?(x),
??
0:4
0 x 1,
094
04
?(x) ?(x) ; $( -% ?? = {(t, x) | ? ? < t < 0, 0 < x < 1}
?+ = {(t, x) | 0 < t < ?, 0 < x < 1}
<"( ( ' % 04 ) ? )% " U (t, x) =
+ $ % 04 %
T (t) и X(x)
T (t) и X(x) ? T (t) и X (x) = ?(sgn t)T (t) и X (x).
7" $ ?(sgn t)T (t) и X(x)=
T (t)
X (x)
X (x)
T (t)
?
и
=
,
?(sgn t)T (t) ?(sgn t)T (t) X(x)
X(x)
" ( '
X (x)
= ?? 2 ,
X(x)
T (t)
X (x)
T (t)
?
и
= ?? 2 ,
?(sgn t)T (t) ?(sgn t)T (t) X(x)
?? 2 ; $ 0 < ? >& % ( %* 094 %
X (x) + ? 2 X(x) = 0, 0 < x < 1,
0:4
X(0) = X(1), X (0) = X (1),
0::4
T (t) ? ? 2 T (t) = 0, 0 < t < ?,
0:84
2
T (t) + ? T (t) = 0, ?? < t < 0,
0:94
2
? 2 = 1 +? ? 2 ?" $ 0:4 0::4 '
X 0 (x) = 1,
X n (x) = {cos ? n x, sin ? n x},
? n = 2?n,
n = 1, 2, . . . .
0:4
+ )! ' --"( %* 0:84 0:94 & a n e ? n t + b n e ?? n t ,
T n (t) =
t > 0,
c n cos ? n t + d n sin ? n t, t < 0,
2
a n , b n, c n , d n ; $"( ( ? n = 1 +? n? 2 n
/"% ' U n (t, x) = T n (t) и X n (x) ,( %
( a n , b n , c n , d n ) )( ( " %
0:4
" %& 0:4 #
0:4
.$ 0:4 % %* 0:4 % c n = a n + b n d n = a n ? b n + -%
0:4 & c n ch ? n t + d n sh ? n t, t > 0,
T n (t) =
0:
4
T n (0 + 0) = T n (0 ? 0),
T n (0 + 0) = T n (0 ? 0).
c n cos ? n t + d n sin ? n t,
t < 0.
+" , $ 0:4204 ) ? ' U (t, x)
+ % -%* 0:4 @ ' % A%" $ (
?
U (t, x) =
? 0 (t) +
[? n (t) cos ? n x + u n (t) sin ? n x] ,
2
n=1
u n (t) = 2
0
1
? n (t) = 2
0
1
U (t, x) sin ? n x dx, n = 1, 2, . . . ,
0:4
U (t, x) cos ? n x dx, n = 0, 1, 2, . . . .
0:64
:6
!"# !$#
/, -% 0:4 0:64 % & % 0:84 0:94 %#
&! %& 0:4 --% t 0:4 0:64 $ %( % 04 %
un (t) = 2
un (t)
Utt sin ? n x dx = 2
0
?n (t)
1
=2
?n (t) = 2
1
Utt sin ? n x dx = 2
0
1
0
0
Utt cos ? n x dx = 2
0
1
=2
Utt cos ? n x dx = 2
0
t > 0,
084
(Uttxx + Uxx ) sin ? n x dx,
t < 0,
08:4
(Uttxx ? Uxx ) cos ? n x dx,
t > 0,
0884
(Uttxx + Uxx ) cos ? n x dx,
t < 0.
0894
1
1
(Uttxx ? Uxx ) sin ? n x dx,
1
0
1
0
.% $ 08420894 % %* 094 %
%&! %
un (t) ? ? n2 u n (t) = 0, t > 0,
084
un (t) + ? n2 u n (t) = 0, t < 0,
084
?n (t) ? ? n2 ? n (t) = 0, t > 0,
084
2
? n (t) + ? n ? n (t) = 0, t < 0,
08
4
2
? n2 = 1 +? n? 2 n
--"( % 084 084 084 08
4 ? = ? n & #
% 0:84 0:94 % %* 0:4 $ 0:4 0:64 %
u n (0 + 0) = 2
U (0 + 0, x) sin ? n x dx = 2
0
? n (0 + 0) = 2
1
1
0
U (0 + 0, x) cos ? n x dx = 2
1
0
1
0
U (0 ? 0, x) sin ? n x dx = u n (0 ? 0),
084
U (0 ? 0, x) cos ? n x dx = ? n (0 ? 0).
0864
--% -% 0:4 0:64 $ t % %* 0:4 un (0 + 0) = 2
?n (0
1
0
U t (0 + 0, x) sin ? n x dx = 2
1
+ 0) = 2
0
U t (0 + 0, x) cos ? n x dx = 2
1
0
1
0
U t (0 ? 0, x) sin ? n x dx = un (0 ? 0),
084
U t (0 ? 0, x) cos ? n x dx = ?n (0 ? 0).
08:4
084 0864 084 08:4 & % 0:4 + $ 084208:4
-% 0:
4 u n (t) =
? n (t) =
c n ch ? n t + d n sh ? n t,
t > 0,
c n cos ? n t + d n sin ? n t,
t < 0,
c? n ch ? n t + d?n sh ? n t,
c? n cos ? n t + d?n sin ? n t,
t > 0,
t < 0.
0884
0894
, ( c n , d n c? n , d?n "$% "( % 04
-% 0:4 0:64
0
?
u n (t) dt = 2
1 ?
0
0
U (t, x) dt sin ? n x dx = 2
:
1
0
?(x) sin ? n x dx = ? n ,
084
?
? n (t) dt = 2
0
0
0
0
??
u n (t) dt = 2
0
??
? n (t) dt = 2
1 ?
U (t, x) dt cos ? n x dx = 2
??
U (t, x) dt sin ? n x dx = 2
1 0
0
??
U (t, x) dt cos ? n x dx = 2
/ t > 0 $ 0884 084 %
?n =
=
?
0
0
1 0
0
1
u n (t) dt =
?(x) cos ? n x dx = ?? n ,
084
?(x) sin ? n x dx = ? n ,
084
?(x) cos ? n x dx = ?? n .
08
4
1
0
1
0
?
0
(c n ch ? n t + d n sh ? n t) dt =
?
dn
cn
dn
cn
sh ? n t +
ch ? n t =
sh ? n ? +
(ch ? n ? ? 1).
?n
?n
?n
?n
0
084
/ t < 0 $ 0884 084 %
?n =
=
0
??
u n (t) dt =
0
??
(c n cos ? n t + d n sin ? n t) dt =
0
dn
cn
dn
cn
sin ? n t ?
cos ? n t =
sin ? n ? +
(cos ? n ? ? 1).
?n
?n
?n
?n
??
08:64
.$ 084 08:64 % % %* $( @--#
c n d n =
c n sh ? n ? + d n (ch ? n ? ? 1) = ? n ? n ,
08:4
c n sin ? n ? + d n (cos ? n ? ? 1) = ? n ? n .
? $ $' " )! %" &)(
0 < ?=
? n (?, ?) = sh ? n ? и cos ? n ? ? ch ? n ? и sin ? n ? + sin ? n ? ? sh ? n ? = 0.
08::4
B $ 0894 084 08
4 % % %* $#
( @-- c? n d?n =
0<?
c? n sh ? n ? + d?n (ch ? n ? ? 1) = ? n ?? n ,
c? n sin ? n ? + d?n (cos ? n ? ? 1) = ? n ?? n .
08:84
. @ $ $' ( % 08::4 /%" ( % 08::4 + $ 08:4 08:84 cn =
?n
? n (cos ? n ??1)?? n (ch ? n ??1) ,
? n (?, ?)
dn =
?n
?? n sin ? n ?+? n sh ? n ? ,
? n (?, ?)
c? n =
?n
?? n (cos ? n ??1)??? n (ch ? n ??1) ,
? n (?, ?)
d?n =
?n
??? n sin ? n ?+?? n sh ? n ? .
? n (?, ?)
/ @ @--( -%( 08:4 08:84 %
?n
? n ch ? n t и (cos ? n ? ? 1) ? sh ? n t и sin ? n ? +
u n (t) =
? n (?, ?)
+ ? n ch ? n t и (1 ? ch ? n ?) ? sh ? n t и sh ? n ? , t > 0,
?n
? n cos ? n t и (cos ? n ? ? 1) ? sin ? n t и sin ? n ? +
u n (t) =
? n (?, ?)
::
08:94
+ ? n cos ? n t и (1 ? ch ? n ?) ? sin ? n t и sh ? n ? ,
08:4
t < 0,
?n
?? n ch ? n t и (cos ? n ? ? 1) ? sh ? n t и sin ? n ? +
? n (?, ?)
+ ?? n ch ? n t и (1 ? ch ? n ?) ? sh ? n t и sh ? n ? , t > 0,
? n (t) =
08:4
?n
?? n cos ? n t и (cos ? n ? ? 1) ? sin ? n t и sin ? n ? +
? n (?, ?)
+ ?? n cos ? n t и (1 ? ch ? n ?) ? sin ? n t и sh ? n ? , t < 0.
? n (t) =
08:4
+" , ?(x) ? 0 ?(x) ? 0 + ? n = ?? n = 0 ? n = ?? n = 0 $ -%
0:4 0:64 08:94208:4 % 1
0
U (t, x) sin ? n x dx = 0,
n = 1, 2, . . . ,
1
0
U (t, x) cos ? n x dx = 0,
n = 0, 1, 2, . . . .
>& % ( ( )( -%* {1, cos ? n x, sin ? nx} L 2[0, 1] $#
& U (t, x) ? 0 x ? [0, 1] t ? [??, ?]
% &'
(
? %* %' % 08::4 /%" ? n (?, ?) = 0 ( ? n = m $ 0:4204 ?(x) ? 0 ?(x) ? 0 " '
d?m sh ? m t и ch ? m ? ? sh ? m ? и (ch ? n t ? 1) и X m (x),
U m (t, x) =
? d? sin ? t и ch ? ? ? sh ? ? и (cos ? t ? 1) и X (x),
m
m
m
m
m
m
?
?
t > 0,
t < 0,
094
d?m ; $"( ( ( %& X m(x) ? {1, cos ? n x, sin ? n x}
? n (?, ?) = 0 @ %
sh ? n ? и cos ? n ? + (1 ? ch ? n ?) и sin ? n ? = sh ? n ?,
2
? n = 1 +? n? 2 ? n = 2?n 7" 0 < ? n < 1 ? n ? 1 n ? ?
n
/ % $( %&! cos(? n ? ? ? n ) = ? n = arccos sh ? n ?
sh 2 ? n ? + (1 ? ch ? n ?) 2
,
% '*=
sh ? n ?
sh 2 ? n ? + (1 ? ch ? n ?) 2
2? n 2?k
4 ? k = 2?k
k ? N C :4 ? k =
+
k ? N N ; , %"( ?n
?n
?n
/ '* ? k = 2?k
% %& ? n (? k , ?) = 0 <% #
?n
2?k
n
+
% %& ? n(? k , ?) = 0
" '* ? k = 2?
?n
?n
% $ ? ( % 08::4 ( $(& % (
/, %!% C 0 > 0 )"' n #
inf ? n (?, ?)
C 0 .
n
:8
09:4
+ ? n ? 1 n ? ? ? = lim ? n = lim arccos n??
= arccos n??
sh ?
sh 2 ? + (1 ? ch ?) 2
sh ? n ?
sh 2 ? n ? + (1 ? ch ? n ?) 2
=
? 2
sh ?
= arccos
th ?.
2
ch ? ? 1
/, %!% C 0 > 0 )"' n
09:4 + % 0 < ? n < 1 ? n ? 1 n ? ? $ 08::4
sh 2 ? + (1 ? ch ?) 2 C0
и cos(? ? ?)
1 +
.
sh ?
sh ?
D ( %&! %
1+
+ ? =
1+
(1 ? ch ?) 2 > 1.
и
cos(?
?
?)
sh 2 ?
(1 ? ch ?) 2
> 1
sh 2 ?
cos(? ? ?)
> ? ?1
0984
' /"% 0 < ? ?1 < 1 &)( $ 0 < ? % &!
% 0984 ( 09:4
% ( $* 0 < ? & -%( 08:94208:4 /@% (#
%* 08::4 09:4 % ( '* 0:4 08:94208:4 ' $
0:4204 ) ? , " ?
U (t, x) =
? 0 (t) +
[? n (t) cos ? n x + u n (t) sin ? n x] .
2
n=1
0994
/, ( % " -%* ?(x) ?(x) % U (t, x)
0994 % % 0:4
/ )"' n ( u n (t)
C 1 ? n + ? n , ? n (t)
C 1 ?? n + ?? n ,
094
u n (t)
C 2 ? n + ? n ,
u n (t)
C 3 ? n + ? n ,
? n (t)
C 2 ?? n + ?? n ,
? n (t)
C 3 ?? n + ?? n ,
094
094
0 < C i = const, i = 1, 3
*" 0 < ? n < 1 ? n ? 1 n ? ? -% 08:942
08:4 % 09:4 *
?
1 ? ?
?
+
?
и
e
и
(ch
2?
+
ch
?)
,
2
?
n
n
?
u n (t)
= C 0
1 ?
?
3 ?n + ?n и 1 + e? ,
?
C0
?
1 ? ?
?
+
??
и
e
и
(ch
2?
+
ch
?)
,
2
??
?
n
n
? n (t)
= C 0
1 ?
?
3 ?? n + ?? n и 1 + e ? ,
?
C0
:9
t > 0,
t < 0,
t > 0,
t < 0.
>& % 094 C 1 = C1 max 3; 2 и e ? ; ch 2? + ch ?; 1 + e ? 0
--% (, 08:94208:4 %
?
1 ? ?
?
?
+
?
и
ch
?
, t > 0,
и
e
и
1
+
e
2
?
?
n
n
u n (t)
= C 0
1 ?
?
3 ? n + ? n и 1 + e ? , t < 0,
?
C0
?
1 ? ?
?
?
+
??
и
ch
?
, t > 0,
и
e
и
1
+
e
2
??
?
n
n
? (t)
= C 0
n
1 ?
?
3 ?? n + ?? n и 1 + e ? , t < 0.
?
C0
% 094 C 2 = C1 max 3; 2 и e ? ; 1 + e ? и ch ? 0
>&
--% (, 08:94208:4 $ %
?
1 ? ?
?
+
?
и
e
и
(ch
2?
+
ch
?)
, t > 0,
2
?
n
n
? C 0
u n (t)
=
1 ?
?
?
3 ? n + ? n и 1 + e ? , t < 0,
C0
?
1 ? ?
?
+
??
и
e
и
(ch
2?
+
ch
?)
, t > 0,
2
??
?
n
n
? n (t)
= C 0
1 ?
?
?
3 ?? n + ?? n и 1 + e ? , t < 0.
C0
>& % 094 C 3 = C1 max 3; 2 и e ? ; ch 2? + ch ?; 1 + e ? 0
+ ,& $ ?(x) ? C 3 [0; 1] ?(x) ? C 3 [0; 1] [0; 1]
-% ? ? & %#(( $( ?(0) = ?(1) =
?(0) = ?(1) ? (0) = ? (1) = ? (0) = ? (1) ? (0) = ? (1) = ? (0) = ? (1) ? (0) = ? (1) =
? (0) = ? (1) ( 4
1
?
2
IV
1
pn
2
?n = ?
p
4
(x)
dx,
?
n
?
n4
0
n=1
4
1
qn
?
n4
4
1
pn
?n = ?
?
n4
4
1
qn
?? n = ?
?
n4
?? n = ?
?
n=1
?
n=1
?
n=1
q n2 4
p 2n 4
q 2n 4
1
2
? IV (x) dx,
0
1
2
? IV (x) dx,
0
1
2
? IV (x) dx.
0
? !"& @ % %)" 0994 ( $ $( @ ) ?
/%" ? n(?, ?) = 0 ( 0 < ? n = k 1, . . . , k s 1 k 1 < k 2 < . . . < k s s ; - %" + $' %* 08:4 08:84
) )( ( " % "
?n = 2
?? n = 2
1
0
1
0
?(x) и sin ? n x dx = 0,
n = k 1, . . . , k s,
09
4
?(x) и cos ? n x dx = 0,
n = k 1, . . . , k s,
094
:
?n = 2
0
?? n = 2
1
1
0
?(x) и sin ? n x dx = 0,
n = k 1, . . . , k s,
0964
?(x) и cos ? n x dx = 0,
n = k 1, . . . , k s.
094
5 @ % ' $ 0:4204 %( ?
U (t, x) =
?
+?
k
1 ?1
n=1
+
k
1 ?1
? 0 (t) ?
+
2
k
2 ?1
n=k 1 +1
+
n=1
+иии +
k
2 ?1
+иии +
n=k 1 +1
?
?
?
? ? n (t) cos ? n x+
n=k s +1
?
? u n (t) sin ? n x +
n=k s +1
m
Cm Um (t, x),
09:4
* % m $ k 1 , . . . , k s Cm ; $"( (
-% Um (t, x) & $ -%( 094
+ )$ $
+ 9 ?(x) ? C 3 [0; 1] ?(x) ? C 3 [0; 1] [0; 1] ?(0) = ?(1) = ?(0) = ?(1) ? (0) = ? (1) =
? (0) = ? (1) ? (0) = ? (1) = ? (0) = ? (1) ? (0) = ? (1) = ? (0) = ? (1) (1.2)(1.5) ? (3.22) (4.2) (4.4) ? n(?, ?) = 0 0 < ? n = k 1 , . . . , k s
(4.2) (1.2)(1.5) ? (4.8)(4.11) ! (4.12)
9
5
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g.NGIR2 EAFKA<B=ARSGKU;PhiNGIRHJN
T. K. Yuldashev
On a mixed type fourth-order di?erential equation
Keywords: boundary value problem, mixed type di?erential equation, fourth-order equation, integral conditions,
unique solvability.
MSC: 35A02, 35M10, 35S05
We consider questions of solvability and constructing the solution to a nonlocal mixed boundary value problem
for a homogeneous mixed type fourth-order di?erential equation. We use the spectral method based on separation
of variables. A criterion for unique solvability of the problem is obtained. We also study questions of existence of
solutions in the case where uniqueness of the solution does not hold.
REFERENCES
1. Baev A.D., Shabrov S.A., Mon Meach. Uniqueness of the solution of the mathematical model of forced
string oscillation with singularities, Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., 2014, no. 1, pp. 50?55
(in Russian).
2. Turbin M.V. Investigation of initial boundary value problem for the Herschel?Bulkley mathematical ?uid
model, Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., 2013, no. 2, pp. 246?257 (in Russian).
3. Benney D.J., Luke J.C. Interactions of permanent waves of ?nite amplitude, Journal of Mathematical
Physics, 1964, vol. 43, pp. 309?313.
4. Akhtyamov A.M., Ayupova A.R. On the resolution of the problem on detection of defects in the form
of a small cavity in a rod, Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 2010, vol. 12, no. 3,
pp. 37?42 (in Russian).
5. Shabrov S.A. Estimates of the in?uence function for a fourth-order mathematical model, Vestn. Voronezh.
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Received 03.05.2016
Yuldashev Tursun Kamaldinovich, Candidate of Physics and Mathematics, Associate Professor, Higher Mathematics Department, Reshetnev Siberian State Aerospace University, pr. im. Gazety Krasnoyarskii Rabochii,
31, Krasnoyarsk, 660014, Russia.
E-mail: tursun.k.yuldashev@gmail.com
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