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Характеристическая задача для системы гиперболических дифференциальных уравнений третьего порядка общего вида с некратными характеристиками.

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?????. ???. ???. ????. ??-??. ???. ???.-???. ?????. 2013. ? 1 (30). ?. 31?36
??? 517.956.3
?????????????????? ?????? ??? ???????
??????????????? ???????????????? ?????????
???????? ??????? ?????? ???? ? ??????????
????????????????
?. ?. ???????, ?. ?. ????????
????????? ??????????????? ??????????? ???????????,
??????, 443100, ??????, ??. ?????????????????, 244.
E-mails: andre01071948@yandex.ru, julia.yakovleva@mail.ru
??????????? ?????????? ?? ??????? ?????????? ?????????????????? ??????
??? ??????? ??????????????? ???????????????? ????????? ???????? ???????
?????? ???? ? ?????????? ????????????????. ??????? ????????? ?????? ????????? ? ????? ????. ???????? ?????? ??????? ?????? ????? ??? ????? ???????
??????????????? ???????????????? ????????? ???????? ???????.
???????? ?????: ??????? ??????????????? ???????????????? ????????? ??????
????, ????????? ??????????????, ?????????????????? ??????, ????????????
?? ???????.
???????????? ????????-??????? ????? ??? ??????????????? ????????? ?
?????? ? ????? ???????????? ??????????? ??????? ???? ??????? ? ??????
??????? ????????????? ????????? ?????? ?????? ???????. ??????? ????????? ?????? ???????????? ?????????? ?????????? ?? ?? ?. ??????? [1?4]. ??
?????????????????? ?????? ??? ?????? ? ????????? ???????????????? ???? ?
??????? ??????????? ? ?????????? ???????????????? ??????????? ????????????. ? ?????????? ?. ?. ??????? [5] ?????????? ???????, ????????????,
??? ??? ??????? ??????? ??????? ? ?????????? ???????????????? ??????
????? ???????? ???????????? ?? ???????.
? ????????? ?????? ?????????????? ? ??????????? ??????????????????
?????? ??? ??????? ??????????????? ????????? ???????? ??????? ?????? ???? ? ?????????? ????????????????. ??????????? ??????????? ??????? ??
????????????.
1. ??????????????? ????????. ?????????? ??????? ????????????????
????????? ? ??????? ??????????? ???????? ??????? ?????? ???? ? ?????
???????????? ??????????? x, y ? R ?? ?????????, ?? ?????????? ??????????? ??????? ?????? ????????
A? Uxxx + B ? Uxxy + C ? Uxyy + D ? Uyyy = 0,
(1)
??? U (x, y) = (u1 (x, y), u2 (x, y)) ? ??????? ????????? ??????-???????, A? ,
B ? , C ? , D ? ? ?????????? ?????????? ??????? ??????? ???????.
? ?????????????, ??? D ? ? ????????????? ???????, ??????? (1) ???????????? ? ?????????? ????:
AUxxx + BUxxy + CUxyy + Uyyy = 0.
(2)
????????? ??????????? ??????? (?.?.-?.?., ???.), ??????, ???. ?????????? ??????????
? ???????????. ???? ???????? ????????, ????????, ???. ?????????? ?????????? ?
???????????.
31
?. ?. ? ? ? ? ? ? ?, ?. ?. ? ? ? ? ? ? ? ?
????? ??????? A, B, C ??????? ????????????, ????? ??? ???????????
???????? ??? ????? ????????? ???:
22
b22 + b12 a11a?a
b12
a11 a12
12
,
A=
, B=
a21 a22
b22
b12 aa21
12
12
c11
c21 aa21
,
C=
22
c21 c11 ? c21 a11a?a
21
aij , bij , cij ? R,
aij , bij , cij 6= 0,
i, j = 1, 2.
??????? ??????????????
(a
?
a
)
b12 ?1?1 t2 b22 +
t1 b22 + ab12
1
22
12
T =
t1
t2
T ?1
a12 ?1 ?2
= 2
b12 dB (a2 ? a1 )t1 t2
t2 ?t2 b22 +
?t1
t1 b22 +
b12
a12 (a2
b12
a12 (a2
b12
a12 (a1
???
dB = b222 + b12 b22
a
a21
? a22 ? b212
,
a12
a12
11
? a22 ) b12 ?2?1
,
!
? a22 ) b12 ?2?1
,
? a22 ) b12 ?1?1
b12
?1 = b22 +
(a1 ? a22 ) b22 ? dB ,
a12
b12
?2 = b22 +
(a2 ? a22 ) b22 ? dB ,
a12
t1 , t2 ? R,
t1 , t2 6= 0,
???????????? ???????? ??????? A, B, C ? ???????????? ?????: T ?1 AT =
= ?A = diag(a1 , a2 ), T ?1 BT = ?B = diag(b1 , b2 ), T ?1 CT = ?C = diag(c1 , c2 ),
??? ???? a1 , a2 , b1 , b2 , c1 , c2 ? ????????? ??????????? ???????? ?????? A, B,
C ??????????????.
??????? (2) ???????????? ????????? ???????:
?A Vxxx + ?B Vxxy + ?C Vxyy + Vyyy = 0,
???
(3)
1
1
1
1
a1 vxxx
+ b1 vxxy
+ c1 vxyy
+ vyyy
= 0,
2
2
2
2
a2 vxxx + b2 vxxy + c2 vxyy + vyyy = 0.
?????? ?????????????????? ????????? ???? ??????? (3) ????? ??? ????????? ?????: ?1 , ?2 , ?3 ? �, �, �, ??????????????.
2. ?????????????????? ??????. ? ?????? [6] ?????????? ??????, ?????????????? ?????????????? ?? ??????? ???????????? ?????????? ?????? ?????
??? ???????????????? ????????? ???????? ??????? ? ?????????? ????????????????, ? ??????? ?????????????????? ??????, ?????????? ?? ???????, ???
???????????????? ????????? ???????? ??????? ?????? ????
avxxx + bvxxy + cvxyy + vyyy = 0
? ?????????? ???????????????? y ? ?1 x + C1 , y ? ?2 x + C2 , y ? ?3 x + C3 .
32
?????????????????? ?????? ??? ??????? ??????????????? ???????????????? ?????????. . .
????? x ? Ic ? [a, b], c = (a + b)/2. ??????? Ic ????? ??????????? ?????????: ?x ? Ic , 2c ? x ? Ic , ????? ??? ????? ??????? f (x) ???????????
?????????
1
f?c = (f (x) ? f (2c ? x)),
2
f (x) = f?c + f?c ,
1
f?c = (f (x) + f (2c ? x)).
2
??????? f?c , f?c ??? c = 0 ????? ?????????? f? , f? ??????????????.
??? ??????? (2) ?????????? ????????? ?????????????????? ??????.
?????? G. ????? ??????? U (x, y) ? C 3 (R � R) ??????? ????????? (2),
??????????????? ????????
hl1 , U (x, ?1 x)i = ?1 (x),
hl2 , U (x, �x)i = ?2 (x),
hl1 , U (x, ?3 x)i = ? 1 (x),
hl2 , U (x, �x)i = ? 2 (x),
hl1 , U (x, ?2 x)i = ? 1 (x),
hl2 , U (x, �x)i = ? 2 (x),
(4)
??? ?i (x), ? i (y), ? i (x) ? C 3 (R), i = 1, 2, ha, bi ? ????????? ????????????;
l1 =
a12 (b22 + ab12
(a2 ? a22 ))?1 a12 ?1 ?2
12
,
,
?
b12 dB (a2 ? a1 )t1
b212 dB (a2 ? a1 )t1
l2 = ?
a12 (b22 + ab12
(a1 ? a22 ))?1 a12 ?1 ?2
12
.
,
b12 dB (a2 ? a1 )t2
b212 dB (a2 ? a1 )t2
??????? 1. ???? ??i (x) = ?i? (?i x) + ??i ((1 ? ?i )x), i = 1, 2, ???
?1 = (?3 ? ?2 )/(?1 ? ?2 ),
?2 = (�? �)/(�? �),
?? ?????? G ????????? ?? ???????.
? ? ? ? ? ? ? ? ? ? ? ? ? ?. ??????? (2) ??????????????? T ???????????? ? ??????? (3). ??????? ??????? ????????? ???? ??????? [6] ????? ?????????
???:
y ? ?2 x
1
1 y ? ?1 x
v (x, y) = ?
+?
? ?1 (0)+
?1 ? ?2
?2 ? ?1
2
1 1 y ? ?3 x
y
?
?
x
(y ? ?1 x)(?2 ? ?3 )
2
1
1
+
? ??
? ??
?
+
2 ? ?1 ? ?3
?1 ? ?2
(?1 ? ?3 )(?1 ? ?2 )
1 1 y ? ?3 x
1 y ? ?1 x
1 (y ? ?2 x)(?1 ? ?3 )
? ??
? ??
?
?
+
2 ? ?2 ? ?3
?1 ? ?2
(?1 ? ?2 )(?2 ? ?3 )
1 1 (y ? ?3 x)(?1 ? ?2 )
1 y ? ?2 x
1 y ? ?1 x
?
? ??
,
?
? ??
2 ? (?1 ? ?3 )(?2 ? ?3 )
?1 ? ?3
?2 ? ?3
1
2
2 y ? �x
2 y ? �x
v (x, y) = ?
+?
? ?2 (0)+
�? ��? �2
1 2 y ? �x
y
?
�
x
(y ? �x)(�? �)
2
2
2
+
? ??
? ??
?
+
2 ? �? ��? �(�? �)(�? �)
1
1
33
?. ?. ? ? ? ? ? ? ?, ?. ?. ? ? ? ? ? ? ? ?
1 2 y ? �x
2 y ? �x
2 (y ? �x)(�? �)
? ??
? ??
?
?
+
2 ? �? ��? �(�? �)(�? �)
1 2 (y ? �x)(�? �)
2 y ? �x
2 y ? �x
?
? ??
.
?
? ??
2 ? (�? �)(�? �)
�? ��? �???? ??????? ?????? G ? ???? ??????? ?????????? ????????? U = T V :
b12
b
+
b22 + ab12
(a
?
a
)
b
(a
?
a
)
b12
22
1
22
12
22
a12 2
12
t1 v 1 +
t2 v 2 ,
u1 (x, y) =
?1
?1
u2 (x, y) = t1 v 1 + t2 v 2 .
???????????????? ???????????? ??????????, ??? ?????????? ??????-??????? U (x, y) = (u1 (x, y), u2 (x, y)) ????????????? ?????? (4). ??????????????? ?????????????? ?? ??????? ???????
A? Uxxx + B ? Uxxy + C ? Uxyy + D ? Uyyy = 0,
? ??????? ??????? A, D ? ???????, B ? ????????? ??????? ??????? ???????, C = ?Q.
? ????????? ??????????? ?????????? x, y ? R ?????????? ???? ???????
??????????????? ????????? ???????? ???????:
(5)
Uxxy ? QUxyy = 0,
??? U (x, y) = (u1 (x, y), u2 (x, y)) ? ??????-???????, Q ? ?????????? ???????
????
p 1?p
Q=
, p ? R.
1+p
?p
?
?
?1
? p?1
2
2
2(p +1) ?
?????????? ??????? T = ? 1
?????, ??? T ?1 QT = ?Q , ???
?
? p+1
2
2(p2 +1)
?
?
p?1
p+1
?
? ?2
1
0
2
?1
q
q
?, ?Q =
.
T =?
2
p2 +1
0 ?1
? p 2+1
2
????????? ??????? (5) ????? ??? ??????? ? ??? ????????? ??????????????.
?????? G1. ????? ??????? U (x, y) ? C 3 (R � R) ??????? ????????? (5),
??????????????? ????????? ????????: U (x, 0) = S?(x), U (0, y) = S?(y),
hl1 , U (x, ?x)i = ? 1 (x), hl2 , U (x, x)i = ? 2 (x), ??? ?(x) = (?1 (x), ?2 (x)), ?(y) =
= (? 1 (y), ? 2 (y)), ?i (x), ??i (y), ? i (x) ? ?
C 3(R), i = 1,
???p2, ha, bi ? ?????????
p
+ 1)/2 ;
?????????; l1 = (p + 1)/ 2, ?(p ? 1)/ 2 , l2 =? ? (p2 + 1)/2, (p2 ?
?1
? p?1
2
2(p2 +1) ?
??????? S ? ?????????? ??????? ???? S = ?
.
p+1
?1
? 2
2(p +1)
2
??????? 2. ???? ??i (x) = ?i? (x) ? ??i (x), i = 1, 2, ?? ?????? G1 ?????????
?? ???????.
34
?????????????????? ?????? ??? ??????? ??????????????? ???????????????? ?????????. . .
? ? ? ? ? ? ? ? ? ? ? ? ? ?. ??????? (5) ???????????? ???????
Vxxy ? ?Q Vxyy = 0
???
(6)
1
1
= 0,
? vxyy
vxxy
2
2
vxxx + vxyy = 0.
??????-??????? V (x, y) = (v 1 (x, y), v 2 (x, y)) ? ??????? ??????? (6), ???
1
1 1
v 1 (x, y) = ?1 (x) + ? 1 (y) ? ?1 (0) ?
?? (x) + ?1? (y) ? ?1? (x + y) ?
2
2
1 1
1 1
?
?? (x) + ??1 (y) ? ??1 (x + y) +
?? (x) + ??1 (y) ? ??1 (x + y) ,
2
2
1 2
1 2
2
2
2
v (x, y) = ? (x) + ? (y) ? ? (0) ?
?? (x) + ?2? (y) ? ?2? (?x + y) ?
2
2
1 2
1 2
2
2
?
?? (x) + ?? (y) ? ?? (?x + y) +
?? (x) + ??2 (y) ? ??2 (?x + y) .
2
2
??????? ?????????????????? ?????? G1 ??? ??????? (5) ???? ? ???? ??????? ?????????? ????????? U = T V :
p?1
1
v2 ,
u1 (x, y) = ? v 1 + p
2
2(p2 + 1)
1
p+1
u2 (x, y) = ? v 1 + p
v2.
2
2(p2 + 1)
?????????? ??????-??????? U (x, y) = (u1 (x, y), u2 (x, y)), ?????????? ? ????? ????, ???????? ???????? ?????????????????? ?????? G1.
????????????????? ??????
1. ?. ?. ???????, ?? ??????? ? ?????????? ?????????????????? ?????? ??? ??????????????? ?????? ??????? ???????? // ????. ?? ????, 1975. ?. 223, ? 6. ?. 1289?1292.
[A. V. Bitsadze, ?On a question on the formulation of the characteristic problem for secondorder hyperbolic systems? // Dokl. Akad. Nauk SSSR, 1975. Vol. 223, no. 6. Pp. 1289?1292].
2. ?. ?. ????????, ???????? ??????? ?????? ?? ???????????? ?????????? ?????????????????? ????? ??? ??????????????? ????????? ???????? ???????? // ?????. ???????,
2003. ?. 74, ? 4. ?. 517?528; ????. ???.: O. M. Dzhokhadze, ?Influence of Lower Terms on
the Well-Posedness of Characteristics Problems for Third-Order Hyperbolic Equations? //
Math. Notes, 2003. Vol. 74, no. 4. Pp. 491?501.
3. ?. ?. ????????, ?. ?. ????????, ???????? ?????? ? ????? ??????????? ????????
?????????? ?. ?. ??? ???????????????????? ????????? ???????? ???????? // ????. ??
????, 1987. ?. 297, ? 3. ?. 547?552; ????. ???.: A. P. Soldatov, M. Kh. Shkhanukov,
?Boundary value problems with A. A. Samarskiy?s general nonlocal condition for higherorder pseudoparabolic equations? // Soviet Math. Dokl., 1988. Vol. 36, no. 3. Pp. 507?511.
4. ?. ?. ?????????????, ?? ???????????? ????? ?????????????????? ?????? ??? ????????????? ?????? ??????? ???????? // ??????. ?????????, 1989. ?. 25, ? 1. ?. 154?162;
????. ???.: S. S. Kharibegashvili, ?Solvability of a characteristic problem for degenerate
second-order hyperbolic systems? // Differ. Equations, 1989. Vol. 25, no. 1. Pp. 123?131.
5. ?. ?. ???????, ????????? ?????? ????????? ? ??????? ???????????. ?.: ?????, 1981.
448 ?. [A. V. Bitsadze, Some classes of partial differential equations. Moscow: Nauka, 1981.
448 pp.]
35
?. ?. ? ? ? ? ? ? ?, ?. ?. ? ? ? ? ? ? ? ?
6. ?. ?. ????????, ????? ?????????????????? ?????? ??? ????????????????? ???????????????? ????????? ???????? ??????? ?????? ???? ? ?????????? ????????????????? // ?????. ???. ???. ????. ??-??. ???. ???.-???. ?????, 2012. ? 3(28). ?. 180?183.
[J. O. Yakovleva, ?One characteristic problem for the general hyperbolic differential equation
of the third order with nonmultiple characteristics? // Vestn. Samar. Gos. Tekhn. Univ. Ser.
Fiz.-Mat. Nauki, 2012. no. 3(28). Pp. 180?183].
????????? ? ???????? 16/XI/2012;
? ????????????? ???????? ? 27/I/2013.
MSC: 35L25
THE CHARACTERISTIC PROBLEM FOR THE SYSTEM OF THE
GENERAL HYPERBOLIC DIFFERENTIAL EQUATIONS OF THE
THIRD ORDER WITH NONMULTIPLE CHARACTERISTICS
A. A. Andreev, J. O. Yakovleva
Samara State Technical University,
244, Molodogvardeyskaya st., Samara, 443100, Russia.
E-mails: andre01071948@yandex.ru, julia.yakovleva@mail.ru
We consider the well-posed characteristic problem for the system of the general hyperbolic differential equations of the third order with nonmultiple characteristics. The
solution of this problem is constructed in an explicit form. The example of the analogue
of Goursat problem for a particular system of the hyperbolic differential equations of
the third order is given.
Key words: system of the general hyperbolic differential equations, nonmultiple characteristics, characteristic problem, Hadamard?s well-posedness.
Original article submitted 16/XI/2012;
revision submitted 27/I/2013.
Aleksandr A. Andreev (Ph. D. (Phys. & Math.)), Associate Professor, Dept. of Applied
Mathematics & Computer Science. Julia O. Yakovleva, Postgraduate Student, Dept. of
Applied Mathematics & Computer Science.
1/--страниц
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