[ОДУ] Методичка ч.2 - А. М. Денисов, А. В. Разгулин
DESCRIPTION
Дифференциальные уравненияTRANSCRIPT
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. ..
.. , ..
2
2008 .
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2 " ". 4 , 3, .
c . .., 2008 .
c .., .., 2008 .
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3
1 51.1 . . . . 5
1.1.1 51.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 . . . . . . . . . . . . . . . . 81.2.1 . . . . 81.2.2 . . . . . . 9
2 122.1 . . . . . . . . 12
2.1.1 . . . . . . . . . . . . . . . . . 122.1.2 . . . . . . . . . . . 132.1.3 . . . . . . . . . . . . . . . . . . . . . . 142.1.4 -
. . . . . . . . . . . . . . 152.1.5 -
. . . . . . . . . . . . . . . . . . . . . . . . 162.1.6 . . . 17
2.2 . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 . . . . . . . . . . . . . . . . . . . 212.3.2 . . . . . . . . . . . . . 222.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.4 . . . . . . . . . . . . . . . . 232.3.5 . . . . . . . . . . . . . . . . . . . . 242.3.6 . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.1 . . . . . . . . . . . . . 262.4.2 : 1, 2 R, 1 6= 2, 1 2 > 0. . . . . . . . . . . . . . . . . . . . . 272.4.3 : 1 = 2 6= 0, dimker(A 1E) = 2. . . . . . . . . 282.4.4 : 1 = 2 6= 0, dimker(A 1E) = 1. . . . . . . . . . 282.4.5 : 1, 2 R, 2 < 0 < 1. . . . . . . . . . . . . . . . . . . . . . . . 292.4.6 : 1,2 = i C, 6= 0, 6= 0. . . . . . . . . . . . . . . . . . . . 292.4.7 : 1,2 = i C, 6= 0. . . . . . . . . . . . . . . . . . . . . . . . 292.4.8 A: detA = 0. . . . . . . . . . . . . . . . 302.4.9 . . . . . . . . . . . . 30
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4
3 323.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.2 . . . . . . . . . . . . . . . 333.1.3 . . . . . . . . . . . . . . . . . . . 343.1.4 . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 . . . . . . . . . . . . 353.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.3 . . . . . . . . . . . . 373.2.4 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.5 . . . . . 41
3.3 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 464.1 . . . . . . . . . . . . . . . . . . 46
4.1.1 . . . . . . . . . . . . . . . . . . . . . . 464.1.2 . . . . . . . . . . . . . 464.1.3 . . . . . . . . . . . . . . . . 474.1.4 . . . . . . . . . . . . . . . . . . . . . . 47
4.2 . . . . . . . . . . . . . . 494.2.1 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.2 -
. . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.3 . 514.2.4 . . . . . . . . . . . . . . 534.2.5 . . . . . . . . . . . . . . . . . 54
5 565.1 . . . . . . . . . . . . . . . . . 56
5.1.1 . . . . . . . . . . . . . . . . 585.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 . . . . . . . 60
5.3.1 , . . . 605.3.2 , . . . . . . . . . 62
5.4 . . . . . . . . . . . . . . . . . 645.5 -
- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.6.1 . . . . . . . . . . . . . . . . . . . . . . . 685.6.2 . . . . . . . . . . . 69
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1.1. 5
1
1.1
1.1.1
, -
y(t) = f(t, y(t)), t [t0 T, t0 + T ], (1.1)y(t0) = y0. (1.2)
f(t, y) y0, (1.1)-(1.2). - , - f(t, y) y0?, - . , .
1.1.1. fi(t, y), i = 1, 2,
Q = {(t, y) R2 : |t t0| T, a y b}
f1(t, y) Q y, .. L > 0,
|f1(t, y) f1(t, y)| L|y y|, (t, y), (t, y) Q., yi(t), i = 1, 2, [t0 T, t0 + T ] {
y1(t) = f1(t, y1(t)),y1(t0) = y01,
{y2(t) = f2(t, y2(t)),y2(t0) = y02,
maxt[t0T,t0+T ]
|y1(t) y2(t)| (|y01 y02|+ T max
(t,y)Q|f1(t, y) f2(t, y)|
)exp{LT}. (1.3)
. , yi(t) C1[t0T, t0+T ], a yi(t) b, i = 1, 2,
y1(t) = y01 +
tt0
f1(, y1())d, t [t0 T, t0 + T ],
y2(t) = y02 +
tt0
f2(, y2())d, t [t0 T, t0 + T ].
-
6 1.
,
|y1(t) y2(t)| |y01 y02|+ t
t0
(f1(, y1()) f2(, y2())
)d
, t [t0 T, t0 + T ].
tt0
f1(, y2())d,
|y1(t) y2(t)| |y01 y02|+ t
t0
f1(, y1()) f1(, y2())d ++
tt0
f1(, y2()) f2(, y2())d , t [t0 T, t0 + T ]. (1.4) , f1(t, y) , t
t0
(f1(, y2()) f2(, y2())
)d
T max(t,y)Q |f1(t, y) f2(t, y)|, t [t0 T, t0 + T ], (1.4)
|y1(t) y2(t)| (|y01 y02|+ T max
(t,y)Q|f1(t, y) f2(t, y)|
)+
+ L
t
t0
|y1()) y2())|d , t [t0 T, t0 + T ].
|y1(t) y2(t)| -, |y1(t) y2(t)|
(|y01 y02|+ T max(t,y)Q
|f1(t, y) f2(t, y)|)exp{L|t t0|}, t [t0 T, t0 + T ],
(1.3). 1.1.1 .
1.1.2
. .
Q+ = {(t, y) : t0 t t0 + T, a y b}. , - .
. f(t, y) Q+ Q+ fy(t, y). (t, y1), (t, y2) Q+
f(t, y1) f(t, y2) =1
0
fy(t, y2 + (y1 y2))d (y1 y2). (1.5)
, .
-
1.2. 7
1.1.2. ( .) fi(t, y), i = 1, 2
Q+ f1(t, y) Q+ f1y
(t, y). ,
yi(t), i = 1, 2, [t0, t0 + T ] {y1(t) = f1(t, y1(t)),y1(t0) = y01,
{y2(t) = f2(t, y2(t)),y2(t0) = y02,
f1(t, y) f2(t, y), (t, y) Q+, y01 y02,
y1(t) y2(t), t [t0, t0 + T ].
. y1(t) y2(t) [t0, t0+T ] , yi(t) C1[t0, t0+T ], a yi(t) b,
y1(t) y2(t) = f1(t, y1(t)) f2(t, y2(t)), t [t0, t0 + T ]. (1.6)
, (1.5),
f1(t, y1(t)) f2(t, y2(t)) = f1(t, y1(t)) f1(t, y2(t)) + f1(t, y2(t)) f2(t, y2(t)) =
=
10
f1y
(t, y2(t) + (y1(t) y2(t))
)d(y1(t) y2(t)
)+ f1(t, y2(t)) f2(t, y2(t)).
v(t) = y1(t) y2(t),
p(t) =
10
f1y
(t, y2(t) + (y1(t) y2(t))
)d,
h(t) = f1(t, y2(t)) f2(t, y2(t)).
f1(t, y1(t)) f2(t, y2(t)) = p(t)v(t) + h(t) (1.6)
v(t) = p(t)v(t) + h(t), t [t0, t0 + T ].
v(t0) = y01 y02,
v(t) = (y01 y02) exp{ tt0
p()d}+
tt0
exp{ t
p()d}h()d, t [t0, t0 + T ].
, (y01 y02) 0 h(t) 0 t [t0, t0 + T ], v(t) = y1(t) y2(t) 0, t [t0, t0 + T ] 1.1.2 .
-
8 1.
1.2
, , - , .
Q = {(t, y, ) : |t t0| T, a y b 1 2}. f(t, y, ) Q, y0() [1, 2].
y(t) = f(t, y(t), ), t [t0 T, t0 + T ], (1.7)y(t0) = y0(). (1.8)
(1.7)-(1.8), , , t, . (1.7),(1.8) y(t, ). y(t, ) ?
1.2.1
1.2.1. f(t, y, ) Q Q - y, -
|f(t, y1, ) f(t, y2, )| L|y1 y2|, (t, y1, ), (t, y2, ) Q, y0() [1, 2].
, y(t, ) (1.7)-(1.8) [t0 T, t0 + T ] [1, 2], y(t, ) t [t0 T, t0 + T ], [1, 2].. y(t, ) t [t0 T, t0 + T ], [1, 2] a y(t, ) b t [t0 T, t0 + T ], [1, 2]. 0 0 + [1, 2]. y(t, 0) y(t, 0+), .
y1(t) = y(t, 0), y2(t) = y(t, 0 +),
f1(t, y) = f(t, y, 0), f2(t, y) = f(t, y, 0 +),
y01 = y0(0), y02 = y0(0 +).
y1(t) y2(t) 1.1.1 . ,
maxt[t0T,t0+T ]
|y(t, 0) y(t, 0 +)| = maxt[t0T,t0+T ]
|y1(t) y2(t)|
(|y01 y02|+ T max
(t,y)Q|f1(t, y) f2(t, y)|
)exp{LT} =
=(|y0(0) y0(0 +)|+ T max
(t,y)Q|f(t, y, 0) f(t, y, 0 +)|
)exp{LT}, (1.9)
-
1.2. 9
Q = {(t, y) R2 : |t t0| T, a y b}., (1.9) y(t, ) 0 .
. , (), t [t0 T, t0 + T ]
|y(t, 0 +) y(t, 0)| (1.10) || () .
[1, 2] y0() , 1() ,
|y0(0 +) y0(0)| exp{LT}2
(1.11)
|| 1() . Q f(t, y, )
, 2() , t [t0 T, t0 + T ] y [a, b]
|f(t, y, 0 +) f(t, y, 0)| exp{LT}2T
(1.12)
|| 2(). (1.9), (1.11) (1.12) , || () = min{1(), 2()}
(1.10), y(t, ) . 1.2.1 .
, 1.2.1 [t0 T, t0+T ] [1, 2] . , y(t, ) (t, ) [t0 T, t0 + T ] [1, 2].
1.2.2
, , (1.7)-(1.8)y(t, ) .
1.2.2. f(t, y, ) Q Q fy(t, y, ), f(t, y, ), y0() [1, 2].
, y(t, ) (1.7)-(1.8) [t0 T, t0 + T ] [1, 2], y(t, ) t [t0T, t0+T ], [1, 2] .
. y(t, ) t [t0 T, t0 + T ], [1, 2] a y(t, ) b t [t0 T, t0 + T ], [1, 2]. + [1, 2]. y(t, ) y(t, +).
v(t, ,) =y(t, +) y(t, )
.
y(t, + ), y(t, ) (1.7) - ,
v(t, ,) = ()1[f(t, y(t, +), +) f(t, y(t, ), )], t [t0T, t0+T ]. (1.13)
-
10 1.
,
()1[f(t, y(t, +), +) f(t, y(t, ), )] =
()1[f(t, y(t, +), +) f(t, y(t, ), +)]+()1[f(t, y(t, ), +) f(t, y(t, ), )].
(1.5),
()1[f(t, y(t, +), +) f(t, y(t, ), +)] =
=
10
fy(t, y(t, ) + (y(t, +) y(t, )), +)d
(y(t, +) y(t, ))()1.
p(t, ,) =
10
fy(t, y(t, ) + (y(t, +) y(t, )), +)d,
q(t, ,) = ()1[f(t, y(t, ), +) f(t, y(t, ), )]. ,
()1[f(t, y(t, +), +) f(t, y(t, ), )] =
= p(t, ,)v(t, ,) + q(t, ,).
(1.13), , v(t, , + )
v(t, ,) = p(t, ,)v(t, ,) + q(t, ,), t [t0 T, t0 + T ]. (1.14)
v(t, , +) ,
v(t0, ,) = ()1[y0(+) y0()]. (1.15)
(1.14)-(1.15),
v(t, ,) = ()1[y0(+) y0()] exp
tt0
p(, ,)d
+
+
tt0
q(, ,) exp
t
p(, ,)d
t [t0 T, t0 + T ]. (1.16) y
(t, ) , -
v(t, ,) 0. , (1.16) 0.
y0() ,
lim0
()1[y0(+) y0()] = dy0d
().
-
1.2. 11
p(t, ,) 0. fy(t, y, ) p(t, ,) ,
lim0
p(t, ,) =f
y(t, y(t, ), ).
f(t, y, )
lim0
q(t, ,) =f
(t, y(t, ), ).
(1.16) 0,
y
(t, ) = lim
0v(t, ,) =
dy0d
() exp
t
t0
fy(, y(, ))d
+
+
tt0
f(, y(, ), ) exp
t
p(, ,)d
t [t0 T, t0 + T ]. (1.17) .
(t, ) = y(t, ), (t, ) (t, )
t. (1.17) , (t, )
(t, ) = fy(t, y(t, ), )(t, ) + f(t, y(t, ), ), t [t0 T, t0 + T ],(t0, ) = y
0().
-
12 2.
2
2.1 . -
t = t0 t [t0; +). t0 = 0.
2.1.1.
y = ay, y(0) = y0,
y0 t [0; +), a R . y(t; y0) = y0 exp{at}. a < 0
|y(t; y0) y(t; y0)| = |y0 y0| exp{at} |y0 y0| 0 y0 y0 0 t 0, |y(t; y0) y(t; y0)| 0 t +.
a = 0 |y(t; y0) y(t; y0)| = |y0 y0| 0
y0 y0 0 t 0, |y(t; y0) y(t; y0)|9 0 t +. a > 0
|y(t; y0) y(t; y0)| = |y0 y0| exp{at}9 0 y0 y0 0 t 0, .
, T > 0 [0, T ]:
maxt[0,T ]
|y(t; y0) y(t; y0)| |y0 y0| exp{|a|T} 0
y0 y0 0. , - t 0.
2.1.1
- y(t) = (y1(t), y1(t), . . . , yn(t))>.
dy(t)
dt= f(t, y(t)), (2.1)
y(t0) = y0, (2.2)
f(t, y) = (f1(t, y), f2(t, y), . . . , fn(t, y))>, y0 = (y10, y20, . . . , yn0)>. -, fi(t, y) fi(t, y)/yj
-
2.1. . 13
= [0,+) Rn i, j = 1, 2, . . . , n. - y0 Rn (2.1)-(2.2) [0, T ] y(t; y0), - y0. - (2.2) y0,
y(t; y0). y =( nj=1
y2j
)1/2
y = (y1, . . . , yn)> Rn.
2.1.1. y(t; y0) (2.1)-(2.2) - , > 0 (, y0) > 0 , y0, y0 y0 < (, y0), - y(t; y0) (2.1) t 0
y(t; y0) y(t; y0) < , t [0,+). (2.3)
, (2.3) t t0, (2.3) sup
tt0y(t; y0) y(t; y0) < .
2.1.2. y(t; y0) (2.1)-(2.2) - , , - 0 > 0 , y0, y0 y0 < 0,
limt+
(y(t; y0) y(t; y0)
)= 0. (2.4)
2.1.2. 2.1.1 y(t) = y0 exp{at} a < 0, ( ) a = 0, a > 0.
2.1.2
f(t, 0, . . . , 0) = , y0 = (2.1)-(2.2) =(0, . . . , 0)>. .
2.1.3. y(t) = (2.1)-(2.2) , > 0 () > 0 , y0, y0 < (), y(t; y0) (2.1) t 0
y(t; y0) < , t [0,+). (2.5)
2.1.4. y(t) = (2.1)-(2.2) , , 0 > 0 , y0, - y0 < 0,
limt+
y(t; y0) = 0. (2.6)
(t) . y0 = y0(0)
-
14 2.
, y(t) = y(t; y0) (t) , - y0, (0). y(t)
dy(t)
dt= F (t, y(t)), y(0) = y0, (2.7)
F (t, y(t)) = f(t, (t) + y(t)) f(t, (t)). (t) - y(t) = (2.7).
2.1.3
2.1.1. B(t) = (bij(t)) , b(t):
|bij(t)| b(t), i, j = 1, . . . , n.
x(t) = (x1(t), . . . , xn(t))>, y(t) = (y1(t), . . . , yn(t))> - y(t) = B(t)x(t),
y(t) nb(t)x(t).
. yj(t) =n
k=1 bjk(t)xk(t), - -,
|yj(t)| =n
k=1
|bjk(t)| |xk(t)| b(t)n
k=1
|xk(t)|
b(t)( n
k=1
12)1/2
( n
k=1
x2k(t)
)1/2= b(t)
nx(t).
j = 1, . . . , n, - 2.1.1.
2.1.2. t 0 y(t) = (y1(t), . . . , yn(t))> t
0
y()d n t
0
y()d.
.
t0
y()d = (I1(t), . . . , In(t))>, Ij(t) =
t0
yj()d, j = 1, . . . , n.
t 0
|Ij(t)| = t0
yj()d
t
0
|yj()|d t
0
y()d.
j = 1, . . . , n, - 2.1.2
-
2.1. . 15
2.1.3. Y (t) - dy/dt = Ay aij R, i, j = 1, . . . , n, 1, 2, . . . n A , p = max
k=1,...,nRe k.
Z(t, ) = Y (t)Y ()1
1. Z(t, ) = Z(t , 0);2. > 0 C > 0 ,
|Zij(t, )| C exp{(p+ )(t )}, t .
.
dZ(t, )
dt= AZ(t, ), Z(, ) = E.
s = t , , Z(s) = Z( + s, ). ,
dZ(s)
ds= AZ(s), Z(0) = E.
Z(s) = Z(s, 0). t, Z(t, ) = Z(t , 0).
Z(s, 0) = Y (s)Y (0)1. , Z(s, 0) s:
Zij(s, 0) = zij(s) exp{s}, deg zij(s) n 1, (2.8)
{1, . . . , n} . > 0 - Cij > 0
|zij(s)| Cij exp{s}, s 0,
| exp{s}| = exp{ Re s} exp{ps},
(2.8)
|Zij(s, 0)| |zij(s)| | exp{s}| C exp{(p+ )s}, C = maxi,j=1,...,n
Cij.
s = t , 2.1.3.
2.1.4
-
dy
dt= Ay, A = (aij) Rnn. (2.9)
1, . . . , n A .
-
16 2.
2.1.1. A,
Re k < 0, k = 1, . . . , n, y(t) = .
. y(t) = y(t; y0)
dy
dt= Ay, y(0) = y0.
y(t) = Z(t, 0)y0. (2.10)
p = maxk=1,...,n
Re k < 0. > 0,
= p+ < 0.
2 2.1.3 C ,
|Zij(t, 0)| C exp{t}, t 0.
2.1.1 B(t) = Z(t, 0) b(t) = C exp{t}
y(t) nC exp{t}y0. () =
2nC, y0 < () -
y(t) < t 0. exp{t} 0 t +.
2.1.5 -
2.1.2. A,
Re k 0, k = 1, . . . , n, , - , Re = 0, .
y(t) = (2.9) , .
. Z(t, 0) = Y (t)Y (0)1 t 0 . Yij(t) , = Re < 0,
|Yij(t)| |yij(t)| exp{t} Cij, t 0.
, Yij(t) , - = iq y(t) = h exp{t}, h
-
2.1. . 17
( ). -, :
|Ykl(t)| = |hl| | exp{iqt}| Ckl, t 0. Y (0)1 -
|Zij(t, 0)| Cij, t 0. (2.10) 2.1.1 B(t) = Z(t, 0) b(t) = C = max
i,j=1,...,nCij
y(t) nCx0.
() =
2nC,
2.1.2. . h Cn - -
, = iq. - , h = 1.
y(t) = 0.50Re h exp{iqt}, 0 > 0, (2.9) h exp{iqt}. t = 0
y(0) = 0.50Re h, y(0) 0.50h = 0.50. q 6= 0 0 > 0 0- y(t), y(t) 6 t +, , , y(tk) = 0.50Re h 6= tk = 2pik/q, k N. q = 0 .
2.1.6
2.1.3. :
1. A .
2. A m , Rem = 0, - , m, .
y(t) = .
. A = p + iq, p > 0. h = hR + ihI , hR, hI Rn. , h = 1.
y(t) = 0.5Re h exp{(p+ iq)t} == 0.5 exp{pt}(hR cos qt hI sin qt), > 0, (2.11)
(2.9) h exp{(p+iq)t}. t = 0
y(0) = 0.5hR, y(0) 0.5h = 0.5.
-
18 2.
q 6= 0 > 0 - (2.11) y(t), t +, , ,
y(tk) = 0.5hR exp{2pikp/q}, y(tk) = 0.5hR exp{2pikp/q} +
tk = 2pik/q, k +, k N. q = 0 . A = iq, q R, -
, > 0 (2.9)
y(t) = 0.5Re (g + th) exp{iqt} == 0.5
((gR + thR) cos qt (gI + thI) sin qt
), > 0,
y(0) = 0.5Re g, y(0) 0.5,
h = hR + ihI , g = gR + igI , g =1. y(t) t = 0 - t +, , , q 6= 0,
y(tk) = 0.5(gR + 2pikp/qhR), y(tk) khR +
tk = 2pik/q, k +, k N. q = 0 .
2.2 . -
dy(t)
dt= f(y(t)), (2.12)
f(y) = (f1(y), f2(y), . . . , fn(y))>. , (2.12) - y(t) = , .. f() = . , , - t = 0 , t 0. , fj(y) (2.12) Rn. .
fj(y) .
f(y) = Ay +R(y), A =
(fiyj
(0, . . . , 0)
), i, j = 1, . . . , n, R(y) = o(y). (2.13)
, R(y) = o(y) ,
> 0 > 0 : y < R(y) < y. (2.14) 2.2.1. (2.13)
A :
Re k < 0 k = 1, . . . , n.
-
2.2. 19
0 > 0 0 0 > 0 , t 0 y(t)
dy(t)
dt= Ay(t) +R(y(t)), y(0) = y0, (2.15)
t = 0 y0, y0 < 0, t 0 y(t) < 0.. , y(t) (2.15) -
y(t) = Z(t, 0)y0 +
t0
Z(t, )R(y())d. (2.16)
, F (t) = R(y()), (2.17)
, y(t) F (t)
dy(t)
dt= Ay(t) + F (t), y(0) = y0,
y(t) = Z(t, 0)y0 +
t0
Z(t, )F ()d.
(2.17), (2.16). (2.16). 2.1.1, 2.1.3 -
2.1.1 , y0 < 0 M > 0 ,
Z(t, 0)y0 M exp{t}y0.
(2.16):
Z(t, )R(y()) M exp{(t )}R(y()).
2.1.1 -
y(t) M exp{t}y0+Mt
0
exp{(t )}R(y())d. (2.18)
> 0 ,
M
|| 1
4.
(2.14) 0 > 0 , y < 0
R(y) < y. (2.19)
-
20 2.
0 = min
{ 04M
,02
}.
, 0 0 . y(t) (2.15) t = 0 y(0) < 0,
y(0) < 0, y(t) < 0 [0, t1). , t1 = +. , t1 (0,+)
y(t) < 0 t [0, t1), y(t1) = 0.
R(y()) y() 0 0 t1. , y0 0 04M
, (2.18)
0 = y(t1) 04exp{t1}+M
t10
exp{(t1)}d 04+04
t10
exp{(t1)}d 02.
2.2.1.
2.2.1. fj(y) - , j = 1, . . . , n.
A = (fi(0, . . . , 0)/yj) (2.13) ,
Rek < 0, k = 1, . . . , n, .
A = (fi(0, . . . , 0)/yj) ,
{1, . . . , n} : Re > 0, .
. . 2.2.1 0 0. 0- y0 y(t) (2.15) (2.16). 2.2.1 t 0 y(t) 0, (2.19)
R(y()) < y(), 0. (2.18)
y(t) M exp{t}y0+M exp{t}t
0
exp{)}y()d,
exp{t} u(t) = exp{t}y(t)
0 u(t) Mx0+Mt
0
u()d.
-
2.3. 21
-, u(t) My0 exp{Mt}. , M /4
y(t) My0 exp{(M + )t} My0 exp{3t/4}. -.
2.3 -.
2.3.1 .
2.3.1. V (y) : Rn R - ( ), :1. V (y) 0 y ;2. V (y) = 0 y = . ,
R > 0 : = {y Rn : y R}. 2.3.1. V (y) -
.
1. 1 > 0 2 > 0 , x , y 1 V (y) 2;
2. 2 > 0 3 > 0 , y , V (y) 2 y 3.
. .1. , . 1 > 0,
2 > 0 y, 1 y R, V (y) < 2. 2 0 < 2,k 0, yk, 1 yk R, V (yk) 0. yk - , , ykm y, 1 y R. V (ykm) V (y) = 0, y = . .
2. , . - 2 > 0, 0 < 3,k 0 yk, yk 3,k, V (yk) 2. V (yk) V (0) = 0, .
, V (y) =2 , y = 3 y = 1.
2.3.1. yk , k +yk V (yk) 0.
y(t) , t 0, t +y(t) V (y(t)) 0.
-
22 2.
, y Rn -. , V (y) = y . , .
2.3.1. V (y1, y2) = y21+y22 , . , .
2.3.2. V (y1, y2) =
y21a2
+y22b2
(a > 0, b > 0, a 6= b) - , . , - a, b.
2.3.2 . .
dy(t)
dt= f(y(t)), ()
f(y) = (f1(y1, . . . , yn), f2(y1, . . . , yn), . . . , fn(y1, . . . , yn))> , fi(0, . . . , 0) = 0, i = 1, . . . , n. V (y) . ()
dV
dt
()
(y) =n
j=1
V (y)
yjfj(y).
2.3.2. V (y) (), () ,
dV
dt
()
(y) 0, y . (2.20)
2.3.3
2.3.1. - (). y(t) = () .. 0 < < R. 2.3.1 2 = 2() , y y ,
V (y) 2. (2.21) V (y) 2 = (2()) , y <
V (y) 22. (2.22)
, . y0 - ,
y0 < , , t 0 y(t) () - y(t) < . t = 0 , y(0) < , (2.22)
V (y(0)) 22. (2.23)
-
2.3. 23
y(t) < - t [0; t1). t1 = +, . 0 < t1 < + , y(t1) , (2.21) V (y(t1)) 2. (2.23),
V (y(t1)) V (y(0)) 2 22
=22> 0. (2.24)
, (2.20)
dV (y(t))
dt=
nj=1
V (y(t))
yjyj(t) =
nj=1
V (y(t))
yjfj(y(t)) 0, t [0, t1].
, V (y(t)) [0, t1], (2.24). , > 0 = () ,
y0 < y(t) < t 0, .
2.3.3. (0, 0) {dx/dt = xy4,dy/dt = yx4.
f1(x, y) = xy4, f2(x, y) = yx4,
A =
f1(0, 0)
x
f1(0, 0)
yf2(0, 0)
x
f2(0, 0)
y
= ( 0 00 0
).
, A 1 = 2 = 0.
V (x, y) = x4+ y4 , ,
dV
dt
()
= 4x3 (xy4) + 4y3 (yx4) 0.
, (2.20). 2.3.1 - .
2.3.4
2.3.2. V (y) - (),
dV
dt
()
(y) W (y), y , (2.25)
W (y) . y(t) = ()
.
-
24 2.
. 2.3.1. , y(t) t +, y(0) .
3.3.2 y(t), - - . V (y(t)), t, -
dV (y(t))
dt=dV
dt
()
(y(t)) W (y(t)) 0,
(2.25).
limt+
V (y(t)) = 0.
, = 0. , > 0, ( - V (y(t))) V (y(t)) . 2 2.3.1 y(t) 3 > 0 t 0, 3 = 3(). 2.3.1 . 1 - W (y), W (y(t)) t 0, = (3) > 0. t + (2.25)
V (y(t)) V (y(0)) = dVdt
()
(y())t W (y())t t ,
V (y). V (y(t)) = 0, 2.3.1
, y(t) t +. 2.3.4. (0, 0) {
dx/dt = y x3,dy/dt = x y3.
f1(x, y) = y x3, f2(x, y) = x y3, A =(
f1(0,0)x
f1(0,0)y
f2(0,0)x
f2(0,0)y
)=
(0 11 0
).
, A 1,2 = i. V (x, y) = (x2 + y2)/2
dV
dt
()
= x (y x3) + y (x y3) = (x4 + y4),
(2.25) -W (x, y) = x4+y4. 2.3.2 .
2.3.5
2.3.3. = {y Rn : y } > 0 D 0 , 0, y = y . D = D 0 U(y), :
1. U(y) = 0 y 0, U(y) > 0 y D;
-
2.3. 25
2. > 0 = () > 0 , y D U(y)
dU
dt
()
(y) ,
dU
dt
()
U ()
y(t) = () .. , .. -. > 0 > 0 , y(t), t = 0 y(0) < , t 0 y(t) < , y(t) . 0, y(0) D, U(y(0)) = u0 > 0, dU
dt
()
(y(0)) > 0.
t = 0 D y(t) -
, y(t) D, dUdt
()
> 0. U(y(t)) ,
,U(y(t)) > U(y(0)) = u0 > 0, t 0.
D0 = {y D : U(y) u0}. = u0 0 > 0 , - y(t)
dU(y(t))
dt=dU
dt
() 0.
[0, t], t + U(y(t)) U(y(0)) + 0t +, y(t) D0,
U(x) - D0. . - .
2.3.5. (0, 0) {dx/dt = xy4,dy/dt = x4y.
f1(x, y) = xy4, f2(x, y) = x4y, A =
(f1(0,0)
xf1(0,0)
yf2(0,0)
xf2(0,0)
y
)=
(0 00 0
).
, A 1,2 = 0. U(x, y) = xy, D , -
, 0 OX OY .
dU
dt
()
= y xy4 + x x4y = xy(x4 + y4) = xy((x y)2 + 2(xy)2) 2(xy)3 23
xy > 0. , 2.3.3 () = 23, .
-
26 2.
2.3.6
y0 Rn ( ) dy(t)
dt= f(y(t)), (2.26)
f(y0) = 0. , f1(y1, . . . , yn) = 0,. . .fn(y1, . . . , yn) = 0.
y0 , y(t) = y0 t (2.26). Rn+1, (y1, . . . , yn) . - y0 ( ) , y(t) = y0 ( ) .
y(t) =y(t) + y0
dy(t)
dt= f(y(t)), f(y) = f(y + y0).
2.2.1 A = (aij):
aij =fiyj
(0, . . . , 0) =fiyj
(y0).
- ( ) .
2.3.4. y0 (2.26), fj(y) y0, j = 1, . . . , n.
A = (fi(y0)/yj) ,
Rek < 0, k = 1, . . . , n, y0 .
A = (fi(y0)/yj) ,
{1, . . . , n} : Re > 0, y0 .
2.4
2.4.1
2.1.1-2.1.3 , : t +, .
-
2.4. 27
, , . (n = 2).
y(t) = (y1(t), y2(t))>
dy
dt= Ay, A =
(a11 a12a21 a22
) R22. (2.27)
(.. (y1, y2)) (2.27). -, - , t (2.27)
dy1dy2
=a11y1 + a12y2a21y1 + a22y2
. (2.28)
(0, 0) (2.28), - . (0, 0) , . , (0, 0) (2.27) (2.28) .
- A. n = 2 1, 2. 1 6= 2, h1 =
(h11h21
) h1 =
(h12h22
) C2. 1 = 2,
, ; , - . A:detA 6= 0.
2.4.2 : 1, 2 R, 1 6= 2, 1 2 > 0.
(2.27)
y(t) =
(y1(t)y2(t)
)= C1
(h11h21
)exp{1t}+ C2
(h12h22
)exp{2t}, C1, C2 R. (2.29)
, :2 < 1 < 0. - . t + :y(t) . , .
dy1dy2
=C1h111e
1t + C2h122e2t
C1h211e1t + C2h222e2t=C1h111 + C2h122e
(21)t
C1h211 + C2h222e(21)t. (2.30)
21 < 0, C1 6= 0 dy1dy2
h11h21
t +, .. h1.
C1 = 0, y(t) = C2(h12h22
)e2t C2 6= 0 ,
h2, t +.
-
28 2.
t . -, , . (2.29) C2 6= 0
dy1dy2
=C1h111e
(12)t + C2h122C1h211e(12)t + C2h222
h12h22
, t , (1 2 > 0),
..
h2. C2 = 0, y(t) = C1(h11h21
)e1t C1 6= 0 ,
h1. , , t.
0 < 1 < 2 , , - , .
: .
2.4.3 : 1 = 2 6= 0, dimker(A 1E) = 2.
= 1 = 2 - h1 h2 A. (2.29)
y(t) = (C1h1 + C2h2) exp{t}
(y1, y2) , < 0 ( ) > 0( ), t +.
2.4.4 : 1 = 2 6= 0, dimker(A 1E) = 1.
= 1 = 2 - h1 A p1. (2.27)
y(t) = C1h1 exp{t}+ C2(p1 + th1) exp{t}.
C2 = 0, y(t) = C1h1 exp{t} , < 0 ( > 0) t + . C2 6= 0, t ,
y(t) = t exp{t}(C2h1 + o(1)), t +., t + < 0, t > 0. t > 0, t + < 0 -, t. .
-
2.4. 29
2.4.5 : 1, 2 R, 2 < 0 < 1.
, . - (2.29). C1 6= 0 t +
y(t) = exp{1t}(C1
(h11h21
)+ C2
(h12h22
)exp{(2 1)t}
)= exp{1t}
(C1
(h11h21
)+ o(1)
).
, (2.30) , dy1dy2
h11h21
, .. t + , h1. C1 = 0, y(t) = C2h2 exp{2t} C2 6= 0 , h2, t +.
t : C2 6= 0 , h2; C2 = 0 y(t) = C1h1 exp{1t} C1 6= 0 , h1, t . .
2.4.6 : 1,2 = i C, 6= 0, 6= 0.
, A . h = h1+ih2 h1,2, 1 = + i. z(t) = h exp{1t} :y1(t) = Re z(t) = exp{t}
(h1 costh2 sint
), y2(t) = Im z(t) = exp{t}
(h1 sint+h2 cost
).
y(t) = C1y1(t)+C2y2(t) = exp{t}(C1 cost+C2 sint
)h1+exp{t}
(C2 costC1 sint
)h2.
C =C21 + C
22 6= 0
sin = C1/C, cos = C2/C,
, h1 h2:
y(t) = 1(t)h1 + 2(t)h2.
1(t) = C exp{t} sin(t+ ), 2(t) = C exp{t} cos(t+ ), , t + < 0( , 21(t) + 22(t) 0) > 0 ( ,21(t) +
22(t) +).
2.4.7 : 1,2 = i C, 6= 0.
, A . , ,
-
30 2.
. h = h1 + ih2 h1 h2 y(t) = 1(t)h1 +2(t)h2
1(t) = C sin(t+ ), 2(t) = C cos(t+ ),
21(t)+ 22(t) = C2. (1(t), 2(t)) , .
2.4.8 A: detA = 0.
. .
1 = 0, 2 6= 0, h1, h2 .
y(t) = C1h1 + C2h2 exp{2t}. , h1, . , h2, t + 2 < 0 t 2 > 0. .
1 = 2 = 0 dimkerA = 2, .. h1 h2. A , (2.27)
y(t) = C1h1 + C2h2.
(. ). 1 = 2 = 0 dimkerA = 1, .. -
h. p. (2.27)
y(t) = C1h+ C2(p+ th) = (C1 + C2t)h+ C2p.
, h, . - , h, , C2 > 0 C2 < 0. .
2.4.9
y0 Rn dy(t)
dt= f(y(t)) (2.31)
,
A = (aij), aij =fiyj
(y0), i, j = 1, . . . , n, (2.32)
-
2.4. 31
n -. 2.3.4. , - (2.31)
dy(t)
dt= Ay(t) (2.33)
. (n = 2) (2.33),
: , . , , (2.33) (2.32).
2.4.1. {dx/dt = x 1,dy/dt = x2 y2.
{x 1 = 0,x2 y2 = 0,
: (1,1)>. f1(x, y) = x 1, f2(x, y) =x2 y2,
f1x
= 1,f1y
= 0,f2x
= 2x,f2y
= 2y.
(1, 1)> A =(
1 02 2
) 1 = 1,
2 = 2. (1, 1)> . (1,1)> A =
(1 02 2
) 1 = 1,
2 = 2. (1,1)> .
-
32 3.
3
3.1
. n-, , - (n 1)- . , . .
y . - F , t, y(t) y(t). , y(t)
y(t) = F (t, y(t), y(t)), t0 t t1. (3.1) ,
y(t0) = y0, y(t1) = y1. (3.2)
, : y(t), - (3.1) (3.2).
, u(x)
d
dx
(k(x)
du
dx
) q(x)u = f(x), 0 x l, (3.3)
u(0) = u0, u(l) = 0. (3.4)
u(0) = u0 , , u(l) = 0 , -. k(x), q(x) f(x) . u(x), (3.3)-(3.4).
, n- , , [0, l], - y(x), , , x = 0, x = l.
.
, , , . ,
y(x) + y(x) = 0, 0 x pi, (3.5)
-
3.1. 33
y(0) = 0, y(pi) = y1. (3.6)
(3.5) c1 sin x+ c2 cosx. y(0) = 0, y(x) = c1 sin x. y1 6= 0, (3.5)-(3.6) . y1 = 0, (3.5)-(3.6) y(x) = c1 sin x, c1- , - . , (3.5) y(x0) = y0, y(x0) = y1 y0, y1 x0 [0, pi].
3.1.1
-
a0(x)y(x) + a1(x)y(x) + a2(x)y(x) = f1(x), 0 x l (3.7)
1y(0) + 1y(0) = u0, 2y(l) + 2y(l) = u1, (3.8)
ai(x), i = 0, 1, 2, f1(x) 1, 1, 2, 2 . (3.7)-(3.8). , ai(x), i = 0, 1, 2,f1(x) , a0(x) 6= 0, 1, 1, 2, 2 , 2i +2i > 0,i = 1, 2.
(3.7). a0(x), -
p(x) = exp( x0
a1(s)a0(s)
ds). ,
d
dx
(p(x)
dy
dx
) q(x)y = f2(x), 0 x l, (3.9)
p(x) C1[0, l], p(x) > 0, q(x) = p(x)a2(x)/a0(x) C[0, l], f2(x) = p(x)f1(x)/a0(x) C[0, l].
3.1.2
(3.8). u0 = u1 = 0, - , - . , (3.9)-(3.8) . y(x) - (3.9)-(3.8). z(x) = y(x)v(x) , v(x) - - , (3.8). (3.9)-(3.8) y(x) = z(x) + v(x), z(x)
d
dx
(p(x)
dz
dx
) q(x)z = f(x), 0 x l,
1z(0) + 1z(0) = 0, 2z(l) + 2z(l) = 0,
f(x) = f2(x) ddx
(p(x)
dv
dx
)+ q(x)v.
v(x), (3.8) , .
-
34 3.
, -
d
dx
(p(x)
dy
dx
) q(x)y = f(x), 0 x l, (3.10)
1y(0) + 1y(0) = 0, 2y(l) + 2y(l) = 0. (3.11)
. (3.10)-(3.11) , f(x) = 0 .
3.1.3
, .
L[y] =d
dx
(p(x)
dy
dx
) q(x)y.
y(x) C2[0, l] z(x) C2[0, l], L[y] L[z],
z(x)L[y] y(x)L[z] = z(x) ddx
(p(x)
dy
dx
) y(x) d
dx
(p(x)
dz
dx
).
z(x)d
dx
(p(x)
dy
dx
) y(x) d
dx
(p(x)
dz
dx
)=
d
dx
[p(x)
(z(x)
dy
dx y(x)dz
dx
)],
z(x)L[y] y(x)L[z] = d
dx
[p(x)
(z(x)
dy
dx y(x)dz
dx
)], 0 x l. (3.12)
. . y1(x), y2(x)
L[y] = 0, - L[y1] = L[y2] = 0. - y1(x), y2(x) (3.12),
d
dx
[p(x)
(y1(x)
dy2dx
y2(x)dy1dx
)]= 0, 0 x l. (3.13)
W (y1, y2) = y1(x)y2(x) y2(x)y1(x) - p(x)W (y1, y2) = c, 0 x l, c - ,
W (y1, y2) =c
p(x), 0 x l. (3.14)
3.1.4
(3.12) 0 l,
l0
(z(x)L[y] y(x)L[z]) dx = p(x)(z(x)y(x) y(x)z(x))x=lx=0
. (3.15)
.
-
3.2. 35
, y(x) z(x) - (3.11),
l0
(z(x)L[y] y(x)L[z]) dx = 0. (3.16)
, ,
p(l)(z(l)y(l) y(l)z(l)) p(0)(z(0)y(0) y(0)z(0)) = 0.
, z(0)y(0) y(0)z(0) = 0. (3.17)
1 = 0, 1 6= 0, y(0) = 0, z(0) = 0, (3.17) . 1 6= 0
1y(0) + 1y(0) = 0, 1z(0) + 1z(0) = 0,
z(0), y(0). -,
1(z(0)y(0) y(0)z(0)) = 0,
(3.17). ,
z(l)y(l) y(l)z(l) = 0.
(3.16) .
3.2 .
L[y] ddx
(p(x)
dy
dx
) q(x)y = f(x), 0 x l, (3.18)
1y(0) + 1y(0) = 0, (3.19)
2y(l) + 2y(l) = 0, (3.20)
p(x), q(x), f(x) , 1, 1, 2, 2 , p(x) C1[0, l], p(x) > 0, x [0, l], q(x), f(x) C[0, l], 2i + 2i > 0, i = 1, 2.
3.2.1. y(x) (3.18)-(3.20), y(x) C2[0, l] (3.18)-(3.20).
3.2.1 .
, - (3.18)-(3.20).
3.2.2. G(x, ) (3.18)-(3.20), [0, l] [0, l] :
-
36 3.
1) (0, l) G(x, ) - x [0, ) (, l]
d
dx
(p(x)
dG(x, )
dx
) q(x)G(x, ) = 0, 0 x l, x 6= .
2) G(x, ) x:
1Gx(0, ) + 1G(0, ) = 0, 2Gx(l, ) + 2G(l, ) = 0, [0, l].
3) G(x, ) [0, l] [0, l], Gx(x, ) = x
Gx(x, x 0) = limx0
Gx(x, ), Gx(x, x+ 0) = limx+0
G(x, ),
Gx(x, x 0)Gx(x, x+ 0) = 1p(x)
, x (0, l).
3.2.2 -
(3.18)-(3.20).
3.2.1. G(x, ).
L[v] = 0, 1v(0) + 1v(0) = 0, 2v(l) + 2v(l) = 0 (3.21)
, (3.18)-(3.20) ,
y(x) =
l0
G(x, )f()d, 0 x l. (3.22)
. , y(x) (3.22) - (3.18)-(3.20).
x (0, l) , - (3.18). (3.22) x [0, x) (x, l], :
y(x) =
x0
G(x, )f()d +
lx
G(x, )f()d.
x. - ,
y(x) =
x0
Gx(x, )f()d +G(x, x 0)f(x) +l
x
Gx(x, )f()d G(x, x+ 0)f(x).
-
3.2. 37
G(x, ) G(x, x0)G(x, x+0) = 0.
y(x) =
x0
Gx(x, )f()d +
lx
Gx(x, )f()d,
p(x)y(x) =
x0
p(x)Gx(x, )f()d +
lx
p(x)Gx(x, )f()d.
x,
(p(x)y(x)
)=
x0
(p(x)Gx(x, ))xf()d + p(x)Gx(x, x 0)f(x)
+
lx
(p(x)Gx(x, ))xf()d p(x)Gx(x, x+ 0)f(x).
p(x)(Gx(x, x 0)Gx(x, x+ 0)) = 1,
(p(x)y(x)
)=
x0
(p(x)Gx(x, ))xf()d +
lx
(p(x)Gx(x, ))xf()d + f(x).
L[y] =
x0
L[G(x, )]f()d +
lx
L[G(x, )]f()d + f(x) = f(x),
.. (3.18). (3.19)-(3.20). 0 < x <
1y(x) + 1y(x) =
l0
(1Gx(x, ) + 1G(x, )
)f()d.
x 0 + 0, (3.19). (3.20).
. y(x) (3.18)-(3.20). v(x) = y(x) y(x) - (3.21) , - y(x) y(x) 0, 3.2.1 .
3.2.3
3.2.2. (3.21) -, (3.18)-(3.20) .
-
38 3.
. y1(x) L[y1] = 0, 0 x l, y1(0) = 1, y1(0) = 1, y2(x) L[y2] =0, 0 x l, y2(l) = 2, y2(l) = 2. , y1(x) (3.19), y2(x) (3.20). y1(x) y2(x) .
G(x, ) =
{c1()y1(x), 0 x ,c2()y2(x), x l,
c1() c2() . , G(x, ) 1) 2) . c1() c2(), 3). G(x, ) x = ,
c1()y1() = c2()y2().
Gx(x, ) x =
c2()y2() c1()y1() =
1
p().
, c1() c2(). ,
c1() =y2()
W ()p(), c2() =
y1()
W ()p(),
W () = y1()y2() y2()y1() . (3.14)
W ()p() = g0 .
G(x, ) =
y1(x)y2()
g0, 0 x ,
y1()y2(x)
g0, x l.
(3.23)
. . -, G(x, ), G(x, ). (0, l). z(x) = G(x, ) G(x, ). [0, l] z(x), Gx(x, ) Gx(x, ) x = . L[z] = 0, x 6= ,
z(x) =q(x)z(x) p(x)z(x)
p(x),
x = - x 0. z(x) x = ,
L[z] = 0, 0 x l, (3.19)-(3.20). [0, l] . z(x) = 0, G(x, ) =G(x, ), 3.2.2 .
-
3.2. 39
3.2.1.
y(x) + a2y(x) = f(x), 0 x l,y(0) = 0, y(l) = 0,
a 6= pinl1, n = 1, 2, . . . . y1(x) = sin ax, y2(x) = sin a(x l). , yi (x) + a2yi(x) = 0, i = 1, 2
y1(0) = y2(l) = 0.
g0 = p(x)W (x) = y1(x)y2(x) y2(x)y1(x) = a sin al.
(3.23) ,
Ga(x, ) =
sin ax sin a( l)
a sin al, 0 x ,
sin a sin a(x l)a sin al
, x l.(3.24)
3.2.4 -
.
y(x) + a2y(x) = F (x, y(x)), 0 x l, (3.25)y(0) = y(l) = 0. (3.26)
3.2.3. F (x, y) x [0, l] y R
|F (x, y1) F (x, y2)| L|y1 y2|, x [0, l], y1, y2 R. lL(a| sin al|)1 < 1, (3.25),(3.26) -.
. y(x) - (3.25)-(3.26). f(x) = F (x, y(x)). y(x)
y(x) + a2y(x) = f(x), 0 x l,y(0) = 0, y(l) = 0,
(3.24). ,
y(x) =
l0
Ga(x, )f()d, 0 x l.
f(x),
y(x) =
l0
Ga(x, )F (, y())d, 0 x l. (3.27)
-
40 3.
, , y(x) (3.25)-(3.26) , (3.27).
. y(x) [0, l] (3.27). (3.24), (3.27) , y(x) (3.26). (3.27) y(x) y(x) (3.25), , y(x) - . (3.27) (3.25)-(3.26). , , (3.25)-(3.26) (3.27).
(3.27), [0, l] . y0(x) = 0,
yn+1(x) =
l0
Ga(x, )F (, yn())d, 0 x l, n = 0, 1, 2 . . . (3.28)
yn(x) [0, l].,
|yn+1(x) yn(x)| M(
lL
a| sin al|)n
, 0 x l, n = 0, 1, 2..., (3.29)
M = max0xl
|y1(x)| = max0xl
l
0
Ga(x, )F (, 0)d
., n = 0 . n = m 1. , n = m. |ym+1(x) ym(x)|.
|Ga(x, )| (a| sin al|)1 0 x, l,
|ym+1(x) ym(x)| l
0
|Ga(x, )||F (, ym()) F (, ym1())|d
(a| sin al|)1Ll
0
|ym() ym1()|d M(
lL
a| sin al|)m
, 0 x l.
(3.29) .
yk(t) =k
n=1
(yn(t) yn1(t)),
yk(t) [0, l] -
n=1
(yn(t) yn1(t)).
(3.29) , [0, l] . yk(x) - [0, l] y(x) . yk(t) ,
-
3.3. - 41
y(x) [0, l]. (3.28) n - , , y(x) (3.27). (3.25)-(3.26).
(3.25),(3.26). , (3.27) . , y1(x), y2(x), - (3.27).
y1(x) y2(x) =l
0
Ga(x, )[F (, y1()) F (, y2())]d, 0 x l.
Ga(x, ),
|y1(x) y2(x)| =l
0
|Ga(x, )|L|y1()) y2()|d < max0xl
|y1(x) y2(x)|, 0 x l.
y1(x) = y2(x). , 3.2.3 .
3.2.5
(3.18)-(3.20) , - .
L[y] = 0 (3.19)-(3.20) (x). , c(x) , c - , - . , . (x) , L[] = 0 (x) (3.19)-(3.20). , (x) = c1(x), c1 . (x), (x) L[] = L[] = 0. W (, )(x) = (x)(x) (x)(x) x = 0. 1 = 0 , (0) = (0) = 0 W (, )(0) = 0. , 1 6= 0, (0) = 1(1)1(0), (0) = 1(1)1(0) W (, )(0) = 0. , W (, )(0) = 0 -, [0, l] (x) (x) , (x) = c1(x).
(3.18)-(3.20), .
3.2.4. (x) L[] = 0 - (3.19)-(3.20), y(x) L[y] =f(x) (3.19)-(3.20).
l0
f(x)(x)dx = 0. (3.30)
. (3.16) y(x) (x):
l0
(y(x)L[] (x)L[y])dx = 0.
L[y] = f(x) L[] = 0, (3.30). 3.2.4 .
-
42 3.
3.3 -
L[y] =d
dx
(p(x)
dy
dx
) q(x)y = y, 0 x l, (3.31)
1y(0) + 1y(0) = 0, (3.32)
2y(l) + 2y(l) = 0, (3.33)
p(x), q(x), f(x) , 1, 1, 2, 2 - , p(x) C1[0, l], p(x) > 0, x [0, l], q(x), f(x) C[0, l],2i +
2i > 0, i = 1, 2 .
, (3.31)-(3.33) y(x) = 0.
3.3.1. 1 (3.31)-(3.33) - y1(x), 1 , y1(x) .
-.
, -, , y(x) - , cy(x) , c - , .
(3.31) - L[y] = y(x). -, (3.32)-(3.33) . - L[y] = y(x) , .
, - . - . (3.31)-(3.33).
-.
3.3.1. - .
. 1 , y1(x) - . , , - 1 = a+ib, y1(x) =u(x) + iv(x). y1(x) (3.31), L[y1] =1y1(x). -
L[u] = au(x) + bv(x), (3.34)
L[v] = bu(x) av(x). (3.35) y1(x) (3.32)-(3.33), u(x),v(x) .
-
3.3. - 43
(3.34) v(x), (3.35) u(x) , 0 l .
l0
(v(x)L[u] u(x)L[v])dx = bl
0
((u(x))2 + (v(x))2)dx.
l0
(v(x)L[u] u(x)L[v]) dx = 0, (3.36)
b
l0
((u(x))2 + (v(x))2)dx = 0.
b = 0. 1 y1(x) .
3.3.2. - .
. - y1(x), y2(x). , (3.31) - (3.32)-(3.33). (3.32) , W (y1, y2)(0) = 0. y1(x), y2(x) (3.31), y2(x) = cy1(x).
v(x) w(x)
(v, w) =
l0
v(x)w(x)dx.
v(x) w(x) , , - (v, w) = 0.
3.3.3. , , .
. 1 6= 2 , y1(x), y2(x) - . y1(x), y2(x) (3.32)-(3.33), (3.16) ,
(L[y1], y2) (y1, L[y2]) =l
0
(L[y1]y2(x) y1(x)L[y2])dx = 0.
L[y1] = 1y1(x) , L[y2] = 2y2(x),
(1 2)(y1, y2) = 1(y1, y2) 2(y1, y2) == (1y1, y2) (y1, 2y2) = (L[y1], y2) + (y1, L[y2]) = 0.
(1 2)(y1, y2) = 0, (y1, y2) = 0 y1(x), y2(x) -.
-
44 3.
3.3.4. 1 = 2 = 0. , ,
min0xl
q(x). (3.37)
. , 1 , y1(x)
1 < min0xl
q(x).
q(x) 1 > 0 [0, l]. (3.31) , d
dx
(p(x)
dy1dx
)= (1 + q(x))y1(x).
0 x,
p(x)y1(x) = p(0)y1(0) +
x0
(q(s) 1)y1(s)ds. (3.38)
y1(x) (3.32), (3.33) 1 = 2 = 0, y1(0) =y1(l) = 0. y1(x) (3.31), y1(0) 6= 0. y1(0) > 0. y1(x) > 0 x [0, l]. , . x0 , y1(x0) = 0. x [0, x0) y1(x) > 0, y1(x) > 0. (3.38) x = x0 q(x) 1,, y1(x0) > 0. y1(x) x [0, l]. y1(x) > 0 x (0, l], y1(l) = 0. (3.37) .
- . 3.3.1. p(x) = 1, q(x) = 0, 1 = 2 = 0, l = pi. -
y(x) + y(x) = 0, 0 x pi, (3.39)y(0) = y(pi) = 0. (3.40)
. . (3.39)
y(x) = c1ex + c2e
x.
x = 0, x = l (3.40), c1 c2
c1 + c2 = 0
c1epi + c2e
pi = 0,
, c1 = c2 = 0. - . , 3.3.4. , = 0 .
. (3.39)
y(x) = c1 sinx+ c2 cos
x.
, c2 = 0. pi sin
pi = 0.
n = n2, n = 1, 2, .... yn(x) = c sinnx, c .
-
3.3. - 45
3.3.1 .
, , , -.
- (3.31)-(3.33). -, . yn(x), n = 1, 2, .... , , - ,
l0
(yn(x))2dx = 1.
. f(x) - [0, l] .
fn =
l0
f(x)yn(x)dx, n = 1, 2, . . .
3.3.5. ( ) f(x) C2[0, l] (3.32)-(3.33),
n=1
fnyn(x)
[0, l] f(x),
f(x) =n=1
fnyn(x), 0 x l.
-
46 4.
4
4.1
4.1.1
n- dx1(t)
dt= f1(t, x1(t), . . . , xn(t)),
...dxn(t)
dt= fn(t, x1(t), . . . , xn(t)),
(4.1)
fi(t, x) D1 Rn+1 - fi(t, x)/xj, i, j = 1, . . . , n.
4.1.1. () (4.1) D1 - v(t, x1, . . . , xn) C1(D1), D1 (4.1).
, x(t) = (x1(t), . . . , xn(t)) (4.1) C ,
v(t, x1(t), . . . , xn(t)) C. (4.2) (, ..).
4.1.2
- . (4.1).
4.1.2. v(t, x1, . . . , xn) C1(D1) (4.1)
dv
dt
(4.1)
=v(t, x)
t+
nj=1
v(t, x)
xjfj(t, x), (t, x) D1.
4.1.1. v(t, x1, . . . , xn) C1(D1) - (4.1) D1 , (4.1) D1:
dv
dt
(4.1)
= 0, (t, x) D1. (4.3)
. v(t, x1, . . . , xn) C1(D1) (4.1) D1. D1 (t, x(t)),
-
4.1. 47
x(t) (4.1), (4.2). (4.2) t dxj(t)/dt (4.1),
0 v(t, x(t))t
+n
j=1
v(t, x(t))
xj
dxj(t)
dt=
=v(t, x(t))
t+
nj=1
v(t, x(t))
xjfj(t, x(t)).
, (4.1) . (t0, x0) D1 (4.1) x(t0) = x0 , (4.3).
, v(t, x1, . . . , xn) C1(D1) (4.3). , (4.3) (t, x(t)) D1.
0 v(t, x(t))t
+n
j=1
v(t, x(t))
xjfj(t, x(t)) =
=v(t, x(t))
t+
nj=1
v(t, x(t))
xj
dxj(t)
dt=
d
dt
(v(t, x(t))
).
v(t, x(t)) t , .. v(t, x(t)) C. v(t, x) (4.1).
4.1.3
4.1.2. v(t, x1, . . . , xn) C1(D1) - (4.1) D1, C0 , D1, j {1, . . . , n} v(t, x)/xj 6= 0 D1.
v(t, x1, . . . , vn) = C0 Rn+1 n- , - (4.1).
. (t0, x0) D1 v(t, x) = C0, .. v(t0, x0) =C0. (4.1) x(t0) = x0 (t, x(t)), (t0, x0). v(t, x) , -
v(t, x(t)) = v(t0, x(t0)) = v(t0, x0) = C0
, v(t, x) = C0 t 6= t0.
4.1.4
v1(t, x), . . . , vk(t, x) (4.1). - Rk (y1, . . . , yk)
(t, x) = (v1(t, x), . . . , vk(t, x))
(4.1).
-
48 4.
4.1.3. v1(t, x), . . . , vk(t, x) (4.1) - D1, k:
rang
(vi(t, x)
xj
)= k, (t, x) D1.
.
4.1.1. D1 n - v1(t, x), . . . , vn(t, x) (4.1). (t0, x0) D1
dx(t)
dt= f(t, x), x(t0) = x0 (4.4)
v1(t, x) = c
01,
...vn(t, x) = c
0n,
(4.5)
c0j = vj(t0, x0), j = 1, . . . , n.
. (4.5) (t0, x0). , - (. 5.6.3 ) (x1, . . . , xn) : det
(vi(t0, x0)/xj
) 6= 0. (. 5.6.1 ) t0 xj(t) = gj(t, c01, . . . , c0n), j = 1, . . . , n,, g(t) = (g1(t), . . . , gn(t)) (4.5) :
v1(t, g(t)) = c01,
...vn(t, g(t)) = c
0n.
(4.6)
x(t) (4.4).
vj(t, x(t)) = vj(t0, x(t0)) = vj(t0, x0) = c0j , j = 1, . . . , n.
, x(t) (4.6), g(t). t0 : x(t) g(t).
, (t0, x0) D1 Rn+1 n (4.1).
4.1.2. (4.1) (.. fj = fj(x), j = 1, . . . , n)
x0, n
j=1
f 2j (x0) 6= 0, (n 1) t (4.1).
-
4.2. 49
4.2
4.2.1
u(x) = u(x1, . . . , xn) (x1, . . . , xn) D0, D0 Rn.
F (x1, . . . , xn, u,u
x1, . . . ,
u
xn) = 0
, - F (x1, . . . , xn, u, p1, . . . , pn) n -.
, , ..
nj=1
aj(x, u)u
xj= b(x, u),
aj(x, u), b(x, u) D1 Rn+1, D1
nj=1
a2j(x, u) 6= 0.
, u, :
nj=1
aj(x)u
xj= 0,
aj(x) D0 Rn, D0 -
nj=1
a2j(x) 6= 0. , .
4.2.1. u = u(x) - D0 Rn, 1. u(x) C1(D0),2. x D0 (x, u(x)) D0,3. u(x)
D0.
4.2.2 - .
- D0 Rn
a1(x)u
x1+ a2(x)
u
x2+ + an(x) u
xn= 0, (4.7)
aj(x) C1(D0), j = 1, . . . , n,n
j=1
a2j(x) 6= 0, x D0. (4.8)
-
50 4.
(4.7) n-
dx1(t)
dt= a1(x1(t), . . . , xn(t)),
...dxn(t)
dt= an(x1(t), . . . , xn(t)).
(4.9)
4.2.2. x(t) = (x1(t), . . . , xn(t)) (4.9) - Rn, (4.7).
(4.9) (4.7) . 4.2.1. u(x) C1(D0)
(4.7) , u(x) t (4.9) D0.
. u(x) t (4.9) D0. 4.1.1 - (4.9) D0:
du
dt
(4.9)
=n
j=1
u(x)
xjaj(x) = 0, x D0.
u(x) (4.7)., u(x) (4.7).
u(x) (4.9), D0. 4.1.1 , u(x) (4.9) D0.
4.2.1. D0 (4.9) n1 t
v1(x1, . . . , xn), v2(x1, . . . , xn), . . . , vn1(x1, . . . , xn).
M0(x01, . . . , x0n) D0 - (4.7)
u(x) = F (v1(x), v2(x), . . . , vn1(x)), (4.10)
F (y1, . . . , yn1) .
. vj(x) (4.9), j = 1, . . . , n 1, F (y1, . . . , yn1) u(x), - (4.10), , t. 4.2.1 u(x) (4.7).
, (4.10) - (4.7) M0(x01, . . . , x0n) D0. u(x) (4.10). v1(x), . . . , vn1(x)
-
4.2. 51
(4.9), 4.2.1) - (4.7). ,
nj=1
aj(x)u(x)
xj= 0,
nj=1
aj(x)v1(x)
xj= 0,
...n
j=1
aj(x)vn1(x)xj
= 0,
x D0. (4.11)
(4.8) x D0 (4.11) - a1(x), . . . , an(x) - . , ,
D(u, v1, . . . , vn1)D(x1, x2, . . . , xn)
= 0, x D0.
v1(x), . . . , vn1(x) - (n 1) . M0 F (y1, . . . , yn1), M0 (4.10).
4.2.3
- D Rn+1
a1(x, u)u
x1+ a2(x, u)
u
x2+ + an(x, u) u
xn= b(x, u), (4.12)
aj(x, u) C1(D), j = 1, . . . , n,n
j=1
a2j(x, u) 6= 0, (x, u) D. (4.13)
(4.12) (n+ 1)- .
dx1dt
= a1(x, u),
...dxndt
= an(x, u),
du
dt= b(x, u).
(4.14)
4.2.3. (x1(t), . . . , xn(t), u(t)) (4.14) - Rn+1, (4.12).
(4.14) (4.12) - .
-
52 4.
4.2.2. v(x, u) t (4.14) D, N0(x01, . . . , x0n, u0) D
v(N0) = C0,v
u(N0) 6= 0. (4.15)
N0
v(x1, . . . , xn, u) = C0 (4.16)
u = u(x1, . . . , xn), (4.12).
. v(x, u) t (4.14). 4.1.1 (4.14) D:
dv
dt
(4.14)
=n
j=1
v(x, u)
xjaj(x, u) +
v(x, u)
ub(x, u) = 0, (x, u) D. (4.17)
(4.16) (4.15) - M0(x01, . . . , x0n), - u = u(x1, . . . , xn), (4.16) :
v(x1, . . . , xn, u(x1, . . . , xn)) C0.
v
xj= u
xj vu, j = 1, . . . , n.
(4.17) v/u 6= 0 n
j=1
aj(x, u)u
xj= b(x, u)
M0. u(x) (4.12).
(4.14) (n + 1). ?? D n t
v1(x, u), . . . , vn(x, u).
F (y1, . . . , yn)
w(x, u) = F (v1(x, u), . . . , vn(x, u))
(4.14). 4.2.2 w/u 6= 0 u(x), -
F (v1(x, u), . . . , vn(x, u)) = 0 (4.18)
(4.12). , (4.18) (4.12) N0. , , [].
-
4.2. 53
4.2.4
u = f(x1, . . . , xn) C1(D0) (4.12) n- (x1, . . . , xn, u). .
4.2.3. u = f(x1, . . . , xn) C1(D0) - (4.12) , - , (4.14) (.. - , ).
.
P = {(x1, . . . , xn, u) Rn+1 : u = f(x1, . . . , xn)}, (4.19) f(x1, . . . , xn) C1(D0), -
= {(x1(t), . . . , xn(t), u(t))} P , . (4.14)
= (dx1(t)
dt, . . . ,
dxn(t)
dt,du(t)
dt) = (a1(x(t), u(t)), . . . , an(x(t), u(t)), b(x(t), u(t))),
u(t) = f(x(t)). P , P . -
n =( fx1
(x(t)), . . . ,f
xn(x(t)),1).
, ( , n)Rn+1 = 0,
a1(x, u)f
x1(x) + + an(x, u) f
xn(x) b(x, u) = 0, (x, u) . (4.20)
, u = f(x) (4.12) . , (4.12) D0.
, u = f(x) (4.12) D0. , M0(x01, . . . , x0n, u0) P P - (x01, . . . , x0n, u0). (x01, . . . , x
0n),
dx1dt
= a1(x, f(x)), x1(t0) = x01,
...dxndt
= an(x, f(x)), xn(t0) = x0n,
(4.21)
x(t) = (x1(t), . . . , xn(t)).
= {(x1 = x1(t), . . . , xn = xn(t), u(t) = f(x1(t), . . . , xn(t)))}. (4.22) P . , , .. (4.14). n (4.21).
-
54 4.
(4.14). , xi(t), i = 1, . . . , n, (4.21), u = f(x) (4.12),
du
dt=
nj=1
f
xj(x(t)) dxj
dt(t) =
nj=1
f(x(t))
xjaj(x(t), u(t)) = b(x(t), u(t)).
, . , , - P .
4.2.5
n = 2, -,
a1(x, y, u)u
x+ a2(x, y, u)
u
y= b(x, y, u), (4.23)
aj(x, y, u) C1(D), j = 1, 2, a21(x, y, u) + a22(x, y, u) 6= 0, (x, y, u) D, D R3.
(4.23) u =f(x, y), (4.23) ` = {(x, y, u) = (1(s), 2(s), 3(s)), s [s0, s1]}, ..
3(s) = f(1(s), 2(s)), s [s0, s1]. (4.24)
4.2.4.
det
(a1(s)
1(s)
a2(s) 2(s)
)6= 0, s [s0, s1], (4.25)
aj(s) = aj(1(s), 2(s), 3(s)), j = 1, 2. `
(4.23)-(4.24).
. (4.23):dx
dt= a1(x, y, u),
dy
dt= a2(x, y, u),
du
dt= b(x, y, u).
(4.26)
(4.26) t = 0 `
x|t=0 = 1(s), y|t=0 = 2(s), u|t=0 = 3(s) (4.27)
x = 1(t, s), y = 2(t, s), u = 3(t, s). (4.28)
(4.27)-(4.28)
1(0, s) = 1(s), 2(0, s) = 2(s), 3(0, s) = 3(s), s [s0, s1]. (4.29)
-
4.2. 55
(4.28) P . ` (4.27).
, ` u = f(x, y), , 4.2.3, f(x, y) (4.23). (4.28) -
x = 1(t, s), y = 2(t, s), (4.30)
(t, s) (x, y). , `, .. t = 0. (4.26)
1t
(0, s) =dx
dt|t=0 = a1(s), 2
t(0, s) =
dy
dt|t=0 = a2(s).
(4.27) ,
1s
(0, s) = 1(s),2s
(0, s) = 2(s).
(4.25)
det
1t 1s2t
1s
(0, s) = det( a1(s) 1(s)a2(s)
2(s)
)6= 0, s [s0, s1].
, (x0, y0) = (1(0, s), 2(0, s)) -
t = t(x, y), s = s(x, y),
(4.30) . (4.28)
u = 3(t(x, y), s(x, y)) = f(x, y).
, 4.2.3 (.. (4.28)), ` - .
(4.25) . = (a1, a2, b) , (1, 2, 3) `, - , (4.25) - (a1, a2) (1, 2) (x, y). , ` .
-
56 5.
5
5.1
M , - C[a, b].
5.1.1. M - .
M C[a, b].
[y(x)] [y(x)] = y(a) + 2y(b). , - ,
[y(x)] =
ba
y(x)dx.
. M - [a, b] , y(a) = y0, y(b) = y1, y0, y1 .
[y(x)] =
ba
[y(x) + 2(y(x))2]dx.
. 5.1.2. y0(x) M
y(x) , y0(x) + y(x) M . , M ,
y(x) y0(x), ty(x) y0(x) t R.
5.1.3. [y0(x), y(x)] [y(x)] y0(x) M
d
dt[y0(x) + ty(x)]
t=0.
, , , .
M = C[a, b].
[y(x)] =
ba
(y(x))2dx.
[y0(x), y(x)] =d
dt[y0(x) + ty(x)]
t=0
=
-
5.1. 57
=d
dt
ba
[y0(x) + ty(x)]2dxt=0
= 2
ba
y0(x)y(x)dx.
[y0(x), y(x)] y0(x).
[y(x)] =
ba
|y(x)|dx
y0(x) = 0, y(x) = 1,
[y0(x), y(x)] =d
dt[y0(x) + ty(x)]
t=0
=d
dt(b a)|t|
t=0,
. 5.1.4. [y(x)] y0(x) M -
() M , y(x) M [y0(x)] [y(x)] ([y0(x)] [y(x)]).
M y(x),
y(x) = maxaxb
|y(x)|.
5.1.5. [y(x)] y0(x) M - () M , > 0 , y(x) M y(x) y0(x) < , [y0(x)] [y(x)] ([y0(x)] [y(x)]).
. - , , .
. 5.1.1. [y(x)] y0(x) M -
M , y0(x) -, [y0(x), y(x)] y(x).
. [y(x)] y0(x) -. [y0(x) + ty(x)], y(x) y0(x). - y0(x) y(x) [y0(x) + ty(x)] t :(t) = [y0(x) + ty(x)]. [y(x)] y0(x) - , (t) t = 0 ., (0) , (0) = 0. - (0) [y(x)] y0(x)
d
dt(t)
t=0
=d
dt[y0(x) + ty(x)]
t=0.
[y0(x), y(x)] =d
dt[y0(x) + ty(x)]
t=0
= 0
y(x). 5.1.1 .
-
58 5.
5.1.1 .
, - , .
Cn0 [a, b], n 1 n [a, b] y(x) , y(m)(a) = y(m)(b) = 0, m = 0, 1, . . . , n 1.
5.1.1. f(x) [a, b] ,
ba
f(x)y(x)dx = 0
y(x) Cn0 [a, b]. f(x) = 0, x [a, b].. , f(x) [a, b]. - x1 (a, b) , f(x1) 6= 0. f(x1) > 0. f(x) > 0 , f(x) f(x1)/2 > 0 x [x1 , x1 + ] (a, b).
y(x) (x(x1))n+1((x1+)x)n+1 x [x1, x1+] [x1 , x1 + ]. y(x) Cn0 [a, b] y(x) > 0 x (x1 , x1 + ).
ba
f(x)y(x)dx =
x1+x1
f(x)y(x)dx > 0,
. 5.1.1 .
5.2
M [a, b] y(x) -, y(a) = y0, y(b) = y1.
[y(x)] =
ba
F (x, y(x), y(x))dx, (5.1)
F (x, y, p) . M . 5.2.1. , x [a, b], (y, p) R2 F (x, y, p) -
. (5.1) y0(x) M , - [a, b], y0(x)
Fy(x, y(x), y(x)) d
dxFp(x, y(x), y
(x)) = 0, a x b. (5.2)
. (5.1) y0(x). -M , y(x) y0(x) - [a, b] , . y(x) C10 [a, b].
-
5.2. 59
,
[y0(x), y(x)] =d
dt[y0(x) + ty(x)]
t=0
=
=d
dt
ba
F (x, y0(x) + ty(x), y0(x) + t(y)
(x))dxt=0
=
=
ba
{Fy(x, y0(x) + ty(x), y
0(x) + t(y)
(x))y(x)+
+Fp(x, y0(x) + ty(x), y0(x) + t(y)
(x))(y)(x)}dxt=0
=
ba
{Fy(x, y0(x), y
0(x))y(x) + Fp(x, y0(x), y
0(x)(y)
(x)}dx
, y0(x) , -
ba
Fy(x, y0(x), y0(x))y(x)dx+
ba
Fp(x, y0(x), y0(x)(y)
(x)dx = 0.
, y(a) = y(b) = 0 ,
ba
{Fy(x, y0(x), y
0(x))
d
dxFp(x, y0(x), y
0(x)
}y(x)dx = 0
y(x) C10 [a, b]. ,
Fy(x, y0(x), y0(x))
d
dxFp(x, y0(x), y
0(x)) = 0, a x b.
y0(x) (5.2) 5.2.1 -.
(5.2) (5.1). - y0(x), (5.1), M ,
Fy(x, y(x), y(x)) d
dxFp(x, y(x), y
(x)) = 0, a x b,y(a) = y0, y(b) = y1.
. , ,
f(x) y(x). , y(x) . . f(x) , f(a) = f(b) = 0. -
ba
(y(x) f(x))2dx+ b
a
(y(x))2dx, (5.3)
-
60 5.
. , - y(x) f(x), - y(x) .
(5.3) y(x) -, y(x) C1[a, b], y(a) = y(b) = 0, (5.3).
F (x, y, p) = (y f(x))2 + p2, Fy(x, y, p) = 2(y f(x)), Fp(x, y, p) = 2p,
2(y(x) f(x)) ddx
(2y(x)) = 0.
y(a) = y(b) = 0, y(x)
y(x) ()1y(x) = ()1f(x), a x b, (5.4)
y(a) = y(b) = 0. (5.5)
, , (5.3) - , (5.4)-(5.5). , (f(x) = 0) (5.4)-(5.5) , - (5.4)-(5.5) f(x). , (5.3).
5.3 -
- .
5.3.1 ,
M y(x) Cn[a, b] ,
y(a) = y0a, y(a) = y1a, y
(a) = y2a, . . . , y(n1)(a) = yn1a , (5.6)
y(b) = y0b , y(b) = y1b , y
(b) = y2b , . . . , y(n1)(b) = yn1b . (5.7)
[y(x)] =
ba
F (x, y(x), y(x), . . . , y(n)(x))dx, (5.8)
F (x, y, p1, . . . , pn) x [a, b], (y, p1, . . . , pn) Rn+1.
(5.8) M .
-
5.3. 61
5.3.1. F (x, y, p1, . . . , pn) x [a, b], (y, p1, . . . , pn) Rn+1 2n. y(x) M , y(x) C2n[a, b], (5.8) M , y(x)
Fy ddxFp1 + + (1)n
dn
dxnFpn = 0, a x b. (5.9)
. (5.8) y(x) y(x) Cn0 [a, b].
[y(x), y(x)] =d
dt[y(x) + ty(x)]
t=0
=
=d
dt
ba
F (x, y(x) + ty(x), y(x) + t(y)(x), . . . , y(n)(x) + t(y)(n)(x))dxt=0.
t, t = 0 ,
ba
(Fyy(x) + Fp1(y)
(x) + + Fpn(y)(n)(x))dx = 0.
, y(x) ,
ba
(Fy d
dxFp1 + + (1)n
dn
dxnFpn
)y(x)dx = 0.
y(x) Cn0 [a, b], - , y(x) (5.9). 5.3.1 .
, y(x) C2n[a, b] - (5.8) M , (5.9), (5.6), (5.7).
- f(x) y(x). , y(x) .
ba
(y(x) f(x))2dx+ b
a
((y(x))2 + (y(x))2
)dx, (5.10)
. , f(x) , f(a) = f(b) = 0, f (a) = f (a) = 0
(5.10) y(x) , y(x) C2[a, b], y(a) = y(b) = 0, y(a) = y(a) = 0.
F (x, y, p1, p2) = (y f(x))2 + p21 + p22,
-
62 5.
(5.9)
2(y(x) f(x)) ddx
(2y(x)) +d2
dx2(2y(x)) = 0.
y(a) = y(b) = 0, y(a) = y(a) = 0, y(x)
y(4)(x) y(x) + ()1y(x) = ()1f(x), a x b,y(a) = y(a) = 0, y(b) = y(b) = 0.
5.3.2 , .
, - . , u(x, y)
[u(x, y)] =
D
F (x, y, u(x, y), ux(x, y), uy(x, y))dxdy, (5.11)
F (x, y, u, p, q) , D , L. , F (x, y, u, p, q) - (x, y) D = D L, (u, p, q) R3 .
M u(x, y), D - L u(x, y) = (x, y), (x, y) L. u(x, y), M , u(x, y), D L, u(x, y) = 0,(x, y) L.
(5.11). - ,
5.3.1. f(x, y) D. D
f(x, y)v(x, y)dxdy = 0
v(x, y), D - L, f(x, y) = 0, (x, y) D.. , f(x, y) D. - (x0, y0) D , f(x0, y0) 6= 0. f(x0, y0) > 0. f(x, y) (x0, y0) ,
S = {(x, y) : (x x0)2 + (y y0)2 < 2}, f(x, y) f(x0, y0)/2 > 0 (x, y) S D. v(x, y),
v0(x, y) =
{ ((x x0)2 + (y y0)2 2
)2, (x, y) S;
0, (x, y) D\S.
D
f(x, y)v0(x, y)dxdy =
S
f(x, y)v0(x, y)dxdy f(x0, y0)2
S
v0(x, y)dxdy > 0,
. , . 5.3.1 .
-
5.3. 63
5.3.2. , F (x, y, u, p, q) - (x, y) D, (u, p, q) R3. (5.11) u(x, y) M , D, -
Fu Fpx
Fqy
= 0, (x, y) D. (5.12)
. (5.11) u(x, y) M , D. , (5.11)
[u(x, y), u(x, y)] =d
dt[u(x, y) + tu(x, y)]
t=0
= 0,
-d
dt
D
F (x, y, w(x, y, t), wx(x, y, t), wy(x, y, t))dxdyt=0
= 0,
w(x, y, t) = u(x, y) + tu(x, y). t t ,
D
Fu(x, y, u, ux, uy)u(x, y)dxdy+
+
D
{Fp(x, y, u, ux, uy)(u)x(x, y) + Fq(x, y, u, ux, uy)(u)y(x, y)
}dxdy = 0. (5.13)
. ,
Fp(x, y, u, ux, uy)(u)x(x, y) =
x
(Fpu
) Fpx
u,
Fq(x, y, u, ux, uy)(u)y(x, y) =
y
(Fqu
) Fqy
u.
D
{Fp(x, y, u, ux, uy)(u)x(x, y) + Fq(x, y, u, ux, uy)(u)y(x, y)
}dxdy =
=
D
( x
(Fpu
)+
y
(Fqu
))dxdy
D
(Fpx
+Fqy
)u dxdy.
D
( x
(Fpu) +
y(Fqu)
)dxdy,
, u(x, y) = 0, (x, y) L, D
( x
(Fpu
)+
y
(Fqu
))dxdy =
L
(Fpudy Fqudx
)= 0.
-
64 5.
D
{Fp(x, y, u, ux, uy)(u)x(x, y) + Fq(x, y, u, ux, uy)(u)y(x, y)
}dxdy =
= D
(Fpx
+Fqy
)u dxdy,
(5.13) D
{Fu(x, y, u, ux, uy)
xFp(x, y, u, ux, uy)
yFq(x, y, u, ux, uy)
}u(x, y) dxdy = 0.
5.3.1 , u(x, y) (5.12). 5.3.2 .
, u(x, y) , u M , D - (5.12),
Fu Fpx
Fqy
= 0, (x, y) D,u(x, y) = (x, y), (x, y) L.
, - . f(x, y), D u(x, y). - f(x, y) L D .
D
{(u(x, y) f(x, y))2 + ((ux(x, y))2 + (uy(x, y))2)}dxdy
(5.12) , , - u(x, y), D L,
uxx(x, y) + uyy(x, y) 1u(x, y) = 1f(x, y), (x, y) D.
5.4
[y(x)] =
ba
F (x, y(x), y(x))dx (5.14)
[y(x)] =
ba
G(x, y(x), y(x))dx, (5.15)
-
5.4. 65
F (x, y, p), G(x, y, p) - .
. y(x), (5.14)
M ={y(x) C1[a, b] : y(a) = y0, y(b) = y1, [y(x)] = l
}. (5.16)
(5.14) - , (5.15) . - .
(5.15)
M = {y(x) C1[a, b] : y(a) = y0, y(b) = y1}.
y(x) , y(x) C1[a, b],y(a) = y(b) = 0. [y(x)] y(x) M
[y(x), y(x)] =d
dt[y(x) + ty(x)]
t=0.
t , t = 0,
[y(x), y(x)] =
ba
{Gy(x, y(x), y
(x))y(x) +Gp(x, y(x), y(x))(y)(x)}dx. (5.17)
y(x) - (5.14) M.
5.4.1. y(x) M, y(x) C2[a, b], - (5.14) M.
y0(x) C1[a, b], y0(a) = y0(b) = 0
, [y(x), y0(x)] 6= 0, , y(x) -
Ly(x, y(x), y(x)) d
dxLp(x, y(x), y
(x)) = 0, a x b, (5.18)
L(x, y, p) = F (x, y, p) + G(x, y, p). (5.19)
. y(x) , y(x) C1[a, b],y(a) = y(b) = 0.
(t, ) = [y(x) + ty(x) + y0(x)],
(t, ) = [y(x) + ty(x) + y0(x)],
t, . (t, ) (t, ) ,
(0, 0) = [y(x)], (0, 0) = [y(x)],
t(0, 0) = [y(x), y(x)], (0, 0) = [y(x), y0(x)],
-
66 5.
t(0, 0) = [y(x), y(x)], (0, 0) = [y(x), y0(x)].
,
D(, )
D(t, )
t==0
= det
([y(x), y(x)], [y(x), y0(x)][y(x), y(x)], [y(x), y0(x)]
)= 0, y(x). (5.20)
, y(x) ,
det
([y(x), y(x)], [y(x), y0(x)][y(x), y(x)], [y(x), y0(x)]
)6= 0.
y(x)
(t, ) = u, (t, ) = v
(u, v), (u0, v0), u0 = (0, 0), v0 = (0, 0).
, , y(x) - - .
(t, ) = (0, 0) = [y(x)] ,(t, ) = (0, 0) = [y(x)] = l.
((0, 0) , (0, 0)) (u0, v0), - t, . ,
(t, ) = [y(x) + ty(x) + y0(x)] = [y(x)] ,(t, ) = [y(x) + ty(x) + y0(x)] = l.
y(x)+ty(x)+y0(x), M - (5.14) y(x). , y(x) . (5.20).
, (5.20),
[y(x), y(x)][y(x), y0(x)] [y(x), y0(x)][y(x), y(x)] = 0, y(x). [y(x), y0(x)] 6= 0. [y(x), y0(x)]
= [y(x), y0(x)][y(x), y0(x)]
,
[y(x), y(x)] + [y(x), y(x)] = 0, y(x).
[y(x), y(x)] [y(x), y(x)]
ba
{Fy(x, y(x), y
(x)) + Gy(x, y(x), y(x))}y(x)dx+
+
ba
{Fp(x, y(x), y
(x)) + Gp(x, y(x), y(x))}y(x)dx = 0, y(x).
-
5.5. - 67
(5.19) L(x, y, p),
ba
{Ly(x, y(x), y
(x)) ddxLp(x, y(x), y
(x))]}y(x)dx = 0, y(x) C10 [a, b].
, , y(x) - (5.18). 5.4.1 .
5.4.1 , , , (5.18). - , , . y(a) = y0, y(b) = y1, [y(x)] = l.
5.5 -
-. , -
d
dx
(k(x)
dy
dx
) q(x)y = y, 0 x l, (5.21)
y(0) = 0, y(l) = 0 (5.22)
. n , - yn(x) -. -.
l0
(yn(x))2dx = 1. (5.23)
[y(x)] =
l0
(k(x)(y(x))2 + q(x)(y(x))2
)dx. (5.24)
, yn(x) - (5.21)-(5.22), n,
[yn(x)] = n. (5.25)
,
l0
k(x)(yn(x))2dx =
l0
k(x)yn(x)yn(x)dx =
= k(x)yn(x)yn(x)x=lx=0
l
0
(k(x)yn(x))yn(x)dx =
l0
(k(x)yn(x))yn(x)dx,
-
68
[yn(x)] =
l0
(k(x)(yn(x))
2 + q(x)(yn(x))2)dx =
= l
0
((k(x)yn(x)) q(x)yn(x)) yn(x)dx = n
l0
(yn(x))2dx = n.
(5.24) , - (5.22) (5.23). (5.23)
[y(x)] = 1, [y(x)] = l
0
(y(x))2dx.
y(x) C2[0, l]. , y(x)
Ly ddxLp = 0, 0 x l,
L(x, y, p) = k(x)p2 + q(x)y2 y2, -
2q(x)y(x) 2y(x) 2(k(x)y(x)) = 0, 0 x l.
, y(x) (5.21) - (5.22). , - (5.23). , y(x) -(5.21)-(5.22). y1(x), 1 . (5.25), [y1(x)] = 1.
, , (5.24)-(5.23) -, - (5.24) -.
5.6
5.6.1
m m+n (u1, . . . , um, x1, . . . , xn) Rm+n:
F1(u1, . . . , um, x1, . . . , xn) = 0,. . .
Fm(u1, . . . , um, x1, . . . , xn) = 0.(5.26)
(5.26) - u1, . . . , um. (5.26) D Rn
u1 = 1(x1, . . . , xn), . . . , um = m(x1, . . . , xn), (5.27)
-
69
, (5.26) :
Fi(u1(x1, . . . , xn), . . . , um(x1, . . . , xn), x1, . . . , xn) = 0, (x1, . . . , xn) D, i = 1, . . . ,m.
F1, . . . , Fm u1, . . . , um -
D(F1, . . . , Fm)
D(u1, . . . , um)= det
F1u1
F1u2
. . .F1um
F2u1
F2u2
. . .F2um
. . . . . . . . . . . .Fmu1
Fmu2
. . .Fmum
,
(u1, . . . , um, x1, . . . , xn). 5.6.1. m
F1(u1, . . . , um, x1, . . . , xn), . . . , Fm(u1, . . . , um, x1, . . . , xn)
N0(u01, . . . , u0m, x01, . . . , x0n), Fi/uj N0, i, j = 1, . . . ,m. -
Fi(N0) = 0, i = 1, . . . ,m,D(F1, . . . , Fm)
D(u1, . . . , um)(N0) 6= 0,
1, . . . , m M0(x01, . . . , x0n), m (5.27), - |ui u0i | < i, i = 1, . . . ,m, - (5.26), M0.
, . 13, 2.
5.6.2
m n u1 = 1(x1, . . . , xn),
. . .um = m(x1, . . . , xn).
(5.28)
, i(x1, . . . , xn), i = 1, . . . ,m, n- D. . k {1, . . . ,m} .
5.6.1. uk D (5.28), x = (x1, . . . , xn) D
uk(x) = (u1(x), . . . , uk1(x), uk+1(x), um(x)), (5.29)
, - . u1, . . . , um - D, D .
-
70
, D (5.29) k {1, . . . ,m}, u1, . . . , um D.
5.6.2. m n m (5.28) M0(x01, . . . , x0n). - m M0, M0.
i(x1, . . . , xn), i = 1, . . . ,m, M0(x01, . . . , x0n), M0. (5.28)
1x1
1x2
. . .1xn
2x1
2x2
. . .2xn
. . . . . . . . . . . .mx1
mx2
. . .mxn
, (5.30)
m n . 5.6.3. (5.30)
1) r- M0(x01, . . . , x0n);
2) (r+1)- M0 (r = min(m,n), ).
r , r- , M0, r .
, . 13, 3.
.
. . . . .
: 1,2 R, 1=2 , 12 >0. : 1=2=0, dimker(A-1E)=2. : 1=2=0, dimker(A-1E)=1.: 1,2 R, 2