Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://mph.cs.msu.ru/stud/2009_ODU-2-RazgulinDenisov.pdf Äàòà èçìåíåíèÿ: Tue Nov 24 15:40:53 2009 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 22:48:00 2016 Êîäèðîâêà:

. ..

.. , ..



2



2009 .


" " , . .. " " . c . .., 2009 . c .. , .. , 2009 .




3



1 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 . . 1.1.2 . . . . . . . . . . . . . . . . . . . 1.2 . . . . . 1.2.1 . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 . . . . . . . . . . . . . . . . 2 2.1 . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 . . . . . . 2.1.2 2.2 . . . . . . . . . . . . . . . . . 2.2.1 . . . . . . . . . . . 2.2.2 . . . . . . . . . . . . . . . . . . . . . 2.2.3 . . 2.2.4

6 6 6 8 10 11 13 16

18 . 18 . 19 21 . 22 . 23

. 25 . 26 . 28


4 2.3 2.4

( ) . . . . . . . . . . . . . . . . . . ( ) . . . . . . . . . . . . . . 2.4.1 . . . . . . . 2.4.2 . . . . . . . . . . . . . . . . . . 2.4.3 . . . . . . . . . . . . . . . 2.4.4 . . . . 2.4.5 . . . . . . . . . 2.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 . 2.5.2 (1 , 2 R, 1 = 2 , 1 · 2 > 0) . . . . . . . . 2.5.3 (1 = 2 = 0, dim ker(A - 1 E ) = 2) . . . . . . . . . 2.5.4 (1 = 2 = 0, dim ker(A - 1 E ) = 1) . . . . . . . . . 2.5.5 (1 , 2 R, 2 < 0 < 1 ) . . . . . . . . . . . 2.5.6 (1,2 = ± i C, = 0, = 0) . . . . . . . 2.5.7 (1,2 = ±i C, = 0) . . . . . . . . . . . . 2.5.8 A (det A = 0) . . . 2.5.9

. 29 . . . . . . . . . . 34 34 36 37 39 41 43 45 45 46

2.5

. 47 . . . . . . 48 49 50 51 52 53

3 3.1 . . . . . . . . . . . . . . . . . . 3.1.1 . . . . . . . . . . . . . . . 3.1.2 . . . . . 3.1.3 . . . . . . . . 3.1.4 . . . . . . . . . . . . 3.2 . . 3.2.1 . . . . . . . . . . . . . . . . . . . . . 3.2.2 . 3.2.3 . . . . . . . . . . . . 3.2.4 . . . . . . . . . . . . 3.3 - . . . . . . . . . . . . . . . . . . 3.3.1 . . . . . . . . . . . . . . . . . . . .

55 55 56 57 58 59 60 60 61 63 64 67 72


4 4.1 . . . . . . . . . . 4.1.1 . . . . . . . . . . . 4.1.2 . 4.1.3 . . . . . 4.1.4 . . . . . . . . . . . 4.2 . . . 4.2.1 . . . . . 4.2.2 . . . 4.2.3 . . . . . . . . . . . . . . . . . . . . 4.2.4 . . . . . . . . . . . . . . . 4.2.5 . . . . . . . . . . . . . . . . . 5 5.1 . . . . . . 5.1.1 . . . . . . . . . . . . . . . 5.1.2 . . . . . . . . . . . . . . . 5.1.3 . . . . 5.2 . . . . . . . . . . . . . . . . . . . . . . 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 , . . . . . . . . . . . . . . . . . . . . . 5.3.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 . . . . . . 5.5 - . . . . . . . . .

5 74 74 74 75 76 76 78

. . . . . .

. 78 . 80 . 82 . 84 . 87 90 90 90 91 93 94

. . . . .

. 97 . 97 . 99 . 104 . 108

A 110 A.1 . . . . . . . . . . . . . . . . . 110 A.2 . . . . 111 114


6

1.

1
1.1.
, y (t) = f (t, y (t)), y (t0 ) = y0 . t [t0 - T , t0 + T ], (1.1) (1.2)

f (t, y ) Q = {(t, y ) : |t - t0 | T, A y B }.

1.1.1. (1.1), (1.2) [t0 - T , t0 + T ] y (t) , y (t) [t0 - T , t0 + T ], A y (t) B t [t0 - T , t0 + T ], y (t) (1.1), (1.2). (1.1), (1.2) f (t, y ) y0 , (1.1), (1.2). , f (t, y ) y0 ? , . , . 1.1.1. 1.1.1. f1 (t, y ) f2 (t, y ) Q f1 (t, y ) Q y ,


1.1. L > 0 , |f1 (t, y ) - f1 (t, y )| L|y - y |, (t, y ), (t, y ) Q.

7

, y1 (t) y2 (t) [t0 - T , t0 + T ] y1 (t) = f1 (t, y1 (t)), y1 (t0 ) = y01 ,
t[t0 -T ,t0 +T ]

y2 (t) = f2 (t, y2 (t)), y2 (t0 ) = y02 ,

max

|y1 (t) - y2 (t)| |y
01

- y02 | + T max |f1 (t, y ) - f2 (t, y )| exp{LT }. (1.3)
(t,y )Q

. , y1 (t) y2 (t)
t

y1 (t) = y

01

+
t0 t

f1 ( , y1 ( ))d ,

t [t0 - T , t0 + T ],

y2 (t) = y

02

+
t0

f2 ( , y2 ( ))d ,

t [t0 - T , t0 + T ].

,
t

|y1 (t) - y2 (t)|

|y

01

- y02 | +
t0

f1 ( , y1 ( )) - f2 ( , y2 ( )) d .

f1 ( , y2 ( )),
t

|y1 (t) - y2 (t)|
t

|y

01

- y02 | +
t0

f1 ( , y1 ( )) - f1 ( , y2 ( )) d +

+
t0

f1 ( , y2 ( )) - f2 ( , y2 ( )) d ,

t [t0 - T , t0 + T ]. (1.4)


8

1.

, f1 (t, y ) ,
t

f1 ( , y2 ( )) - f2 ( , y2 ( )) d
t0

T max |f1 (t, y ) - f2 (t, y )|,
(t,y )Q

t [t0 - T , t0 + T ], (1.4) : |y1 (t) - y2 (t)| |y
01

- y02 | + T max |f1 (t, y ) - f2 (t, y )| +
(t,y )Q t

+L
t0

y1 ( ) - y2 ( ) d ,

t [t0 - T , t0 + T ].

|y1 (t) - y2 (t)| - ??, t [t0 - T , t0 + T ] |y1 (t) - y2 (t)| |y
01

- y02 | + T max |f1 (t, y ) - f2 (t, y )| exp{L|t - t0 |},
(t,y )Q

(1.3). 1.1.1 . 1.1.2. , . . Q+ = {(t, y ) : t
0

t

t0 + T ,

A

y

B }.

, . 1.1.1. f (t, y ) Q+ Q+ fy (t, y ). (t, y1 ), (t, y2 ) Q+
1

f (t, y1 ) - f (t, y2 ) =
0

fy (t, y2 + (y1 - y2 ))d (y1 - y2 ).

(1.5)


1.1. . 1.1.2. ( ) f1 (t, y ), Q+ f1 (t, y ) Q+ f1 (t, y ). , y1 (t), y2 (t) [t0 , y y1 (t) = f1 (t, y1 (t)), y1 (t0 ) = y01 , f1 (t, y ) f2 (t, y ), (t, y ) Q+ , y
01

9 , f2 (t, y ) t0 + T ]

y2 (t) = f2 (t, y2 (t)), y2 (t0 ) = y02 , y02 ,

y1 (t) y2 (t), t [t0 , t0 + T ].

. y1 (t) y2 (t) [t0 , t0 + T ] , [t0 , t0 + T ], A yi (t) B , i = 1, 2, 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)) =
1

=
0

f1 t, y2 (t) + (y1 (t) - y2 (t)) d y1 (t) - y2 (t) + y +f1 (t, y2 (t)) - f2 (t, y2 (t)).

v (t) = y1 (t) - y2 (t),
1

p(t) =
0

f1 t, y2 (t) + (y1 (t) - y2 (t)) d, y

h(t) = f1 (t, y2 (t)) - f2 (t, y2 (t)).


10

1.

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
t t t

v (t) = (y01 - y02 ) exp
t0

p( )d +
t0

exp


p( )d h( )d , t [t0 , t0 + T ].

, y v (t) = y1 (t) - y2 (t) 1.1.2 . 0, t [t0 , t0 + T ],
01

-y

02

0,

h(t)

0,

t [t0 , t0 + T ],

1.2.
, , µ, . Qµ = {(t, y , µ) : |t - t0 | T, A y B, µ1 µ µ2 }.

f (t, y , µ) Qµ , y0 (µ) [µ1 , µ2 ]. y (t) = f (t, y (t), µ), y (t0 ) = y0 (µ). t [t0 - T , t0 + T ], (1.7) (1.8)


1.2.

11

µ (1.7), (1.8), , , t, µ. (1.7), (1.8) y (t, µ). y (t, µ) µ ?

1.2.1. 1.2.1. f (t, y , µ) Qµ Qµ y , |f (t, y1 , µ) - f (t, y2 , µ)| L|y1 - y2 |, (t, y1 , µ), (t, y2 , µ) Qµ ,

y0 (µ) [µ1 , µ2 ]. , y (t, µ) ­ (1.7), (1.8) [t0 - T , t0 + T ] µ [µ1 , µ2 ], y (t, µ) µ t [t0 - T , t0 + T ], µ [µ1 , µ2 ].

. y (t, µ) t [t0 - T , t0 + T ], µ [µ1 , µ2 ] A y (t, µ) B t [t0 - T , t0 + T ], µ [µ1 , µ2 ]. µ0 µ0 + µ [µ1 , µ2 ]. y (t, µ0 ) y (t, µ0 + µ), . y1 (t) = y (t, µ0 ), f1 (t, y ) = f (t, y , µ0 ), y01 = y0 (µ0 ), y2 (t) = y (t, µ0 + µ), f2 (t, y ) = f (t, y , µ0 + µ), y02 = y0 (µ0 + µ).

y1 (t) y2 (t) 1.1.1 .


12

1.

,
t[t0 -T ,t0 +T ]

max

|y (t, µ0 ) - y (t, µ0 + µ)| =
01 (t,y )Q

t[t0 -T ,t0 +T ]

max

|y1 (t) - y2 (t)|

|y

- y02 | + T max |f1 (t, y ) - f2 (t, y )| exp{LT } = = |y0 (µ0 ) - y0 (µ0 + µ)|+

+ T max |f (t, y , µ0 ) - f (t, y , µ0 + µ)| exp{LT }, (1.9)
(t,y )Q

Q = {(t, y ) : |t - t0 | T , A y B }. , (1.9) y (t, µ) µ0 . ­ . , () , t [t0 - T , t0 + T ] |y (t, µ0 + µ) - y (t, µ0 )| (1.10)

|µ| () . [µ1 , µ2 ] y0 (µ) , 1 () , |y0 (µ0 + µ) - y0 (µ0 )| 2 exp{LT } (1.11)

|µ| 1 () . Qµ f (t, y , µ) , 2 () , t [t0 - T , t0 + T ] y [A, B ] |f (t, y , µ0 + µ) - f (t, y , µ0 )| 2T exp{LT } (1.12)

|µ| 2 (). (1.9), (1.11) (1.12) , |µ| () = min{1 (), 2 ()} (1.10), y (t, µ) µ. 1.2.1 . 1.2.1. 1.2.1 [t0 - T , t0 + T ] â [µ1 , µ2 ] µ. , y (t, µ) (t, µ) [t0 - T , t0 + T ] â [µ1 , µ2 ].


1.2. 1.2.2.

13

, , y (t, µ) (1.7), (1.8) µ. 1.2.2. f (t, y , µ) Qµ Qµ fy (t, y , µ), fµ (t, y , µ), y0 (µ) [µ1 , µ2 ]. , y (t, µ) ­ (1.7), (1.8) [t0 - T , t0 + T ] µ [µ1 , µ2 ], y (t, µ) t [t0 - T , t0 + T ], µ [µ1 , µ2 ] µ.

. y (t, µ) t [t0 - T , t0 + T ], µ [µ1 , µ2 ] A y (t, µ) B t [t0 - T , t0 + T ], µ [µ1 , µ2 ]. µ µ + µ [µ1 , µ2 ]. y (t, µ) y (t, µ + µ). v (t, µ, µ) = y (t, µ + µ) - y (t, µ) , µ t [t0 - T , t0 + T ].

y (t, µ + µ), y (t, µ) (1.7) [t0 - T , t0 + T ] , v (t, µ, µ) = f (t, y (t, µ + µ), µ + µ) - f (t, y (t, µ), µ) . µ (1.13)

, f (t, y (t, µ + µ), µ + µ) - f (t, y (t, µ), µ) = µ f (t, y (t, µ + µ), µ + µ) - f (t, y (t, µ), µ + µ) = + µ f (t, y (t, µ), µ + µ) - f (t, y (t, µ), µ) + . µ


14

1.

(1.5), f (t, y (t, µ + µ), µ + µ) - f (t, y (t, µ), µ + µ) = µ
1

=
0

fy (t, y (t, µ)+(y (t, µ+µ)-y (t, µ)), µ+µ)d·

y (t, µ + µ) - y (t, µ) . µ


1

p(t, µ, µ) =
0

fy (t, y (t, µ) + (y (t, µ + µ) - y (t, µ)), µ + µ)d, f (t, y (t, µ), µ + µ) - f (t, y (t, µ), µ) . µ

q (t, µ, µ) =

, f (t, y (t, µ + µ), µ + µ) - f (t, y (t, µ), µ) = µ = p(t, µ, µ)v (t, µ, µ) + q (t, µ, µ). (1.13), , v (t, µ, µ + µ) [t0 - T , t0 + T ]: v (t, µ, µ) = p(t, µ, µ)v (t, µ, µ) + q (t, µ, µ). (1.14)

v (t, µ, µ) , y0 (µ + µ) - y0 (µ) v (t0 , µ, µ) = . (1.15) µ (1.14), (1.15) v (t, µ, µ) =
t

y0 (µ + µ) - y0 (µ) exp µ
t

t

p( , µ, µ)d +
t0

+
t0

q ( , µ, µ) exp


p( , µ, µ)d d ,

t [t0 - T , t0 + T ]. (1.16)


1.2.

15

y (t, µ) µ , v (t, µ, µ) µ 0. , (1.16) µ 0. y0 (µ) , lim y0 (µ + µ) - y0 (µ) dy0 = (µ). µ dµ

µ-0

p(t, µ, µ) µ 0. Qµ fy (t, y , µ) p(t, µ, µ) , f lim p(t, µ, µ) = (t, y (t, µ), µ) µ0 y (t, µ) [t0 - T , t0 + T ] â [µ1 , µ2 ]. fµ (t, y , µ) lim q (t, µ, µ) = f (t, y (t, µ), µ) µ

µ0

(t, µ) [t0 - T , t0 + T ] â [µ1 , µ2 ]. , (1.16) , µ 0, y dy0 (t, µ) = lim v (t, µ, µ) = (µ) exp µ0 µ dµ
t t t

fy ( , y ( , µ), µ)d +
t0

+
t0

fµ ( , y ( , µ), µ) exp


fy ( , y ( , µ), µ)d d . (1.17)

1.2.2 . y (t, µ), z (t, µ) µ z (t, µ) t. (1.17) , z (t, µ) [t0 - T , t0 + T ]: z (t, µ) = z (t, µ) = fy (t, y (t, µ), µ)z (t, µ) + fµ (t, y (t, µ), µ), z (t0 , µ) = y0 (µ). (1.18) (1.19)


16

1.

1.2.3. y (t) = f (t, y (t), µ), y (t0 ) = y0 (µ) (1.20)

µ [µ1 , µ2 ], µ = µ0 (µ1 , µ2 ) (, f (t, y , µ0 ) y ). u0 (t). u0 (t) = y (t, µ0 ) u0 (t) = f (t, u0 (t), µ0 ), u0 (t0 ) = y0 (µ0 ). (1.21)

, u0 (t) (1.21) - , y (t, µ) (1.20) µ, µ0 1.2.2. t [t0 - T , t0 + T ] y (t, µ) µ µ0 . ( µ0 ) : y (t, µ) = y (t, µ0 ) + y (t, µ0 ) (µ - µ0 ) + o(µ - µ0 ). ¯ µ

y (t, µ0 ) µ y (t, µ) - , µ = µ0 , (1.18), (1.19) u1 (t) , u1 (t) = u1 (t) = a(t)u1 (t) + b(t), u1 (t0 ) = y0 (µ0 ) (1.22)

a(t) = fy (t, u0 (t), µ0 ), b(t) = fµ (t, u0 (t), µ0 ).

µ - µ0 0 y (t, µ) (1.20): y (t, µ) = u0 (t) + u1 (t)(µ - µ0 ) + o(µ - µ0 ), ¯ t [t0 - T , t0 + T ], (1.23)


1.2.

17

u0 (t) u1 (t) (1.21) (1.22). o(µ - µ0 ) ¯ y (t, µ) u0 (t) + u1 (t)(µ - µ0 ). , (1.23) y (t, µ) µ - µ0 y (t, µ0 ) fy (t, y , µ) fµ (t, y , µ). f (t, y , µ) y µ , (1.23) . 1.2.1. µ 0 y (t) = y (t) + 3µy 4 (t) + µ2 t, y (0) = exp{2µ}.

t0 = 0, µ0 = 0, y0 (µ) = exp{2µ}, y0 (µ) = 2 exp{2µ}, f (t, y , µ) = y + 3µy 4 + µ2 t, fy (t, y , µ) = 1 + 12µy 3 , fµ (t, y , µ) = 3y 4 + 2µt, f (t, y , 0) = y , fy (t, y , 0) = 1, fµ (t, y , 0) = 3y 4 , y0 (0) = 1, y0 (0) = 2. (1.21) µ = 0 u0 (t) = y (t, 0) u0 (t) = u0 (t), u0 (0) = 1, : u0 (t) = exp{t}. fy (t, u0 (t), 0) = 1, fµ (t, u0 (t), 0) = 3 exp{4t}. (1.22) u1 (t) u1 (t) = u1 (t) + 3 exp{4t}, u1 (0) = 2

u1 (t) = 2 exp{t} + exp{4t}. (1.23) µ 0: y (t, µ) = exp{t} + (2 exp{t} + exp{4t})µ + o(µ). ¯


18

2.

2
2.1.
t = t0 t [t0 ; +). t0 = 0. 2.1.1. y = ay , y (0) = y
0

y0 t [0; +), a R ­ . y (t; y0 ) = y0 exp{at} (. . 2.1). a < 0 |y (t; y0 ) - y (t; y0 )| = |y0 - y0 | exp{at} y0 - y0 0 t t +. a = 0 |y0 - y0 | 0

0, |y (t; y0 ) - y (t; y0 )| 0

|y (t; y0 ) - y (t; y0 )| = |y0 - y0 | 0 y0 - y0 0 t t +. a > 0 0, |y (t; y0 ) - y (t; y0 )| 0

|y (t; y0 ) - y (t; y0 )| = |y0 - y0 | exp{at} +, t +, .


2.1.

19

a<0

a=0

a>0

. 2.1. 2.1.1: y (t; y0 ) = y0 exp{at} a.

T > 0 [0, T ]:
t[0,T ]

max |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) = f (t, y (t)), dt y (t0 ) = y 0 , f (t, y ) = (f1 (t, y ), f2 (t, y ), . . . , fn (t, y )) , y 0 = (y10 , y20 , . . . , yn0 ) . (2.1) (2.2)

, fi (t, y ) fi (t, y )/ yj = [0, +) â R
n


20

2.

i, j = 1, 2, . . . , n. ?? y 0 Rn (2.1), (2.2) [0, T ] y (t; y 0 ), y 0 . (2.2) y0 ,
n

y (t; y0 ). y =
j =1

y

2 j

1/2

-

y = (y1 , . . . , yn ) Rn . 2.1.1. y (t; y 0 ) (2.1), (2.2) , > 0 (, y 0 ) > 0 , y0 , y0 - y 0 < (, y 0 ), y (t; y0 ) (2.1) t 0 y (t; y0 ) - y (t; y 0 ) < , t [0, +). (2.3)

y (t; y 0 ) . , (2.3) t 0, (2.3) sup y (t; y0 ) - y (t; y 0 ) < .
t0

2.1.2. y (t; y 0 ) (2.1), (2.2) , 0 > 0 , y0 , y0 - y 0 < 0 ,
t+

lim

y (t; y0 ) - y (t; y 0 ) = 0.

(2.4)

. 2.2. 2.1.2. 2.1.1 y (t; y0 ) = y0 exp{at} a < 0, ( ) a = 0, ­ a > 0.


2.1.

21

.

.

. 2.2. y (t) = y (t; y 0 ): . y (t) = y (t; y0 ) - y (t) ( y - y (t) < , t 0); . y (t) - y (t) 0 t +.

2.1.2. f (t, 0, . . . , 0) = , y 0 = (2.1), (2.2) = (0, . . . , 0) : y (t; ) = , t 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) , -


22

2.

0 > 0 , y0 , y0 < 0 , (2.6) lim y (t; y0 ) = 0.
t+

y (t; y 0 ) (2.1), (2.2) . (2.1) , x(t) = y (t) - y (t; y 0 ). y (t) ­ (2.1), x(t) x(t) y (t) y (t; y 0 ) = - = f (t; y (t)) - f (t; y (t; y 0 )) = dt dt dt = f (t; x(t) + y (t; y 0 )) - f (t; y (t; y 0 )). , x(t) x(t) = f (t; x(t) + y (t; y 0 )) - f (t; y (t; y 0 )). dt x(t; ) x(0) = 0. : x(t; ) = , t y (t; y 0 ) . , , , .

2.2.
dy = Ay , dt A = (aij ), aij R, i, j = 1, . . . , n. A , .


2.2. 2.2.1.

23

2.2.1. 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) .
n

. yj (t) =
k=1

bj k (t)xk (t), ,

-,
n n

|yj (t)| =
k=1

|bj k (t)| · |xk (t)|
n

b(t)
k=1 1/2 2

|xk (t)|
n 1/2

b(t)
k=1

1

·
k=1

x2 (t) k

= b(t) n x(t) .

j = 1, . . . , n, 2.2.1. 2.2.2. t y (t) = (y1 (t), . . . , yn (t))
t

0 -

y ( )d
0



t

n
0

y ( ) d .

. -
t t

y ( )d = (I1 (t), . . . , In (t)) ,
0

Ij (t) = yj ( )d ,
0

j = 1, . . . , n.

t

0
t t t

|Ij (t)| =
0

yj ( )d
0

|yj ( )|d
0

y ( ) d .


24

2.

j = 1, . . . , n, 2.2.2 2.2.3. Y (t) ­ dy /dt = Ay R, i, j = 1, . . . , n, 1 , 2 , . . . n ­ , p = max aij A Re k .
-1

k=1,...,n

Z (t, ) = Y (t)Y 1. Z (t, ) = Z (t - , 0);

( ) -

2. > 0 C > 0 , |Zij (t, )| C exp{(p + )(t - )}, t .

. dZ (t, ) = AZ (t, ), Z ( , ) = E . dt s = t - , ­ , Z (s) = Z ( + s, ). , dZ (s) = AZ (s), Z (0) = E . ds Z (s) = Z (s, 0). t, Z (t, ) = Z (t - , 0). Z (s, 0) = Y (s)Y -1 (0). - , Z (s, 0) (. ??): Zij (s, 0) = qij (s) exp{k s}, (2.7)

k ­ , qij (s) ­ deg qij (s) n - 1. > 0 Cij > 0 , |qij (s)| Cij exp{ s}, s 0.


2.2. p = max Rek ,
k=1,...,n

25

| exp{k s}| = exp{ Re k s} , (2.7) |Zij (s, 0)| |qij (s)| · | exp{k s}|

exp{ps}.

C exp{(p + )s},

C =

i,j =1,...,n

max

Cij .

s = t - , 2.2.3. 2.2.2. : dy = Ay , dt (2.8)

A = (aij ), aij R, i, j = 1, . . . , n. 1 , . . . , n ­ A . 2.2.1. A : Re k < 0, k = 1, . . . , n.

y (t; ) = (2.8) . . y (t) = y (t; y 0 ) ­ dy = Ay , dt y (0) = y 0 .

, , y (t) = Z (t, 0)y 0 . (2.9)


26 p =

2. max Re k < 0. = p + < 0.

k=1,...,n

> 0, 2 2.2.3 C , |Zij (t, 0)| C exp{t}, t 0.

2.2.1 B (t) = Z (t, 0), b(t) = C exp{t} x(t) = y 0 (2.9) y (t) () = nC exp{t} y 0 .

, y 0 < () 2nC y (t) < t 0. exp{t} 0 t +. 2.2.3. 2.2.2. A , Re
k

0,

k = 1, . . . , n

, , Re = 0, . y (t; ) = (2.8) , . . Z (t, 0) = Y (t)Y
-1

(0)

t 0 . Yij (t) ,


2.2.

27

, 2.2.1, |Yij (t)| Cij exp{t}, t 0, Cij ­ , < 0. , |Yij (t)| Cij , t 0.

, Ykl (t) , = iq , - y (t) = hl exp{t}, h = (h1l , . . . , hnl ) ­ ( ). , : |Ykl (t)| = |hkl | · | exp{iq t}| Ckl , t 0.

, Y (t) . Y (t) Y -1 (0) . , |Zij (t, 0)| Cij , t 0.

(2.9) 2.2.1 B (t) = Z (t, 0), b(t) = C = max Cij x(t) = y 0 i,j =1,...,n y (t) nC y 0 . . . h Cn ­ - , = iq , q > 0. , h = 1. - y (t) = 0.50 Re h exp{iq t}, 0 > 0,

(2.8) h exp{iq t}. t = 0 y (0) = 0.50 Re h, y (0) 0.50 h = 0.50 .


28

2.

0 > 0 0 - y (t), y (t) - t +, , , y (tk ) = 0.50 Re h = tk = 2 k /q , k N. q = 0 . 2.2.4. 2.2.3. : 1. A ; 2. A Re
m m

,

= 0,

, m , . y (t; ) = . . A = p + iq , p > 0, q > 0. h = hR + ihI , hR , hI Rn . , h = 1. - y (t) = 0.5 Re h exp{(p + iq )t} = = 0.5 exp{pt} hR cos q t - hI sin q t , > 0, (2.10)

(2.8) h exp{(p + iq )t}. t = 0 y (0) = 0.5 hR , y (0) 0.5 h = 0.5.

> 0 - (2.10) y (t), t = tk = 2 k /q , k N, k + : y (tk ) = 0.5 hR exp{2 k p/q }, y (tk ) = 0.5 h
R

exp{2 k p/q } +.


2.3.

29

q = 0 . A = iq , q > 0, , > 0 (2.8) y (t) = 0.5 Re (g + th) exp{iq t} = = 0.5 (g R + thR ) cos q t - (g I + thI ) sin q t , y (0) = 0.5 Re g , y (0) 0.5, > 0,

h = hR + ihI ­ , g = g R + ig I ­ , g = 1. y (t) t = 0 , t = tk = 2 k /q , k N, k + : y (tk ) = 0.5 (g R + tk hR ), y (tk ) k h
R

+.

q = 0 .

2.3. ( )
dy (t) = f (y (t)), dt f (y ) = (f1 (y ), f2 (y ), . . . , fn (y )) . , f () = . (2.11) y (t) = . . 2.4 , , t = 0 , t 0. , fj (y ) (2.11) Rn (. ??). . (2.11)


30

2.

fj (y ) . f (y ) = Ay + R(y ), A= fi (0, . . . , 0) , yj i, j = 1, . . . , n, R(y ) = o( y ). ¯ (2.12)

¯ , R(y ) = o( y ) , > 0 > 0 : y < R(y ) < y . (2.13)

2.3.1. (2.12) A : Re k < 0, k = 1, . . . , n.
0

0 > 0 y (t; y 0 )

0 > 0 ,

dy (t) = Ay (t) + R(y (t)), dt y
0

y (0) = y 0 ,

(2.14)

< 0 , y (t; y 0 ) < 0

t

0.

. , y (t; y 0 ) (2.14)
t

y (t; y 0 ) = Z (t, 0)y 0 +
0

Z (t, )R(y ( ; y 0 ))d .

(2.15)

, F (t) = R(y (t; y 0 )), (2.16)

, y (t; y 0 ) F (t) dy (t) = Ay (t) + F (t), dt y (0) = y 0 .


2.3.

31

(??), ?? ??,
t

y (t) = Z (t, 0)y 0 +
0

Z (t, )F ( )d .

(2.16), (2.15). (2.15). 2.2.1, 2.2.3 2.2.1 , y 0 < 0 M1 > 0 , Z (t, 0)y 0 M1 exp{t} y 0 . (2.15): Z (t, )R(y ( ; y 0 )) M2 exp{(t - )} R(y ( ; y 0 )) .

2.2.2 -,
t

y (t; y 0 )

M exp{t} y

0

+M
0

exp{(t - )} R(y ( ; y 0 )) d , (2.17)

M = max{M1 , M2 n}. > 0 , M 1 . || 4 (2.13) 0 > 0 , y < 0 R(y ) < y . (2.18) , 0 0 . , 4M 2 , 0 0 . y (t; y 0 ) (2.14) t = 0 y 0 < 0 , y 0 < 0 , 0 = min


32

2.

y (t; y 0 ) < 0 [0, t1 ). , t1 = +. , t1 (0, +) y (t; y 0 ) < 0 , (2.18) R(y ( ; y 0 )) , y (2.17)
t1 0

t [0, t1 ),

y (t1 ; y 0 ) = 0 .

y ( ; y 0 )

0 , 0 , 4M

0



t1 .

0

0 = y (t1 ; y 0 )

0 exp{t1 } + M 4

0 0

exp{(t1 - )}d
0

0 M + 4 ||

1 - exp{t1 }

0 . 2

2.3.1. 2.3.1. fj (y ) , j = 1, . . . , n. A = fi (0, . . . , 0)/ yj : Re k < 0, k = 1, . . . , n,

(2.11) . A = fi (0, . . . , 0)/ yj : {1 , . . . , n } : Re > 0, . . . 2.3.1 0 0 . 0 - y 0 . y (t; y 0 ) ­ (2.14)


2.3.

33

(2.15). 2.3.1 t 0 y (t; y 0 ) 0 (2.18) R(y ( ; y 0 )) < y ( ; y 0 ) , (2.17) t 0.

0
t

y (t; y 0 )

M exp{t} y

0

+ M exp{t}
0

exp{- )} y ( ; y 0 ) d .

exp{-t} u(t) = exp{-t} y (t; y 0 ) ,
t

0

u(t)

My

0

+ M
0

u( )d ,

t

0.

-, u(t) My
0

exp{M t}.

, M y (t; y 0 ) My
0

|| , 4

exp{(M + )t}

My

0

exp{3t/4}.

. 2.3.1. (0, 0)
4 dy1 /dt = -y1 - ay2 + y2 , 5 3 dy2 /dt = y1 - y1 + y2 .


34

2.

4 5 3 f1 (y1 , y2 ) = -y1 - ay2 + y2 , f2 (y1 , y2 ) = y1 - y1 + y2 ,

A=

fi (0, 0) yj

=

-1 -a 1 0

.

A -a = 2 + + a. - 1,2 = 0.5(-1 ± 1 - 4a). a > 0 Re1,2 < 0. a = 0 1 = -1, 2 = 0. a < 0 1 < 0, 2 > 0. , , a > 0, a < 0. a = 0 . M () = det(A - E ) = -1 - 1

2.4. ( )
2.4.1. 2.4.1. V (y ) : Rn R ( ), : 1. V (y ) 0, y ;

2. V (y ) = 0 y = . , R > 0 : = {y Rn : y R}.

2.4.1. V (y ) ­ . : 1. 1 > 0 2 > 0 , y , y 1 V (y ) 2 ;


2.4.

35

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 < 2k 0, y k , 1 yk R, V (y k ) 0. y k , , y R. V (y km ) V (y ) = 0, y km y , 1 y = . . 2. , . 2 > 0 , 0 < 3k 0 y k , y k 3k , V (y k ) 2 . V (y k ) V (0) = 0, . , V (y ) = 2 , y = 3 ­ y = 1 (. . 2.3). 2.4.1. y k , k + y k , V (y k ) 0. t 0 - y (t) , t +

y (t) , V (y (t)) 0. , y Rn . , V (y ) = y y . , .


36

2.

. 2.3. V (y ), y = (y1 , y2 ).
2 2 2.4.1. V (y1 , y2 ) = y1 +y2 , . , . 2 2 y2 y1 + 2 (a > 0, b > 0, a = b) a2 b , . , a, b.

2.4.2. V (y1 , y2 ) =

2.4.2. dy (t) = f (t, y (t)), dt y (0) = y 0 , (2.19)

f (t, y ) = (f1 (t, y1 , . . . , yn ), f2 (t, y1 , . . . , yn ), . . . , fn (t, y1 , . . . , yn )) , fj (t, y1 , . . . , yn ) [0; +) â , fj (t, 0, . . . , 0) = 0, j = 1, . . . , n, t 0. , (2.19) y (t; ) = .


2.4.

37

2.4.2. V (y ) (2.19),
n

j =1

V (y ) fj (t, y ) yj

0,

y , t

0.

(2.20)

2.4.3. 2.4.1. (2.19). y (t; ) = (2.19) . . 1 (0, R). 2.4.1 2 = 2 (1 ) , y y 1 , V (y ) 2 . (2.21) V (y ) 2 (1 ) = (2 (1 )) , y < V (y ) 2 . 2 (2.22)

, 1 . y 0 - ( y 0 < ) , t 0 y (t) = y (t; y 0 ) (2.19) y (t) < 1 . t = 0 , y (0) (2.22) 2 V (y (0)) . 2 y (t) < t [0; t1 ). . , y (t1 ) 1 , =y
0

<

1 , (2.23)

1 t1 = +, t1 (0, +) -


38

2.

. 2.4. 2.4.1.

(2.21) (. . 2.4) V (y (t1 )) 2 .

(2.23), V (y (t1 )) - V (y (0)) , (2.20) dV (y (t)) = dt
n

2 -

2 2 = > 0. 2 2

(2.24)

j =1

V (y (t)) dyj (t) = yj dt

n

j =1

V (y (t)) fj (t, y (t)) yj

0,

t [0, t1 ].

, V (y (t)) [0, t1 ], (2.24). , 1 > 0 = (1 ) , y 0 < y (t; y 0 ) < 1 t 0, . 2.4.3. (0, 0)
4 dy1 /dt = -y1 y2 , 4 dy2 /dt = y1 y2 .


2.4.
4 4 f1 (y1 , y2 ) = -y1 y2 , f2 (y1 , y2 ) = y1 y2 ,

39

A=

fi (0, 0) yj

=

0 0

0 0

.

, A 1 = 2 = 0. 4 4 V (y1 , y2 ) = y1 + y2 , V (y1 , y2 ) V (y1 , y2 ) 3 4 3 4 f1 (y1 , y2 ) + f2 (y1 , y2 ) = 4y1 · (-y1 y2 ) + 4y2 · (y1 y2 ) 0. y1 y2 , (2.20). 2.4.1 . 2.4.4. 2.4.2. V (y ) (2.19),
n

j =1

V (y ) fj (t, y ) yj

-W (y ),

y ,

t

0,

(2.25)

W (y ) ­ . y (t; ) = (2.19) . . 2.4.1. , y (t) = y (t; y 0 ) (2.19) y (t) t +,

y 0 . 2.4.1 y (t), 1 - .


40

2.

V (y (t)), t, dV (y (t)) = dt
n

j =1

V (y (t)) fj (t, y (t)) yj

-W (y (t))

0,

(2.25).
t+

lim V (y (t)) =

0.

, = 0. , > 0, V (y (t)) V (y (t)) . 2 2.4.1 y (t) 3 > 0 t 0, 3 = 3 (). 2.4.1 . 1 W (y ), W (y (t)) t 0, = (3 ) > 0. t + (2.25) V (y (t)) - V (y (0)) = dV (y ( )) t dt -W (y ( ))t - t -,

V (y ). V (y (t)) = 0 , 2.4.1, , y (t) t +. 2.4.4. (0, 0)
3 dy1 /dt = -y2 - y1 , 3 dy2 /dt = y1 - y2 . 3 3 f1 (y1 , y2 ) = -y2 - y1 , f2 (y1 , y2 ) = y1 - y2 ,

A=

fi (0, 0) yj

=

0 -1 10

.

, A 2 2 1,2 = ±i. V (y1 , y2 ) = (y1 + y2 )/2 V (y1 , y2 ) V (y1 , y2 ) f1 (y1 , y2 ) + f2 (y1 , y2 ) = y1 y2 3 3 4 4 = y1 · (-y2 - y1 ) + y2 · (y1 - y2 ) = -(y1 + y2 ).


2.4.

41

, V (y1 , y2 ) , (2.25) 4 4 W (y1 , y2 ) = y1 + y2 . , 2.4.2, . 2.4.5. 2.4.3. = {y Rn : y < } > 0 D 0 , 0 , y = y . D 0 U (y ), : 1. U (y ) = 0 y 0 , U (y ) > 0 y D; 2. > 0 = () > 0 , y D U (y )
n

j =1

U (y ) fj (t, y ) yj

,

t

0.

y (t; ) = (2.19) . . , . > 0 > 0 , y (t) = y (t; y 0 ) (2.19), t = 0 y 0 < , t 0 y (t) < , y (t) . (2.26)

0 , y 0 D, U (y 0 ) = u0 > 0. U (y (t)). dU (y (t)) = dt
n

j =1

U (y (t)) fj (t, y (t)). yj dU (y 0 ) > 0. dt

(2.27)

t = 0


42

2.

. 2.5. .

t = 0 D y (t) (y (t) D), dU (y (t)) > 0. dt U (y (t)) , , U (y (t)) > U (y 0 ) = u0 > 0. (2.28)

D 0 ( U |0 = 0), ( (2.26)). y (t) D, t 0, (2.29) (2.28) t > 0. U (y (t)) D0 (. . 2.5), D0 = {y D 0 : U (y ) u0 }.

(2.27), (2.28) (2.29) = u0 0 > 0 , y (t)


2.4.

43

dU (y (t)) 0 . dt [0, t] t +, U (y (t)) U (y 0 ) + 0 t +, y (t) D0 , U (y ) D0 . . . 2.4.5. (0, 0)
4 dy1 /dt = y1 y2 , 4 dy2 /dt = y1 y2 . 4 4 f1 (y1 , y2 ) = y1 y2 , f2 (y1 , y2 ) = y1 y2 ,

A=

fi (0, 0) yj

=

0 0

0 0

.

, A 1,2 = 0. U (y1 , y2 ) = y1 y2 . D ­ , , 0 OY1 OY2 . V (y1 , y2 ) V (y1 , y2 ) 4 4 f1 (y1 , y2 ) + f2 (y1 , y2 ) = y2 · y1 y2 + y1 · y1 y2 = y1 y2 4 4 2 2 = y1 y2 (y1 + y2 ) = y1 y2 ((y1 - y2 )2 + 2(y1 y2 )2 ) 2(y1 y2 )3 2

3

y1 y2 > 0. , 2.4.3 () = 23 , . 2.4.6. y 0 Rn ( ) dy (t) = f (y (t)), (2.30) dt


44

2.

f (y 0 ) = . , f1 (y1 , . . . , yn ) = 0, ... fn (y1 , . . . , yn ) = 0. y 0 ­ , y (t) = y 0 t (2.30). (t, y1 , . . . , yn ), (y1 , . . . , yn ) ­ . y 0 , , y (t) = y 0 , . y (t) = y (t) + y 0 dy (t) = F (y (t)), dt F (y ) = f (y + y 0 ).

2.3.1 A = (aij ): Fi fi (0, . . . , 0) = (y ). aij = yj yj 0 ( ) . 2.4.4. y 0 ­ (2.30), fj (y ) y 0 , j = 1, . . . , n. A = fi (y 0 )/ yj : Re k < 0, k = 1, . . . , n,

y 0 . A = fi (y 0 )/ yj : {1 , . . . , n } : Re > 0, y 0 .


2.5.

45

2.5.
2.2.1-2.2.3 , : t +, . , , . (n = 2). 2.5.1. - y (t) = (y1 (t), y2 (t)) dy = Ay , dt A= a11 a21 a12 a22 . (2.31)

( (y1 , y2 )) (2.31). , , t (2.31) a11 y1 + a12 y2 dy1 = . dy2 a21 y1 + a22 y2 (2.32)

(0, 0) (2.32), . (0, 0) , . , (0, 0) (2.31) (2.32) . A. n = 2 1 , 2 . 1 = 2 , h1 = h h
11 21

,

h2 =

h h

12 22


46

2.





. 2.6. : a ­ , ­ .

C2 . 1 = 2 , , ; , . A (det A = 0). 2.5.2. (1 , 2 R, 1 = 2 , 1 · 2 > 0) (2.31) y (t) = y1 (t) y2 (t) = C1 h h
11 21

exp{1 t} + C2

h h

12 22

exp{2 t},

C1 , C2 R. (2.33) , : 2 < 1 < 0. . t + : y (t) . , . dy1 C1 h11 1 e1 t + C2 h12 2 e2 t C1 h11 1 + C2 h12 2 e( = = 1 t + C h e2 t dy2 C1 h21 1 e C1 h21 1 + C2 h22 2 e( 2 22 2 2 - 1 < 0, C1 = 0 dy1 h dy2 h
11 21 2 -1 )t 2 -1 )t

. (2.34)

t +,


2.5.

47

h1 . C1 = 0, y (t) = C2 h h
12 22

e2 t .

, , h2 , t +. t -. , , . (2.33) C2 = 0 dy1 C1 h11 1 e( = dy2 C1 h21 1 e(
1 -2 )t 1 -2 )t

h12 + C2 h12 2 , h22 + C2 h22 2

t -,

(1 - 2 > 0),

h2 . C2 = 0, y (t) = C1 h h
11 21

e1 t ,

, h1 . . 2.6, , t. 0 < 1 < 2 , , , . : , . 2.5.3. (1 = 2 = 0, dim ker(A - 1 E ) = 2) = 1 = 2 h1 h2 A. (2.33) y (t) = (C1 h1 + C2 h2 ) exp{t}


48

2.





. 2.7. : a ­ , ­ .





. 2.8. : a ­ , ­ .

(y1 , y2 ) , < 0 ( ) > 0 ( ), t + (. . 2.7). 2.5.4. (1 = 2 = 0, dim ker(A - 1 E ) = 1) , 1 = 2 < 0, , 1 = 2 > 0. = 1 = 2 h1


2.5.

49

. 2.9. .

A p1 . (2.31) y (t) = C1 h1 exp{t} + C2 (p1 + th1 ) exp{t}. C2 . = 0, y (t) = C1 h1 exp{t} , < 0 ( > 0) t + C2 = 0, y (t) = t exp{t}(C2 h1 + o(1)), t +.

, t + < 0 t - > 0. t - > 0 t + < 0 , t. 2.8. 2.5.5. (1 , 2 R, 2 < 0 < 1 ) , . (2.33). C1 = 0


50

2.





. 2.10. : a ­ , ­ .

t + y (t) = exp{1 t} C1 h h
11 21

+ C2

h h

12 22

exp{(2 - 1 )t} = h h
11 21

= exp{1 t} C1 , (2.34) ,

+ o(1) .

dy1 h11 , dy2 h21 t + , h1 . C1 = 0, y (t) = C2 h2 exp{2 t}, , h2 , t +. t - : C2 = 0 , h2 . C2 = 0, y (t) = C1 h1 exp{1 t}, , h1 , t -. 2.9. 2.5.6. (
1 ,2

= ± i C, = 0, = 0)

, A


2.5.

51

. h = h1 + ih2 ­ h1,2 , 1 = + i . z (t) = h exp{1 t} : y 1 (t) = Re z (t) = exp{ t} h1 cos t - h2 sin t , y 2 (t) = Im z (t) = exp{ t} h1 sin t + h2 cos t . y (t) = C1 y 1 (t) + C2 y 2 (t) = = exp{ t} C1 cos t + C2 sin t h1 + exp{ t} C2 cos t - C1 sin t h2 . C =
2 2 C1 + C2 = 0

C1 C2 , cos = , C C , h1 h2 : y (t) = 1 (t)h1 + 2 (t)h2 . sin = 1 (t) = C exp{ t} sin( t + ), 2 (t) = C exp{ t} cos( t + ),

, t + 2 2 < 0 ( , 1 (t) + 2 (t) 0) 2 2 > 0 ( , 1 (t) + 2 (t) +). 2.10. 2.5.7. (
1 ,2

= ±i C, = 0)

, A . , ­ , . h = h1 + ih2 h1 h2 y (t) = 1 (t)h1 + 2 (t)h2 1 (t) = C sin( t + ), 2 (t) = C cos( t + ),


52

2.


. 2.11. .



2 2 1 (t) + 2 (t) = C 2 . (1 (t), 2 (t)) , (. . 2.11).

2.5.8. A (det A = 0) . . 1 = 0, 2 = 0, h1 , h2 ­ . y (t) = C1 h1 + C2 h2 exp{2 t}. , h1 , . , h2 , t + 2 < 0 t - 2 > 0. 2.12 2.12b. 1 = 2 = 0 dim ker A = 2, h1 h2 . A , (2.31) y (t) = C1 h1 + C2 h2 .


2.5.

53




. 2.12. .



. 1 = 2 = 0 dim ker A = 1, h. p. (2.31) y (t) = C1 h + C2 (p + th) = (C1 + C2 t)h + C2 p. , h, . , h, , C2 > 0 C2 < 0. 2.12. 2.5.9. y 0 Rn dy (t) = f (y (t)) dt , A = aij , aij = fi (y ), yj 0 i, j = 1, . . . , n (2.36) (2.35)


54

2.

n . 2.4.4. , (2.35) dy (t) = Ay (t) dt (2.37)

. (n = 2) (2.37), : , . , , (2.37) (2.36). 2.5.1. dy1 /dt = y1 - 1, 2 2 dy2 /dt = y1 - y2 . y1 - 1 = 0, 2 2 y1 - y2 = 0, : (1, ±1) .
2 2 f1 (y1 , y2 ) = y1 - y2 , 2 2 f2 (y1 , y2 ) = y1 - y2 ,



f1 = 1, y1

f1 = 0, y2

f2 = 2y1 , y1

f2 = -2y2 . y2

10 2 -2 1 = 1, 2 = -2. (1, 1) ­ . 10 (1, -1) A = 22 1 = 1, 2 = 2. (1, -1) ­ . (1, 1) A =


3.1.

55

3
3.1.
. n- , , (n - 1)- . , . . y . F , t, y (t) y (t). , y (t) y (t) = F (t, y (t), y (t)), t
0

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)


56 u(0) = u0 ,

3. 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, y (0) = 0, y ( ) = y1 . (3.6) 0 x (3.5)

(3.5) c1 sin x + c2 cos x. y (0) = 0 , y (x) = c1 sin x. y1 = 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, ]. 3.1.1. a0 (x)y (x) + a1 (x)y (x) + a2 (x)y (x) = f1 (x), 1 y (0) + 1 y (0) = u0 , 0 x l, (3.7) (3.8)

2 y (l) + 2 y (l) = u1 ,


3.1.

57

ai (x), i = 0, 1, 2, f1 (x) 1 , 1 , 2 , 2 . y (x) C 2 [0, l], (3.7), (3.8). , ai (x), i = 0, 1, 2, f1 (x) 2 2 , a0 (x) = 0, 1 , 1 , 2 , 2 , i + i > 0, i = 1, 2. (3.7).
x

a0 (x), p(x) = exp , d dx p(x) dy dx
0

a1 (s) a0 (s)

ds . -

- q (x)y = f2 (x),

0

x

l,

(3.9)

p(x) ­ [0, l], p(x) > 0, q (x) = - p(x)a2 (x) , a0 (x) f2 (x) = p(x)f1 (x) a0 (x)

[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,

1 z (0) + 1 z (0) = 0, f (x) = f2 (x) - d dx

2 z (l) + 2 z (l) = 0, p(x) dv dx + q (x)v .

v (x), (3.8), , .


58

3.

, d dx p(x) dy dx - q (x)y = f (x), 0 x l, (3.10) (3.11)

1 y (0) + 1 y (0) = 0,

2 y (l) + 2 y (l) = 0.

. (3.10), (3.11) , f (x) = 0 . 3.1.3. , . Ly = d dx p(x) dy dx - q (x)y .

y (x) C 2 [0, l] z (x) C 2 [0, l], Ly Lz , z (x)Ly - y (x)Lz = z (x) z (x) z (x)Ly - y (x)Lz = d dy dz p(x) z (x) - y (x) dx dx dx , 0 x l. (3.12) d dx p(x) dy dx - y (x) d dx p(x) dz dx = d dy dz p(x) z (x) - y (x) dx dx dx , d dx p(x) dy dx - y (x) d dx p(x) dz dx .

. . y1 (x), y2 (x) ­ Ly = 0, Ly1 = Ly2 = 0. y1 (x), y2 (x) (3.12), d dy2 dy1 p(x) y1 (x) - y2 (x) dx dx dx = 0, 0 x l. (3.13)


3.2. , W [y1 , y2 ](x) = y1 (x)y2 (x) - y2 (x)y1 (x)

59

p(x)W [y1 , y2 ](x) = c, 0 x l, c ­ , c W [y1 , y2 ](x) = , 0 x l. (3.14) p(x) 3.1.4. (3.12) 0 l,
l x=l

(z (x)Ly - y (x)Lz ) dx = p(x) z (x)y (x) - y (x)z (x)
x=0 0

.

(3.15)

. , , y (x) z (x) (3.11),
l

(z (x)Ly - y (x)Lz ) dx = 0.
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 = 0, y (0) = 0, z (0) = 0, (3.17) . 1 = 0 1 y (0) + 1 y (0) = 0, 1 z (0) + 1 z (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) .


60

3.

3.2. .
Ly d dx p(x) dy dx - q (x)y = f (x), 0 x l, (3.18) (3.19) (3.20)

1 y (0) + 1 y (0) = 0, 2 y (l) + 2 y (l) = 0,

p(x), q (x), f (x) ­ , 1 , 1 , 2 , 2 ­ , p(x) C 1 [0, l], p(x) > 0, x [0, l], q (x), f (x) 2 2 C [0, l], i + i > 0, i = 1, 2. 3.2.1. y (x) (3.18)-(3.20), y (x) C 2 [0, l] (3.18)-(3.20). 3.2.1. , (3.18)-(3.20). 3.2.2. G(x, ) (3.18)-(3.20), [0, l] â [0, l] : 1) (0, l) G(x, ) x [0, ) ( , l] d dx p(x) dG(x, ) dx - q (x)G(x, ) = 0, 0 x l, x = .

2) G(x, ) x: 1 Gx (0, )+1 G(0, ) = 0, 2 Gx (l, )+2 G(l, ) = 0, (0, l).


3.2.

61

3) G(x, ) [0, l] â [0, l], Gx (x, ) x = Gx ( + 0, ) = lim Gx (x, ),
x +0

Gx ( - 0, ) = lim Gx (x, ),
x -0

Gx ( + 0, ) - Gx ( - 0, ) = 1 , p( ) (0, l).

3.2.2. 3.2.1. Lv = 0, 1 v (0) + 1 v (0) = 0, 2 v (l) + 2 v (l) = 0 (3.21)

, (3.18)-(3.20) . . y1 (x) Ly1 = 0, 0 x l, y1 (0) = -1 , y1 (0) = 1 ,

y2 (x) Ly2 = 0, 0 x l, y2 (l) = -2 , y2 (l) = 2 .

, y1 (x) (3.19), y2 (x) (3.20): 1 y1 (0) + 1 y1 (0) = 0, 2 y2 (l) + 2 y2 (l) = 0. (3.22)

y1 (x) y2 (x) , . : G(x, ) = c1 ( )y1 (x), 0 c2 ( )y2 (x), x x , l,

c1 ( ) c2 ( ) . , G(x, ) 1) 2) . c1 ( ) c2 ( ) , 3). G(x, ) x = , c1 ( )y1 ( ) = c2 ( )y2 ( ).


62

3.

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 ­ . y1 (x)y2 ( ) , 0 x , g0 G(x, ) = (3.23) y ( )y (x) 1 2 , x l. g0 . . , : G(x, ) G(x, ). ­ (0, l). z (x) = G(x, ) - G(x, ). [0, l] z (x), Gx (x, ) Gx (x, ) x = . Lz = 0, x = , z (x) = q (x)z (x) - p (x)z (x) , p(x)

x = x ± 0. z (x) x = , Lz = 0, 0 x l,

(3.19), (3.20). [0, l] . z (x) = 0, G(x, ) = G(x, ), 3.2.1 .


3.2. 3.2.1. y (x) + a2 y (x) = f (x), y (0) = 0, y (l) = 0, 0 x l,

63

a = nl-1 , n = 1, 2, . . . . y1 (x) = sin ax, y2 (x) = sin a(x - l). , yi (x) + a2 yi (x) = 0, g0 = p(x)W (x) = y1 (x)y2 (x) - y2 (x)y1 (x) = a sin al. (3.23) , sin ax sin a( - l) , 0 x , a sin al (3.24) Ga (x, ) = sin a sin a(x - l) , x l. a sin al 3.2.3. (3.18)-(3.20). 3.2.2. (3.21) , (3.18)-(3.20) ,
l

i = 1, 2,

y1 (0) = y2 (l) = 0.

y (x) =
0

G(x, )f ( )d ,

0

x

l.

(3.25)

. , y (x), (3.25), (3.18)-(3.20). (3.23) ,
x l

y2 (x) y (x) = g0
0

y1 (x) y1 ( )f ( )d + g0
x

y2 ( )f ( )d .

(3.26)


64

3.


x l

y (x) = d dx p(x) dy dx

y2 (x) g0
0

y1 ( )f ( )d +

y1 (x) g0
x

y2 ( )f ( )d .

(3.27)

=

y1 (x)y2 (x) - y2 (x)y1 (x) p(x) f (x)+ g0
x l

1d + g0 dx

dy2 p(x) dx
0

1d y1 ( )f ( )d + g0 dx

dy1 p(x) dx
x

y2 ( )f ( )d .

Ly1 = Ly2 = 0, y1 (x)y2 (x) - y2 (x)y1 (x) p(x) = g0 , Ly =
x

d dx

p(x)

dy dx

- q (x)y (x) =
l

= f (x) +

Ly2 g0
0

y1 ( )f ( )d +

Ly1 g0
x

y2 ( )f ( )d = f (x).

, y (x) (3.18). (3.19), (3.20). (3.26), (3.27) (3.22) ,
l

1 y1 (0) + 1 y1 (0) 1 y (0) + 1 y (0) = g0
0

y2 ( )f ( )d = 0.

(3.20). . y (x) (3.18)-(3.20). v (x) = y (x) - y (x) (3.21) [0, l] , y (x) - y (x) 0, 3.2.2 . 3.2.4. -


3.2. . y (x) + a2 y (x) = F (x, y (x)), y (0) = y (l) = 0. 0 x l,

65

(3.28) (3.29)

3.2.3. F (x, y ) x [0, l] y R y : |F (x, y1 ) - F (x, y2 )| L|y1 - y2 |, x [0, l], y 1 , y 2 R.

lL(a| sin al|)-1 < 1, (3.28), (3.29) . . y (x) - (3.28), (3.29). f (x) = F (x, y (x)). y (x) y (x) + a2 y (x) = f (x), y (0) = 0, y (l) = 0, 0 x l,

(3.24). ,
l

y (x) =
0

Ga (x, )f ( )d ,

0

x

l.

f (x),
l

y (x) =
0

Ga (x, )F ( , y ( ))d ,

0

x

l.

(3.30)

, , , y (x) ­ (3.28), (3.29), (3.30). . y (x) [0, l] (3.30). (3.24), (3.30) , y (x) (3.29). (3.30) y (x)


66

3.

y (x) (3.28), , y (x) . , (3.30) (3.28), (3.29). , , (3.28), (3.29) (3.30). (3.30), [0, l]. y0 (x) = 0,
l

y

n+1

(x) =
0

Ga (x, )F ( , yn ( ))d ,

0

x

l,

n = 0, 1, 2, . . . . (3.31)

yn (x) [0, l]. , |y
n+1

(x) - yn (x)|

M

lL a| sin al|

n

,

0

x

l,

n = 0, 1, 2, . . . , (3.32)


l

M = max |y1 (x)| = max
0xl

0xl 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,
l

|y

m+1

(x) - ym (x)|
0 l

|Ga (x, )||F ( , ym ( )) - F ( , y lL a| sin al|

m-1

( ))|d

L a| sin al|
0

m

|ym ( ) - y

m-1

( )|d

M

,

0

x

l.

, (3.32) .
k

yk (t) =
n=1

(yn (t) - y

n-1

(t)),


3.3. -

67

yk (t) [0, l]


(yn (t) - y
n=1

n-1

(t)).

(3.32) , [0, l]. , yk (x) [0, l] y (x). yk (t) , y (x) [0, l]. (3.31) n , , y (x) (3.30). , (3.28), (3.29). (3.28), (3.29). , (3.30) . , y1 (x), y2 (x), (3.30).
l

y1 (x) - y2 (x) =
0

Ga (x, ) F ( , y1 ( )) - F ( , y2 ( )) d ,

0

x

l.

Ga (x, ),
l

|y1 (x) - y2 (x)|
0

|Ga (x, )|L|y1 ( )) - y2 ( )|d < < max |y1 (x) - y2 (x)|,
0xl

0

x

l.

y1 (x) = y2 (x). , 3.2.3 .

3.3. -
Ly = d dx p(x) dy dx - q (x)y = -y , 0 x l, (3.33)


68

3. 1 y (0) + 1 y (0) = 0, 2 y (l) + 2 y (l) = 0, (3.34) (3.35)

p(x), q (x) ­ , 1 , 1 , 2 , 2 ­ , p(x) C 1 [0, l], p(x) > 2 2 0, x [0, l], q (x) C [0, l], i + i > 0, i = 1, 2 ­ . , (3.33)-(3.35) y (x) = 0. 3.3.1. 1 (3.33)(3.35) y1 (x), 1 , y1 (x) . -. , , , y (x) ­ , cy (x), c ­ , . (3.33) L. , (3.34), (3.35) . , Ly = -y (x) , . , . -, , . (3.33)-(3.35). -. 3.3.1. - . . 1 ­ , y1 (x) ­ . , , 1 = a + ib, y1 (x) = u(x) + iv (x).


3.3. -

69

y1 (x) (3.33), Ly1 = -1 y1 (x). , Lu = -au(x) + bv (x), (3.36) Lv = -bu(x) - av (x). (3.37)

y1 (x) (3.34), (3.35), u(x), v (x) . (3.36) v (x), (3.37) u(x), 0 l .
l l

v (x)Lu - u(x)Lv dx = b
0 0

u2 (x) + v 2 (x) dx.


l

v (x)Lu - u(x)Lv dx = 0,
0

(3.38)


l

b
0

u2 (x) + v 2 (x) dx = 0.

, b = 0. 1 y1 (x) . 3.3.2. . . y1 (x), y2 (x). , (3.33) (3.34), (3.35). (3.34) , W [y1 , y2 ](0) = 0. y1 (x), y2 (x) ­ (3.33), y2 (x) = cy1 (x).


70

3. v (x) w(x)
l

(v , w) =
0

v (x)w(x)dx.

v (x) w(x) , , (v , w) = 0. 3.3.3. , , . . 1 = 2 ­ , y1 (x), y2 (x) ­ . y1 (x), y2 (x) (3.34), (3.35), (3.16) ,
l

(Ly1 , y2 ) - (y1 , Ly2 ) =
0

y2 (x)Ly1 - y1 (x)Ly2 dx = 0.

Ly1 = -1 y1 (x) , Ly2 = -2 y2 (x), (1 - 2 )(y1 , y2 ) = 1 (y1 , y2 ) - 2 (y1 , y2 ) = = (1 y1 , y2 ) - (y1 , 2 y2 ) = -(Ly1 , y2 ) + (y1 , Ly2 ) = 0. , (1 - 2 )(y1 , y2 ) = 0, (y1 , y2 ) = 0 y1 (x), y2 (x) . 3.3.4. 1 = 2 = 0. , ­ , min q (x). (3.39)
0xl

. , 1 ­ , y1 (x) ­ 1 < min q (x).
0xl

q (x) - 1 > 0 [0, l]. (3.33) , d dx p(x) dy1 dx = (-1 + q (x))y1 (x).


3.3. - 0 x,
x

71

p(x)y1 (x) = p(0)y1 (0) +
0

(q (s) - 1 )y1 (s)ds.

(3.40)

y1 (x) (3.34), (3.35) 1 = 2 = 0, y1 (0) = y1 (l) = 0. y1 (x) ­ (3.33), y1 (0) = 0. y1 (0) > 0. y1 (x) > 0 x [0, l]. , . x0 , y1 (x0 ) = 0. x [0, x0 ) y1 (x) > 0, y1 (x) > 0 x (0, x0 ). (3.40) x = x0 q (x) - 1 , , y1 (x0 ) > 0. y1 (x) x [0, l]. y1 (x) > 0 x (0, l], y1 (l) = 0. , (3.39) . -. 3.3.1. p(x) = 1, q (x) = 0, 1 = 2 = 0, l = . - y (x) + y (x) = 0, 0 x , (3.41) (3.42)

y (0) = y ( ) = 0.

. = -µ . (3.41) y (x) = c1 exp{ µx} + c2 exp{- µx}. x = 0, x = l (3.42), c1 c2 c1 + c2 = 0, c1 exp{ µ } + c2 exp{- µ } = 0, , c1 = c2 = 0. . , 3.3.4. , = 0 .


72

3.

. (3.41) y (x) = c1 sin x + c2 cos x. , c2 = 0. sin = 0. n = n2 , n = 1, 2, . . . .

yn (x) = c sin nx, c ­ . 3.3.1. , -. - (3.33)(3.35). , . yn (x), n = 1, 2, . . . . , , , ,
l

(yn (x))2 dx = 1.
0

f (x) [0, l] .
l

fn =
0

f (x)yn (x)dx,

n = 1, 2, . . . .

, . 3.3.5. ( ) f (x) C 2 [0, l] (3.34), (3.35),


fn yn (x)
n=1


3.3. - [0, l] f (x),


73

f (x) =
n=1

fn yn (x),

0

x

l.


74

4.

4
4.1.
4.1.1. n dx1 (t) = f (t, x (t), . . . , x (t)), 1 1 n dt . . (4.1) . dx (t) n = fn (t, x1 (t), . . . , xn (t)), dt fi (t, x) D1 Rn+1 fi (t, x)/ xj , i, j = 1, . . . , n. C 1 (D1 ) D1 . 4.1.1. (4.1) D1 v (t, x1 , . . . , xn ) C 1 (D1 ), D1 (4.1). , x(t) = (x1 (t), . . . , xn (t)) (4.1) C , v (t, x1 (t), . . . , xn (t)) C. (4.2)

(, ..).


4.1. 4.1.2.

75

(4.1). 4.1.2. v (t, x1 , . . . , xn ) C 1 (D1 ) (4.1) dv dt =
(4.1)

v (t, x) + t

n

j =1

v (t, x) fj (t, x), xj

(t, x) D1 .

4.1.1. v (t, x1 , . . . , xn ) C 1 (D1 ) (4.1) D1 , (4.1) D1 : dv dt = 0,
(4.1)

(t, x) D1 .

(4.3)

. v (t, x1 , . . . , xn ) C 1 (D1 ) (4.1) D1 . D1 (t, x(t)), x(t) ­ (4.1), (4.2). (4.2) t dxj (t)/dt (4.1), v (t, x(t)) + t
n

0

j =1

v (t, x(t)) dxj (t) = xj dt = v (t, x(t)) + t
n

j =1

v (t, x(t)) fj (t, x(t)). xj

, (4.1) . (t0 , x0 ) D1 (4.1) x(t0 ) = x0 , (4.3) D1 . , v (t, x1 , . . . , xn ) C 1 (D1 ) (4.3). , (4.3) -


76

4.

(t, x(t)) D1 . 0 v (t, x(t)) + t
n

j =1

v (t, x(t)) fj (t, x(t)) = xj
n

=

v (t, x(t)) + t

j =1

v (t, x(t)) dxj (t) d = v (t, x(t)) . xj dt dt

v (t, x(t)) t , v (t, x(t)) C . v (t, x) ­ (4.1). 4.1.3. v (t, x1 , . . . , xn ) C 1 (D1 ) (4.1) D1 , C0 ­ , D1 , j {1, . . . , n} v (t, x)/ xj = 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 , , t = t0 v (t, x) = C0 . 4.1.4. v1 (t, x), . . . , vk (t, x) ­ (4.1). Rk (y1 , . . . , yk )


4.1. (t, x) = (v1 (t, x), . . . , vk (t, x))

77

(4.1). 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 x(t) = (x1 (t), . . . , xn (t)) dxk (t) = fk (t, x1 (t), . . . , xn (t)), dt k = 1, . . . , n, x(t0 ) = x0 (4.4)

v1 (t, x) = c0 , 1 . . (4.5) . 0 vn (t, x) = cn , c0 = vj (t0 , x0 ), j = 1, . . . , n. j . (4.5) (t0 , x0 ). , (. 4.1.3 k = n) (x1 , . . . , xn ) : vi (t0 , x0 ) = 0. det xj (. A.1.1 ) t0 xj (t) = gj (t, c0 , . . . , c0 ), 1 n j = 1, . . . , n


78

4.

, g (t) = (g1 (t), . . . , gn (t)) (4.5) : v1 (t, g (t)) = c0 , 1 . . (4.6) . 0 vn (t, g (t)) = cn . x(t) ­ (4.4). vj (t, x(t)) = vj (t0 , x(t0 )) = vj (t0 , x0 ) = c0 , j j = 1, . . . , n.

, x(t) (4.6), g (t). t0 : x(t) g (t). , . 4.1.2. (4.1), fj = fj (x), j = 1, . . . , n,

x0 ,
n 2 fj (x0 ) = 0, j =1

(n - 1) t (4.1).

4.2.
4.2.1. u(x) = u(x1 , . . . , xn ) ­ x = (x1 , . . . , xn ) D0 , D0 ­ Rn . F x1 , . . . , xn , u, u u ,..., x1 xn =0


4.2.

79

, F (x1 , . . . , xn , u, p1 , . . . , pn ) n . , ,
n

aj (x1 , . . . , xn , u(x))
j =1

u(x) = b(x1 , . . . , xn , u(x)), xj

aj (x, u) = aj (x1 , . . . , xn , u), b(x, u) = b(x1 , . . . , xn , u) D1 Rn+1 ,
n

D1
j =1

a2 (x, u) = 0. j

, u, :
n

aj (x)
j =1

u(x) = 0, xj

aj (x) D0 Rn ,
n

D0
j =1

a2 (x) = 0. , j

. 4.2.1. u(x) D0 Rn , 1. u(x) D0 ( u(x) C 1 (D0 )); 2. x D0 (x, u(x)) D1 ; 3. u(x) D0 .


80

4.

4.2.2. D0 Rn a1 (x) u u u + a2 (x) + · · · + an (x) = 0, x1 x2 xn
n

(4.7) (4.8)

aj (x) C 1 (D0 ), j = 1, . . . , n,
j =1

a2 (x) = 0, j

x D0 .

(4.7) n- dx1 (t) = a (x (t), . . . , x (t)), 11 n dt . . (4.9) . dx (t) n = an (x1 (t), . . . , xn (t)). dt 4.2.2. x(t) = (x1 (t), . . . , xn (t)) (4.9) Rn , (4.7). (4.9) (4.7) . 4.2.1. u(x) C 1 (D0 ) (4.7) , u(x) t (4.9) D0 . . u(x) t (4.9) D0 . 4.1.1 (4.9) D0 : du dt
n

=
(4.9) j =1

u(x) aj (x) = 0, xj

x D0 .

u(x) ­ (4.7).


4.2.

81

, u(x) ­ (4.7). u(x) (4.9), D0 . 4.1.1 , u(x) (4.9) D0 . 4.2.1. D0 (4.9) n - 1 t v1 (x1 , . . . , xn ), v2 (x1 , . . . , xn ), ..., vn
-1

(x1 , . . . , xn ).

M0 (x0 , . . . , x0 ) n 1 D0 (4.7) u(x) = F (v1 (x), v2 (x), . . . , vn F (y1 , . . . , y .
n-1 -1

(x)),

(4.10)

) ­

. vj (x) ­ (4.9), j = 1, . . . , n - 1, F (y1 , . . . , yn-1 ) u(x), (4.10), , t. 4.2.1 u(x) ­ (4.7). , (4.10) (4.7) M0 (x0 , . . . , x0 ) D0 . u(x) ­ n 1 (4.10). v1 (x), . . . , vn-1 (x) (4.9), 4.2.1 (4.7). , n u(x) = 0, aj (x) xj j =1 n v1 (x) aj (x) = 0, xj x D0 . (4.11) j =1 . . . n vn-1 (x) aj (x) = 0, xj
j =1


82

4.

(4.8) x D0 (4.11) a1 (x), . . . , an (x) . , , D(u, v1 , . . . , vn-1 ) = 0, D(x1 , x2 , . . . , xn ) x D0 .

v1 (x), . . . , vn-1 (x) (n-1) . M0 F (y1 , . . . , yn-1 ) , M0 (4.10). 4.2.3. D Rn+1 a1 (x, u(x)) u u + a2 (x, u(x)) + ... x1 x2 · · · + an (x, u(x)) aj (x, u), b(x, u) C 1 (D),
n

u = b(x, u(x)), (4.12) xn

j = 1, . . . , n,

a2 (x, u) = 0, j
j =1

(x, u) D.

dx1 dt = dxn dt = du = dt

(4.12) (n + 1)- . a1 (x, u), . . . an (x, u), b(x, u). (4.13)


4.2.

83

4.2.3. (x1 (t), . . . , xn (t), u(t)) (4.13) Rn+1 , (4.12). (4.13) (4.12) . 4.2.2. v (x, u) ­ t (4.13) D, N0 (x0 , . . . , x0 , u0 ) n 1 D v (N0 ) = C0 , v (N0 ) = 0. u (4.14)

N0 v (x1 , . . . , xn , u) = C0 (4.15)

u = u(x1 , . . . , xn ), (4.12). . v (x, u) t (4.13). 4.1.1 (4.13) D: dv dt
n

=
(4.13) j =1

v (x, u) v (x, u) aj (x, u) + b(x, u) = 0, xj u (x, u) D. (4.16)

(4.15) (4.14) M0 (x0 , . . . , x0 ), n 1 u = u(x1 , . . . , xn ), (4.15) : v (x1 , . . . , xn , u(x1 , . . . , xn )) C0 . v v u =- · , xj u xj j = 1, . . . , n.


84

4.

(4.16) v / u = 0
n

aj (x, u(x))
j =1

u = b(x, u(x)) xj

M0 . u(x) ­ (4.12). (4.13) (n + 1). , 4.1.2 , D n t v1 (x, u), ..., vn (x, u).

F (y1 , . . . , yn ) w(x, u) = F (v1 (x, u), . . . , vn (x, u)) (4.13). 4.2.2 w/ u = 0 u(x), F (v1 (x, u), . . . , vn (x, u)) = 0, (4.17)

(4.12). , (4.17) (4.12) N0 . 4.2.4. u = f (x1 , . . . , xn ) C 1 (D0 ) (4.12) n- (x1 , . . . , xn , u). .


4.2.

85

. 4.1. 4.2.3.

4.2.3. u = f (x1 , . . . , xn ) C 1 (D0 ) (4.12) , , (4.13) , ). . P = {u = f (x1 , . . . , xn ), (x1 , . . . , xn ) D0 },

( -

(4.18)

f (x1 , . . . , xn ) C 1 (D0 ), = {(x1 (t), . . . , xn (t), u(t))} P , . (4.13) = dx1 (t) dxn (t) du(t) ,..., , = dt dt dt = (a1 (x(t), u(t)), . . . , an (x(t), u(t)), b(x(t), u(t))),

u(t) = f (x(t)). P ,


86

4.

P . n= ( , n)R a1 (x, u)
n+1

f f (x(t)), . . . , (x(t)), -1 . x1 xn

= 0, (x, u) . (4.19)

f f (x) + · · · + an (x, u) (x) - b(x, u) = 0, x1 xn

, u = f (x) (4.12) . , (4.12) D0 . , u = f (x) ­ (4.12) D0 . , M0 (x0 , . . . , x0 , u0 ) P P n 1 (x0 , . . . , x0 , u0 ). n 1 (x0 , . . . , x0 ), n 1 dx1 = a (x, f (x)), x (t ) = x0 , 10 1 1 dt . . (4.20) . dx n = an (x, f (x)), xn (t0 ) = x0 , n dt x(t) = (x1 (t), . . . , xn (t)). = {(x1 = x1 (t), . . . , xn = xn (t), u(t) = f (x1 (t), . . . , xn (t)))}. (4.21)

P . , ­ , (4.13). n (4.20). (4.13). , xi (t), i = 1, . . . , n, (4.20), u = f (x) (4.12), du = dt
n

j =1

f dxj (x(t)) · (t) = xj dt

n

j =1

f (x(t)) aj (x(t), u(t)) = b(x(t), u(t)). xj

, ­ . , , P .


4.2.

87

4.2.5. n = 2, , a1 (x, y , u) u u + a2 (x, y , u) = b(x, y , u), x y (4.22)

b(x, y , u), aj (x, y , u) C 1 (D), j = 1, 2, D ­ R3 , a2 (x, y , u) + a2 (x, y , u) = 0, 1 2 (x, y , u) D.

(4.22) u = f (x, y ), (4.22) = {(x, y , u) = (1 (s), 2 (s), 3 (s)), s [s1 , s2 ]} D, 3 (s) = f (1 (s), 2 (s)), s [s1 , s2 ]. (4.23) 4.2.4. det a1 (s) 1 (s) a2 (s) 2 (s) = 0, s [s1 , s2 ], (4.24)

aj (s) = aj (1 (s), 2 (s), 3 (s)), j = 1, 2. (4.22), (4.23).

-

. (4.22): dx = a (x, y , u), 1 dt dy (4.25) = a2 (x, y , u), dt du = b(x, y , u). dt


88

4.

. 4.2. 4.2.4.

(4.25) t = 0 x|t=0 = 1 (s), y |t=0 = 2 (s), u|t=0 = 3 (s) (4.26) x = 1 (t, s), (4.26), (4.27) 1 (0, s) = 1 (s), 2 (0, s) = 2 (s), 3 (0, s) = 3 (s), s [s1 , s2 ]. (4.28) y = 2 (t, s), u = 3 (t, s). (4.27)

(4.27) P . (4.26) (. . 4.2). , u = f (x, y ), , 4.2.3, f (x, y ) ­ (4.22). (4.27) x = 1 (t, s), y = 2 (t, s), (4.29)

(t, s) (x, y ). ,


4.2.

89

, t = 0. (4.25) dx 1 (0, s) = t dt = a1 (s), 2 dy (0, s) = t dt = a2 (s).

t=0

t=0

(4.26) , 1 (0, s) = 1 (s), s 1 1 t s det 1 2 t s 2 (0, s) = 2 (s). s

(4.24) (0, s) = det a1 (s) 1 (s) a2 (s) 2 (s) = 0, s [s1 , s2 ].

, (x0 , y0 ) = (1 (0, s), 2 (0, s)) t = t(x, y ), s = s(x, y ),

(4.29) . (4.27) u = 3 (t(x, y ), s(x, y )) = f (x, y ). , , 4.2.3, ( (4.27)), . (4.24) . = (a1 , a2 , b) , (1 , 2 , 3 ) , , (4.24) (a1 , a2 ) (1 , 2 ) (x, y ). , (. . 4.2).


90

5.

5
5.1.
M , C [x0 , x1 ]. 5.1.1. M . . M C [x0 , x1 ]. [y (x)] : [y (x)] = y (x0 )+2y (x1 ). , ,
x1

[y (x)] =
x0

y (x)dx.

. M [x0 , x1 ] , y (x0 ) = y0 , y (x1 ) = y1 , y0 , y1 ­ .
x1

[y (x)] =
x0

y (x) + 2(y (x))2 dx.

5.1.1. 5.1.2. y0 (x) M y (x) , y0 (x) + y (x) M . , M , y (x) ­ y0 (x), t y (x) y0 (x) t R.


5.1. 5.1.3. [y0 (x), y (x)] [y (x)] y0 (x) M d [y0 (x) + t y (x)] dt .
t=0

91

, , , . M = C [x0 , x1 ].
x1

[y (x)] =
x0

(y (x))2 dx.

[y0 (x), y (x)] =
x1

d [y0 (x) + t y (x)] dt
x1

=
t=0

=

d dt
x0

[y0 (x) + t y (x)]2 dx

=2
t=0 x0

y0 (x) y (x)dx,

[y0 (x), y (x)] y0 (x).
x1

[y (x)] =
x0

|y (x)|dx

y0 (x) = 0, y (x) = 1, [y0 (x), y (x)] = d [y0 (x) + t y (x)] dt =
t=0

d (x1 - x0 )|t| dt

,
t=0

. 5.1.2. 5.1.4. [y (x)] y0 (x) M () M , y (x) M [y0 (x)] [y (x)] ([y0 (x)] [y (x)]).


92

5.

M y (x), y (x) = max |y (x)|.
x0 x x1

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) + t y (x)], y (x) y0 (x). y0 (x) y (x) [y0 (x) + t y (x)] t : (t) = [y0 (x) + t y (x)]. [y (x)] y0 (x) , (t) t = 0 . , (0) , (0) = 0. (0) [y (x)] y0 (x) d (t) dt , [y0 (x), y (x)] = d [y0 (x) + t y (x)] dt =0
t=0

=
t=0

d [y0 (x) + t y (x)] dt

.
t=0

y (x). 5.1.1 .


5.1.

93

. 5.1. 5.1.1.

5.1.3. , , . , C n [x0 , x1 ], n N n n [x0 , x1 ] . C0 [x0 , x1 ] n y (x) C [x0 , x1 ] , y
(m)

(x0 ) = y

(m)

(x1 ) = 0,

m = 0, 1, . . . , n - 1.

5.1.1. f (x) ­ [x0 , x1 ] ,
x1

f (x)y (x)dx = 0
x0

y (x)

n C0

[x0 , x1 ]. f (x) 0 [x0 , x1 ].

. , f (x) [x0 , x1 ]. x2 (x0 , x1 ) , f (x2 ) = 0. f (x2 ) > 0. f (x) > 0 , f (x) f (x2 ) > 0, 2 x [x2 - , x2 + ] (x0 , x1 ).

y2 (x) (. . 5.1): y2 (x) = (x - (x2 - ))n 0,
+1

((x2 + ) - x)n

+1

, x [x2 - , x2 + ]; x [x2 - , x2 + ].


94

5.

n y2 (x) C0 [x0 , x1 ] y2 (x) > 0 x (x2 - , x2 + ). , x1 x2 +

f (x)y2 (x)dx =
x0 x2 -

f (x)y2 (x)dx > 0,

. 5.1.1 .

5.2.
M [x0 , x1 ] y (x) , y (x0 ) = y0 , y (x1 ) = y1 .
x1

[y (x)] =
x0

F (x, y (x), y (x))dx,

(5.1)

F (x, y , p) ­ . M . 5.2.1. , x [x0 , x1 ], (y , p) R2 F (x, y , p) . (5.1) y0 (x) M , [x0 , x1 ], y0 (x) Fy (x, y (x), y (x)) - d Fp (x, y (x), y (x)) = 0, dx x0 x x1 . (5.2)

. (5.1) y0 (x). M , y (x) y0 (x) [x0 , x1 ] , 1 (. . 5.2). y (x) C0 [x0 , x1 ]. , [y0 (x), y (x)] = d [y0 (x) + t y (x)] dt =
t=0


5.2.

95

. 5.2. 5.2.1.
x1

d = dt
x0 x1

F (x, y0 (x) + t y (x), y0 (x) + t( y ) (x))dx

=
t=0

=
x0

Fy (x, y0 (x) + t y (x), y0 (x) + t( y ) (x)) y (x)+ =
t=0

+Fp (x, y0 (x) + t y (x), y0 (x) + t( y ) (x))( y ) (x) dx
x1

=
x0

Fy (x, y0 (x), y0 (x)) y (x) + Fp (x, y0 (x), y0 (x))( y ) (x) dx

, y0 (x) ,
x1 x1

Fy (x, y0 (x), y0 (x)) y (x)dx +
x0 x0

Fp (x, y0 (x), y0 (x))( y ) (x)dx = 0.

, y (x0 ) = y (x1 ) = 0,
x1

Fy (x, y0 (x), y0 (x)) -
x0

d Fp (x, y0 (x), y0 (x)) y (x)dx = 0. dx


96

5.

1 y (x) C0 [x0 , x1 ]. ,

Fy (x, y0 (x), y0 (x)) -

d Fp (x, y0 (x), y0 (x)) = 0, dx

x0

x

x1 .

, y0 (x) (5.2) 5.2.1 . (5.2) (5.1). y0 (x), (5.1), M , Fy (x, y (x), y (x)) - y (x0 ) = y0 , d Fp (x, y (x), y (x)) = 0, dx y (x1 ) = y1 . x0 x x1 ,

. , , , f (x) y (x). , y (x) . . f (x) , f (x0 ) = f (x1 ) = 0.
x1 x1 2

(y (x) - f (x)) dx +
x0 x0

(y (x))2 dx,

(5.3)

­ . y (x) f (x), , y (x) . (5.3) y (x) , y (x) C 1 [x0 , x1 ], y (x0 ) = y (x1 ) = 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)) - d (2y (x)) = 0. dx


5.3.

97

, y (x) y (x) - ()-1 y (x) = -()-1 f (x), y (x0 ) = y (x1 ) = 0. x0 x x1 , (5.4) (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) C n [x0 , x1 ] , y (x0 ) = y00 , y (x0 ) = y01 , y (x0 ) = y02 , . . . , y y (x1 ) = y10 , y (x1 ) = y11 , y (x1 ) = y12 , . . . , y
x1 (n-1) (n-1)

(x0 ) = y (x1 ) = y

0n-1 1n-1

, .

(5.6) (5.7)

[y (x)] =
x0

F (x, y (x), y (x), . . . , y

( n)

(x))dx,

(5.8)

F (x, y , p1 , . . . , pn ) x [x0 , x1 ], (y , p1 , . . . , pn ) Rn+1 . (5.8) M .


98

5.

5.3.1. F (x, y , p1 , . . . , pn ) x [x0 , x1 ], (y , p1 , . . . , pn ) Rn+1 2n. y (x) M , y (x) C 2n [x0 , x1 ], ¯ ¯ (5.8) M , y (x) ¯ Fy - d F dx
p1

+ · · · + (-1)n
( n)

dn F dxn

pn

= 0,

x0

x

x1 ,

(5.9)

F = F (x, y (x), y (x), . . . , y

(x)).

. (5.8) y (x) ¯ n y (x) C0 [x0 , x1 ]. [y (x), y (x)] = ¯
x1

d [y (x) + t y (x)] ¯ dt

=
t=0

d = dt
x0

F (x, y (x) + t y (x), y (x) + t( y ) (x), . . . , y (n) (x) + t( y )(n) (x))dx ¯ ¯ ¯
t=0

.

t, t = 0 ,
x1

Fy y (x) + Fp1 ( y ) (x) + · · · + Fpn ( y )(n) (x) dx = 0.
x0

, y (x) ,
x1

Fy -
x0

dn d Fp1 + · · · + (-1)n n Fpn y (x)dx = 0. dx dx

n y (x) C0 [x0 , x1 ], , , , y (x) (5.9). ¯ 5.3.1 .


5.3.

99

, , , y (x) ¯ C 2n [x0 , x1 ] (5.8) M , (5.9), (5.6), (5.7). f (x) y (x). , , y (x), .
x1 x1

y (x) - f (x) dx +
x0 x0

2

(y (x))2 + (y (x))2 dx,

(5.10)

­ . , f (x) , f (x0 ) = f (x1 ) = 0, f (x0 ) = f (x1 ) = 0 (5.10) y (x) , y (x) C 2 [x0 , x1 ], y (x0 ) = y (x1 ) = 0, y (x0 ) = y (x1 ) = 0. F (x, y , p1 , p2 ) = (y - f (x))2 + p2 + p2 , 1 2 (5.9) d d2 (2y (x)) + 2 (2y (x)) = 0. dx dx y (x0 ) = y (x1 ) = 0, y (x0 ) = y (x1 ) = 0, y (x) 2(y (x) - f (x)) - y
(4)

(x) - y (x) + ()-1 y (x) = ()-1 f (x), x0 x y (x0 ) = y (x0 ) = 0, y (x1 ) = y (x1 ) = 0.

x1 ,

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)


100

5.

. 5.3.

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 (. . 5.3). (5.11). , 5.3.1. f (x, y ) D. f (x, y )v (x, y )dxdy = 0
D

v (x, y ), D L, f (x, y ) = 0, (x, y ) D. . , f (x, y ) D. (x0 , y0 ) D , f (x0 , y0 ) = 0. f (x0 , y0 ) > 0. f (x, y )


5.3.

101

. 5.4. 5.3.1.

(x0 , y0 ) , S = {(x, y ) : (x - x0 )2 + (y - y0 )2 < 2 } f (x0 , y0 ) > 0 (x, y ) S D. 2 v0 (x, y ) , (. . 5.4) , f (x, y ) v0 (x, y ) = f (x, y )v0 (x, y )dxdy =
D S

(x - x0 )2 + (y - y0 )2 - 0,

22

, (x, y ) S ; (x, y ) D\S.

f (x, y )v0 (x, y )dxdy f (x0 , y0 ) 2
S

v0 (x, y )dxdy > 0,

. , . 5.3.1 . 5.3.2. , F (x, y , u, p, q ) (x, y ) D, (u, p, q ) R3 . (5.11) u(x, y ) ¯ M , D, Fu - Fp Fq - = 0, x y (x, y ) D. (5.12)


102

5.

. (5.11) u(x, y ) M , ¯ D. , (5.11) [u(x, y ), u(x, y )] = ¯ d dt
D

d [u(x, y ) + t u(x, y )] ¯ dt

= 0,
t=0

F (x, y , w(x, y , t), wx (x, y , t), wy (x, y , t))dxdy

= 0,
t=0

w(x, y , t) = u(x, y ) + t u(x, y ). t ¯ t , Fu (x, y , u, ux , uy ) u(x, y )dxdy + ¯¯ ¯
D

+
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 ) = ¯¯ ¯ Fp Fp u - · u, x x Fq Fq u - · u. Fq (x, y , u, ux , uy )( u)y (x, y ) = ¯¯ ¯ y y

, Fp (x, y , u, ux , uy )( u)x (x, y ) + Fq (x, y , u, ux , uy )( u)y (x, y ) dxdy = ¯¯ ¯ ¯¯ ¯
D

=
D

Fp u + Fq u x y

dxdy -
D

Fp Fq + u dxdy . x y

(Fp u) + (Fq u) dxdy x y
D


5.3. , u(x, y ) = 0, (x, y ) L, Fp u + Fq u x y
D

103

dxdy =
L

Fp udy - Fq udx = 0.

, Fp (x, y , u, ux , uy )( u)x (x, y ) + Fq (x, y , u, ux , uy )( u)y (x, y ) dxdy = ¯¯ ¯ ¯¯ ¯
D

=-
D

Fp Fq + u dxdy , x y

(5.13) Fu -
D

Fp - Fq u(x, y ) dxdy = 0, x y

Fu , Fp , Fq (x, y , u(x, y ), ux (x, y ), uy (x, y )). ¯ ¯ ¯ u(x, y ), , 5.3.1, , u(x, y ) ¯ (5.12). 5.3.2 . , u(x, y ) , u M , D ¯ ¯ (5.12), : Fu - Fq Fp - = 0, (x, y ) D, x y u(x, y ) = (x, y ), (x, y ) L.

, . f (x, y ), D u(x, y ). , f (x, y ) L D . (u(x, y ) - f (x, y ))2 + (ux (x, y ))2 + (uy (x, y ))2
D

dxdy


104

5.

(5.12), , , u(x, y ), ¯ D L, uxx (x, y ) + uyy (x, y ) -
-1

u(x, y ) = -

-1

f (x, y ),

(x, y ) D.

5.4.

x1

[y (x)] =
x0

F (x, y (x), y (x))dx

(5.14)

[y (x)] =

x1

G(x, y (x), y (x))dx,
x0

(5.15)

F (x, y , p), G(x, y , p) ­ . . y (x), ¯ (5.14) M = y (x) C 1 [x0 , x1 ] : y (x0 ) = y0 , y (x1 ) = y1 , [y (x)] = . (5.16)

, (5.14) , (5.15) . . (5.15) M = {y (x) C 1 [x0 , x1 ] : y (x0 ) = y0 , y (x1 ) = y1 }.

y (x) ­ M , y (x) C 1 [x0 , x1 ], y (x0 ) = y (x1 ) = 0.


5.4. [y (x)] y (x) M ~ [y (x), y (x)] = ~ d [y (x) + t y (x)] ~ dt .
t=0

105

t t = 0, [y (x), y (x)] = ~
x1

=
x0

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) C 2 [x0 , x1 ], ¯ ¯ (5.14) M . y0 (x) C 1 [x0 , x1 ], y0 (x0 ) = y0 (x1 ) = 0

, [y (x), y0 (x)] = 0, , ¯ y (x) ¯ Ly (x, y (x), y (x)) - L(x, y , p) = F (x, y , p) + G(x, y , p). (5.19) . y (x) , y (x) C 1 [x0 , x1 ], y (x0 ) = y (x1 ) = 0. (t, ) = [y (x) + t y (x) + y0 (x)], ¯ (t, ) = [y (x) + t y (x) + y0 (x)], ¯ t, ­ . (t, ) (t, ) , (0, 0) = [y (x)], ¯ t (0, 0) = [y (x), y (x)], ¯ (0, 0) = [y (x)], ¯ (0, 0) = [y (x), y0 (x)], ¯ d Lp (x, y (x), y (x)) = 0, dx x0 x x1 , (5.18)


106

5. t (0, 0) = [y (x), y (x)], ¯ (0, 0) = [y (x), y0 (x)]. ¯

1 , y (x) C0 [x0 , x1 ]

D(, ) D(t, )

= det
t= =0

[y (x), y (x)], [y (x), y0 (x)] ¯ ¯ [y (x), y (x)], [y (x), y0 (x)] ¯ ¯

= 0. (5.20)

, y (x) , ~ [y (x), y (x)], [y (x), y0 (x)] ¯ ~ ¯ det = 0. [y (x), y (x)], [y (x), y0 (x)] ¯ ~ ¯ , y (x) = 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)] = , ¯ ­ . ((0, 0) - , (0, 0)) (u0 , v0 ), t , . , (t , ) = [y (x) + t y (x) + y0 (x)] = [y (x)] - , ¯ ~ ¯ (t , ) = [y (x) + t y (x) + y0 (x)] = . ¯ ~ , y (x) + t y (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 ¯ ¯ ¯ ¯


5.5. -

107

1 y (x) C0 [x0 , x1 ]. [y (x), y0 (x)] = 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)], ¯ ¯ :
x1

Fy (x, y (x), y (x)) + Gy (x, y (x), y (x)) y (x)dx + ¯ ¯ ¯ ¯
x0 x1

+
x0

Fp (x, y (x), y (x)) + Gp (x, y (x), y (x)) y (x)dx = 0. ¯ ¯ ¯ ¯

(5.19) L(x, y , p),
x1

Ly (x, y (x), y (x)) - ¯ ¯
x0

d Lp (x, y (x), y (x))] y (x)dx = 0, ¯ ¯ dx

1 y (x) C0 [x0 , x1 ].

, , y (x) (5.18). 5.4.1 . ¯

5.4.1 , , , (5.18). , , , . y (x0 ) = y0 , y (x1 ) = y1 , [y (x)] = .


108

5.

5.5.
-. , d dx k (x) dy dx - q (x)y = -y , y (l) = 0 0 x l, (5.21) (5.22)

y (0) = 0,

. n , yn (x) ­ -. . , :
l

(yn (x))2 dx = 1.
0

(5.23)


l

[y (x)] =
0

k (x)(y (x))2 + q (x)(y (x))2 dx.

(5.24)

, , yn (x) ­ - (5.21), (5.22), n , [yn (x)] = n . ,
l l 2

(5.25)

k (x)(yn (x)) dx =
0 x=l x=0 0

k (x)yn (x)yn (x)dx =
l l

= k (x)yn (x)yn (x)

- (k (x)yn (x)) yn (x)dx = -
0 0

(k (x)yn (x)) yn (x)dx,


5.5. -
l

109

[yn (x)] =
0 l

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 0

(yn (x))2 dx = n .

(5.24) , (5.22) (5.23). (5.23)
l

[y (x)] = 1,

[y (x)] =
0

(y (x))2 dx.

y (x) C 2 [0, l]. ¯ , y (x) ¯ Ly - d Lp = 0, dx 0 x l, (5.26)

L(x, y , p) = k (x)p2 + q (x)y 2 - y 2 . (5.26), L(x, y , p): 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) .


110

A

A
A.1.
m m + n (u1 , . . . , um , x1 , . . . , xn ) Rm+n : F1 (u1 , . . . , um , x1 , . . . , xn ) = 0, ... (A.1) Fm (u1 , . . . , um , x1 , . . . , xn ) = 0. (A.1) u1 , . . . , um . (A.1) D Rn u1 = 1 (x1 , . . . , xn ), . . . , um = m (x1 , . . . , xn ) (A.2) , (A.1) : Fi (u1 (x1 , . . . , xn ), . . . , um (x1 , . . . , xn ), x1 , . . . , xn ) = 0, (x1 , . . . , xn ) D, i = 1, . . . , m. F1 , . . . , Fm F1 F1 ... u1 u2 F F2 2 D(F1 , . . . , Fm ) ... = det u1 u2 ... D(u1 , . . . , um ) ... ... Fm Fm ... u1 u2 u1 , . . . , u F1 um F2 um ... Fm um ,
m

-

(u1 , . . . , um , x1 , . . . , xn ).


A.1.1. m F1 (u1 , . . . , um , x1 , . . . , xn ), ..., Fm (u1 , . . . , um , x1 , . . . , xn )

111

N0 = N0 (u0 , . . . , u0 , x0 , . . . , x0 ), 1 m 1 n Fi / uj N0 , i, j = 1, . . . , m. , Fi (N0 ) = 0, i = 1, . . . , m, D(F1 , . . . , Fm ) (N0 ) = 0, D(u1 , . . . , um )

1 , . . . , m M0 (x0 , . . . , x0 ), n 1 m (A.2), |ui - u0 | < i , i = 1, . . . , m i (A.1), M0 . [2], . 13, §2.

A.2.
m n u1 = 1 (x1 , . . . , xn ), ... um = m (x1 , . . . , xn ).

(A.3)

, i (x1 , . . . , xn ), i = 1, . . . , m, n- D. . k {1, . . . , m} ­ . A.2.1. uk D (A.3), x = (x1 , . . . , xn ) D uk (x) = (u1 (x), . . . , u
k-1

(x), u

k+1

(x), . . . , um (x)),

(A.4)


112

A

­ , . u1 , . . . , u
m

D, D . , D (A.4) k {1, . . . , m}, u1 , . . . , um D. A.2.1. m n m (A.3) M0 = M0 (x0 , . . . , x0 ). 1 n , - m M0 , M0 . i (x1 , . . . , xn ), i = 1, . . . , m M0 (x0 , . . . , x0 ), n 1 M0 . (A.3) 1 1 1 ... x1 x2 xn 2 2 2 ... (A.5) x1 x2 xn , ... ... ... ... m m m ... x1 x2 xn m n . A.2.2. (A.5) 1) r- M0 (x0 , . . . , x0 ); n 1 2) (r + 1)- M0 ( r = min(m, n), ).




113

r , r- , M0 , r . [2], . 13, §3.


114


1. .. . .: - , 2007. 2. .., .., .. . 1. .: - , 1985. 3. .. . .: , 2003. 4. .. . .: , 1983. 5. .., .., .. . .: , 1985. 6. .. . .: , 2004. 7. .. . : - , 2000. 8. .. . .: , 2002.