Линейные интегродифференциальные уравнения Фредгольма: Учебное пособие по спецкурсу и спецсеминару

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    -

    2007

  • 1

    ..

    -

    , 010100 (010101)

    -

    2007

  • 2

    517.948 655

    - -

    .. - -. , . , .. - -. , . -

    .

    .. 655

    : . . -: , 2007. 195 .

    -- . , - -, , .. 1934 .

    , , , - , , , .

    , , , .

    .., 2007

    , 2007

  • 3

    C , , XVIII-XIX -.

    , - , :

    1)

    =+1

    121 ;)(),()()( dttytxKxyxy

    IVIV

    2)

    [ ] ['0

    ( ) ( ) ( ) ( , ) ( ) ( ) ;t

    t k f t m t K t s f s m s ds = + 3)

    .)(211

    )()(

    1

    1

    +

    =

    xt

    dtdt

    tdgx

    xg

    1903 . 1934 . - .. [31]. .. [6]-[11], .. [12]-[14], .. [3]-[5] .

    - , - , . , - . - , , .

    - :

    [ ] [ ] . )()(),...('),(,,)( ),...,('),(, )()( =b

    a

    mn xfdyyzyzyzyxKxzxzxzxF

    - . [35], .. [3]-[5], .. [6]-

  • 4

    -[11],..[12]-[14], . [36]-[41], .. [22]-[23], .. [26]-[28], .. [30] .

    - .. [16], .. [17], . [18], .. [21] . - .. [2], .. [23] .

    I , .

    , , .

    I , [1]-[31] [35]-[41], , . .. 1970-1990 .

    II , III IV .

    IV , .

    II-IV [22]-[32] .

    V [43]-[49] [24]-[25] .

  • 5

    I

    1.

    1. ,

    =+b

    amn dyyzPyxKxzL ,0)]([),()]([ (1)

    , )()(...)()()()()]([

    ),()(...)()()()]([

    1

    1

    10

    1

    1

    1

    yzybdy

    yzdybdy

    yzdybyzP

    xzxadx

    xzdxadx

    xzdxzL

    mm

    m

    m

    m

    m

    nn

    n

    n

    n

    n

    +++=

    +++=

    , K(x,y) , b, n>m.

    -. z1(x), z2(x),, zn(x)

    [ ] , 0)( =xzLn (2)

    z(x)=c1z1(x)+c2 z2(x)+.+ cn zn(x). (3) , (1) [ ] , )()( xFxzLn = (4)

    F(x) . z(x) (4)

    F(x) , .. - (4)

    z(x)=C1(x) z1+ C2(x) z2+.+ Cn(x) zn. (3*) i(x)

  • 6

    =+++=+++

    =+++=+++

    ).(...,0...

    ,0...,0...

    ')1('2

    )1(2

    '1

    )1(1

    ')2('2

    )2(2

    '1

    )2(1

    '''2

    '2

    '1

    '1

    ''22

    '11

    xFCzCzCzCzCzCz

    CzCzCzCzCzCz

    nn

    nnn

    nn

    nnn

    nn

    nn

    LLLLLLLLLLL (5)

    )(x - nizi ,1, = (2).

    .0

    ...

    .........

    ...

    ...

    )(

    )1()1(2

    )1(1

    ''2

    '1

    21

    =

    nn

    nn

    n

    n

    zzz

    zzzzzz

    x (6)

    nix

    zzxFzz

    zzzzzzzz

    xCn

    nn

    in

    in

    nii

    nii

    i ,1,)(...)(...

    ...0...

    ...0...

    )()1()1(

    1)1(

    1)1(

    1

    ''1

    '1

    '1

    111

    ' =

    =

    +

    +

    +

    LLL

    , F(x) - nixi ,1),( = , :

    nixFxx

    xC ii ,1 ),()()(

    )(' =

    = . (7)

    (7) i(x):

    ix

    ii cdFxC +

    =

    )()()()( (8)

    , i(x) (3*), (4)

    =

    +=n

    i

    xi

    ii dFcxzxz1

    .)()()()()(

    (9)

    F(x) , (9)

  • 7

    (1). , z() (9) n - (5), :

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

    1

    )()( =

    += =

    nkdFcxzxzn

    i

    xi

    ik

    ik

    (9k)

    ).()()()()()(

    1

    )( xFdFcxzxzn

    i

    xi

    in

    in +

    += =

    (9n)

    , (9) (9k) (9n) (1). (1)

    ,0),()(

    )(),()()( =

    ++ dyyxKdF

    yHygxFb

    a

    y

    (10)

    F(x) ,

    )]([)(1

    yzPcyg in

    imi

    =

    = )]([)(),(1

    yzPyH in

    imi

    =

    = .

    2.

    (10)

    .)()(

    1

    =

    =r

    rr xxF (11)

    (10) (11), , (10) ,

    ,),()()(1 =b

    a

    dyyxKygx

    ,)(),()(

    ),()( 1 =b

    ar

    y

    r dydyxKyHx

    ,...3,2=r

  • 8

    (12): ,)(

    ),(,),( MyHAyxK

    ,,,,),()( yxNdyyxKyg

    b

    a

    byxa ,, .

    .,...2,1,2

    )(

    ,2

    )(,)(

    1

    12211

    22

    21

    =

    =

    rab

    NAMx

    abMANdydMANxNx

    r

    r

    rrr

    b

    a

    y

    (11) -

    12 21 1

    11 2

    r

    rr rr

    r

    b aM A N

    =

    . (13)

    , (13) -

    2

    22 abMAq =

    q

  • 9

    (2) z1(x)=e3x, z2(x) =e -2x, , - z(x)=c1e3x+ c2e-2x.

    (4) -

    ==+

    ),(23,0

    '2

    2'1

    3

    '2

    2'1

    3

    xFcececece

    xx

    xx

    -

    = )(x 5ex, = )(1 x e-2x, = )(2 x e

    3x, 2

    ' 311

    ' 222

    3 21 1 2 2

    ( ) 1( ) ( ) ( ) ( ),( ) 5 5( ) 1( ) ( ) ( ),( ) 5

    1 1( ) ( ) , ( ) ( ) .5 5

    xx

    x

    x

    x x

    x eC x F x F x e F xx exC x F x e F xx

    C x e F d C C x e F d c

    = = =

    = =

    = + = +

    g(y) H(,y) - (10)

    g(y)= c1 e3y+ c2e -2y, H( ,y)= -e-2 e3y+e3e-2y. (1) F(x) -

    (9) :

    ++=x

    xxxx dFeeeeececxz )()(51)( 223322

    31 ,

    (10)

    .0)1(2

    )1(3

    )()(5

    )( 22311

    0

    2233 =+++ ececdFeeeedyxF

    yyy

    (11) , -

    ( ) ( )

    += 22311 12

    13

    )( ececx , ( ) ( )3 21 22 1( ) 1 16 3 2c cx e e = +

    ,

    , ( ) ( )

    +=

    22311 12

    136

    1)( ececx rr , , ,

    (10)

    .)1(2

    )1(36

    6)( 2231

    +

    = ececxF

  • 10

    (9), -

    -

    3 23 2

    1 2( 1) (1 )( ) .

    3(6 ) 2(6 ) e ez x e

    = + + +

    l

    . , - 1 , =6. , (14) - - (11).

    . (14) 1 .

    3. .

    (11), 1() 2(), 1 1 - )(2 x . , , -:

    dydyHdyygxb

    a

    yb

    a

    = 111

    1

    1112 ),(),()(

    ),()()(1

    . (15)

    1(,) =(,), (151)

    ,),(),()(),(),(

    1

    111

    1112

    dyHdyxb

    a

    y

    = (152)

    : dyygxb

    a= ),()()( 22 .

    )(3 x

  • 11

    dyygxb

    a= ),()()( 33 ,

    11211

    1113 ),(),()(

    ),(),(1

    dyHdyx

    b

    a

    y

    = ; (153)

    dyygxb

    arr = ),()()( ,

    ,...3,2 , ),(),()(

    ),(),( 11111

    111

    1

    =

    = rdyHdyx

    b

    a

    y

    rr . (15r)

    )(xr (11) , , - ,

    [ ] ....),(...),(),(),()()( 1322 dyxxygxF rrb

    a

    +++++=

    R(x,y; ) =K(x,y)+ K2(x,y)++ 1r Kr(x,y)+ . (16) -

    dyRygxFb

    a= );,()()( . (11*)

    -, , (11) .

    r() , -

    ,...3 ,2 , ),(),()(

    ),(),( 11111

    111

    1

    =

    = rdyHdyx

    b

    a

    y

    rr , (17r)

    .

  • 12

    . (16) (15r), r = 1, 2 .

    = 1111

    111

    1

    ),(),()(

    ),(),();,( dyHdyxyxR

    b

    a

    y

    ....),(),()(),(

    11211

    111

    2 1

    dyHdyb

    a

    ....),(),()(

    ),(1111

    1

    111

    11

    +

    dyHdy

    b

    a

    y

    rr

    -

    [

    ] ,...),(...),(

    ),(),()(

    ),(),();,(

    1112

    12

    111

    111

    1

    d

    yHdyxyxR

    rr

    b

    a

    y

    ++++

    +

    =

    ,

    1111

    111 );,(),()(

    ),(),();,(1

    dyRyHdyxyxRb

    a

    y

    = . (181) (16) -

    (17r), , , -

    1111

    111 );,(),()(

    ),(),();,(1

    dyRyHdyxyxR

    b

    a

    y

    = . (182)

    4. ..

    (11) -

    (10), (14). , - . - 1.

  • 13

    (10) .

    )()()(

    ),( xdFx

    = (19)

    , F(x) (10) (19),

    ddyyyKdyygyKx

    x b

    a

    b

    a

    = )(),()(),(

    )(),()( ,

    , , -