линейные краевые задачи интегродифференциальных уравнений вольтерра с функциональными запаздываниями

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  • . .

    -2015

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  • 517.948 22.161.61

    - 655

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

    . . 655

    : . - -: , 2015. - 78 . ISBN 978-5-9793-0724-4

    . , , .

    , , , .

    Shishkin G. A.The Linear Boundary Value Problems For Volterra Integer-

    differential Equations with Functional Delays: monograph. - Ulan- Ude: Buryat State University Publishing Department, 2015. - 78 p.ISBN 978-5-9793-0724-4

    In the monograph the authors researches have been stated, they consider the transmission of boundary value problems for Volterra linear integer-differential equations with retarded argument to resolving integral equations with ordinary argument. The classes of such equations are defined, the possibilities of solution in the closed type with the help of one modification of function of flexible structure are considered, as well as the variant of approximate solution.

    The monograph will be useful for specialists who solve the problem with deviating argument and also for postgraduate and undergraduate students, specializing in the field of functional equations.

    ISBN 978-5-9793-0724-4 , 2015

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  • . (1903). . . (1771) , . ( , ).

    1913 . [32]. [33]-[38].

    20- , , - [1]-[6], [9]-[15], [18]-[20] . , . . [2] - . , , , . . [3]-[6], . . [12]-[15], . . [18]-[20] .

    - , . . [22], . . [28], . . [25], . [39], . . . [30] ., .

    . [39] .

    - . . [27] . . [29]. .

    . . [12] Zi (x), i = 1, n

    3

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  • - n > m n > k :

    L m [ ) ] - [ ( - h ) ] K ( X t ) (i) ( t ) d t = f ( X ) 0 i=0

    y (l> (a) = r , y ! (x) = (x), a - h < x < a, (a) = r Vi = 0, n -1, Lm [(x)] Pk [y (x - h)] - m .

    y (n)(x) = 0, Rj [ ] = 0, j = 1, n , [12] , :

    () (x) = (x) = () (a)Z() (x), a - h < x < a, z ( )(a) = 1. . . [7] -

    , , , , :

    '(x) + (x - ) + j y(t)dt = 0,0

    (x0) = 0, (x) = (x) x0 - < x < x0, x0 0. , . . [16]-

    [17], , - , .

    [40] - .

    , . ,

    4

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  • .

    [41] [42] - .

    , , , , .

    , , .

    , . .

    [45] , [47] . , , , - -.

    [48]-[53] .

    5

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  • 1

    1.1.

    :

    Z Z f ( ) ( ( u j ( x ) ) + ^ j ( , ) ( i) ( u j ( ) ) = f ( ) ( I )I=0 i=0 a

    M0 () = , Uj () < , Uj () = Vj = 1, l, f i}. (), f () Uj ()

    , K ij (,) a < ,< b. u j ( ) < , j = 1, l (1) -

    .

    f ij. () K ij ( , ) . , . , , .

    (1) , - [7].

    1 (1),

    U- () Vj = 1, l (-1)- , . . (1):

    6

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  • f n] () = 0 K nj(, ) = 0 Vj = 1,l , a f n0(x) = 0 , ,

    , f n0( ) = 1. 2 (1),

    Uj () , j = 1, l , n- , , . . (1) f n0() = 0 3j 0,

    f nj () = 0 K nj ( ,^) = 0 .

    3 (1),

    Uj (), j = 1, l , j, , , . . (1) f n0 () = 0 3J 0 , f nj () = .

    1.2.

    n [16]-[17], , , , (1) :

    () = D_1[ ^ (*-1)(0 ),( - ^ + J An( - *)dt], (2)5=1 0

    D = D(r1, r2, ..., rn) - , r1, r2, . , rn, , :

    1 1 . 1

    r r2 rnD =

    n-1 n-1 n-1r1 r2 rn

    7

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  • As (x - 1), s = 1, n D s- expr(x-t),expr,(x-t),..,expr,(x-t) (x) - .

    , (2), :

    (

  • - (, ) , .

    1.3.

    (1) :

    y (i )(u, (x)) = ( )(x V ! )(u (x )) = n - 1 x G EXn,, - -.,(iVv VV)' > V \ j (5)

    E x = ^ , E 1x - , -j=

    Uj (x) < x x > x Vj = 1,l , E = [a, x0]. (1)

    :

    Z [)(x) + P )(xi )] = 7, = 0, - 1, a < x < xi < . (6)i=

    , (1), (5), (6) , x [ x, b] . Cj Uj (x) = x

    x [ x,b] , ,

    cj = b. (1)

    (5), :

    (n)(Uj (x)) = (n-1)(x)p'n-1(Uj(x)), j = ,l, x e Exa,

    +

    Z Z [ (x) (i)(Uj (x)) + ^K j (x,r )y (i)(x )q>i(Uj (r))d r +J= = a

    x \K 4 (x ,r )y (i )(u; (r ))d r] = f(x). (1*)

    c

    9

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  • , (5), (6), (2) (3)-(4), ()(0) () ,

    : 1) 0 < X1 < Cj Vj = 0,1;

    2) 0 < Cj < X1 Vj = 0,l; 3) X1 , 3j = 0, l ,

    X0 < X1 < Cj , - X0 < Cj < X1. .1. , , x = X1

    (5) j = 0 y(jq) =!)(,) (6), :

    X (') (0 ) [ + Ph V, (X1)] = ,i =0 (7)

    = 0, n - 1 ()(X0).

    : = det[ + PhVi(X1)], , = 0,n - 1

    , :

    ( i)(X0) = * , = 0, - 1. (8)=0

    , [41], (i)(X0) (8). :

    n-1

    V * (uj ( ) ) = V (uj ( ) ) -1 . (5*) = 0

    , , , :

    () = D-1 [X -1 X (,-1)s( - X0) + j n( - t)v(t)dt], (2*)

    10

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  • n n-1y(i)(x) = D [Z 0 Z yt(0(s-1) T ( x- x0) + J 5 a ^ ^ u(t)dt]

    s=1 T=Q dx'(3*)

    ' =1 n - 1,

    y (n) (x) = D 1 [ o -- s-dtA? (x - Xq) + J(n) - x ) + J 5 An (x t) n(t)dt] + u(x). (4*)dxns=1 T =Q2. ,

    (2) (3)-(4), (i X1),i = Q, n - 1 y(x) ( )(xQ) :

    n(i) T-V-I rV1(l)(xi) = D-1E (s-1)(xq)AS')(xi - xq)

    s=1

    + J 5 (x; t) ju(t)dt], i = Q, n -1 .xq 5xl

    (9)

    (l)(x1) (6)n-1Z {' ( ' (xQ ) + P'TD-1 [Z (S-1) (xQ )} (x1 - xQ ) +'=Q s=1

    ? 5' (x - 1)+5xl

    u(t)dt]} = r T, T = Q, n -1

    , (l)(xQ):

    Z [a'T + P1Td 1(+1(x1 - xQo)]y (')(xQ) =

    = n-1 -n

    D 1 Z Pt J 5' An (xI - t)5xlu (t)d t, T = Q, n -1 .

    (1Q) :o = det \_ + PitD -1 (+1 (xi - xq)] , ' , = Q,n - 1,

    (1Q)

    (11) o iT, :

    11

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  • n-1 X k(ijCx0) = X [, - D - X P , J

    ,=0 k=GakAnU - )

    Sxkp(t)dt] ,i = 0,n - 1. (12)

    (12) (5) (2), (3), (4) 2 3 :

    n-1 Xl (u, (x)) = (u, (x ))X [, - D - X J

    =0 =0SkAnCx - 1)

    Sxkp(t )dt ], (5**)

    i = 0, n - 1, j = 0, l , x e Exn-1

    (, (x)) = D -1 {X A, (uj (x) - Xg) X [ , -TT TT

    - D X P ,, Jx1 Sk A n (x, - 1)

    Uj (x)

    =GkSx1

    p(t)dt ] + J00

    An (uj (x) - 1M tMtK

    (i ) (u , ( x )) = D - 1I Xd A s (u j (x ) - x o )

    dx1 X [ r , -

    (2**)

    -1 xL ^k- D 1 X P , J

    SkA_nU - 1) Sxk

    Uj(x) S1A n (u, (x) - 1) p(t)dt]+ J - ^ j W '

    Sx1p(t)dt}, (3**)

    i = 0, n - 1 ,, j = 0, l , x e [ c , , b],

    , , -U SnAs(u ,(x )- x0) n-1 "' (u, (x)) = D -1IX ----- , " X [ r , -dx"

    - Dn-1 x1

    1 X Pk, Jk=G x0

    S k A (x, - 1)kSx1

    p(t)dt ] +

    +u, (x)

    Jx 0

    Sn An (u, (x) - 1)Sxn

    p ( t ) d t } + u ' " (x )p (u , (x )). j = 0 ,l . (4**)

    2 3 , , .

    12

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  • (2**), (3**), (4**) (1*):

    X X f (*)-{j=0 i=0 s=1

    d ' As ( Ui (x) - 0 ) n -" jW X [, -=0

    ' An (Uj (x) - 1)-1 xL- D 1 -X Pb J

    A^ixl - 1)

    u(t)dt ] + J

    dx'

    'u ( t )dt} +

    + X fn j (x)uj (x)u (w(x)) +x X X J K J (x ,^)% (uj r n X [r-. - - ,=0 j =0 j=0 i=0 -1 xL k

    - D 1- X Pb J k A n (x1 - 1)

    dxu(t )dt]dr/ +

    I n x+ X X X S K j (x , r ) H X X [r, -

    j=0 i=0 c

    -1 \ \k- D 1 - X Pb J

    xO

    A^ix1 - 1) xk

    4 r) 'A n(Uj ( r ) - 1) ju(t)dt] + J ------- j ------- M t)dt} +

    +x X J Knj(x ,r) Wjn(r)u(Uj(r))dr = f (x).j=0 c

    , , , , :

    X ifnj (x)ujn(r)u(Uj (x)) +X J Knj (x,r)u'jn(r)u(Uj (r))dr +j =0

    {1}d A s ( u j ( x ) - x 0 ) n -1 xf A n ( x - 1)

    + X [ - f i j (x )D - X ~ s j X X P , , T k > ( t ) d +,=0 k=0 x dxl

    {2}

    +T An (Uj (x ) - 1)

    f u (x )D J --------- j ---------MOdt -xQ

    {3}

    !

    k

    0

    13

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  • a -1 *x j K11 (XTj)q,, (ut (t ))X -JL D-1 X P j

    a T=O a k=0 X't $ A,, (X ,-1)

    a x-q t)dtdv +

    {4}x ujiTi a a n (Uj (T) - 1)

    + x j K j (x ,T )D 1 J ------- j ------- /u(t)d t-

    {5}

    -x j K j (x,T)D1ri a As ( u j ( t ) X o)X a Ta

    {6}-I xi Tska A, (x, - 1)

    F (x) = f (x) - X X

    X PkT j --- ---- ' ^ (t)dtdT } = F(x),

    a A s(uj ( x ) - x0) ^ a

    k=O

    j=O 1=OCj

    f j (x )D-i X -s=1 ax 1

    Z 1st n +T=O a

    j n-1 a+ x J K ,j (x, T q ( uj T ) )Z ~ ^ Td T +

    T=O a

    c

    0

    x a As (Ui (t ) - 1) a ~+x J K ,j (x ,T)D X ------- j , ------ - Z - Y q T

    J T=O a _s=1(13)

    , q(x), {1}-{6}. {1} , Uj () = t , = Uj 1 (t), Uj 1 (t) - Uj ( ) :

    {1} j Knj (x,T)U j (T)q Uj (T))dT =cj

    Uj(x) Uj(x)= j K n j (x ,Uj ->(t)U n(Uj-1(t)(Uj-1(t)) 'q (t)dt = j H j (x ,t)