. .

. 2014. . 2. . 103-121
=
511.3
. .
*
*
. .
.
. . (^ (, , ) () =
= /(X) () | = >(), (),/(),<^() () .
: , , .
1.
[5] . . :
2 2 ..
+... + = /() (1)
(),/() %, (2)
/ (, . . . , ], . . . ,) = -/(, . . . ,] , . . . ,) ( = 1, ... ,),
() = 0 (1 ) ... (1 ) = 0. (3)
,
() \ =0, = [0; 1).
. . = ({ ^ } ,,{ ^ }) ( = 0,..., N 1) ,
( 11-01-00571).
/(X). , , , , , .
:
= /() (4)
() 1 = ^() , (5)
(),/(),^() %, (6)

^ ( 9 9 \ = ^ ^ 911 1 (7)
,'"7 3) = =0 1= 11- !1 '? ( )
, . , /(X) <^(), , .


() .
, Q (-, , ),
(X), ^ (X) = 1 ^() / (X) = 1 / (X) ().
2.

Q( - >> ). 3 Q(m)
- {)
)= ^ $^(2)11+--- +1{- 3/ = ^ ()(2)!, (8)
31=0 1=0 1 = 0
1( )= ^2 3-,...,3 1- 1 (9)
3-+. . . +1 =1
, 11;. ..;1 , Q(m) = 0 , 3- () = 0, ] = 0, , ^).
1. 1 2(>) -) Q(m),
^() = 0, , Q(m) = 0.
. ,
... 2() = (2)1+- + 1... 2(>). - 1

^ ^ 2(?,;) =
^9X1
- 3
-+ + 2(?,) =
...
= ^ -. (2)
1=0 3=0
= Q(m )2(),
.
, , 2(,) .
5 1 . (5)
(5) = | (X) = :
I ^
2(,)
, 5
, /(X) /(X) = 0,
(:)=/ ^ (10)
. , (5)
(5) = (5 \ ) ^|^
(10) ,
/(X) (5 \ )
1. (5)
/ (X) = X] 2(,) (5 \ ) , - < XV < (V = 1, ,)
* 5\
(11)

()= I] 2() + ^ 2(), (12)
\ ^
_ .
. () 0(5):
() = > 2(>).
1

' < ї2"*-** =
41 8 7 5
= ^ <2( )2(-) = / (X) = 2^
5\ 5\
, ,

()
5 \ , 5 | , .
= = (,?1, ,,78), , 1, 2, , , 1 ^ 7 < <7* ^ , 1 ^ < < ^ . 17|-4| =
= ^8 8 = {(1, 2, , )}. 8-1:
,71 -1 = {(2,.", 5, 1), (1, 3,.",5, 2),..., (1,.",5 - 2,5,5 - 1), (1,.. .,*)}.
|7* | = 5.

^ 7;% ^>() 0() (';.) 0(5,/8,*),
0 (3, /,.,) =
= { ((! , . . . ,,)) = (..... )2"(>. + +ї**)
(,... .,,)S(jSt)
(.. ,*) ( , . . . , 3 (/;*)
3 (.) = {( ,.",4 )|(,..., 8) 3},
(/;)^() = ^ 2( ++) ^ .
(,. ^, )eS(ja,t) , *63 >
,
(,0) = ^ ,
raS

(/5.5)^() = ^().
2. (),^() 0()
() = <^() ( ;) (13)
, 0 ^ <5 ^ 7|4
(/5.*)() = (/;*).
. ^^) 5- , ^ = 0 + 1 ^ V ^ 5, ^(^) 5-
, ^ = 1 + 1 ^ V ^ 5. ,
= (0(/.*) )) .
0<*<; ~^*,
0^*) 1 (;, *), () = (;.*)(), , ^>() = (;.)^(). .
3. (X), ^>(X) (5):
(X) = ^2 *2(), ^) = ^2 2()

(13) , 7,8-1 ^8-1 (^1, , ^3_1) (78-8_1)
^2 = ^ (14)
5, 5,
( . . ,^3_1) = ( . . ,_1> ( ,"^_1) = (^1 . . ;;_1)
. ,
Pr(js,s-l)(X) =
= X] 2( *1+ +^3_1 *;_1) ^2 ,
(^ ,,*, 1)5(7.._1) ,
V ;1 > ;3_1'^ \, ^ (;.....; ) = (; ,. . . ,,' _ )
Pr(Js,s-l)^(X) =
2*(;1 * ;1 + +;3_1 *;3_1 ) ^ ^
2(;1 *;1 + +;3_1 *,5_1 )
(;1 , ,,5_1 ^(,^)
2 :
2*(;1 *;1 + +;3_1 *;_1 ) ^2 =
(;1 , 1 )5'(.?3,3_1) ;
*
5,
(,... ,;3_1)=(,... ,;3_1)
(,... ,;3_1)=(,... ,;3_1)
2*(;1 *;1 + +;3_1 *;_1 ) ^2
2(;1 *;1 + +;3_1 *;3_1 )
( - ,;_1 )5(75,5_1) (;1 ,. . .
.
2. (5)

4^X7 ;!;) / (X)
/(X) = X] 2(,*) (5 \ ) , < XV < (V ^^^)
5\ (15)

(X) = ^>(X) (X 8) , ^() (5), (16)

^>()
<^()=^ 2( ), (17)
1
, ( -),
^ , 5-1 8-1 (^,..........,3-) 5 , 5-1)

- . V-^
Q(m )
nGS\Kerg, nGSflKerg,
Cn
X) dn, (18)
nGS,

u(X)= e2ni(mX) + 2 cm e2ni(mx). (19)
m GS\Kerg m ESHKerg
.
u(X) = ^ e2ni(mx) + ^ cm e2ni(mx) (20)
m GS\Kerg m ESHKerg
(15). 3 .
, ^P|Kerg = {(0,..., 0)}, (11)
u(x)= Q(m ^+.
mGS\{0}
.
Q ( ,..., gf-)
Q , Q (, , gf-), KerQ = {(0,..., 0)}.
,
. , Q.
, Q (^,..., gf-) (7) , 0 ... 0 0.
(nji ,
(nji , nj--i)=(mh , ,mj--i)
(nji nje-i)_(mji mjs-i)
3.
, () (). (4)
(6) , , 5- , () 5- . (()).
:
, * * () (, *) ./. , *),
/.) = { = (,..., ^ ) 4+ = = , = 0} ,
(/. ,*)() = 1 = (, , .)
() (, , 4) = (, , ^
^+

-
^ = ^

2( , X)
= ^ /() = ^ ()
^\
= ^ /() = ()2(,)

/ () .
/ ()
,
) () = / ()
/() ^
.
0
1. '
() = /() () (21)
/() (22)
' () (* ()), (21) ( 22).
2.
;')() = /() () (23)
/() (24)

() = ^>(), ( 5) , (25)
<^() ^ ( > (^) + 1), /^() ,
^() = ^ ( )2( ),
*()

() = N ^()-2(?7), N = | ()|.
()
() (*()), (23) (24) (25).
, (4) (6).
(^,..., -) 0 = = {0} , 2( , ) . * = ^ . , * /() ^, (0) = 0.
3.
(-)<*> =/()-* /(), (26)
41 .7 ()
" (27)

(*> = ' 2'(-) + , (28)
*() ^( )

X] /() 2(> ), = 0;
() = ^ ()
0, = 0\
. / ()
1/() = ^ ( )2-( ),
*()

() = N /()-2( ,)-
()
(0) = N X] /($),
()
/*() = /() - N /()
()

1/*() = ' ( )2( > ). *()
1 5 = *() .
4.
()() =/()-N /, (), (29)
41 ^ ()
( '/ () (30)

() = /^(), ( 5) , (31)
^>() ^ ( > () + 1), /^() ,
^() = ^ ( )2(),
* ()
1
() = 7 ^)-2(), N = | ()|
()
, , .8)8- 1 ^*8- 1
= .
* (), ' * (),
( , ,7--1)=(71 ,7--1) ( , ,7--1)=(71 , ,7--1)
2! + ... + 2-_1 = 0 (32)
() + '
()
= ,
Q(ft)
* (), * (), 4 7
(^1 , -_1) = (.,1 , ,^-_1) (^1 , >,-_1) = ( , >^-_1)
21 + ... + 2-_1 = (33)
1
<) = 57 /()-2'(.
()
(29) (31)
(*)= ' 2"(' + , (34)
* () ^( )
. 2 5 = * () = {0} .
*(*()), ()
() = ( )2(>) = ^ ( )2(>).
* ()\{0> * ()
*(*()) ,
() = N " 2(>-), (),
* ()

() 1 - N, = ,
() \ -2^(>), = , ().
() *(* ())
/ \
^ ^ -,..., ) () = (), (). (35)
1
,
1 . 2(?,-)
= N
* () ^ 1

^() = /() - NN /( () (36)
41 ^ ()

() = / ()() + ,
()
.
4.

^^),
\ ^ ^ 71 7-
7
1 ^ ;1= ;-="71'"7-1 7
= + ... + 8.
3. ^(1,..., 8) 1 ^^),
^(x) =
m eZs
p2ni(m,X)
cm,e

IMlEf(Q)
(mi.. mS)a Q(m)

. (37)
Ef(Q) = sup |cm (mi...ms)a Q(m)| < to, (38)
m Zs ni ns
Q(m ) = E aji-",js (2ni)j1+--- +js mi1... mSs. (39)
ji=0 js=0
Ea(Q), .

Q*(m) = Q(n)-1.
()^()
, ^ Q* () ^ 1.
4. ^(1,..., 8) 1 +^),
^(x) =
m ezs
p2ni(m,X)


|Cm| < _______lE++a(Q)______________________________________. (40)
m (mi... ms)a Q*(m)
E+a(Q) = sup |cm (mi...ms)a Q*(m)| < to. (41)
s mezs
E+a(Q), .
5. ^^)
) (*) =/ <*)
< ^ < (V = 1,..., ),
/()= 2(), /() +^) (42)
-\{0}

() = " 0^ 2() + , () ^), (43)
- ∞()
(42) (43) .
. () ^^):
() = ^ 2(), , .
-

__ _
1
<*>=1>)2"(>
41 5 7 -
, , ^ \ {0} Q(m) = , = 0() .
,

| 1 < (.. .^) Q*(m)

I
] |2('") < = (! + 2())' (44)
- - 3
2(,) < _____________ < (1 + 2^())5
(?) ^ / (1 _) Q(m)1 Q*(m) ^ ( ( )) ,
-
(..) ^()| Q*(m)
.
6. *(^)

--:) ,1() =/ (),
< ^ < (V = 1, .. . , ),
/() = 2() (46)
2

() = <^() ( :) , <^() *, (47)
<^()
^() = 2(>) (48)

, 0,
_7:,:-1 ^|:-1 (^,..........,^3_1)

V -^ = V ^
( . . ,_) = ( . . >3_) ( , ,",_) = ( , 73-)
+ .. + _ = ∞ (49)

()
2, 2, '
= ,
+ ... + - = ∞. (50)
(46) (48)
.
()
'<*>=' 2"(') + ^ (51)
2
. ()
(/:,:-1)() =
' 2( + +3-) \^'
/ ^ ') +
( >3- )^(.?,-) ( ,,. ,)1( ,. _)
(,. ,3_)-(,. ,3_) (,. ,3_ )-(,. ,_)
<^()
(/)_1)<() =
= ^ 2( + +,-_1 --1) ^ .
(,-, , ,, 1)^-(,;--_1) , 6^-
V > ,-_^ ' ^ (",1 ...,-_1) = (,1 . ..,-_1)
.
5.
*() = () Zs . 4 (. . 112) (29) (31)
(,)
Q(m)'
()= ' ) 2-(-) + , (52)
* ()

() = N /(/)-2(
N
()
()
= ' ,
() + '
()
* (), * (), ^ 4 7
=(,0,. . . ,0) =( ,0,. . . ,0)
() () (46) (47). .
7. (51) (46) (47)
<) 211 ((-1)!(1>) +
+ ^ ( 2 ()-2 + )2 <0
=0 = \)=&+2 )=+1
. ,
()=' (') + , () = ^'2"( + .
23 ( ) * () ( )

() - ()= 1 2*(') + 2*(') + - .
* () Q( ) -\ * ()Q( )
, .
:
Si =
' bm () 2(,)
Q(m)
mM * () 1
EV"^' |bm+n| __
^ |Q(m )l
|Q(m)| ' |Q(m (n, ))|
mM*() '^v 7 1 neZs\M *()'^ V ,n
m(n, ) M*() n m(n, ) , n = m(n, ) (mod ). ,
S < y^ ___________________II/Q^IIe+^q)___________ <
^ neZS\M*() (nT---nS)a Q*(n) -|Q(m(n ))| "
< I|f (X)lle+(Q) E (nr...ns) ,
neZs\M * () v 1 sy
Q*(n) ^(,(,))| ^ 1 Q*() ((, )) ^ ().
:
S2 _
m ezs\M * (^()
bm 2(,)
< _________llf(x)|Eg(Q)_________ <
ezS\M *() (mT---mS)a -|Q(m)|- Q(m)-i "
< llf()IU;j)_
m eZs\M * ()

/ \
_ e dn + E' Ok E d(n) + E'
Q(n)
neZs, n6Zs, '
b(n) Q(n)
* (), * (), ^ V 7 I
(ni ,0,. . . ,0) n=(ni,0,. . . ,0) \n = (ni,Q,. . . ,0) n=(ni,0,. . . ,0) /
d(n))+ dn + ' it+ b
Q(n) ' Q(n)
* (), 2-\ * (), * (), 7 -\ * (),^4 7
=(10. . . ,0) n=(nl0. . . ,0) n=(nl0. . . ,0) n = (nl0. . . ,0)
, 51 2. , .
,
*> - ї + <&( s z <>j-2 + es 2j ,
m=0 * k=m ( > \j=fc+2 j=k+l ) )
.

1. Rodionov A. V. Number theoretic metods for solving partial differential equations // Algebra and Number Theory: Modern Problems and Application: proc. of XII Iintern. confer., dedicated to 80-th anniversary of professor V. N. Latyshev. Tula: TGPU, 2014. P. 159-161.
2. . . // . . . . . . 2003. . 9. . 1. . 82-90.
3. . ., . ., . . // . : , 2002. . 3. . 2(4). . 43 - 59.
4. - / .. [ .] // . : , 2004. . 5. . 1(9).. 122-143.
5. . . - . .: , 1963.
6. . . - . ( ) .: , 2004.
7. . ., . . // . . . 2014. . 1. . 1. . 50-63.
8. . . . . - . . // . : , 2009. . 10. . 3.
9. . . . . // . . 2013. . 4. . 2 . 120-124
10. . . - // : : . XII . ., 80- . .. . : , 2014. . 297-300.
(rodionovalexandr@mail.ru), , , , . .. .
Korobovs method of approximate solution of partial differential
equations
Abstract. The paper discusses the generalization of N. M. Korobovs method of approximate solution of the Dirichlet problem for equations of the form
with boundary condition u(x) |dGs = <^(x), where the functions u(x), f (x), ^>(X) belongs to the class of periodic functions Ef in case of using generalized paral-lelepipedic nets M(A) integer lattices A.
Keywords : parallelepiped nets, partial differential equations, the Dirichlet problem.
Rodionov Alexander (rodionovalexandr@mail.ru), assistant, department of mathematical analysis, algebra and geometry, Leo Tolstoy Tula State Pedagogical University.
A. V. Rodionov
17.04.2014