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Аппроксимация краевой задачи термодесорбции системой ОДУ ускорение сходимости.

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????? ?????????? ???ч???? ?????? –јЌ
? 8. 2016. ?. 11?23
DOI: 10.17076/mat395
”ƒ 519.6:539.2
????????????? ??????? ??????
?????????????? ???????? ???:
????????? ??????????
?. ?. ?????, ?. ?. ?????????
??с???у? ?????????х м???м???ч?с??х ?сс????????й
????ьс???? ??уч???? ц????? ?јЌ
¬ ?????х ??х??????ч????х з???ч ??????????? ???????????????? (???юч?? ?????? ITER) ??????? ??????????й ????? ??з??ч??х ?? ??з??ч???ю ??????????????х ?????????? ? з?????? з???????? ????????? ?????????????????????.
ќ???? ?з э???????????????х ??????? ???????? ?????????????????? ????????????? (???). ќ???з??, ????щ????й ?????????, ????з??????? ? ???????х ?????????????? ? ??????????? ???????. ? ????щ?ю ????-???????????? ?????????????? ????????????й ?????, ??з????ющ?й ?????? ? х???????? ?з??????й?????
?з?????? ???????? ? ??????? ?????. »?????? ??????????ю? ????? ????????? ????????, ??? ??эфф??????? ??фф?з??, ???????????, ?????????. ¬ ??????
???????????? ?????????????? ??????? з???ч? ?????????????? ? ч???????й ????? ????????????? ???-???????, ?????ющ?й ???? ?????????????? ??????й??й ??????? ???????????х ??фф???????????х ????????й (ќ?”) ??????????
???????. »з??ж?? ????? ????????? ?х???????? ?? ?????? ????????? ????????????й ?????й ???????????.
? ю ч ? ? ? ? c ? ? ? ?: ?????????????????????; ??????????????; ??????й???
??????? з???ч?; ??????ч????? ?????ч??? ???????; ч???????? ?????????????.
Yu. V. Zaika, E. K. Kostikova. APPROXIMATION OF THE
BOUNDARY-VALUE PROBLEM OF HYDROGEN THERMAL
DESORPTION BY
ODE SYSTEM: CONVERGENCE
ACCELERATION
One of the technological challenges for hydrogen materials science (including ITER
project) is the currently active search for structural materials with various potential
applications that will have predetermined limits of hydrogen permeability. One
of the experimental methods is thermodesorption spectrometry (TDS). A hydrogensaturated sample is degassed under vacuum and monotone heating. The desorption
ux is measured by mass spectrometer to determine the character of interactions
of hydrogen isotopes with the solid. We are interested in such transfer parameters as
the coe cients of di usion, dissolution, desorption. The paper presents a distributed
boundary value problem of thermal desorption and a numerical method for TDSspectrum simulation, where only integration of a non-linear system of low order
ordinary di erential equations (ODE) is required. The method of convergence
acceleration based on the allocation of integrable weak singularity is explained.
This work is supported by the Russian Foundation for Basic Research (project
?15-01-00744).
K e y w o r d s: hydrogen permeability; thermal desorption; nonlinear boundary-value
problems; dynamical boundary conditions; numerical simulation.
11
¬в?д????
»?????? ? ?????????????? ???????? ???????? ? ?????????? ??????????? ????? ????????????? ???????? [1, 2, 5, 6, 10, 1220].
Ё????????? ??????? ?? ?????? ? ?????????? ??????????, ?? ? ? ?????????? ????????? [19]. ????????? ??????? ?????? ??????????? ????????????????, ????????? ? ????? ??????, ??????????? ? [2124]. Ё???????????????? ???? ??????????, ??? ????????????? ???????? ?? ?????? ????????, ?? ? ??????-?????????? ???????? ??
??????????? [1, 2]. ??????????? ?? ????????????? ??????????????, ???????? ???? ???????? ??????? ? ??????????????? ???????????????? ???-????????????. ?????? ????????
???????????? [4], ??????? ?????????? ?????????? ?????? ? ?????? ?????????.
ћ?т?м?т???с??¤ мод??ь п?р??ос?
?????????? ??????? ???????? ?????? ??????? ???????????? ????????? (???????? ???????? ` ). ?????? ?????????, ???????????
???????????, ??? ??? ???????? ???????????
???????????? ??? ? ???????????? ?????
????? ??????? ???????????????? ????????? ????????????. ???????????? ????????????? ???????? ????. ???????? ?????????? ?????????, ????? ?????????? ??????????????? H
? ????????? (?????????????? ?????????, ??????? ????? ?????????? ???????). ??? ???????? ?????? ??????????? ?????????:
ct (t, x) = D(T )cxx (t, x),
(t, x) 2 Qt? ,
(1)
??? t ?????, Qt? = (0, t ) (0, `); c(t, x) ???????????? ??????????????? ???????? (??????????). ??????????? ???????????? ???????? D ?? ??????????? T (t) ????????????? ?????? ј????????: D = D0 exp ED /[R T (t)] .
????????? ?????????? ??????? ??????? ??
? ?????????? (1), ? ? ????????????? ??????????? ?????????? ?????????. ????? ???????? ???????????? ? ???????????? H2 ?
??????????? ???????? ????????????? ???????? (??. [1, ?. 177206; ?????, ?????????, ????????]). ????? ? ?????? (??)??????????? ????????? ??????? ??????? ?????????:
c(0, x) = c?(x), x 2 [0, `], t 2 [0, t ],
(2)
c0 (t) = g(T )q0 (t), c` (t) = g(T )q` (t),
(3)
q?0 (t) = µs(T )p0 (t)
b(T )q02 (t) + Dcx (t, 0), (4)
q?` (t) = µs(T )p` (t)
b(T )q`2 (t)
12
Dcx (t, `), (5)
b(T ) = b0 exp Eb [RT ] 1 , s(T ) = . . ., ???
c0 (t) c(t, 0), c` (t) c(t, `) ????????? ??ъ????? ???????????? ??????????????? ????????; q0 (t), q` (t) ???????????? ?? ???????????? (x = 0, `); g(T ) ???????? ??????????
?????????? ????? ?????????????? ?? ??????????? ? ? ???????????????? ??ъ???; µ ???????????? ???????????; s(T ) ????????, ?????????? ??? ????, ??? ?????? ????? ?????
Ђ???????????ї ???????? ???????? ? ?????
?????? ?? ???????????; p0 (t), p` (t) ????????
???? (H2 ); b(T ) ??????????? ?????????. ?????? ? ??????? (? ?????????, ?????????????
2 ), ????????????? ?? ?????????.
J0,` = bq0,`
? ?????? ?????????? ??? ????????????,
??? ? ????????? ???????. ??? ????????????
??????? ????? ????? ? ??????: [c] = 1/??3 ,
[q] = 1/c?2 , [Dcx ] = [J] = 1/c?2 c J = bq 2 . ? ???????????? ?????? ????? ???????? µp ?????????? ????? ?????? (? ?????? ?????? ??????? H2 ), ????????????? ? ????????? ????????? ??????????? ? ??????? ???????. ?? ??
???? ????????? s ?????? ???????????? ????????? µsp ??? ????????? ?????? ??????, ????????? ?? ???????????. Ё?? ???????????? ??????????, ??? ?????????? ???????? ?? ??????.
?????? s ????? ???????? 2s ? ???????????????? s ??? ???? ????????????? ?????? H.
? ???????? ???????????????? ????? ??????????????? ?????? ??????? [p] = Topp,p?????? µ(T ) = (2?mkT ) 1/2
2, 484 1022 / T .
2
????? [µ] = 1H2 /(Topp c? c), [T ] = K, k ?????????? ?????????, m ????? ????????
????????. ???????????? ????????????
???p
?????? ?? ??????????? ( µ / 1/ T ) ????? ???????????? ?? ???? ?????????? ? s(T ).
«?м?ч?ни?. ? ?????? ??????? ??????? ?????????? ??????? ????????? (3)(5) ???????? ??
µs(T )p0,` (t) b(T )c20,` (t) = D(T )cx 0,` . (6)
????????? ?????? ????????? µsp (? ???????????????? ????) ??????????????? ???????? H2 ???????. ????????? (6) ?????????? ?? ????????? (4), (5) ??? ????? ????????
?????????? ?? ??????????? (q?
0). ??????????? ????????? ???????????? ????? ??????, ???? ? (6) ? (4), (5) ??? ?????? ????????. »? ??????????????? ???????? ? ??????
bsurface = g 2 bvolume (??. (3)). ? [10] ??????????
????????? ?????? ????????? ?????? ????????????? ???????? ?? ???? ? ?????? ? ???????.
??????????? b ? ??????? (6) ???????? ??????????? ????????????? ????????????.
“ƒ—-эксп?рим?н? и м???ль изм?р?ний
? ?????? c ???????? ????????? ?? ???????????? ??????? ??? ?????? ???????? ??????? ?
??????? ???? ??? ???????????? ??????? ????????. ????? ???? ??? ??????? ???????? ??????????? ?????????? ???????? (?? ?????????
???????????? ?????????), ?? ?????? ??????????? (??????????? ??? ???????). ? ?????? ???????????? ??????????? ?????????????? ?????? ????? ????? ???????????. ? ??????? ????-???????????? ?????????? ????????
????????????? ???????? ? ????????? ??????, ????????????? ?????????? J(t) = b(t)q 2 (t).
Ђ??????????ї p( ) 7! J( ) ???????????? ???????????????? ?????????, ????????? ???????
??????? ??????? J(t). ??? ????????? ??????????? ?????? ??????????? ??????:
b(t)
b(T (t)), D(t)
D(T (t)), s(t)
s(T (t)).
??????????? g ???????? ??????????? (?????????? ?????????? Ђ?????????????ъ??ї)
??????? ?????????? (Ek? = Ek+ ) ? ????????
???????????????? ???-???? ?????????.
??? ?????? ??? ????????? ?????????:
p(t) = p0,` (t), q(t) = q0,` (t), c0 (t) = c` (t),
D(t)cx (t, 0) =
(7)
D(t)cx (t, `), c?(x) = c? = const.
????? t ????????? ???????????? ????????? ???????? ??????????? ?????? ?????????:
p(t) 0, t > t , c(t , x) 0, x 2 [0, `].
»?т?гро-д?фф?р??ц???ь?о?
ур?в????? т?рмод?сорбц??
???????? ?????? ???-?????????:
ct = D(t)cxx , c(0, x) = c?, c0,` (t) = gq(t),
q?(t) = r(t)
r(t)
b(t)q 2 (t) + D(t)cx (t, 0),
µs(t)p(t),
J(t)
b(t)q 2 (t).
????????? ??????? ??????? ?????????? ??????, ????? ????? ?????????, ?? ????? ????????? ?????????? ?????????? (r(t) = 0) ? ???????????? ????????? ??????? (q? = . . .). ???????????? ??????? ?????? (? ? ????? ?????
?????? ????? ?????????? ??????? ? ??ъ???)
?????????? ? [3]. ??????????? ?????? ??????? ?????????? ????????????? ???-???????
J = J(T ). Ё??? ???? ????????? ??? ???????????? ???????????? ????????? ?????????????. ?????? T (t) ?????? ????????? ????????
(T (t) = T 0 + vt). ???????? ??????? v ????????
(< K/c). ?? ?????????? ???????????? ??????????? (???? ????????? ??? ?? ???????????)
?????? ????????????: T (t) = T max .
?????? ?? ? ????????? ?????????????
?????????????? ??????. ? [3, 22] ???????????? ?????????? ????? ??????? ??????? ?????? (? ??? ????? ? ? ?????? ??????? ?????????
?????). ?? ????? ?????????? ????????? ? ????????????????? ???-???????, ????? ??????
????????????? ???????????? (J = bq 2 ). ??????????? ?????????? ???????? ?????????????
??????? ??????? ?????? ??? ??????? ???????????? ?????????? ?????? D0 , ED , b0 , Eb , s0 ,
Es , g. ? ???? ????? ???????? ??????????????,
????? ? ????? ? ????????????? ?????????????
???? ??????? ??? ?????????? ???????.
??????? ??????? R??????????? t, ????????
t
?????? ??????? t0 = 0 Dds :
ct (t, x) = cxx (t, x), c(0, x) = c?, c0,` = gq(t), (8)
cx j0 =
cx j` = q?(t) + [J(t)
r(t)]D
1
(t).
(9)
??????? q(t) ?????????????? ??????????,
? (9) ?????????????? ???????????? ? ???????? ?????? (8). ??????? ??????, ?????????? ??????? ??????? ? (8) ? ??????????:
c? = c(t, x)
f (t) =
gq(t), c?t (t, x) = c?xx (t, x) + f (t),
g q?(t), c?(0, x) = ??(x) = 0, c?j0,` = 0.
??????? ??????? ? ??????? ??????? ??????????? ????????? (??????? ?????) [8, ??. 2]:
Z tZ `
c?(t, x) =
G(x, ?, t ? )f (? ) d? d?,
0
G(x, ?, t) =
0
1
2X
`
exp
n=1
n n2 ? 2 o
n?x
n??
t sin
sin
.
2
`
`
`
? ???????????? ????????? ??????? ??????
??????????? c?x (t, 0):
Z
n n2 ? 2
o
X0
4g t
c?x j0 =
q?(? )
exp
(?
t)
d? ,
` 0
`2
P
P0
n=1,3,5... . ??? ? = t ??? ??????????,
??? ??? ??????????????? ????????? ??????????????. ? ???????? ??????? t ?????
cx (t, 0) = c?x (t, 0) = cx (t, `) = c?x (t, `), cx (t, 0) =
Z
n n2 ? 2 Z t
o
4g X0 t
=
q?(? ) exp
D(s)
ds
d? .
`
`2 ?
0
???????????? ???????????? ????????? ??????? ????????? ? ?????
b(t)q 2 (t)
q?(t) = r(t)
4gD
`
XZ
0
0
t
q?(? ) exp
(10)
n n2 ?
`2
13
2Z
?
t
o
D(s) ds d? .
????????? ???????????? ???????? ???????
??????: ??????? q(t) ?????????? ?????????? c(t, x). ??? ?????????? ??????? ???????
[t1 , t2 ]
(0, t ), ??????????????? ??????????? ???? ?????????????? (????????? ??? t
??????? ? 0, t ????????????????). ??? ?????? ????????? ??????? ?????????? ???????????? (r(t) = 0), ??? ???????? ?????????.
Ѕ?зр?зм?рн?¤ ф?рм? з???чи
??? ?????????? ????????????? ?????? ??????? ? ????????????
??????????, ????????
Rt
??????: t0 = 0 D(s)ds/`2 , x0 = x/`, v = q/q?
(c? = g q?). ?? ????? ??????????? t, ????????:
b?(t)
b(t)`2 q?
, v?(t) = b?(t)v 2 (t)
D(t)
X0Z t
2 2
4g`
v?(? ) exp
n ? [t
(11)
? ] d? ,
0
v(0) = 1 (????????? ??????). ???????????
??????? k ?????????? ????. ?????????:
Z t
zi (t) =
v?(? ) exp
(2i 1)2 ? 2 [t ? ] d? .
0
????????????????? zi ?? t ? ????????? ?????? v? ????????? (11) (????? ?? n = 2k 1):
z?i = b?v 2
(2i 1)2 ? 2 + 4g` zi
4g`(z1 + ... + zi
1
+ zi+1 + ... + zk ).
? ????? ???????? ????????? ??????? ???????????? ???????????????? ?????????:
? ?
? ?
v
1
z
1?
?
?
?
d ? 1?
2
?.?
(12)
=
b?(t)v
(t)
.
? .. ?
dt ? .. ?
1
zk
?
?? ?
1 1 ... 1
z1
??1 1 ... 1 ? ?z2 ?
?? ?
4g` ?
? ... ... . . . ... ? ? ... ?,
1 1 ... ?k
zk
?i
1 + (2i 1)2 ? 2 /(4g`), v(0) = 1, zi (0) = 0
(??????????? ??????? (k + 1) k).
„?с????о? мод???ров????
??? ????????? ????????????? ?????????????? ?????? ?? ?????? ? ??????????? ?????
????? 12?18?10?. ???????????? ??????????
????????? ??????? ??????? ????????????????
(Ђ???????????ї) ???-??????? ? ??????????
???????????? ????????. ??? ??????????
??????????? ? ???????? ???????? ??????? ?????? ?????????? ??????? [22] ?????????????
14
?????????????? ??????? (12) ??????? 1015
(?????? ???????????? ??????? Scilab 5.5). ?????????? ????????????? ???????????? ? [4].
?????? ?? ?????? [5]: [E] = ??/????;
D = 7, 5 10 3 expf 40 000/RT g [??2 /?]; bvol =
1, 53 10 18 expf 43 200/RT g [??4 /?]; s = 1, 8
10 2 expf 61 400/RT g; p? = 37, 4 [????]; ` =
0, 02 [cm]; T = 770 [K], T (0) = 295 [K]; T? =
0, 5 [K/c]. ??? ?????????????? ???????????
b0 surf = 1, 53 10 14 [??2 /?]; g = 100 [1/?].
?????? ?? ????? ????? 12?18?10? [5, 18]:
D = 3, 09
10 4 expf 27 780/RT g; bvol =
5, 05
10 13 expf 97 140/RT g; s = 7, 05
2
10 expf 59 510/RT g; T (0) = 300 ?. ???
?????????????? b0 surf = 5, 05 10 9 ; ` = 0, 1;
g = 100; p? = 100; T = 850, T? = 0, 5 [K/c].
¬ыд?????? особ???ост?
д?¤ ус?ор???¤ сход?мост?
??????????????, ?????? ??? ??????????
?????? ????????????? ???????-????????????????? ????????? (11) ???????? ??? (12)
????????? ?????? ???????????? ??????? ?????????? ?????????? zi (t). ???????
X0
2 2 X0 X
?(s) = 4
exp
n ? s ,
n=1,3,5,...
????? ???????? ???????? ??? s > 0. ??? ?????? ???????? ??? ??????? s, ? ??? ??????????
??????????? s = 0 (????? ?????????? ?????????????? ? ????????? ???????? ??????? t) ???????? ???????????? ???. Ђ???????ї ????????? ??????????????. ??? ???? max jzi (t)j =
O(n 2 ) (n = 2i 1, i > 1), ??? ???????? ? ????????? ??????????. ???????-?????? ??? ?????????? ???? ???? ?? ???????? ?????????????? ?????????, ?? ???????? ???????????? ???
????????????? ?????? ????????????? ? ??????? ??????????? ????????????????. ????????
?????? ????? ??????????: ??? ?????? ????
??????? (45), ????? ????? ???? ???????????? ??????????? ?????? ???????? (????????, ???????? ???????????????? Scilab). ??????? ????????????? ??????????? ? ??????????????? ??????? Ђ?????????ї ?????????
? ??????????. ??????? ????? ????? ???? ? ?????? ??????????? ?????????????.
???????? ??????????????, ????????? ??????????? я????. ??????, ??? ??????????
X1
?3 (t, x) = 1 + 2
expf n2 ? 2 tg cos(2n?x).
n=1
??? x = 0, t > 0 ????? ?????????????? ????
????????? ?3 (t, 0) ??. [6, c. 179]; [11, c. 353] :
1
X
1
n
1 X
exp
1+2
expf n ? tg = p
?t 1
n=1
2 2
n2 o
.
t
??? ????? ?????? ???????? ??? ??????? t,
? ??? ?????? ??? ????? ????????? t (??????? ??? ??????
???? ????P ? ??????????).
?????? ?(t) =
expf ?n2 tg (n 2 Z, t > 0),
?? ??????? ?????????????? ?????????
???
p
t?(t),
???
????-???????
[7,
c.
261]
?(1/t)
=
p P
P
t expf ?n2 tg =
expf ?n2 /tg (n 2 Z).
???????? ??????????????? ??????????????, ???????? ????????????? ????? ??
???????? ? ?????? ?????????:
1
1
n n2 o
X
1 X
p
exp
=1+2
expf n2 ? 2 tg =
t
?t 1
n=1
=1+2
X0
+2
1
X
expf k 2 ? 2 4tg =
k=1
1
n
X0
1 X
=2
+p
exp
4?t 1
n2 o
.
4t
?????? (????? ????????? ?? ??????? ???? ?????????? ? ????????) ???????? ??????? ?????????? ??? s > 0:
X0
2 2 1+S
?(s) 4
exp
n ? s = p ,
?s
1
n
X 2
1o
S(s) 2
qn , q
exp
.
4s
n=1
??? S ????? ?????? ???????? ??? ????? s.
??? s ! +0 ????? p
S ! 0 ? ?????????????
??????????? ?
1/ ?s. ??? ??? ? ???????????????, ??? ?????? ??????????? S < 0 (?????? S 2 ( 1, 0)) ? jSj 6 2jqj. ????????? ?????? ?????? (?? ????????? ? S2m > S > S2m+1
? ???????? ????????? ????), ?? ??? ?? ?????
?????????? ??? ???????????? ????? s. ?????? ??????? S(s) ????? S-???????? ???
?????? ????????? ? ??? S(1)
0, 999. ? ?????????? ? ??????? ?????????? ??????????
??? ?(s) ????????? t > 0, v(0) = 1 v?(t) =
Z t
1 + S(t ? )
p
= b?(t)v 2 (t) ?
v?(? ) d? (13)
?[t ? ]
0
????? (?????? ??????????? ??? ??????????)
?????? ??????? ?? ????????????? (? = t) ??
??????????? ???? (? = t 1). ?? ?? ????????? ????? ?????????? ??????????? ??????,
????????? ? ????? ?????? (b?
1, ?
1)
??????? jv?(t)j ?????? ??????? ? ?????? t.
??????, ????????? ?????????? (? ????????? ???-????????????)
???????? ????????p
??? ?(s) 1/ ?s, ??????? ???????????? ???????? ???????? ???????????? ??? ??????????? (jSj
1 ??? ????? s). ??? ?????????????? ????????? ??????? 2% ?????? ? ?????????: 1 + S 1, jSj 6 2 expf 1/4sg < 2 10 2 .
?????????? ???????? s 6 1/20. ? ????????????????? ??????? ???????? ????????? ? ??????????????? ???????? ??????????? ? ???????? 1020 %. Ё?? ????????? ? ?????? ????????
?????? ????????? (11) ?????????? ? ?????
v?(t) =
b?(t)v 2 (t)
X0 Z t
4?
h
...
4?
0
X0 Z
t
. . . (14)
t h
t > h, b?
q?b`2/D, ?
g` ? ? ?????????
????? (Ђ??????? ?????ї)p??????????????? ?????????????? ?(s) 1/ ?s (s = t ? ) ? ?????????? ???????????? 0 6 s 6 h = 1/20. Ё??
?????? ? ???????????? (Ђ????????????ї) ???????. ??????????? ? ?????????
???????????
Rt
2
0
???????, ????? h
t h D d?/` . ????????
7
???????? ` = 0, 02 ??, D = 10 (10 6 ) ??2 /c,
??? ???????? ? ?????? ?????? h
200(20) c.
????? h ?? ???????? ????????? ????? ? ???????? ????? ???-????????????. ????? ???????, ?????????? ?????? ????????? ???????
????? ????????? ?? ???????? ? ?????????????? ???????????? ??????????????? ?????????
??? ????????? ????????????????? ??????.
????? эфф?к?и?ных к?эффици?н???
? ???????? ????????? (11) ??????? ? ????? ????????? ????? ????????? (??? ?????????????? ??? ? ?????? ???????? ????? ?????????? ? ?????????? n2 = 1, 9, 25):
X0 X
X
=
+
,
2
X
X1
X
X
,
.
1
n=1,3,5
2
n=7,9,...
P
??? 1 ??????, ??? ? ?????, ??????????
Z t
2 2
zi (t) = v?(? ) exp
n ? [t ? ] d? , n = 2i 1,
0
i = 1, 2, 3; zi (0) = 0, t > 0 ) zi < 0. ?????
P
2 ??????????????, ??????????? ?? ??? ??????????. ??? ???? ???????
?????????????
p
??????????? ?(s)
1/ ?s (s = t ?
1) ?
????? ?????????? ?????????? ?????? ? ???????????. ÷???: ???????? ? ?????? ??? ?????????? ??????? (????? ??????????????? ????????????? ????????????? ?????????).
????????? ???????? h Ђ???????? ?????????ї ? ? t (t ? 6 h) ? ?????????? ??????p
??? ??????? ??????? t 2 [0, h]. ?????? 1/ ?s
??????? ???????? (S < 0, jSj
1), ????????????? ???????????? ????????? ??????? ???
????????? ????????? (v? < 0). ???????
Z t
X
X0
1
4?
6 4?
6 ? v?(? )p
d?. (15)
2
?[t ? ]
0
15
?????? ??????????? ?????? ?????, ??????
????????? (11) ????????
X
v?(t) = b?(t)v 2 (t) 4?
zi (t)
i=1,2,3
t
Z
?
0
v?(? )
p
d? , t 2 [0, h]. (16)
?[t ? ]
?????????? zi (t) ??????? ??? ????? Ђ??????? ??????????ї ? ???????????? ?????????????? ?? ????????? t > h. ??? ????? h ?
????????? ?????? zi (0) = 0 ??? ????? ?????????? (???? ???????? ???????????, ????????? ????? ?????? ??????????? ? (15)):
Z t
v?(? )
2
d? , (17)
v?(t) = b?(t)v (t) ? p
?[t ? ]
0
v(0) = 1, t 2 [0, h]. ???
?????????? w(t) = v?(t)
Rt
v(t) = 1 + 0 w d? ???????? ?????????? ???????????? ????????? ? ??? ?????????? ?????? ????????????. ???? ????? ? ???? ????????
v?0 (t) = b?(t)v02 (t) (v0 (0) = 1), w0 = b?v02 ,
Z t
wk+1 (? )
wk+1 (t) = b?(t)vk2 (t) ? p
d? ,
?[t ? ]
0
?? ????? ??????????????? ???????? ???????
???????? ???????????? ????????? ?? ??????
???????????? [9] ??? ????????????
p ??????????? (?????? ??????? wk+1 (t) 1/ ?t ). ?????
????????, ??????? ??????????? ????? ?????????????? ????????, ?????? ???????????????? ?????????. ? ???? p
?? ??????????????
?????????? ?????? ? 1/ ?s ? (15).
???????, ??? ?? ???? ??????????? ??????????? Ђ??????????
?????ї ? ??????? ????p
??? v?(t) 1/ ?t ?????? ???????? v?(? ), ?
t. ??????? ??? ??????????
p v?(? ) ?? v?(t)
(? ???????????? ????? O( t ? )). ? ?????? ???????? ?????????? ???????? ????????
????????? ????????????? ???????????? ???
???????? ? ????????? ????????????? ??????????, ??? ???????? ???????????? ???????????? ??????????????? ?????????. ????? ????????, ??? ????????? ?????????????? ???????????? ??? J = bq 2 , ??? ??? ??????????
?????????? q(t) (v = q/q?) c ?????????????
????? ??????????? ?????? b(T (t)) ?? ???? ??????? ???????? ? ????????????? ???-????.
? ?????, ?????????? ???????????, ?????? (16)
???????? ? ???
X
p zi (t).
v?(t) 1 + 2? t/? = b?(t)v 2 (t) 4?
i
? ?????? (17) ????? zi ????? ?????????????.
???????, ??? ????????? ???????????? ??? v?
16
????????????? ??????????? ??????: ? ???????? ????????? (11) ?? ?????? ? ?????? ?????
????????????? ?????????, ??? ????????? v?,
?? ????????? [. . .] ??? ????????????.
? ?????? z?i (t) = n2 ? 2 zi (t) + v?(t) ????????
????????? ??????? ??? (v = v(t), zi = zi (t)):
?
?
v? = b? v 2 4?? z1 4?? z2 4?? z3 ,
?
?
?
?
?z?1 = b? v 2 4?? + ? 2 z1 4?? z2 4?? z3 ,
(18)
2 4?? z
2 z
?
z?
=
b?
v
4
??
+
9?
4
??
z
,
?
2
1
2
3
?
?
?
?
z?3 = b? v 2 4?? z1 4?? z2
4?? + 25? 2 z3 ,
??? ??????? v(0)
p = 1, zi (0) = 0, b? =pb?(T (t))
b?(T (t))/[1 + 2? t/?], ?? ? /[1 + 2? t/?] (????? h ???????????? ???? [. . .]
1). ?? ????????? ? ???????? ???????? ????????? ????
????????????? ????????????? ?? ???? ??????? ??????????? ??? ? ! t ? ??????????.
??? ???????? ? ????????? t > h ?????????? ????????? (14) ??? t > 0, ??? ???????????? ????????? ??????? v?(? ) = 0, ? < 0.
? ???? ????????????? ??????? h ?? ??????
?????
p ????????????? ????????????? ?(s)
1/ ?s. ? ?????? ????? ??????? Ђ???????ї
?????? ?????????. ??? ???? ??????????? ????????????? ????????. ? ??????? ??????????? ??????? ?????? ?????????????? t h ??????? ?? ??????? t. ???????? ? ?????????
X
v?(t) = b?(t)v 2 (t) 4?
zi (t)
i=1,2,3
t
Z
?
t h
p
v?(? )
?[t
?]
d? , t > 0.
????? ? ?????? ??????????????? ???????????
???????? v?(? ) ?? v?(t) (??????? ? v?(? ) = 0,
? < 0) ? ???????? ???????
(18). ?????? ???
p
t > h ???????? [1+2p
? t/?] ?? Ђ??????????????ї ???????? [1+2? h/?]. ??? ?????????????
??????? ???????? (????????, Scilab) ?????
???????????? ????????????? ?????? ???????
? ????? (18) ??? t > 0 c ???????????
????p
??? ????????? sat: [. . .] = 1+2? sat(t/h)h/?.
??????????? ??????????? ? ????????????
? ????????? ??????-??????????? ??????????? ????? ?????????? ? ????????????: ?????????? ?????????? ?????? (18) ?????????????? ? ?????? ????????????? ??????????????
????????????? ????????? ? ?????????. ??
??????? v(t) = q(t)/q? (t = t0 ) ????? ??????????? ? ??????????? ??????? ???????? ????????? ???-?????? J(T ) = b(T )q 2 (t) (T (t) $ t).
??????????????? ??????? 1. ??????????
????????????? ????????? h ???????? ????????????? ???????????, ?? ???????? ????????-
??? ??????????? Ђ??????ї (t ? ? > h) ????????? ?????? ?????????? zi (t). ??????????
????? ???-??????? ???????????????? ? ??????? ????????? ? ??????? ????????? h, ??????????? ????????????? ? ??????????? ????. ???????? ????????????? ???? ??????????? ?? h (???????? ??????? ?? ????????
max). ??? ????, ????????, h = 0, 1 ?????
Ђ???????? ?? ????ї, ? h = 1 ?????? ?? ????? (T > 600 K). ? ?????? ???????????? ??????????? ??????????? ????????? ?????? ?????????????? ??????????? ?????-???? ??????? ???????????????. ?????????????? ???????????? ???????? ? ??????????? ????????? 0, 1 < h < 0, 8. ??????? 2 ????????????? ??????? Ђ???????? ?????ї: ? h = 0, 3 ??
???, ? i 6 3 ????????. ????? ????, ??????? (18) ????????? ???????????? ??????????????? ????????????? ? ????????? ??????.
??????? ?????????? ???????? h = 10 (? ???
J(T(t)) .10-14
t
Ni
h = 0.3
i<5
i<3
i<1
T
???. 1. ??????? (18), ??????? h (??????)
???. 2. ??????? ???? (18) ??????? 2?6
J(T(t)) .10-14
J(T(t)) .10-14
t
1.5
material 12X18H10T
material
12X18H10T
h = 0.1, 1, 10
i<3
1.2
h = 10, 5, 0.5, 1
i<5
h = 10
i<7
0.9
0.6
0.3
T
T
???. 3. ??????? ???? (18), ??????? h (?????)
???. 4. ??????? ???? (18), ?????
17
????? ? ??? ????????? ?? ???????? ????? ???????????????). ???? ?? ??????? 1, ??? ??????????? ????????? ???????????? ??????????? ????????? ??????? ????????? ????? ???????? ? ??????? (????????? c?). ?????????? ?
?????????? ???????? ???????? ???????? ???????? ?????????? (????????? ??????????????
b0 ?? 12 % ??? ????????? g ?? 11 %), ????? ???????? ?????????????????? ??????????? ???-???????. ?? Ђ??????ї ?????? ??????,
?????????? ???????? ???????? ??????? ?????? ?????????? ??????? [22].
??????? 3, 4 ????????????? ???????????
????????????? ? ????? ??????? ??????, ????? ??????????? ??????????? ????????? ??????????. ???????, ??? ?????? ???? ??????
???????????????? ??? ??????? ??????? ? ?????????? ????????? ????? (? ?????? ????? ???,
???? ?????????? ???????? ???????? ??????????????? ????????? ? ????????? ????????).
J(T(t)) .10-13
????? ?????
t
Ni
???????? ? ????????? (13), ??????? (???
????????????) ????????? v?(? ) = 0 ??? ? < 0.
????????? S(1)
?0, 999, ?? ??????????
??????????? Ђ?????ї ????? ??? h0 = 1 ?????????? ? ????? ??????? ? ?????????
Zt
2
Ni
h=1
h = 0.1
h = 0.05
1+S(t?? )
p
v?(? ) d? . (19)
?[t ? ? ]
v?(t) = ?b?(t)v (t)? ?
t?h0
??????????? ??????? ???????? t 2 [0, h],
h 6 h0 . ??????????? ??? ?
t (S(+0) =
0) ????????? ? ?????? ??????????? ????????
v?(? ) v?(t). ???????? ???:
p
p v?(t) 1 + f?(t)+2 tg? / ? = ?b?(t)v 2 (t), (20)
Z t
S(+0)
S(s)
p ds, p
?(t)
= 0.
s
+0
0
P
n2 (q = ? expf?1/(4s)g)
? ???? S(s) = 2 1
n=1 q
??????????? ? ???? ??????????????? ???????
?????????????. ????? ???????? ????????????? ?? s 2 [?, t], ??????? ?????? ?????? ??
????? ? > 0 (??????,
10?4 ). ??????? ????p
??? S(s) ? S(s)/ s ?????????? ?? ??????? 5.
??? S ?????? ???????? ??? s 2 [0, 1], ?????????? ??????? ?????????. ??? ???????? ? t > h,
???????? ? (19) ?????? ???????? ?? ? 2 [t ?
h, t], ??????? ?????? Ђ??????????ї
p
p ??????????? [. . .] ??? v?(t): ?(t) ! ?(h), t ! h. ????????? ????????????? ???????????? ??????? 6, 7. ???? ???????????? ??????, ??????????????? ??????????? ?????? ???-????, ??
??? ?????? ?????????? ????????? (20) ?????????? ??? ?????? ????????????? ??????. ???
???? ?????? ????? ???????????? ????? ?????? ????????????, ??? v?(? ) v?(t).
s
(s) =
S( )
d
0
N
n2
S(s) = 2
q ,
q = exp{ 1/(4s)},
N > 10
n=1
S(s)
{
(s) + 2 s }
s
S(s)
s
p
???. 5. ??????? S(s) ? S(s)/ s
18
T
???. 6. ???-?????? (??????), ????? ?????
J(T(t)) .10-13
1200
1400
t
material
12X18H10T
h=1
T
???. 7. ???-?????? (?????), ????? ?????
????????? ??????????????? ????????????? ????? ???????? ????????? ???????.
????????? ??? h, ???????? h = h0 . ? ????????? (19) ????????????? ??? ?????????? ???????? ????????????? v?(? )
v?(t) + A[t ? ? ].
???????? A ???????? ?????????? ?? ????????? ? ?????????? ?????????????? ? . ?? ??????? [kh, (k + 1)h] (k > 0) ???????? A =
?(v?(t) ? v?(kh))/(t ? kh), ????? ??? ? = kh
???????? Ђ??????????ї ???????? v?(kh).
?????? ???????? ???????????? ????? ??????????????? ???????????? ????????????
v?(? )
v?(t) + A[t ? ? ] + B[t ? ? ]2 . ???????
? = t ) v?(t), ? = kh ) v?(kh) ???????? ? ?????????: t 2 [kh, (k + 1)h], ? 2 [kh, t],
v?(? )
v?(t) ?
v?(t) ? v?(kh)
[t ? ? ]?
t ? kh
?B[t ? kh][t ? ? ] + B[t ? ? ]2 .
????? B < 0 ????????? ?????? ??????????
??????? v? (v? < 0). ???????? B ???????
??????? ??????? ?????????: v?((k + 1)h 0) =
v?((k +1)h+0). ????? ?? ????? ????????? ?????????? ?????????????? ????????.
? ?????????? ???????? ?????????? ?????? ????? ??? ????? h, ????? ?????
p ???????????? ???????????? ?(s)
1/ ?s ?? ?????? ????????? ? ?????? ????? ????????? (14).
??????????? ???????? ????????????? v?(? )
v?(t) + A[t ? ], A = (v?(t) v?(kh))/(t kh).
????????? ??? h ????????????? ???????
(t = t0 ) ??? ??????????????????
????????P
p
????? ?(s)
4 0 exp
n2 ? 2 s
1/ ?s
s 2 (0, h], s = t ? . ????????????? ???? ??????? ?? ????????? ??????? t 2 [0, h] ? ????????
???????-???????????????? ????????? (11):
XZ
0
t
????????? ? ??????? t 2 [kh, (k + 1)h]
(k > 1), ??????? ??? ? ! t ???????????:
Zt
v?(? )
d? 4? ?k (t), (21)
v?(t) = b?(t)v (t) ? p
?[t ? ]
kh
X0Z kh
... =
?k (t)
2
=
0
Z
k
1
X0 X (j+1)h
v?(? ) exp
j=0
n2 ? 2 [t
? ] d?.
jh
? ????????? (21) ? ?????? ???????? ???????????? ???????? (t 2 [kh, (k + 1)h])
v?(? )
v?(t)
v?(t)
t
v?(kh)
[t
kh
? ].
(22)
... )
0
Z t
v?(? )
2
v?(t) = b?(t)v (t) ? p
d? .
?[t ? ]
0
p
? ??????? v?(t) 1/ ?t ?????? ??????? ???????? ???????, ??? ??? ??????? ?????? Ђ???????ї ??????? ???????????????? ? ??????
???????. ????????? ??-?? ??????????? ?????????? ??? ????? ???????? v?(? ) ??? ?
t,
?? ????? ?????? v?(? ) ?? v?(t) ???????? ? ??2
????
v?(t)p= b?(t)v (t) (v(0) = 1), ??? b? =
b?/ 1+2? t/? . ? ??????????? ???????? ?????????? ??????? v1 , v?1 . ?????????? v?1 ???
????????, ???????? ????????? ??????? v? =
b?v 2 + f1 (t). »??????? ????? ?????????. ?? ?
????? ??????? ????????, ????????? ?? Ђ?????????????ї ? ???????? ???????? ??????? ?????? ??????????? ????????. ? ????? ????????? ????????????? ??? ?????????? ???????? ????????????? v?(? )
v?(t) + A[t
? ].
? ?????? v?(0) = b?(0) ???????? ????????
A = [v?(t) + b?(0)]/t ? ????????? ???????
p
p
h
4? t i
2? b?(0) t
2
p
v?(t) 1 + p
= b?(t)v (t) +
,
3 ?
3 ?
???????, ??? ? ???? ??????????? ??????????
? ???? v? < 0 ?????? ????? ?????? ?????. „???? ?????????????? ?????????? ?????? ?(s)
? v?(? ) ? (21), ??? ????????????? ???????
?k (t) ????? ???????????? ??? ?????????? ???????? ?????? ???????????????:
v(0) = 1, t 2 [0, h]. ? ?????????? ????????
????????? v? = b?v 2 ????????????? ??????? ????????? ???????? ? ??????????? ??? ???????????? Ђ????????ї ?? ??ъ???. ????? ???????????? ????????? ? ?????? ??????????? ? ???????? (? = g`, t = t0 (D)): ????????? ??????????? ??????????? ????????? ? ??????????
????????????? ????????? ? ?????? ?????.
wk v?(kh), t 2 [kh, (k +
1)h], v(0) = 1
?0 ( ) = 0, w0 = b?(0) .
??????? 8, 9 ??????????, ??? ????????
h ????? ??????????? ????????? ??? ?????
????? ???????????? ??????????? ????????? ?k (t). ?? ???????? 10, 11 ?????????? ???????? N = 0 ????????????? ?????????? ?????
???????????.
v? =
b?v 2
4g`
?k (t)
?k (t)
?(?)
Xk
j=1
X 1
1,3,...,N
jh), t > kh,
v?(jh)?(t
2
e n?
n2 ? 2
2
h
e
n2 ? 2 ?
, ? > 0.
??????? ? ???????????? ??????????????,
N
1 (????????? N = +1). ??????? ??????????? ?????? ?????????, ??? ???
?????????? ?????????? ????????? ?????????
????????? ?? j (j = k, j = k 1, . . . ). ?????????? ??????? ???????? ?? ???????? ? ????????????? ?????????? ???????????.
? ????? ? ???????? ?????????? ?????? ?
?????? ??? ????????????
p
h
4? t kh i
p
v?(t) 1 +
= b?(t)v 2 (t)
3 ?
p
2? t kh
p
wk 4? ?k (t), k > 0, (23)
3 ?
19
J(T(t)) .10-14
J(T(t)) .10-14
t
1.5
material
12X18H10T
Ni
1.2
h=6
h = 50
h=3
h = 30
2
h=1
0.9
h = 10
0.6
1
0.3
0.5
T
???. 8. ?????? (23), ??????? h
T
???. 9. ?????? (23), ??????? h
J(T(t)) .10-14
Ni
t
J(T(t)) .10-14
1.5
t
material 12X18H10T
h = 10
N = 39
1.2
N = 19
N=3
N=5
N=1
N=3
0.9
N=0
N=1
N=0
0.6
h=1
0.3
T
???. 10. ?????? (23), ??????? N
???. 11. ?????? (23), ??????? N
???????ч??? с??????????. ????????????? ? (21) ?????? ???????? ???????????? (22)
????????????: v?(? ) v?(t) + A[t ? ] + B[t ? ]2 .
? ???? ? = t ) v?(t), ? = kh ) v?(kh) ????????
v?(? )
v?(t)
v?(t) v?(kh)
[t ? ]
t kh
B[t kh][t ? ] + B[t
? ]2 , (24)
t 2 [kh, (k + 1)h], ? 2 [kh, t]. ????? B < 0 ???????? ?????? ?????????? ??????? v? (v? < 0).
???????? ???????? ????????? B < 0 ?? ??????? Ђ??????? ???????ї ????????? (23).
??????????? ?????? (22) ???????????? ????????????? (24) ? (21) ???????? ? ?????????
? ?????? ????? ????????? (23) ?????????????p
?? ?????????? Qk (t) 4B ? [t kh]5/2 /[15 ?]:
p
h
4? t kh i
p
v?(t) 1 +
= b?(t)v 2 (t)
3 ?
p
2? t kh
p
wk 4? ?k (t) + Qk (t). (25)
3 ?
20
T
??????? ?????? k-? ????????? ??? ????????
t = (k + 1)h = (k + 1)h 0 ? (k + 1)-? ?????????? ??? t = (k + 1)h = (k + 1)h + 0:
p
p
h
4? h i
2? h
2
p wk +
v? 1 + p
= b?v
3 ?
3 ?
4B ?
+ p h5/2 4? ?k ((k + 1)h),
(26)
15 ?
v? (k + 1)h = b?((k + 1)h v 2 ((k + 1)h
4? ?k+1 ((k + 1)h).
(27)
????????, ??? ?k+1 ((k + 1)h) =
= ?k ((k + 1)h) + v? (k + 1)h ?(h),
X0 1 e n2 ?2 h
?(h)
.
n2 ? 2
????? ??? ?(0) ???????? N ! +1. ?????p
p
??? ????? h ? B ????????: 3 ??(h) = h,
2Bh2 = 5wk . ????? ????????? (26) ? (27) ????????. Ё?? ????????, ??? ??? ???????? ? ??-
?????????? h-???? ?? ??????? ????? ??????????? ?? ?????? ?????????? v(t) (?? ??????????), ?? ? ??????????? v?(t). ????? ????, ?????????? ???????? Bk B ????????????? ? ????
v?(t).
wk < 0, ?. ?. ??????????? ??????????
p
p
???????? ?????????? ?(h)/ h = 1/[3 ?].
??? h ! +1 ?????
X0 1 ? e?n2 ?2 h
n2 ? 2
??????? ????????????? ??????? t = t0 2
[kh, (k + 1)h], ?????????? ????????????? ???????????, ?????????????? ? [22], ??????? ???????? Ђ??????ї ???????? ???????? ???????
?????? ?????????? ??????? (??? ??????? (12)
??????? 10).
X0 1
?(h)
1
= ) p ! 0.
2
2
n ?
8
h
!
???? h p! +0, ??
????????
p ??
p
P0 ???????
?n2 ? 2 h = lim 2 h
lim ?(h)/
h
=
lim
2
h
e
p
p
p
1/[4 ?h] = 1/[2 ?] > 1/[3 ?]. ?????????? ???????? h > 0 ???????????? ????????.
?????? h < 1 ??? ??????????? ???????????
Ђ?? ????? ?????ї ??????????? ? ????????????
???? ??????????? ?????????????.
??????? 12 ????????????? ?????????????
??????
h ?? ??????? ??????????? ?(h) =
p
p
h/[3 ?]. ??????? 13 ? 14 ??????????, ???
?????? (25), ????????? ???? ?????????? ?????????????? ?????????? ??? ?? ???????
_1
8
h
h
???. 12. ?????? (25), ????? h
J(T(t)) .10-13
J(T(t)) .10-14
t
[Eb] = ???/????
b0 = 9 .10-15
b0 = 2 .10
Eb = 45
-9
[b0] = ??2/?
Ni
Eb = 100
Eb = 43.2
b0 = 1.53.10-14
material
12X18H10T
b0 = 10 -8
Eb = 35
Eb = 90
b0 = 1.53 .10-12
T
???. 13. ?????? (25), ??????? b0 , Eb
??????????
? ?????? ???????????? ??????? ?????? ?????????????? ? ??????????? ????????????? ?????????? ????????? ??? ????????????? ???-??????? ????????? ???????????????? ?????????, ?????????????? ???????????
?????????. ??????? ?????? ?????? ?????????? ??? ????????????? ????????????? ? ?????????????????? ??????? ? ??? ???????????? ???????????? ????????? ??????????????? ?????????????. ???????? ??????????
????????? ??????? ???????????????? ?????????? ? ????????? ?????????? ???????????? ????????. ??? ????????????? ?????-
???. 14. ?????? (25), ??????? b0 , Eb
?????????????? ?????? ????????? ?????????????? ????????, ????????? (?????? ????????????? ??????? ?????????????? ???????
?????? ? ???????? ????????????? ??????????) ???? ?????????????? ?????????? ??????? ??? ?????????? ???????. ????????????
????????? ??????????? ? ????? ?????????
??????? ??????? ?? ?????? ????????? ????????????? ???????????. ????????? ?????????? ?????????????? ?????????????, ???????????? ????????????????? ?????? ?? ??????
? ????? 12?18?10?.
?????? ????????? ??? ????????? ????
(????? ? 15-01-00744).
21
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studies, mechanism and modelling. International
Journal of Hydrogen Energy. 2010. Vol. 35.
P. 9060?9069. doi: 10.1016/j.ijhydene.2010.05.092
14. Gabis I. E. The method of concentration
pulses for studying hydrogen transport in solids.
Technical Physics. 1999. Vol. 44, no. 1. P. 90?94.
doi: 10.1134/1.1259257
15. Handbook of hydrogen storage: new materials
for future energy storage. Ed. M. Hirscher. Wiley?
VCH, 2010. 353 p.
16. Indeitsev D. A., Semenov B. N. About a model
of structure-phase transfomations under hydrogen
in uence. Acta Mechanica. 2008. Vol. 195. P. 295?
304. doi: 10.1007/s00707-007-0568-z
17. Lototskyy M. V., Yartys V. A., Pollet B. G.,
Bowman R. C. Jr. Metal hydride hydrogen
compressors: a review. International Journal of
Hydrogen Energy. 2014. Vol. 39. P. 5818?5851.
doi: 10.1016/j.ijhydene.2014.01.158.
18. Popov V. V., Denisiv E. A. Inhibition
of hydrogen permeability by TiN: evaluation
of kinetic parameters. Hydrogen Materials Science
and Chemistry of Carbon Nanomaterials Eds
T. N. Veziroglu et al. Springer. 2007. P. 671?680.
doi: 10.1007/978-1-4020-5514-0
19. The hydrogen economy. Eds M. Ball,
M. Wietschel. Cambridge Univ. Press, 2009. 646 p.
20. Varin R. A., Czujko T., Wronski Z. S. Nanomaterials for solid state hydrogen storage.
Springer, New York, 2009. 338 p. doi: 10.1007/9780-387-77712-2
21. Zaika Yu. V., Bormatova E. P. Parametric
identi cation of a hydrogen permeability
model
by
delay
times
and
conjugate
equations. International Journal of Hydrogen
Energy. 2011. Vol. 36, no. 1. P. 1295?1305.
doi: 10.1016/j.ijhydene.2010.07.099
22. Zaika Yu. V., Kostikova E. K. Computer
simulation of hydrogen thermodesorption.
Advances in Materials Science and Applications.
World Acad. Publ. 2014. Vol, 3, iss. 3. P. 120?129.
doi: 10.5963/AMSA0303003
23. Zaika Yu. V., Rodchenkova N. I. Boundaryvalue problem with moving bounds and dynamic
boundary conditions: di usion peak of TDSspectrum of dehydriding. Applied Mathematical
Modelling. Elsevier. 2009. Vol. 33, no. 10. P. 3776?
3791. doi: 10.1016/j.apm.2008.12.018
24. Zaika Yu. V., Rodchenkova N. I. Hydrogensolid boundary-value problems with dynamical
conditions on surface. Mathematical Modelling.
Nova Sci. Publishers. 2013. P. 269?302.
Received May 30, 2016
—¬≈ƒ≈Ќ»я ќЅ ј¬“ќ–ј’:
CONTRIBUTORS:
????? ???? ??????????
???????????? ???., ?. ?.-?. ?.
???????? ?????????? ??????????????
???????????? ??????????? ???????? ?????? ???
??. ??????????, 11, ????????????,
?????????? ???????, ??????, 185910
??. ?????: zaika@krc.karelia.ru
???.: (8142) 766312
Zaika, Yury
Institute of Applied Mathematical Research,
Karelian Research Centre, Russian Academy of Sciences
11 Pushkinskaya St., 185910 Petrozavodsk, Karelia,
Russia
e-mail: zaika@krc.karelia.ru
tel.: (8142) 766312
????????? ????????? ??????????????
??????? ?????????, ?. ?.-?. ?.
???????? ?????????? ??????????????
???????????? ??????????? ???????? ?????? ???
??. ??????????, 11, ????????????,
?????????? ???????, ??????, 185910
??. ?????: kostikova@krc.karelia.ru
???.: (8142) 766312
Kostikova, Ekaterina
Institute of Applied Mathematical Research,
Karelian Research Centre, Russian Academy of Sciences
11 Pushkinskaya St., 185910 Petrozavodsk, Karelia,
Russia
e-mail: kostikova@krc.karelia.ru
tel.: (8142) 766312
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