Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems
Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer...
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| Date: | 1993 |
|---|---|
| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
1993
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5932 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512158591221760 |
|---|---|
| author | Berezovsky, A. A. Kerefov, A. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. |
| author_facet | Berezovsky, A. A. Kerefov, A. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. |
| author_sort | Berezovsky, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:21:13Z |
| description | Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric. |
| first_indexed | 2026-03-24T03:24:20Z |
| format | Article |
| fulltext |
Y,UK512.949.8
M. X. lliXAHYKOB, n-p cpH3.-MaT. 11ayK,
A. A. KEPEcJ>OB, KaHl\:_,qJH3.-MaT. uayK (Ka6apwrno-6a.nK. yu-T),
A. A. EEPE30BCKHII, n-p cpH3.-MaT. 11ayK (H11-T MaTeMaTHKH AH YKpaHHbl, KHes)
KPAEBhIE 3AJ].A qu J].JHI YPABHEHU.SI
TEilJIOilPOBOJ].HOCTH C J].POEHOH IlPOU3B0J].HOH
B rPAHHqHbJX YCJIOBU.SIX U PA3HOCTHhIE
METOJ].bl HX qucJIEHHOH PEAJIU3AUHU
The boundary-value problems for the heat equation are considered in the case where the boundary
conditions contain a fractional derivative. The problems of these type arise when the heat processes are
estimated by a nonstationary heat fl o w by using the one-dimensional thermal model of a two-layer
system (coating - base). It is proved that the problem considered is correct. A one-parameter family of
difference schemes is constructed and the stability and convergence of difference schemes in the uni
fonn metric is proved.
Po3fJUl/lalOTbCR KpaAoBi 3a/la'li /lJlR pimlRHHll TennonposinHOCTi 3 npo6oDOIO noXi/lHOIO B KpaAODHX
YMOBaX. 3ana'li TaKoro THny onep)KyEMO npH 01.liHIOBaHHi TenJIOBHX npoi1ecin 3 nonOMoro10 O/lHODH
Mip,mi Ten,1ocpi3H'IHOi ~l0/lCJli /lBOWaposoi CHCTCMH (noKpHTTR--OCHOBa) HecTauiouap11010 TenJIOBOIO
Te•1iEto. ,lloseneua KOpeKTHiCTb po3r JIRAysa,mi 33/la'li, no6ynosaua om10napaMerpH'IHa ciM' R pi3HH
ues11x cxeM, BCTaHOBJICHa CTiAKiCTb i 36i)KHiCTb pi3HHI~eBHX CXCM y piBHOMipHiA MCTPHf.li.
1. TiyCTb Q = { (x. t): 0 < X < +00 , 0 < t < T} - 06JiaCTb He3aBHCHMbIX nepeMeHHblX X
H t H3 R 2 . PacCMOTpHM CJieAylOIJ..(ylO 3aAalfy: HaATH orpam-1qeIrnoe B o6JiaCTH Q
pewe,rne u(x. t) H3 KJiacca C(Q) n C1(Q) ypasHeHHH
YAOBJieTBOpHIOIJ..(ee ycJIOBH>IM
u(x, 0) = 0, 0 ~ X < +oo,
t E (0, T],
(1)
(2)
(3)
rAe f (x, t) H (j) 1 (t) - 3Mam-1hIe HenpepbIBHhie <PYHKLJ;HH, <J) 1(0) = 0, a D~,'l:J
onepai-op Apo6uoro AHq:>q:>epem~HposaHHH (1) nopHAKa a = 1/2. l1cnoJih3YH
CBOACTBa onepaTOpOB AP06noro HHTerpHpOBaHHH H Apo6noro AHq:>q>epeHu;HpOBaHH51
AJI51 v(t) E L(O, T) , HMeeM
(4)
TipeAnOJIO)KHM, lfTO peweH1-1e JaAalfH (1)-(3) cylJ..(ecmyeT. TorAa peweutte cMewaH
HOA KpaesoA 3aAalfH u(x, 0) = 0, ux<0, I) = v(t) AJI51 ypasueHH51 (1) AOnycKaeT
npeACTasJie1rne (2)
u(x, t) = - c J (t-ll)- 1I2 exp x v(11)dr] + 1
1
(
2
)
-vn O 4(1-11)
+ 1 +-f dE,f' (t - 11)-112[exp( (x -f,}2) + exp( (x + f,}2 )]/(f,, '7)drJ.
2✓7t o o 4(1 - 11) 4(t - 11)
TiepeXOA51 B nony lfeHHOM npeACTaBJieHHH u(x. t) K npeAeJiy npH X ➔ 0 + H BBOA51
o6o3HalfeHHe u(O, t) = 't(t), uaxoAHM
© M. X. IIIXAHYKOB, A. A. KEPE<DOB, A. A. 6EPE30BCKH'1, 1993
ISSN 0041-6053. YKp. Mam . ;;..ypH., /993, m. 45 , N" 9
(5)
1289
1290 M. X. llIXAHYKOB, A. A. KEPE<I>OB. A. A. 6EPE30BCKJ-llt
,llettCTBYJI Ha o6e 'l8CTH paBeHCTBa (5) oaepaTOpoM npo6Horo n11ct>¢epeH11,lipOBaHHJI
IlOPJIAKa ex.= 1/2 H HCil0Jlb3YJI paseHCTBO (4), HMeeM
D}j;2 -c(t) = - V(t) + <I>(t), (6)
B CHJIY rpaHH•moro ycJIOBHJI (3) 3aKJ1IO'laeM, 'ITO
Dl/r2 -c(t) = a-1[v(t) + cp 1(t)] ,
CJ1eAOBaTeJ1bHO, npH cx. 1 "lc- -1 cpyHKUHJI v(t) = ux(0, t) cymecTsyeT H oapeile
J1ReTCJ1 OAH03Ha,mo, a 3anaqa (1) - (3) peilyUttpyeTCJI K xopowo H3yqeHHOtt aepson
Kpaesott 3anaqe t1.J1J1 ypasHeHHJI (1) B aonyaoJioce Q.
TiyCTb Q = { (x, t): 0 < x < I, 0 < t < T} H ll,JlJI ypasHeHHJI (1) paCCMOTpHM 3anaqy:
HOOTH peweHHe yprumeHHR (1) B o6J1aCTH Q H3 KJiacca C( Q) () C1(Q), Yll.OBJ1eT
BOpJ110w;ee YCJIOBHRM
u(x, 0) = <I>0(x), 0:::; x:::; I,
a 11 u/0,t)-a 12 D~2u(0,t) = -cp1(t), o:::;i:::;T,
a 21u/l,t) = cx.22 Dl/r2 u(l,t) = cpi(t), 0~t~T.
(7)
(8)
(9)
rll,e CX.;1, i = 1, 2; j = 1, 2, - IlOCTOJIHHhie, <l>o(X) E C1[0, /], npuqeM He HapywaJI
o6ll(HOCTH cqHTaeM <1>0(0) = 0, <I>o(/ ) = 0, <?;(t) - HenpepbIBHhre cpyHKQHH, ytJ.OBJieT
BOpRIOIUHe YCJIOBHJIM cornacoBaHHJI: all <I>o(0) = - Cf>1 (0), 0.21 <I>o(L) = - cpz(0)
PeweHHe ypasneHHJI (1), Yll.OBJieTBOpJIIOU{ee HaqaJibHOMY YCJIOBHIO (7). ,a,onycKa
eT apeaCTasneHHe [3]
u(x, t) = - 1,::- J (t -11)-112 exp( x ) [ui; (0, 11)- x u(0, 11)]d11 +
2-v 1t O 4(t -11) 2(t -11)
+ _l_J , - 112 exp(- (x- ~)2 )<I> (S)ds +
2../i O 4t O
+ 1,::-J(t-ll)-112 exp( (x-/)
2
)[u1;,(/,T])- x-l 11(l.11)]d11+
2-v7t O 4 (t -11) 2(1 -T])
} / I ( (. s>2)
+ 2 cfdsJt(s,'ll)(t - 11r112 exp - x - dT).
-v1to o 4(( -11)
(10)
X d 2 )
4(t -11) ll
npH X ➔ 0 +. ITpeasapHTeJlbHO BbIITOJIHHB 3aMeHy nepeMemrott mrrerpHpOB8HHjl no
ISSN 004/ -6053. Y,;p. ~WIil. JKYP"· • /993, Ill. 45 . N' 9
KPAEBblE 3A)lA'-IH ,Ufl>I YPABHEHHSI TEnJ10nPOBO,UI-IOCTH ... 1291
cpopM)' Jle 2✓ ( - 1) 13 = X, nOJIY'IHM
xc f (t - TJ )- 312 u(O, TJ) exp(- x 2 )dri =
4-v7t O 4(/-TJ)
l +J~ ( x 2 ) _132 = c uO,t-~ e df3,
'V 1t x/(2,.fi) 413
H, cne.a.oBaTeJibHO,
lim -~J(t-TJ)-312 u(0,TJ)exp(- x
2
)dri = -2
1 u(O,I). (11)
x➔ O+ 4-v7t O 4(1 -TJ)
A1iaJ I01 H'IHO MO)KHO noKa3aTb, 'ITO
Jim - ~ J(t-TJ)- 312 u(l, TJ ) exp(- (x-1)2 )dri = -2
1 u(l, t) . (12)
x➔ I- 4-V7t O 4(t -TJ)
IlepeXOll..ll B Bblpa)KeHHH (10) K npe.a.eny npH. x ➔ 0 + H x ➔ I - nocne.a.orutTeJibHO
C y'leTOM (11) H (12), H8X0ll.HM
1 I
u(O, t) = - c J (t - TJ)- 11 2 ui;(O, TJ)dTJ +
-v7t 0
+ ~ f (t -TJ)- 112 exp(- I )uc-,(1, TJ)dTJ +
-V7t O 4(! - TJ)
+ c J (t -TJ)-312 u(I, TJ) exp --- dTJ + [
1
( /2 )
2-v7t O 4(t -TJ)
l 1 ( r-2) + -f 1-112<1> (S) exp -~ dS + -Ji O O 4t
l I
1
( S2 ) + ~ J dsJ f(S, TJ)(t-TJ)-112 exp --"- dTJ,
-V7t O O 4(t-TJ)
u(l,t) = - cf(t-TJ)- 112 exp ---- tti;(O, TJ)dl)+ 1 I ( /2 )
-y7t O 4(t -TJ)
l 1 ( (/ _ r-)2 )
+ - J t - 112 exp - ':, <I> (S)ds +
../io 4(t-TJ) o
+ cf (t -TJ)-312 u(O, TJ) exp --- dTJ + c J (t -TJ)-112 ui;(/, TJ)dTJ + [
1
( /2 ) l 1
2-v7t O 4(t -TJ) -V7C 0
1 / I ( (/ - r-)2 )
+-JdsJ/(S,TJ)(t-TJ)-112 exp --~"'- dTj.
-Ji O O 4(t - TJ)
C y'leTOM rpaHH'IHbIX YCJIOBHA: (8) a (9) Bblpa)KeHHJI. (13) u (14) □pHHHMalOT BHll.
ISSN 0041-6053. Y,cp . Mam. JK)'p1t., 1993, m . 45, N• 9
(13)
(14)
1292 M. X. IIIXAHYKOB, A. A. KEPEOOB, A. A. 6EPE30BCKHit
+ ~J(t -1))-112 exp --- 112 u(l, l)}dl) + F (t), I ( /2 f
..fir, 0.21 0 4(t -1)) 011 I
(15)
+ -;:;--r=J(t -1))-312 u(O, TJ) exp --- dT} -/
1
( /2 )
2-vrc O · 4(t - 11)
(16)
F,,(t) == ,=--J{t-TJ)- 112 exp --- <p1(TJ)dT]+ 1 ' ( ,2 )
- -v1ta.110 4(t-T})
PaCCMOTpHM HtITerpa.llbl
ISSN 004/-6()53. YKp. Nam . JKYP"·· 1993, 111. 45. N• 9
KPAEBblE JAJlA 411 .[{Jl.SI YPABHEmUI TEITJIOITPOBOJ.lHOCTH ... 1293
TaK KaK
I I
= - /;12 Ju(0.11)d11~JU-111)-l/2 (111-ll)-112dTJ1
-v1t a.11 o dT) TJ . ,
TO, BbinOJiffj(j( no.u 3HaKOM BHyTpeuuero HHTerpana JaMeny nepeMeuuoi:t HHTerpH
poBaHHj( no <POPMYJie 11, = 11 + (t-T))z H IlOJibJyj(Cb onpe.uei,eHHeM 6ern-cpyHKQHH,
nonyqaeM / I = 0. AuanorH\fHO uaXO.UHM, \fTO H / 2 = 0.
BBH.UY Toro 'fTO
K 1 (t, 11) = a,22,= (t - 11)- l/2 exp( 12 ) E C([O, T] x [0. T]).
0.21-v1t 4(t-11)
Kz(t, 11) = a, 12 (t - 11)· 1/2 exp( 12 ) E C([O. T] x [0. T]).
0.11 .Ji 4(t -11)
0.11 "# 0, 0.21 "# 0 H r(z)- raMMa-cpyHKQHj(.
C yqeTOM H3JIO)KeHHOrO Bblllle paseucrna (15) H (16) npHimMalOT BH.U
I
u(0,t) = Ju(l,T))11 1(t,T))dT)+F 1(t).
0
I
11(/.t) = Ju(O,T))112(t , 11)d11+F2(r).
0
_ -, 112K I _312 ( (2 ) 11 1(1, 11) - r(l/_)DTJ, 1(1. 11) + ✓- (t-T)) - exp - E
2 1t 4(1 - T])
E C(fO. T] x (0. Tl) .
ISSN 0041-6053. Y1-p . \UJ/11 . "'"'"'"' ·· J<)'J3 . II/ . .J5. N " 'J
(17)
1294 M. X. lllXAHYKOB, A. A. KEPE<I>OB, A. A. EEPE30BCK11t1
n 2(t,ll) = -r(l/2)D~~2Ki(t,ll)+ 2~(1-11)-312 exp( 4(11~ 11)) E
E C([0, T] x [0, T]).
lfa CHCTeMbl (7) e CH.TI)' HenpepbIBH0CTH F/1) 1-1 fl;(t, 11), i = 1, 2, q>yHKL\HH u(0. 1) 1-1
u(l. t) orrpeAeJUUOTCJ'I 0AH03Ha•mo H, CJieA0BaTeJibH0, 3ari,a•-rn ( I). (7). (8) . (9)
peAYUHpyeTCJ'I KO BT0poA KpaeeoA 3Ma'le AJIJ'I ypaenemu (1). K0rAa na nexapaK-
TepHCTHtJ:eCKHX y'laCTKax rpaHHQbl X = 0 H X = I 06JlaCTH Q 3MaIOTCJ'I npoH3-
B0AHble u(x. t) no uanpaBJieHHIO BnyTpenneA H0pMaJIH.
2. Ha no.ny6ecK0He'IH0A n0JIOCe X > o. 0 < t < T. pacCMO-rpHM 3aAa'ly
k, ux(0, t) = 13 1 (t)u(0, t) - µ! (t ),
u(x, 0) = 0, I u<x, t) I::,; M, 0 < x < oo, Q::,; t::,; T.
k(x) = { k1 , x::,; x,.
k2 , x>x1•
(18)
(19)
(20)
B T0tJ:Ke pa3pbIBa K03q>q>HqHenTa TenJI0np0B0AH0CTH k(x) Bbin0JIHeHbl YCJI0BHJ'I
HenpepbIBHOCTH TeMnepaTypbl H TenJI0B0r0 n0T0Ka
(21)
Peweuue 3aAa'-{H ( 18) - (20) u = u+ s o6JiaCTH x > x I• t > 0 HMeeT BHA
OTCIOAa npH x = Xi no.ny'laeM
Ha OCH0BaHHH (21) H3 llOCJieAHero HaX0llHM
(22)
u-(x, t) - pemeuue 3all.a'IH (18) - (20) B o6;iacTH O < x < x 1, 0 < t ~ T. O6pa~aJ1
HHTerpaJibnoe ypastteHHe A6eJIJ1 (22), no.ny'laeM
I
-k -( -0 )-lf2.!!_Ju-(x1,'t)d
1UxX1 ,t - ✓ 't,
1t dt t - -r
(23)
0
TaKHM o6pa30M, npH Bbl'-{HCJieHHH TeMneparypttoro n0JIJ'I s o6JiaCTH O < x < x,.
t > 0 BJIHJIHHe noJiy6ecK0He'IH0A 06JiaCTH X > X1, t > 0 M0)KH0 y'leCTb, nocTaBHB
npH X = X1 HeJI0KaJibHOO y'CJIOBHe (23).
ISSN 0041 -6053. YKp. -'Wm. J1CypH., 1993, m. 45. N• 9
K7AERblE 3A}lA l.Uf .[{JI.SI. YPABHEHIDI TEITJIOIIPOBO,UHOCTl1 . . . 1295
ttcxoaH HJ HJJIO)KeHHor·o, B O6J1acTH D = {0 < x < I, 0< t < T} 6yaeM
p accMarpHnaTb Jaaaqy
f3 1 (t)u(0, t)- µ 1 (t),
Fi_!!_ s' u(l, Tl) dT\ ,
.Ji dt ✓r -ri
0
u(x, 0) = 0.
(25)
(26)
(27)
ECJIH cpyHKIJ,HH u(l , t ) a6comonm HenpepbIBHa Ha o-rpe3Ke (0, T], TO HeJIO
KaJibHOe ycJIOBHe (26) c y'leTOM (27) MO)KHO npeacTaBHTb B BHtle [4]
_ .Jk; J1 u'(l, Tl)
-k,ux(l,t ) - c r;-;;dll,
-vrc o -..,t - ri
(28)
3 . Pa3HOCTHblR MeTOtl peweHHH 3a,ua'{ BHJ].a (8), (9) 6y11.eM paccMaTpHBaTb Ha
npHMepe Ja,ua,m (24) - (27), XOTH Bee paccy )KJJ.eHHH cnpaBMJllIBbl H JlJIH KpaeBbIX
YCJIOBHfi BHtla (8), (9 ) H JJ.JlH 6onee OOll.(HX y paBHeHHR napa6OJIH'{eCKOro THna c
nepeMeHHblMH K03Cpcpm:,utetrraMH.
lITaK, a D aseaeM ceTKy 00 1r1:= ooh x ooT= {(ih.j't), i = 0, 1, ... ,N;j = 0, 1, ... ,
j 0}, rt1.e ooh= {x; =ih, i= 0, 1, . .. ,N}, ooT = {tj =j't,j = 0, 1, ... , j 0}, h, 't - warn
ceTKH no npOCTpaHCTBeHHOR H BpeMemIOR KoopaHHaTaM. Tort1.a
'sJ u'(l , Tl ) dri - "t' tf, u'(l, Tl) d =
o (t - 1'1)172 - s"':'1 1,_1 (t j - ll)112 Tl
rae u} N = (11N - it"ti1) /'t = 1~(ts) + O ('t).
B aaJib'iethueM 6y 11.eM npe11.noJiaraTb, '{TO peweHHe HCKOMOR 3a,ua'IH
Tpe6yeMYIO r Jla.llKOCTb. Ha OCHOBaHHH (29) ycJIOBHe (28) nepenHWeM TaK:
(29)
HMeeT
(30)
3aaa'{e (24 ). (25). (30), (27) nocntBHM B COOTBeTCTim:e OAHOnapaMeTpH'-leCKOe
ceMeti:CTBO pa3HOCTHbfX cxeM:
(32)
(33)
y(x , 0) = 0. -1
Ay = Y.u:, ; = h (y; + I - 2y, + Yi - ,),
(34)
I SSN 004/-6053. Y,-;p. "'""'· ;,.,yp11., 1993, m . .J5 , N" 9
1296 M. X. llIXAHYKOB, A. A. KEPE<DOB, A. A. EEPE30BCKHft
<J - npoH3BOJibHblfi BeIUeCTBeHl-lblfi napaMeTp.
O6o3tta'IHM z = y - u. Iloa.crasmu1 y = z + u s (31) - (34), no11yqaeM 3aa.aqy
AJIJI norpernHOCTH :
z = Az(crl + \f', \f' = {O (h~ + 't), cr =/:- 1 / 2, (3S)
1 O(h- + 't2), cr = l / 2,
{
(k(1z;,o -131z;; =t5hz,, o -Vi (t), )(cr) (36)
- k1zx,N + .fii,2 s~l zf,N(,Jt;- - ts-1 - -Jt1 -t,.) = 0,5hz,. N - V2(t),
z(x, 0) = 0, v1, v2 = 0(h2 + 't). (37)
qTOObl Il0JIY'IHTb 0QeHKy AJIJI norpewH0CTH, 3aa.aqy (35) - (37) npusea.eM K Ka
H0HH'lecK0MY s~y [5]
(.!. + 2crk1 )z = crk1 (z 1 + z i ) + (.!.- 2(1-cr)k1 )z- +
't h2 I h2 1+ , - 't h2 '
(38)
(39)
(40)
ra.e
1- (J 2.._ff; 1 [( 0 [;, r:-:::i) 1 ( ,,, ) j-1] V2 - -- c """"c ,y J - 2-y J - l + 'VJ - 2 z N + .. . + '\/ 2 - 2 z N + - . ( 41)
K -vn 'V't K
.[lm1 0QeHKH z(x,-. t) 6ya.eM np0H3BOAHTb "pacc11oett1-1e" ceTKH [5] Ha Il0AMH0-
)1(:eCTBa MeHblliero 'IHCJia H3MepeHHfi H 0QemmaTb ua CJtoe z1"+ 1.
OQeHHM cyMMbl, CT0Jlll{He s KBaApaTHbIX CKo6Kax s COOTIIOIIIeHHH (41):
cr[ ... ] ~ ~132 ~(-fj --f]+I + --/2 -1) max jzt j,
K -V't l:5.l::5j- 1
(42)
(1-cr)[ ... ) ~ (l - cr)j32 ~(✓J-1--fj + 1) max lz~I-
K -v't l:5.l: :5j-1
TaKHM o6pa30M, 3aa.aqy AJIJI norpeIIIIiOCTH Mb( npHBeJIH K BHJ.(Y
A(P )z(P) = ""I,B(P, Q)z(Q) + F(P ),
Q E lll'(P)
ISSN 004/-6053 . YKp. Mam. )l(ypH. , /993 , m. 45, N• 9
Kl'AEBblE 3AJlAYH JlJI.SI YPABHEHHSI TEnnonPOBO}lHOCTH ... 1297
rAe lll'(P)- 0Kpecn,oCTb y3na P (no11.Mt10)KeCTB0 ceTKH ro, uecoAep)Kaw.ee y3en
P).
Peweime 3alla'-IH (38) - ( 40) umeM n BHHe z = z0 + z •, ni.e z O - pewettue 3Ma'-IH
(38)-(40)c 'P0 ='PN=0,a z*-pewet-111eTOtt)KeJaAa'-IHC 'P1 =0,i= 1,2, ... ,
N-1. TaK KaK
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7 . P11.x111>1al1ep P ., Mopmou K. Pa3HOCTHw: MCTO/ll,I peme11HJ1 Kpaemix 3a;1a'I. - M .: M1-1p, 1972. -
418 C.
IloJIY'IC!IO 11. 06 . 92
ISSN 0041 -605.3 . YKp . Ma111. »:yp11 .. 1993 , m . 45 . N" '.I
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| id | umjimathkievua-article-5932 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:24:20Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d7/93d5e914ba79d800ed4d63b86f96e0d7.pdf |
| spelling | umjimathkievua-article-59322020-03-19T09:21:13Z Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems Краевые задачи для уравнения теплопроводности с дробной производной в граничных условиях и разностные методы их численной реализации Berezovsky, A. A. Kerefov, A. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric. Розглядаються крайові задачі для рівняння теплопровідності з дробовою похідною в крайових умовах. Задачі такого типу одержуємо при оцінюванні теплових процесів з допомогою однови- мірної теплофізичної моделі двошарової системи (покриття-основа) нестаціонарною тепловою течією. Доведена коректність розглядуваної задачі, побудована однопараметрична сім’я різницевих схем, встановлена стійкість і збіжність різницевих схем у рівномірній метриці. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5932 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1289–1398 Український математичний журнал; Том 45 № 9 (1993); 1289–1398 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5932/8547 https://umj.imath.kiev.ua/index.php/umj/article/view/5932/8548 Copyright (c) 1993 Berezovsky A. A.; Kerefov A. A.; Shkhanukov-Lafishev M. Kh. |
| spellingShingle | Berezovsky, A. A. Kerefov, A. A. Shkhanukov-Lafishev, M. Kh. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. Березовский, А. А. Керефов, А. А. Шхануков-Лафишев, М. Х. Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems |
| title | Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems |
| title_alt | Краевые задачи для уравнения теплопроводности с дробной производной в граничных условиях и разностные методы их численной реализации |
| title_full | Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems |
| title_fullStr | Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems |
| title_full_unstemmed | Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems |
| title_short | Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems |
| title_sort | boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. difference methods for numerical realization of these problems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5932 |
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