Linear singularly perturbed problems with pulse influence
We establish the closeness of solutions of a linear singularly perturbed problem with asymptotically large pulse influence and the corresponding degenerate problem.
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4592 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510740428881920 |
|---|---|
| author | Mel'nik, T. A. Мельник, Т. А. Мельник, Т. А. |
| author_facet | Mel'nik, T. A. Мельник, Т. А. Мельник, Т. А. |
| author_sort | Mel'nik, T. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We establish the closeness of solutions of a linear singularly perturbed problem with asymptotically large pulse influence and the corresponding degenerate problem. |
| first_indexed | 2026-03-24T03:01:48Z |
| format | Article |
| fulltext |
1)
2)
CTBy
Y/IK 517.9
T. A. Me.abHHK (Once. y.-T)
JII4I.-I ~ I t i b I E C H H F Y . . r l B . P H C o B O 3 M Y l I ~ H I - I b I E 3 A , ~ , k L I H
C HMIIYJIbCt'I~IMH BO3~EHCTBILqMH
We establish the closeness of solutions of a linear singularly perturbed problem with asymptotically
large impulse influences and of the corresponding degenerate problem.
BcTaHo~rlena 6YlH3bKiCTb po3B'g3zia YliHi~hioi cHHryJl~lpno s6ypenoi 3anaqi a aCHMrlTOTHqHO BeYiH-
KHMH iMHyJIbCHHMH/~i#IMrl Ta Bi/~noni/~HOi BHpo~TKeHO[ ~t~ta~i.
B CTaTbe HCC.rle/~ylOTC~ JIHHeflHhle CHHFy.rI$1pHO Bo3MylUeHI-n,le 3a~aqH c HMHyYI~CHbl-
~ a BO3ae~lCTBnm, m, ~onycKaromHe B OTna~ne OT pa60T [1, 2] acnMrrroTnqeCKH 6onb-
tune pa3pnma 6nCTp~X nepeMeHnrax.
1. IIoeTaHOnKa 3a~am~. PaccMaTpnBaeTca petueHne YmHei~HO~t CHHry.napno BO-
3MymeHnOit CHCTeMbI
I~Z = Ali(t)Z + A2i(t)y + f l i( t) ,
= A3i(t)Z + A4i(t)y + f2i(t), t ~ ti, (1)
C Haqa/IbHI:,IMn yc..qOBn..qMn
z(0, It) = z 0, y(0 , It) = y0, (2)
/~onycKmomee B MomenT BpeMenn t = ti, i = 1, n , pazpr.rma riepBoro po/~a:
~Az[ t=tiOt ) = Bli(kt)x(ti, kt) + B2i(P,),
Aylt=ti(g ) = B3i(~l.)x(ti, It) + Bai(I.t ). (3)
g > 0 - H a n ~ napaMerp; t e 1 = [0, T]; z ~ R M, y ~ R m, x = ( ~ ) 3~ecb
Axltffiti =x(ti + 0, g ) - x( t i - O, It); petueHne x (t, g ) 3a~a,m (1) - (3) ~ana onpe~e-
neHHOCTH 6y~en CanTaT~ HenpeprJBH~n~t czeBa; ti(g) ~ (0, T) - - MOMeHT~ BpeMeHn
3aHyMepOBaHHbIe B Bo3pacTaiomeM nopazme.
Hyc'n, B~no.nI-LalOTC~ yCJXOBH~
lim t i (~ ) = ~, lim B i i ( g ) = B-~, i = l, n , k = I, 4 ;
g--->O I~-~u
C06CTBeHHHe 3HaqeHa~ ~ri(t) MaTpHU~ All(t)y/~OB~eTBOpJ~eT HepaBeH-
R e X r / ( t ) < - ~ x < O, t a li, r = 1, M ,
r/~e li = [ ~ / - 8, ti+l + E], e > 0, m neKoTopaa KOHCTaHTa;
3) MaTprmr~ Ata(t), k = 1, 4, fji(t), j = f, 2, i = 1,"n, aenpepbmHO zmqbqbepeH-
mrpyema Ha li.
BI~3K,~eHHa,,'I ~a~aqa, COOTBCTCTB~Oma~ 3a~aqe (1) - (3) , I4Me~DT B I ~
0 = Ali( t )Z + A~(t)-y + f l i( t) , t ~ ~i,
= A31(t)z + A4i( t)y +f21(t ) ,
~(0) = yo,
(4)
C53
�9 1". A. ME.rIbHHK, 1999
I$SN 0041-6053. Y~. ~am. ~'ypn., 1999, m. 51,1~ 1 133
134 T . A . MF_.HbHHK
A ~ - [ ~3i - Aoi ( ti ) Bli ] ~ ( ti ) + [~4/-A0(Ii)B2i], (6)
r'/le Aoi(t)= A3i(t)A~l(t), npu~eM n~apoxc~eHuy~o 3a~aqy (4) - (6) Heon~a
rtonyma-n, H3 aa~a,m (I) - (3) cImpMa.rmHO, nos[arax napa~erp Ix pa~nraM nymo.
2. OCHOBHO~ peay~sTaT. Bsm3oc'rb pemcmat x(t, Ix) H �9 ( t ) 3,a~aq (1 ) - (3) H
(4) - (6) coo'rBerc'meHHO ycraHa~mnaeT coae~:flomax reopeMa.
TeopeMa. Ilycrab abmonnxtomca yc~osu~ I - 3. Tozaa cnpa6ea~uea or a
C n... {" i
IIz(t, Ix) - z ( t ) l l --- ~tCIx)+ - 2 _ ~ e x p l - k ~ l H ( t , ti) +
Ixifl k IX J
+ Cexp - k + C ~ l H ( t , t i ) -H(t ,~i) l , (7)
iffil
I ly( t , i x ) - Y( t ) l l <
n : t _ t i ~ n
7(IX) + C ~ e x p / - k - - T { - - , / H ( t , t i ) C ~ l H ( t , t i ) -H( t , ? i ) l + <
i=t x I'~ / i=1
n
+ C ~ l n ( t , t i ) -u( t ,? i ) [ ,
i--I
zae T(IX)--->0 npu Ix ---> 0; H(t, ti)=l a ~ t>ti; H(t, ti)=O npu t<t i ;
k >0; C>0 .
3. qacTriste c.ayqaH. Ylyerb StanO:IHaIOTCJ~ yc:xom~a 1 - 3. PaccMo'rpaM 3azmqy
(1), (2), y KO'ropott 6tac'rp~e riepeMeHntae I, xMeIOT paaptan n o p a ~ a 0(1), T. e.
Azlt=tiO~) = Bli(ix)x + B21(IX), Aylt-~ti(rt) = B3i(Ix) x + B4i(Ix) . (8)
Tor/Ia ~tonoJmrrre3n,mae yC.rlOBH~I B BI:dpoTK]~CHHOfl 3a~aqe (4), (5) HMeuYr Brta
AyI,=, i = Y~i~(?i) + B4, (9)
3~ecs srapo~eHHyio 3a~a~xy (4), (5), (9) MO~KrIO noay'IHT)- a3 3a~aqrI (1), (2), (8),
c1~pMa.tl1,HO rlO.)lar'a~l r lapaMeTp Ix paBHHM Hy.rlIO.
Caeacmeue 1. ~.a~ peutenua x(t , Ix) u ~( t ) saaa~ (1), (2), (8), u (4), (5), (9)
obtno/ltLqlomcYl ol~eIocu
Ilz(t, Ix)- ~(t)ll--- F ~ ) , I ly( t ,v)- Y(t)ll ~ F2(IX), t~ I,
zcge to = 0,
F~(g)--7(IX) + C ~ e x p - k I-I(t, ti) + C~lH( t , t~ ) - I ' l ( t ,~ ) l ,
i--I
r'~(ix) - 7(ix) + c~lIl( t , ti)- I-I(t,~)l.
i=l
Eena s ~,z~a~e (1), (2) ~onoJlmlTe.rlLHtae yc.rlOeH.a I~elOT ~ttlI
~zlt . . t / )o = Bt~(ix)x(t~, it) + B~(IX).
Ayltffi~(g~ = Ix[B3i(ix)x(t~, IX) + B4/(Ix)], (10)
"to s coo-r~-'remymmctt za~aqe (4), (5)~tonosnmTenmH~e ycstonHa OTC~rC'I'~TIOT.
ISSN 0041-6053. Yxp. ,uam. .wylm., 1999, m. 51, Iv ~ 1
/IHHE~HI~IE CHHI'T/I~IPHO BO3MYU/EHHhIE 3A/1AHH ... 135
CneOcm6ue 2. ]ln.~ petuenu~ x ( t , IX) u ~ ( t ) zaOa~ (1), (2), (10) u (4), (5)
ablno,/u-t~iomc~ ltepa6eHcmoa
IIz(t, IX) - z ( t ) l l s r ~ t ) , Ily(t, IX) - y ( t ) l l < v(IX), t ~ t . (11)
PaccMoTpHM ~tz~aqy (1), (2), B KOTOpOI~
AZlt=ti(lO ~ Z , Ay]t=ti(l~) ~ P'Y, (12)
rne Z c R ~ Y c R m - - KOMnaKTnrae MnO~KecrBa.
CaeOcm~ue 3. ~Tn~ nto6~x petuenu~ x ( t, IX) saSa,cu (1), (2), (12) u petuenu.~
3acga~u (4), (5)cnpaneO.auea ot~enxa (11)
HyCTb B ~ a q e (1), (2) ~onosmrrresmHrae yc.ao~rt~ m~exoT BH~
AZ[t=ti(~t)E Z , Aylt=tiOt)E Y, (13)
B COOTBCTCTBylOtUe~ BbIpO~t(.~CHHOl~ 3a~aqe (4), (5) ~OrlOJIHHTeJIbHrae yCJ'IOBH$I FIpH-
MyT BH~
~YI,=~, e Y. (14)
Cneacm~ue 4. ~naMno:~ecTB X(t , IX) H X (t) pemeHn~t x(t , It) n H (t) aa-
za'~ (1), (2), (13) a (4), (5), (14) cnpa~e~tsmnra Hepanencrna
h(Z( t , ix), Z ( t ) ) < FI(IX), h(Y ( t , lX), Y ( t ) ) < 1"2($t),
r~e h (., .) ~ paCCTOZHHe no Xayc~opqby.
4. ~ogaaaTe.rlbCTllO weopeg-~. B cnny yC~OBHJt 2) MarpHua A2i(t) aen~rpo~K-
/~eHHaSl. TaKHM o6pa3OM, no.ayqHM Ha CHCTeI~,I (4) rlprl t ~ ( t i , ti+ 1 ]
~ ( t ) = - A~l ( t ) [A2i ( t )y ( t ) + f~i(t)] ,
= [A4i ( t ) - Aoi( t )A2i( t )] y + [f2i(t) - A o i ( t ) f l i ( t ) ] (15)
n Ha CHc'reM~ (1) n p a t ~ (ti , ti+l]
z(t , IX) = - A~l ( t ) [A2i(t)y(t, IX) + f u ( t ) - IX~(t, IX)],
= [A4i(t) - Aoi(t)A2i(t)]y + [f2i(t) - Aoi(t)f l i( t)] +
+ pAoi(t)~(t ' IX). (16)
CorYlacno c13opl~lyne Komrt pemenne z(t , IX) nepBoro ypaBHennsl CHGTelvI~I (1) rl
pemeHae y(t, IX) SToporo ypa~aeHHJ~ CHerema (16) Ha npo~e~Kyaxe (ti, ti+ 1] MO~K-
HO HI~CTaBHTb B BH~e
z(t , IX) = Zi(t, ti, IX)KIi(IX) + Zi(t,s, IX)[A2i(s)y(s,g) + f l i ( s ) ]ds ,
y(t, IX) -- Yi(t , tl)[y(ti, IX) + BNx(ti , IX) + B4i(IX)] +
t t
+ I ~ ( t , s ) [ f ~ ( s ) - A~(s)f~i(s)]d~ + I X I ~ ( t , s ) ~ ( s ) i ( s , IX)d$, (17)
t~ t~
r ae Zi( t, $, IX) H ri( t, s) - - cl~YH,HalvleHTa.rlbHl/tr MaTpHI~ CHG'rcM
ISSN 0041-6053. Yxp. ~am. agypn., 1999, m. ~1, bl ~ 1
136 T.A. MEdIbHHK
I X ~ = Au( t )Z i , ~Yi~t = [A4i(t) - Aoi(t)A2i(t)] Yi'
zi(s , s, Ix) = E M, ~ (s, s ) = E m, (18)
KU(IX) = z( ti, Ix) + ~ [ Bu(ix )x( ti, Ix) + B2/(IX)] �9
H3 BTOpOrO TpaBHCHH,,q CHC"rCMH ( 1 7 ) c noMom~m HHTCl'~HpOBaHH.,q nO qacTm~ 3a-
K.moqaeM
f
y(t, it) = Y(t, t/)K21(IX) + f g( t ,s) [ f 2 ( s ) - Ao(s)y(s)]ds +
li
t d
+ gAo(t)Z(t , IX) - gS~[Y(t,s),%(s)lz(s, ~.)ds, (19)
tt
r~Ie tE (ti, t i . l] .
K2i(IX) = y(ti, IX) + B3i(P.)x(ti, IX) + B4i(Ix) - IxAo(ti)z(ti + O, g ) =
= y(ti, ~) + [B3i(IX) - Ao(ti)Bu(g)]x(ti , IX) +
+ [B4i(IX) - Ao(ti)B2i(ix)] - IX Ao(ti)B~(g).
I'IO~CTaBHM rtoytTqeHHoe Hpe/~CTaBYleHHe (19)E~J~ y (t, IX) s nepBoe ypaBHeHHe
c r l c r ema (17). T o r ~ a np~ ti < t <_ ti + l
3
z(t, g ) - Z(t, ti, g ) g u ( i x ) + ~ l k i ( i x ) +
k=l
t
+ IZi( t , s , IX)A2(s)Ao(s)z(t, I x ) d s -
ti
_ iz,(,,,,~)~t r,~,.~)Ao(,)j~(,.~)~,. (20)
t~
3l~ecb
1 t
I l i ( ix) = ~t I Zi ( t,s, ix ) A2( s ) Yi (s, n)K2i(ix )ds,
0
t $
12,(ix)-- ~Iz,(,,,,~) A~(,) Ir,(,:)t:2('~) - Ao(,):i(,)le~,,
t~ t~
t
[3i( ix)- -~!Zi( t , s , ix)fli($) d$.
B mrrerpanr 12i(IX), IIOMeaI~B M ~ Ilpo~edIIM HHTeI'pllpoBaltR~, II021~
tt
~,,(ix)- ~IIz,(,.,,ix),~,c,)r,c,:)t,-,,(~)- ,~,(,)~,(,)1~,~.
J'IHHEI~IHBIE CHHI~JLqPHO BO3MYlI-IEHHME 3AI IAqH ... 137
Bocnon~yer~cz TCM, HTO B CH.lly CHCTeMM (18)
�9 ~Zi(t ,s ,g)A-lrs ~ Z i ( t , s , l ~ ) = - ~ , ~ ~ , ,,
T o r ~ a
~ c
I I i ( P ) = - i OZi(-~-: '!~ ) A~l ( s)A2i( s) Yi( s'ti ) K2'(g )ds =
ti
= A~l(t)A2i(t)Yi(t, tl)K2i(P.) + Zi(t , t i ,P)A~l(t i)A2i(t i)K2i(P.) + 15i(11),
i i ~Zi( t,s,~t ) A-I /2i( l ' t ) = - ~ li ($)A2i(s)Yi(s,q~)[f2i('c) - Aoi(x)f l i(z)]dsd'c =
t
= - fA~l( t )A2i( t )Yi i ( t , 'c )[ f2i ( 'Q - Aoi('Qfli('Q]d'r + 15i(1.1.),
ti
13,(}.t) = - i '~Zi(~sS'12)A~il(s)f l , (S)ds=
ti
= - A~il( t) f l i ( t) + Zi(t , til~)A~il(ti)fli(ti) + 16i(P') ,
t ~ - l s Y st
0
15i(P) =
t
f Zi ( t,'cp ) A~il ('Q A2i('Q[ f2i ('C ) - Aoi('Q fli ('Q ld'[ +
ti
t t ~ l
B cnny ouemca [3]
t d _ i s s ds. I61(I-t) = ~Zi(t,s,l~)-~[Au ( ) f l i ( )1
ti
IIz (t,s, t)ll < Cexp s),
g
Taxa~ o6pa3oM, COOTnOmerme (20) nprn~eT m,,a
zCt, p ) -- zict, ti,lx)[guCIs)A~t(ti)A~Cti)(g~i(Ix) + fuCti))] +
+ ~ J ( p ) - A ~ l ( t ) A 2 i ( t ) Y i ( t , t i ) K 2 i ( I x ) -
t
- A~(t )[A2t( t )~Y~( t , s )[ f~(s) - Aoe(S)fu(s)]ds + ~ i ( t ) ] +
ti
(21)
ISSN 0041-6053. Yrp. ~am. ,vcypn., 1999, m. 51, N e I
138 T.A. MF.dlbHHK
t
+ $ziCt, s, i x ) A 2 i ( s ) A o i ( S ) Z ( s , ~ t ) d s -
t~
t
- I z iCt , s, ix)~-s[~(t ,s)Aoi(s)]zCs, g ) d s . (22)
Cor~Iacao ~opMyole Korea as npe/ICTaBOleHHA (15) caIeffyer, tlTO y ( t ) = Yi (E),
E(t) = ~i(t) npa t ~ ( t i , tt+l], r~e
t
Yi(t) = Y(t , ti)F,2i + ~ Y i ( t , s ) [ f 2 i ( s ) - A o i ( s ) f l i ( s ) ] d s ,
t~
zi ( t ) = - Al]l(t) A2i(t) Yi(t, t/)~'2i -
t
- A~l(t)[A2i(t) f Yi(t, s)[f2i(s ) - Aoi (S) f l i (S)]ds "t" 3~i(t)],
ti
g21 = Y(~/) .4- [&, -- A 0 ( t / ) B I i ] X ( ~ ) +[W4i- Aoi ( t / )B2 , ] "
(23)
(24)
CpaBHrIBa~ n p e ~ c ' r a ~ e m m (22) ~ a a z (t, g ) c npezIc'ra~sienneM (23) ~sxu ~/(t),
aKtg4M, qyorlprl t r ( t i , ti+l]
z(t , g) - ~i(t) = Zi(t, t i, g)[gli(ix) + All(ti)A2(ti)(K2i(P.) + 3~(ti))] + [~1 (~t) +
t
+ fZi( t , s, gl[A2i(s)Aoi(S) - ~ (Y i ( t , s)Aoi(s)l][z(s, •1 - # i ( s ) lds , (25)
ti
Fl~e
~31 (It) = ~3 (3t ) - A~l(t)[A2i(t)[Yi(t, ti) Kli(~t) - Yii(t, t /)Kli(g)] -
- i z , ( ' . ' , -
ti
- A~il(t) A2i( t l f Yi(t, $)[f2i(s) - Aoi(s)] fli(s)]d~ -
ti
O6oaaaqaa nepe3 r ( t ) =exp (o~t/g) II z(t , Ix) - "zi(t)II, Ha (25) a (21) noayaaeM
t
r ( t ) • h ( t ) + C I r ( s )ds , (26)
t~
r~e
IX
IIpHMenaa a e ~ y FpoHyo.n_~a - Br [4] x HepaBeaca~y (26), npHxo~nM X
oReHxe
I$SN 0041-6053. Yr,#. uam. ucylm., 1999, m, 51 ,1~ 1
0"IHHEflHHE CHHr'YJI$IPHO BO3MYII]EHHHE 3A/]AqH ... 139
t
r(t) <_ h(t) + C lh(s)eC(t-S)ds,
to
OTKy~a npti t ~ (t i, ti+ l ]
I I z ( t , g ) - Z~(t)ll
t ~ (ti" " t i+ l ] ,
--- II 131(~)11 + c:~c , - , , ) /~ . g
H3 COOTHOUIeHHfl (19) H (23) BHTeKaeT
y ( t, g ) - Yi(t) = [Yi(t, t i )K2i ( IJ , ) - Y i ( t ,? i )g2i (g)] +
t
+ g A o i ( t ) Z ( t , l . t ) - [ . t f ~ - ~ [ Y i ( t , S ) A o i ( S ) ] Z ( & ~ ) d s -
0
ti
- ~ ~(t, s)[f2i(s) - &i(s )Ai (s ) las .
CoIeROBaTeJmHO,
i l y ( t , g ) - Y/(t)ll < II1~,0~)11 + c: ~'-'v~.
Ot~eHKa (7) B~JTeKac'r ~a HepaaeHcTaa
II ~(t) - ~i(t)II -<
<_ C[H(t, t i ) -H(t ,~) l + clH(t, t i+l)--H(t,~+l) [, t~
H O~eHOK (27), (28).
(ti, ti+O
(27)
(28)
1. llepecmlor H. A., llucoecraa B. 11. rleptto~HqecKHe pemeHH~ CHI'U'yJl.~lpHO Bo3MymCHHI:,IX CHCTeM
He.aHHeI:tHblX ~ttlqbqbepetlUHa.abHldX ypaBHeHHtt, no~aepx'c.eHHblX HMny.~bCHOMy eo3~ettCTBmO//
~Hcl~epemx. ypanHeHHa c Ma, ntaM napaMeTpo~.-KHee: HH-T MaTeMaTtlXH AH YCCP, 1989. -
C. 85 - 9 3 .
2. Bainov D.D, Hekimova 114. A., Veliov V. M. Boundary value problem for a singularly perturbed
system of linear differential equations with impulses / /Proc . Edin. Math. Soc. - 1988. - 31. -
P. 107-126.
3. Bacu,~beea A. E., Eymy3oo B. O. ActlMnTOTtlqeCKHe paa.qoxeHHa cnHry.qapHo BOaMy~eHutax
ypaeHemltt. - M.: HayKa, 1973. - 272 c.
4. O,le~tunz Y., Pume.a P. OrlTHMa.rlbHOC ynpaaaTeHne /IeTepMHHHpOBaHHIdMH H CTOXaCTHqeCKHMH
cHc'reMaMH. -M.: MHp, 1978. - 3 1 6 c .
l'IoJlyqeHO 14.06.96
I$SN 0041-6053. YKp. ~tam. ~xypn., 1999, m. 5], N e I
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| id | umjimathkievua-article-4592 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:01:48Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/02/d4348ab9802e27c568d2cb1037f8ee02.pdf |
| spelling | umjimathkievua-article-45922020-03-18T21:09:14Z Linear singularly perturbed problems with pulse influence Линейные сингулярно возмущенные задачи с импу льсными воздействиями Mel'nik, T. A. Мельник, Т. А. Мельник, Т. А. We establish the closeness of solutions of a linear singularly perturbed problem with asymptotically large pulse influence and the corresponding degenerate problem. Встановлена близькість розв'язків лінійної сингулярно збуреної задачі з асимптотично великими імпульсними діями та відповідної виродженої задачі. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4592 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 133–139 Український математичний журнал; Том 51 № 1 (1999); 133–139 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4592/5888 https://umj.imath.kiev.ua/index.php/umj/article/view/4592/5889 Copyright (c) 1999 Mel'nik T. A. |
| spellingShingle | Mel'nik, T. A. Мельник, Т. А. Мельник, Т. А. Linear singularly perturbed problems with pulse influence |
| title | Linear singularly perturbed problems with pulse influence |
| title_alt | Линейные сингулярно возмущенные задачи с импу льсными воздействиями |
| title_full | Linear singularly perturbed problems with pulse influence |
| title_fullStr | Linear singularly perturbed problems with pulse influence |
| title_full_unstemmed | Linear singularly perturbed problems with pulse influence |
| title_short | Linear singularly perturbed problems with pulse influence |
| title_sort | linear singularly perturbed problems with pulse influence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4592 |
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