On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument
We consider the application of the Krylov-Bogolyubov-Mitropol’skii asymptotic method and Runge-Kutta methods to the investigation of oscillating solutions of quasilinear second-order differential equations with random deviations of argument. For specific equations, we obtain approximate numerical so...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4744 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510911149637632 |
|---|---|
| author | Kolomiets, O. V. Коломієць, О. В. |
| author_facet | Kolomiets, O. V. Коломієць, О. В. |
| author_sort | Kolomiets, O. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We consider the application of the Krylov-Bogolyubov-Mitropol’skii asymptotic method and Runge-Kutta methods to the investigation of oscillating solutions of quasilinear second-order differential equations with random deviations of argument. For specific equations, we obtain approximate numerical solutions and characteristics of random oscillations. |
| first_indexed | 2026-03-24T03:04:31Z |
| format | Article |
| fulltext |
YJ2K 517.9
O. B. Ko.rioMir (Iu-T Ma'reMaTl.lgrl HAH Ygpahm, Ka/n)
IIPO 3ACTOCYBAHH,q qHC.J'IOBI4X METOLIIB
4 0 PO3B'R3YBAHH$I HEdlIHII~I, IX
~H(DEPEHIIIAd'IbHHX PIBHflHb ~[PYFOFO IIOP~I~KY
3 BHIIA~KOBHMI4 BI~XHdIEHH~IMI/I APFYMEHTY
We consider the application of Krylov-Bogolyubov-Mitropol'skii asymptotic method and Runge-
Kutta methods when investigating oscillating solutions of second-order quasilinear differential equations
with random deviations of an argument. For certain equations, we obtain approximate numerical
solutions and characteristics of random oscillations.
Po31"21RIl[a~','rbC,~l 3aCTOCyBRIIII~I aCHMIITOTHqlIOI'O Me'l'O}Iy Kpaotol,a- ~Ol'OJIlO~oBa- MH'l~OrIOJIbebKOl'O
ra Mc'roltia Pym'e - KyTra Ro l[OCJli/t~KelnlJ, I KOJ|HBIIHX p03a'a3gfi~ gna3iafiIfifllmx l][Hc~pellltiaJIhllHX
pinuam,/tpyroro nop~/Igy 3 BHna/IKOBI.IMH BiItxHJIelIIDIMH apvyMemT. s gOllgpCTlIHX piBll~lllh
311a$1/[ellO Ila6JIH~Kelli qHcJIOBi pO3B'~I3KI4 Ta xapaKrepHc'l~lKl, i BI41"IaI[KOBI4X KOJIHI~.IIh.
BeTyn. CrlCTeMrI 3 BHrlaRKOBI~MH BiRxHHeHHa~n apryMeHTy CTaaOBHaTb anaqnnit
aayKoBnfl i npaKTnqna~ iHTepec. Kpi~ aHa.niTr~aHoro R0cHia~eHHa, TagO~ a e m b
KOFO 3HaqeHH~q Ha6yBaRgTb IIH'I'aHHJt MOJKHHBOCTi iX Ha6JIH~KeHOFO pO3B'Jt3yBaHH$1 3a
ROIIOMOFOIO e.qeKTpOHHHX 06qI4CYlIOBaHbHHX bialIIHH. OC06HHBO Ba~'<.rlHI~I4MH e
ROCHi~t~eUHa yMOB mmHKHeHHa y UHX CnCTer, mx Bnnaz~goaax Ko~aaauh Ta 3Ha-
xoR~enna iXHiX qnCHOBHX xapaK-repncTaK. Y ~taniI:l po6o-ri tti nnTanna p o a r ~ a n y r i
CTOCOaHO ttaqbepenttia~HnX pianaHb Rpyroro nopa~tKy 3 Bnrla~goa~tMr~ a iaxa~en-
tL,qMH apryMeHTy. HaBeReHO HpHKHa/~ KOHKpeTHOFO piBHgHH~, po3B'.q3aHOFO Ha
EOM. PeaynbTaTn noRaHi y rpa~i,~noray Brtrnaai.
I locTanOnKa aatlaqi. Ilpea~,teTOM ROC.aiRxenHa r gaaaiHiHiflna CaCTeMa, ~ o
onHcyerbca RUL13epeHLIiaHbHo-piaHrlUeBHM piananaaM Rpyroro nopaRKy a BrlnaRKO-
BHMI, I Bi~xHHeHHJ~MH apryMeHTy
d2x(t) dx ( t ) + k, d x ( t - A l ( t ) ) + k3x(t ) + k 4 x ( t - A l ( t ) ) =
+ k~ dt - dt
dx( t ) , dx ( t - A, (t))],
= Ef x(t), x ( t - A2(t)), - ~ "dr " J (1)
~e k~, k2, k3, k4 ~ ReaKi CTaHi goeqbiaieHTH, e ~ ~a.~H~ ;K0~aTHntt napaMe'rp, f ( x ,
y, u. v ) ~ HeHiai~Ha dpyagraia. ~ 0 ~ar R0CTaTHIO giHbKiCTh noxi~lmtx no r~cix
3MiHHUX, A~(t) i A 2 ( 0 ~ ~iRxnHeHHa apryMenTiB, ttto r cTattioaapm~Ma ~rIna~-
g0BnM~ npottecaMH. Po3rHaz~acMo ~HrlaROK Matrix @nyKTyattitt Bi~xnHeHHa Haago-
nO CTa2IHX BeHI, IqHH
A l(t) = A |0 + wf'~Pl~( t, bt),
(2)
A2(t) = A20 + ~/~P2~(t,P-),
Re A/0 -=- MAi, Pi, i = 1, 2 , ~ R o R a T n i nocvi~lHi 8eJmqriHrt. ~(t, It) ~ cTatlioHapm~t
Bana/x~oar~fl npottec, mo nepeT~opmerbca npn ~t --~ 0 B CTaH/Iaprm~fl 6iHr~fl myM
fie(t) a xapagTepr~crr~gaMa M r = 0, Mvk(t)w(t+x) = ~ ( X ) (TyT CHMBOH M
noaaaqar MaTe~aTrtqHe crlo/~iBarlaa, $('~) ~/le.rlbva-c~yHKtfiS ~2ipaga). Y pO6oTi [1]
noKaaaao, tuo ,ao piauaHHa (1) rao~Ha 3aCTOCyBaTa acm~rrroxaq~ti UeTOaa Kp~.rIoBa
- BoroH~O6OBa -- MnTponozbcbKoro [2].
I Iopo/I~yIo '~e p i a u a n n a . I'lpr~ 8 = 0 p iaaanua (1) neperaop~oerbca a aiHi~ne
Rri~epeHttiaHbao-piannttcae p i~aanna a CTanar.~ aaniaaenaaM
+ dx(t)dt + kg_ dx(t.dt- AI0) (t) + + t q x ( t - = 0. (3) AI0)
�9 O. B, KOJ]OMICLIb. 1999
ISSN O~ l-6053. Y~p. ~u~m, :~.~ptt,, 1999,m. 51,1~ 10 1433
1434 O.B. KOTIOMIEI.[B
~oc.ni~tmdo BI4HaI~KH, KOJ'IH l.[e piBH)IHH$1 Mae KO~IHBHH~I pO3B'~IaOK, j~.rlJt LIboro
posrsw, HeMO floro xapaKTepnCTHqne piBH$11tH~l
H()Q -~.z + kl)~ + k2Le -a'~ + k3 + k4 e-a'~ = 0 (4)
i mtaHaqnrao, y aznx sHnaaKax SOHO Mac ~ntue napy '~ac'ro yaSHnX Kopenir~ X = +i CO.
Po3B'~q3ylOqH CHC'TeMy piBH.,qHb ./~.rl.,q BH3HaqeHH~ CO
H~(C0) -- -CO ~ + k2c0 sincoA~o + k 3 + k4cosOAlo = 0,
aHaxo~x~o [31
H2((o) - kl 0) + k20)coso)Alo - k4sin(oAlo = 0,
(5)
1 k4 k2 -k lk2k -k3k4
co = arccos 9 , (6)
A~0 k2 k2 +k~
k = ~ 2k2_k21+2k3+3](k~_kZ+2k3)2_4(k3_k4),
npnqoMy nepio/Iaqnritt po3a'~laOK piBHZHHH (3) MO)KTIHBH~'I nprl BHKOHaHHi o/Iniei 3
I~aOX nacrynnHx rpyn yMos:
2 9 k~ -kg < 0,
| k4 k2 - k~k2k - k3k4.
k = arccos
A~0 k2 k2 + k 4 '
(7)
k 2 - k l + 2 k 3 > 0 , ( k 2 - k l + 2 k 3 ) 2> 4(k 3 - k 4 ) ,
1 k4k 2 - k~k~k - k3k4.
k = arccos
AIo k2 k2 +k 4 '
qaCTOTa Ko.,rIHBaHB y ttb0My BHHa~Ky
2 2 2 2 2 k 3 , 2 4 , , 2 k2,. ) ~ K 3 4 ) (6")
Or:~e, n p ~ S~KOHa~i yMo~ (7) l~a~'~aoK piB~mm~ (3) Mac B ~ r ~
x ( t ) = a cos (cot + 0 ) , (8)
~e a i 0 ~ t o s i J u , n i cTa~i a e ~ q n H n .
PO3B~$130K 3aradlbHOFO p iml~Ima y nepmoMy na6.~Hatemfi. ,qK noKa3aHo y
pO6OTi [1], poas'aaOK piBHaHHa (1) myKae~to 3a nonoMoro~o aC~IMnTOTHqHOrO
~eTo/~y Kpn~ona - I3oro~a~6oBa - MnTponoz~,Ct,Koro y neptuo~y na6~m~eHni
x( t ) = a ( t )cos (Cot + 0(t)) ; (9)
da(t) = s l ( a ( t ) , ~(t, It), ~),
dt
(10)
dO(t) = ~B i (a ( t ) ' ~(t, It), ~).
dt
KoeqbiuieHTH y npaailt qaca~mi piSHaH~ (10) si/~myzy1OT~Ca TaKHM qHHOM, too6
mlpaa (9) a C~yHKI~iSIMH a(t) Ta 0 (t), BH3HaqOHHMH 3 CHCTOMH (10), 3a~OBOJIBH~qB
piSH~HH~ (1).
l'IicJDi BHKOHaHHSi CTaH~apTHHX ~JIJl aCHMHTOTHqHOFO MeTO/Iy nepeTBopeH~
ISSN 0041-6053. YKp. ~tam. ~'vpn., 1999, m. 51. N'-' 10
HPO 3ACTOCYBAHHfl qHCJIOBHX METO~IB ... 1435
(niztcTaaoa~a anpaain (9), (10) y piaHaHna (1), poaKnan npasoi qaCTaaa y pan Oyp'r
Ta NpHpiBnlOaaHn~l KOeC1~illieHria npa O/IHaKOBHX rap~onixax) OTpa~yeMO sHpa3H
~nJ~ ~oeqbitfieHTiB A ~ Ta B ~. CltCTeMa pianJm~ (10) Ha6npae Barnz~ay
d_.aa = eAI l (a ) + . f ~ A l 2 ( a ) ~ ( t , p . ) '
d t
d__00 = s ) + x/-~Bl2(a)~(t , l . t ) ;
d t
H~ (to)r I (a) + H{(to)ql (a)
A 11 = H~2(to) + H~2 (to) �9
AI2 =
(11)
n.~ (to)(k 4 sin toA~o - k2tocos toA~o) - Hf(o~)(k4 cos toAlo + k2tosin toA~O)atop~ '
H;2 (to) + H-~2(to)
(12)
Hf(to)r 1 (a) - H~ (co)q~ (a)
BII -- a(Hf2(to)+H.~_2(to)) '
BI2 =
H~ (to)(k2tosin:toA~ 0 + k4 cos cOA~o) - Hr cos toad0 - k4 sin to.,'ho)top~,
a (Hf:(to) + H~2 (to))
ne r~ (a) i q 1 (a) ~ zoe~itt ie}ml poaz.na~y qbynKuii
f (acos v , acos (V - toA2o), - ato sin V, - ato sin 0g - cOA2o)),
Xl/(t ) = tot + 0(t)
a pa~t Oyp'e; H~2(to) i H~2(to) ~ n o x i ~ H i nisux nacTria pisHan~ (5).
t lHe~oni MeTO/1H poan 'aaynam~a pinning, /X~la a~taxotI~enna a~ln~liTy~rl Ta
qbaan KO~nna}m. MeTo/1 P y n r e - KyTTa ,~eTnepToro nopa~aKy. P o a r n a H e ~ o
piBHaHH~I/x.rDl aMnnia-y~ri Ta Opaarl KOJ1HBattb (11). Bpaxoaymqa y~osy, HaK,rianeHy
na mma~aoBHt, t npouec ~(t, It), B rpaHmti npri Ix ~ 0 aaMicT~, piaaJ~m, (1 1) MO~KHa
poar.nznaTH Bi~anoainrti IM aaaaa~mi CTOXaCTaani/xHdpepeauian~-ai piaHZHrt~t
da = s ) + f l -~Al2(a)Vv,
dt
(13)
dO = e/~l l (a)+ 4-~Sn(a)w.
dt
Tonep/I~a aa6nHaceuor6 ~aCnOBOrO poan'aaynaaH~ waxax piaaaH~ aacTocyeMo
o~na 3 MeT0ZfiB PyHr0 -- KyTTa [4], ~Ki nerKo az~anwymwbca no CTOXaCTHqHHX
~aadpepeuttia~m~nx piauaab [5]. HaflKpame cniBBi~H0tUeHH~ Mi~ WOqaicTm o6~nc-
nen~ i a.nrOpHTMi~Hom 8qbeKTaBHiCTm aa6eane~ye MCT0n P y H r ~ - KyTra ~eTnepToro
nopa/~Ky, OCKiJlbKH ~IJIfl MeTO/~iB BHI.UOFO n o p ~ y cxna~HiCTb 06qHCJICHb 3pocTar
tUBH~me, niar ToqHiCTb OTpH~yBaHHx pc3yJlbTaTiB. ToMy CKOpHCTaeMocb ILHM MC-
TO~[OM.
]~e3 06Me:~CHH~ 3araJlbaocTi, po3rn~HeMo CHCTOMy pisaam, (13) Ha Bi/Ipi3Ky
uacy t ~ [t3, T] . Hexa~ ~iHcpi~c~KH~ npot~cc w(t) nar nHcncpcim Dw(z) = c.
Po3iS'eMo Bi~piaoK [0, T] Ha N pimlnX qacTHrl TOHKaMH t i ----i~[, A t ~ T / N , i =
= 0, N - I. HOKna~cMO
a 0 =- a (O) , a i - - a ( t i ) ; 0 0 - O(0), e i =_- o( t i ) . (14)
ISSN 0041-6053. YKp. ~unn. ~.'~pu.. 1999, m: 51. IV'-' 10
1436 O.B. KOJ'IOMI~LIB
To~ai 3HaqeHa.q a~n~iTy~H Ta qbaan o6,~ricameMo aa qbop~y~aMn
ai+ 1 = a i + s + 2P2 + 2P3 + P4) + fl-~Al~(ai)Awi,
6
0i+! = 0 i + 8(ql +2q2 +2q3+q4) + af~Bl2(ai)Awi;
6
Pl = All(ai)At, q! = Bll(ai)At,
P2 = al l (a i+~] At, q2 = Bll(ai+~] At,
P3 = A l l ( a i + - ~ ) At,
P4 = A I l ( a i + P 3 ) At ,
q3 = B ~ [ ( a i + ~ ) A t ,
q4 = B I l ( a i + q 3 ) A t ;
(15)
(16)
i = 0, N - 1 .
TyT A w i ~ npnpocTn aiHepiacbKOrO npouecy w(t ) Ha Biz~piaKaX [ t i, ti+ x ], aKi e
HOpMaJIbnO poal'IO/~iJIeHrlMH HeaaJIeT,(HHMH BHHa/~KOBHIVIH qncnaMn 3 MaTeMaTHqHHM
CIIo/~iBaHH~IM M A w i = 0 i aucnepcielo M ( A w i ) 2 = At. 3a llOqaTKOBe 3HaHCHH$1 a 0
MOCHa Ba.qTH aalt6i~btu iMoaipuy aMn~iTy~y cTauioHapmlx BHIIa]~KOBHX KOJIHBaHb,
3Hal~/~eHy 3a ]~OIIOMOFOIO MeTO/~y piBH~IHb (I)OKKepa - I'[~atlKa - Ko~MorOpOBa [1 ].
FIOqaTKOBe 3HaqeHHg qba3H KOYIHBaHb MOT, ella 3Ha~TH aHa~oriqHo, a6o rIOK~aCTH
00=0 .
Mamqn Ha6JaH~xerfi qHCJIOBi 3HaqeHH~l / ~ I aMn.~iTy/~n i ~a3n BHFIa/IKOBHX
Ko.rmBam, y TOqgax t i Bi~apiaza [ 0, T] , raomeMo o6,mczrmi Ha6mI~xeHi aHaqeHHa
nepeMimeHHa x( t ) y ttax TOaKaX, BrlXO~a,~rl 3 qbopMy.an (9), TO6TO
X i = a i c o s (0~ t i + Oi). (17)
3ayr~a~xHMO, mo o/~epmaHi TaKnM qHHOM qHC~OBi pe3y.rlbTaTrl 3aJIe~xaTb Bi//
Ha6opy (nceB~aO-) BHHa~KOBHX qHCeJI, i TOMy e ~mue 3pa3KOM OZIHiei a MO~4~JIHBHX
pea~i3atfifl BHIIa~KOBHX npoReciB, ttlo onncyIOTb aMn~iTyay, qba3y Ta rlepeMimeHaJ~
BHIIaIIKOBHX KOJ-IRBaHb. FIpOTe tIi pe3yJ1bTaTH /lalOTb y$1BJIeHH.q n p o xapagTep
nepe6iry KOJIHBaHb, a TaKO:K MO~KyTb 6yTH B~KOpncrani /Ina Ha6~n~enoro 3Ha-
xo/~meHna ixnix qaC~OBr~X xapaKTepHCTHK (TaKHX, ~qK cepe~ne 3HaqeHH.q, ]~ncnepcia
Ta imui) 3a MeTO/~aMH MaTCMaTHqHOi CTaTHCTHKH, npnqoMy THM TOqHiuIe, qHM
6inbme TOqOK pO36HTT~I ai~pi3Ka BHKoprlc'I'OByBaYI0Cb np~ o6qrlcJIeHH~IX.
IlpoinmcTpyeMo Bce B~KJm~eHe BHme Ha n p n ~ a a i piBHJIHH~I THrly BaH-/~ep-
I'[odIJt 3 BHIIa~KOBHMH Bi/~XHYleHHJtMrl apryMeHTy
d2x
+ klX(t ) + k 2 x ( t - AI0 - ~ p ~ ( t , It)) =
= ~ ( 1 - x 2 ( t ) ) d x ( t - A 2 ~ (18)
dt
,,,roomy 3Ha'aem-m i ~r~ac'ra~cri Bcix Koe~itt ierrrir, "raKi ~ , az i y pir~Hamd (1).
A~a.ai~ym~ta e~po~xemt~ sHnaRom piBaaHHa (18) np~ ~ = 0, a~axo~Mo, tuo
KOJIHBHi pO3B'$13KH MO~Kaaai npa BHKOHaHHi o/mid 3 TaKrtX ~aBOX rpyn yMoa:
~21
k l + k 2 > O , A~0= ~ , / e N U{0},
(19)
qaCToTa KOJIHBaHb CO ---- N/kl -i- k 2 ;
ISSN 0041-6053. YKp. :uam. :~. pn.. 1999, m, 51, N ~ 10
HPO 3ACTOCYBAHHg HHc.rIOBHX METO~IB ... 1437
n(2 /+ 1)
k l - k 2 > 0 , A~0 = ~ , l~ N U {0},
(20)
qaCTOTa KOJIHBaHb CO = 3/"kl - k 2 .
~a.ni poar.rt.q/IaeMo Brtna/gOK cepejartbOrO 3naqertHg ni/Ixrt2IeHHJl i KoeqbittieEriB
piBH~IHHZ, l/IO 3a~oBoJlbrta[o'rb yMOBy (19) nprt I = I. 3aCTOCOByIOqH /~O piBHaHHa
(I 8) aCHMIITOTICqHrti~ MeTOR K p n n o a a - B o r o n t o 6 o a a - MnTpononbC~KOrO aX onnca-
HO aHllle, o/IepzcyeMo piBH.qnH.,q (9) Ta (13), y .RKHX qbyHKRii A I I , A 12, B 11, .B 12
Brtpa~Ka~OTbC~ qbopMy.na~rt
Al l = y l a + T 2 a3, A~ 2 = T3a,
(21)
BII = K I + K2 a2, BI2 = I(3,
Re
- u + v - (20} 4 sin a + 2k2~c02 cos a )
'YI = ~ ' KI =
d d
3u - v 6o} 4 sin a + co 3 cos a
~2 = ~ , ~2 = d 4d
-(2rl:o)2k2Pl ) 2k2(kl +k2)Pl g 3 = - ; (22)
Y3 = d ' d
u = 2 k 2 ( k ~ + k 2 ) n s i n a , v = 2 ( k ~ + k 2 ) 2 c o s a ;
d = 4(k l+k2) 2 * o + 4k] r t ' , a = ~ k I + k 2 A20.
Poarna~tamqrt piBH~IHHSl (18) Ha sigpiazy t e [0, T] , 3aCTOCOByCMO ~O HbOrO
OTprtMaHi mltue ~opMynn MeTOIIy Pyrite -- Kyan'a (15), (16) Ta (17), mIzopnCTOBytO-
art qbopMynrt (21), (22).
3a noqaTKoBe 3rtaqcnnJ~ a 0 y qbopMynax (14) Mo~cna aaaTn nafl6Lrmm iMoaipHy
aMnsfiTy~y CTaLdonapHrtX BrtIIa~KOBrtX KOJInBanb, ~ J I 06qrtcYlenH~l JIKOi cKopncTar
MOCb C~opMy:loIO, 3Ha~ReHOIO y pO6OTi [1 ] 3a ROnOMOFOIO MOTO~y piBn~ra~ (DOKKepa --
HnaHKa - KO~MOFOpOBa:
7g -71
act = (23)
IIpoaHaniayeMo qbopMyny (23). Y BrtnaaKy, Konrt k I = 1, k 2 = 0, piBH~HH~ (18)
nepeTaOplOeTbC~ B 3Brtqa~H0 piBn,ClHrt,q BaH-lXcp-Ilon~. Hpn u~or, iy OTpmayeMo
BiROMHI~ [2] FpaHrtqrtrt~:l peaySlbTaT:
act = 2. (24)
KoJ~I k I + k2 = l, &me k 2 r 0, MaCMO TaK07~ RiKaBH~ artnaaOK. Po3raaneMo
nCnKi MOT~JIHBi BHna~KH nOBC~iHKH ae,.(p I ) gK cl~yHKI~ii iHTCHCHBHOCTi p I 36ypIo-
tOHOFO BHHa~KOBOFO npoLlecy.
1. f lXmO A 2 0 = ~ . TO3HaqeHrtS a c , . ( p l ) > 2 arts Pl > 0 , i nprt P] - ~ 0
E1DyHKI~iJt ac r (p I ) -4 2, MOHOTOHMO CHa~aIOqH. l'[pH 36i31bLIICHHi i~TCHCrtBnOCTi Pl
3HaHCHHH ac. r HeO6MCTKeHO 3pocTa~.
2. J;[KIRO A 2 0 = 2 ~ , TOaHaqeHH~I acT(P 1 ) < 2 ~ I ~ p i > 0, i npH p l " ~ 0
qbyHK~ig ae r (p ! ) --~ 2, MOHOTOHHO 3pocTaIOqH. l'IpH 36i.rl~IIICHtti iHT~HCHBHOCTi p l
3I-IaqeHH.q a c r cna~a~T~ no THX nip, HOKH He ~OCRFHyTb Hy~-oao ro 3Haq0Hn~l,
I$$N 0041-6053. Yr#. Juam. a~ylm., 1999. m. 51. IV" 10
1438 O.B. KOYlOMIEI2B
TO6TO y UbOMy paai BHnaAKoBi KOJIHBaHHYl 3HHKaIOTb. I~e BiA6yBaeTbC~, KOKIH
iHTeHCHBHiC'rb a6ypm~oqoro npottecy /Ioc~l'ar KpHTHqHOFO 3HaqeHH.q p [Kp' ,qKe
o6aacmoea~cz Ha nittcTaBi (23) za ~opMyzoIo
2(i +
P ~ o = k~ (25)
~ J t aHaqeas P l > P lKp pOaFYI~I/IaTH CHCTeMy (18) neMar CeHcy.
3ayBa_~,m, io, mo npa Ha~ro Be.qHKrlX aHaqeHHaX Pi a60 A20 onrmaHi ~eroAn He
AmOTb 3a~oBi~bHnX peay~t, raTiB [6].
HpuKJmAH poall'~laynamt~ pinHam, na EOM. C z z ~ e H o nporpaMy, aKa pea-
~iaye onrmann~ Bnme a~tropi~rM Aria piBHaHHa (18) Ha nepcoHa~Hor, ty zo~ri ' ]orepi i
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roqoz pOa6rlTTa BiApiaKa aacy. l-lporpa~a poapo6~eHa y cepez~oBami Visual Basic
for Applications; no'~aT~oBi ttaHi Ta peay~TaTa po6ora poa~ituy~or~,ca y po6oqoMy
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no/iizeHrix Brlna/~KOBrlX qnceYl 3 naKeTy Analysis ToolPak. Pe3yylbTa'rH pO6OTri
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MittIeana KOYiriBanb/lYla ]1BOX pi3HnX 3HaqeHb iHTeHCHBHOCTi Pl = 6 T a p I = 12 no-
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ISSN 0041-6053, Yxp, nan). ~.'vpn.. 1999, m. 51, N e I0
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ISS N 0041-6053. Yup. .uam. ~..'Vl:)U.. 1999, m. 51, IV'-' 10
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ISSN 0041-6053. Yxp. ,yam. ~.'vp,,. 1999. m. 51. N ~ I0
|
| id | umjimathkievua-article-4744 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:04:31Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ac/0fe30619a27e72656237a7c0056654ac.pdf |
| spelling | umjimathkievua-article-47442020-03-18T21:12:54Z On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument Про застосування числових методів до розв'язування нелінійних диференціальних рівнянь другого порядку з випадковими відхиленнями аргументу Kolomiets, O. V. Коломієць, О. В. We consider the application of the Krylov-Bogolyubov-Mitropol’skii asymptotic method and Runge-Kutta methods to the investigation of oscillating solutions of quasilinear second-order differential equations with random deviations of argument. For specific equations, we obtain approximate numerical solutions and characteristics of random oscillations. Розглядається застосування асимптотичного методу Крилова - Боголюбова - Митропольського та ме тодів Рунге - Кутта до дослідження коливних розв'язків квазіліпійних диференціальних рівнянь другого порядку з випадковими відхиленнями аргументу. Для конкретних рівнянь знайдено наближені числові розв'язки та характеристики випадкових коливань. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4744 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1433–1441 Український математичний журнал; Том 51 № 10 (1999); 1433–1441 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4744/6189 https://umj.imath.kiev.ua/index.php/umj/article/view/4744/6190 Copyright (c) 1999 Kolomiets O. V. |
| spellingShingle | Kolomiets, O. V. Коломієць, О. В. On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument |
| title | On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument |
| title_alt | Про застосування числових методів до розв'язування нелінійних диференціальних рівнянь другого порядку з випадковими відхиленнями аргументу |
| title_full | On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument |
| title_fullStr | On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument |
| title_full_unstemmed | On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument |
| title_short | On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument |
| title_sort | on the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4744 |
| work_keys_str_mv | AT kolomietsov ontheapplicationofnumericalmethodstothesolutionofnonlinearsecondorderdifferentialequationswithrandomdeviationsofargument AT kolomíêcʹov ontheapplicationofnumericalmethodstothesolutionofnonlinearsecondorderdifferentialequationswithrandomdeviationsofargument AT kolomietsov prozastosuvannâčislovihmetodívdorozv039âzuvannânelíníjnihdiferencíalʹnihrívnânʹdrugogoporâdkuzvipadkovimivídhilennâmiargumentu AT kolomíêcʹov prozastosuvannâčislovihmetodívdorozv039âzuvannânelíníjnihdiferencíalʹnihrívnânʹdrugogoporâdkuzvipadkovimivídhilennâmiargumentu |