- -
I ," ,
I BO GIAO DUCVADAO TAO
I . .' .
I. TRUONG D~ H<;>CT6NG HQP THANH PHO HO CHi MINH
I
TRAN VAN LANG
sir DI)NG PHUONGPHA?56 VAo M9T 56 BAI rOAN CO HQC
Chuyen nganh : Cd'HQCV~TRAN81(HD~G
}riaso : 1.02.21
r
I
I
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TOM TAT LU!N AN
Ph6TienSiKhoaHQcTDanLy -
Thanhph6H~ChiMinh
- 1995 -
.'
"
\ LuAnan nay duoc ho3n thanh tai Khoa Toan-Tin hoc
" .' .' . .
Twang D~i Hc;>cT6ng Hqp Thanh ph6 H6 Chi Minh
Ngum hu(mg dAn
- Ph6 Giao su Ph6 Ti€n 81Ng6 Thanh Phong
-Ph6 Ti€n 81Tran Thanh Trai
Ngum nh~n-K-et-l
Ngum nh*n xct 2
Ca quaD nh~n ~ct

Chlldng 1: T6ng qUaDve mOhlnh va phuong phap giro mQts(fhi\i
tmlh co hQc.
Chlldng 2: MQtsObili loan dao dQngva bi<!ndl,ll1gcUathanh dan
hOi.
Chl(dng 3: MQt sO bai toan dl)ng Il!c hQc mO ta bCliphlfO'ng tdnh
parabolic phi tuy<!n.
Chuang 4: Ml)t sOkel qua Hnh tOtin.
Cuoi C\)ngla phau tai li~u tham khao cUa lu~ ~n. .
'/, I . i-"
";' ,. '
-1- iHLf\:.i[;,~:'-
'
CHlJONGI
T6ng quaDv~m()blnb va pbll<mgpbap giaim()lso bai
toaD cO' bQc
Trong chuang n~y chUngWi triOObay mQt s6 kl!t qua nghien CUll
hen Tht gi6i va hong nuercv~ cac bAiloan d~tfa hong lu~n an. Cling nlll!
mQts6 ktt qua ciiachung wi diid~t dugc so veriOOUngkef qua cii~1.1cgh\
hong va ngoai mt6c. Nhihlg bAitmin chUngtOikMo sat trong lu~ an nay
bao gbm:
1. Bai loan bien d~g eua mQtthanh dan hbi phi luytn dugc nhung
trong moi fIuemgcha:tlong. Cae kef qua n~y, cluing Wida'dl1og!rong [l]l2]
[21][22][23][30].
2. Bai loan thief kt bua may d6ng C9c.Cae kef qua dii dugc dl1ng
trong t24][25][26][27][28] [29]. I'
3. Bai loan dQngl,!chQcbien t,!a I-chien. .Cac kef qua etadugc d~
~p dtn trong [3][4].
4. Bat loan etQngl"c hQcbien va d~i duang. Car kef qua cua mO
h1OObai loan n~y da dugc dilng trong [5][6][7][8][9][10]fll][12]ft3]f14]
[15].

hQi t\! cua lai giai xap xi v~ nghi~m ctla phuClng trinh xuat phat. C:~ckef
qua da dttqc dAng trong [21][22][30].
(1.2)
San d6 chUngwi da xet sl! ph\! thuQcclla lo-igiai vao cac dit kien
cho ban dhu bJ,b2,/,g cila bai toan va da.chti'ngminh duqc sv ph\! lhu<>c
nftylitlien tl!cva dClndi~u,van loi giMduy nh11'1cua bi'tiloan (1.2), (1.3).
II. Bi\i toan boa may dong c<;'c.Co h~ ky thu~t cua bUa may d5ng
c9c c6 t116xcI mQt each t6ng quat nlnt san: M<,\tbUa c6 kh6i htqng ,17/(,
dltqc n6i liCn v6i C9c va de c6 kh6i 1ttqng n72' bhng Ie)xo v6i h~ 86 oan h()j
- 3-
Ct. Gina bUa va c<,)ccon dllgc gaB them mQt bQ giarn cMn' c6 h~ s6 ma sat
Iii bl" Khmmg cach ban d'au (& trlpl.g tllai ct\n bhng) giUa bua va c9C Hi <\.
L1,£cngoai (tuan bean) taG dQng len bua ml la F. G<,)iUI'u2 Ian lugt la dQ
dich chuyen clla bua va C9C.L1,£ccan clla dat sinh fa trong qua trlnh chuy~n
dQng baa gbm l1,£ema sat kilo gifra <hItva c<,)cP'nsd'I1,£ccan 1<1'aucge ~. Cae
I1,£cn'ay n6i chung, phI,!thuQe VaGdQ dich chuy~n eua e9c. v~n t6e Gilacqc
va cac d~c trung cua Mt v.V
Dl!a VaGnguyCn uk rung va, voi cac gh\ thic"t ma sM gifta C~)cvii
<1:1'1.la ma sat khO, c9c drug hly~t d6i, va ch~m gilla boa viI c9c 111lue 1110i,
chUng tOt xet ba giai do~ co the Kayfa tfang qua trlnh dong C9Cnhl1sail:
- Giai do~n rung, day la giai dolpl.elma xtiy ra va chl;\mgilla Ma va
C<,)c,00 h~ dao dQng d\10i taG d\lng eua h;rcngo1\i tu'an hoim F, Ivc ma sal
giUa bua va daL H~ phl1O11gtrinh dao dQng co d~g nlllr saIl:
(2.1) 11z.~+ bt(if.l - uJ+ Ct(u, - It'}) =F
(2.2) ~~- bl(i4 - "2)- CI(UI-'~) = -Frn.<d
De ket thUGgiai <1o~nrung nay chung ta co di~uki~:nv~sl!va ch~m
gillaMa vac9c.
(2.3) '4- 'uz=80
- Giai dOff1Jva dqp, thl!c ra day chi IiimQt 8\1tac dl}l1gchu khOl1g11\
mOt giai d9an, hat qua t1"1l1hva chl;\tnx~y fa 1.(1Cthai, co ngh!a Hithai gian

11m hai thai diCm1»12be nhtIt, 10< II < 12va hai ham vector
;;(t),
10S I S It,
U(I),
10s I ~; II
sao cho
(2.12)
du=Au+~,
dl
'0 < It < '2.
- 5 -
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
-
( )
-0
u to =u ,
tl =min{t > to / Ul(t)- uit) =80}
dv ~
- = A"+ Fx+ F2' to < tl < t2
dt
V(tl) = BU(II)
'2= min{t > tl
/ "4(1).(111(1)-1I2(t) 8J =O}.
TrltO"nghf/P khOng va dljp "1(t2)- "2(t2):f:-80
Khi do chUng ta giai bili loan (2.12) - (2.14) vui
(2.18) to=tv ;;0=V(t2)'

. -,
(1.2) V(x,y,z,t)= V/nI(x,y,t),
(1.3) p(x,y,z, t) = ~q (x,y, t)
Pl1\n duai day bi~n, giii su co ma sat 16nva mrac khOng ngam xu6ng, trong
tn! emghgp nlly <lieuki~n bien sc11\:
(1. 1) V(x,y,z,t) = O.
Ph \1]bien xung quanh, c6 th~ cho dieu ki~n nu0c khOng tach khoi bo:
. oV
(1.5) - =0,
. an
ho~(;dieu ki~n ma sat lao
(1.6) V(x,y,z,t)= O.
Die\] ki~n d1\u
(1.7)
- -
V(x,y,z,O):= Vo(x,y,z),
p(x,y,z,O)= Po(x,y,z)
- 7-
De girobai to~n n~y, chUngtoi dii xet sI!hQit\l ciia nghi~m phuong
trlnh sai phan xap xi cap hai tl1cothai gian va khOnggian. Tuy theo m(,\!s6
truOnghgp, chUngtoi ClIngch(rngminh duqc sI!6n djnh da Jai gi:\idp xi
tit h~phuong trlnh (l~is6.
II. Bai toaD dqng h.rch«:,cbi~n va d~i duong. Ngoai gi:\ thi~t v~
cMt long nInt di'ineu trong bai loan n~utr~n, d6i v6i bi\itoan n~y ch~ng (Oi
xet tMm cae hi~u 1h1gr6i thoo phUC1Ilgnfun ngang Ox, OJ, ding nhu die
thi\nhphM nh6t theo ba Inr6ng.Cac Il!Cbell ngoai baa g<lml,!c Coriolis va
lllc tr<;mgtruOng"cOnthi\nh pMn v~ t6e gi6 thi hi~n qua di~u ki~n biCn.
M~t thming b trl!-llgthili ban dhu duqc gic\thi~t c6 dao dQngso v6i mQtm~!
chuan nao d6 (cae k~t qua cila bai loan nay chUngtOjdiitrlnh bay trong Ill]
[12][13][14][15]).H~pln1ongtrinh mOta bai loan c6 d~ng;

-8-
- -
V(x,y,z,O) = Yo(x,y,z), p(x,y,z,O)= po(x,y,z)
Sit dl.mg phl1ang pMp phfin ra theo t<;>adQ, bhng each klu10 sat !.fuh
nita xac d!nh duang va hennit. cia tOaDtit vi phnn. ChUng tOi Om nghi~m
thco tang IlltOOg,GOngthC1ichUng minh dltgC nghi~m elm biii lotio tl1cOqufi
trlnh pMn ra ehfnh Iiinghi~m eua phuang trinh xuat pMt (D!nh Iy 1,MI!G II,
Chuang 3 cia Lu~n an).
Buoc tiep thoo, b~ng each sai phan theo thOng gian, chUng toi nh~n
dllgc cae xlfp xi h~e n cua phuong trinh sai pMn so voi phuang triah x\l1ft
phill ban dhu
III. Mo hinh d9ng h9C tt.ra ph.tong tring Saint- Venant. 1 chn~QI.
'I110ng thuong, dtl.Hnh dong eh?y khOng 6n dinh tren h~ thong sOng ktnh
ciia vung anh hui':1ngthl1y trieu, ngtf()i ta su dl!ng phuong trlnh Saint- Venwl(
I-chiCu, trong lntemg hqp 1111y?lnh 11ltal1gcua ma sat nh6l b! (Iii bo qua 'lit
xcm SI.tma sat clJa eMt long vii th1'mhr:\n 11'1nan!,-kf (j (T1\y,do mu()n ~d
hi~lI (cngnh6l LacdQng I~n dong eMy, u\jng t1J~jjmuOn xua1 pIlat lir phlJ'OTlg
trinh dQng h!e l19c Navier-Stokes, phuong trlnh baa to1\n kh6i lugng, chung
16i dua fa dUllCmOt mO hinh ty:a pllltO'11gtrinh Saint- Vcnant l-chi~u, trang,
<16 e6 s~r tham gia elm thal1hph110nhc)ttrong pluto'ng trInh, ket qlla cluing wi
cIatrlnh bay trong [16].
V6i cae gia thiet san day:
- Chlft lr'mg dOng ehftt, khOng nen lhlc;1e,dAnghu6ng,
- Ap suit li\ tll"y Hnh.
Khi eto fir h~ phuong trlnh chuyCn etQng t6ng quat baa g'orn die
tensor (tng suft'tnhot 't~ cluing ta e6:
(2.7)
(3.1)
Olt au au au 1 op ]
(

l

-h -hax -h Oy
ChUng ta gh\ thWl them rhng dQ sftu 1/ =11+ h cua m,!c ntt6c
khOng dang ke so v6i m~t phang nhm ngang, khi d6 chUng ta c6 th~ (ltta V8.0
cac df;liItt<1l1gd~c trung cho sv phftn bO v~n tOe trung blnh theo chi~u thang
dUng:
(3.6)
(3.7)
11
V(x,y,t) =J u(x,y,z,t)dz,
-h
11
V(x,y,t) = J v(x,y,z,t)dz.
:-h
va cae gia thief atia M~nh d~ 1 (Mvc III, Chttcmg3 eua Lu~n an) va C1'13
M~nh ~ san, chUngWinh~ dttqe h~ phttC1.tlgtrlnh lien h~cchuycindQngde
xac dinh ehi~u cao Ii cling nhtt phftn b6 Wong v~ Wc V, V trong m~\t
phang Oxy. H~ cac phttong trlnh n'a.ykhac h~ phttong trlnl1Saint-Venant hai
ehi~u(jeM, c6 tbam gia eae tbanh ph~ d~oham b~ehai cua v~ t6c.
M~nh~ 2: Gia sit
I
(i) Moi Wemg 1ftdang htt6ng 't~ = 't;,
(3.8)
"
- 10-
(ii) H~ s6 nh6\: eua eMt long thee cae hl1(mg 11\ nhl1 nhau
v.=v.xy=v,
(Hi) Tensor N} duClcxap xi dltai dl;Ulg:
(3.9)

trlnh Jit:nt\lc c6 dC;lng:
au a u2 a UV 8rl a
(
au ()V'
)
(3.12) ot -+-fy-;/i-+- By II = gH ax -+-lV-l- v~U -+-v a; ax -I [~1;,I
g ~(~:_:~_V~1~+1g2P~~2.coseg,
1 It- p .
c
av iJ UV iJ \/2 8rl iJ
(
iJU <11/
)
(3.13) &-+- ij-; H -+- By H :co -gl! B.y-LU - v~V -+-v By Ox -I-~?}; -
)
1/
V
(
u7 I v2 /7 - 2 Fa W2sin8g
g +"(1'. 8
(/ [(J. p
- I I -
all aU oV
(3.14) + + =0
at ox 8y
San d6, gi<\thiel d6y cua long dan c6 d~g ntta hloh tang tl1;l,SI!thay
d6i b'e mM va da
y
theo true O
y

l1(X,t,&) = L l1";(X,t)!>'"
",=0
-12-
v6i E all nho lit!c6 thl! cui Sign(Q) nhula Sign(Q)J. Trong .:16die h~ 86
Qo' 110thou h~ phuong trlnh:
(3.19) B~llo+ ~ = 0
at ox '
1 0{1 1 a ,~:! Urlo QoIQ,1
+ + + =,0
gA at gAox.l1 at c2A2R
con cae h~ 86 Q"" 11"" f11= 1, 2, thoa h~ ph\1t:mgtrlnh
(3.20) B ?J", + ~g", - 0
at ox ,
~~_.?f2'!!+.3_!. ~q",.,: <7r.b.+~~q~:: l~'I(Q(I,QI, ,Q",I)
gA at gAox A ax cAR
IV. Hai foan Ian fruY4!:nva khulch tan cua l.gu;~11ga)/ ~i}II1Ihij~l1l.
Chllng fa ghi sl'r(p(x,y,z,t) bi6u dj~n luqng nhi~m ban du<;rcIan truy:~n vit
!JlU~ch t{mc!(?ctheo quy d~o cua de h<;1tm{)i tntang chuy(!n d<>ngv6'i v~n
t6c V(x,y,z,t). Baj loan m() l<\51!Ian truy'en va khut!ch t{in (lJa ngul'in gay
(\ nhiCm c6 o,.\ngsan:
(4.1)
o<p O(P 8<p - O(P 0 o<p
+ /1-+V + w -" J-tLl<p+ -v-+ f
at ox oy oz 8z 8z
\I(x,y,z) EO, \It E(O,T]
Dr~u ki~nullu
(4.2) <p=<Po trong 0, khi t = 0
Di'eu ki~n bien
(4.3)
<p:=<Pstr~n L, t E( O,TJ

as V.[(Zt-z(J/~KIVz,IVS]
(5.2) 0 == . + KVzSS+ Q
, Ot Zt Zd
tren <X1sa (tinh lu~t th:1m Darcy
(5.3) V(x,y,r) =-KVz,(x,y,t)
va mQt s6 gic\thi(ll [17J: m~t tlJ do cua mtoc duoi dtI't z,(x,y,t) n;!m thap
han so voi m~t dtI't,chuyen dQng cua nuoc dttoi d:1tgan m~t (%1:Iii chuy&n
dQng khOng ap, lOp dtI'tsetz(,(x,y)dttoi mi~n chuy~n dQng cua mr6e thtlm
thay d6i ft, nttoc dttoi dtI'tla GMt long GOngeMf, khOng ncn dugc:, 1:1:11: 1:\
mOi tntang kllOng ntn dttqc va dang htt6ng.
ChUng tOi pMn ra pai plltrang trlnh tr~n 111Qtcckh lil~ntWp dbng
thoi trCn cling mQt kh<n\nglhhi ginn. 'f'{nhIlIla xac dinh dllctng din die: lolin
ta vi phfin xuff'thi~n trong p)ntctng trlnh dIng dugc khan 8M d€n. Sa" rf6
bang phuong pMp sai phfin fin, chUng Wi dua v~ h? phl1ong tflnh ft~j :><5
tuyen Hnh, d~ gic\ih~ n'!\ycMng wi con khan sat tht'h1 !fnh 6n dinh clla lai
gic\ivoi die raub bu(!c v~ v~n tOed()ng thatn cling n]n( cua bl1~5Chr6i,
- 15 -
CHtJONGIV
M<'ts6klt quaHnhto{m
Trang chuang n1\ychUngwi neu ml>ts6 ket qua Hnh s6 du6i d~g
dOthi cua cac mOhlnh bai Loan,rung nhu phuong pMp tfnh Loan,da d~t ra
!rang Chttdngl va Chlfdllg/ll. Cac ket qua n1\ydt!aLIenmQts6 s6 li~uOWe
te ding nhu ghi dinh, de qua d6 danh gia ve m~t dinh tfnh cua mOhlnh va
phttong pharo
I. Tlnh toan dao d(mg va biln d~.mget'm f.hanl. d~lI1hoi. CluIng
Wi11\nluc;tf.Hnhloan cae bai loan gall:
1. Bd; loan
u6fJ lhafJh dafJ /Wi phi luyefJ, Cqung wi tinb loan cho ml;)ts6
truem.ghgp neu trong Chttdng/l, M~4CI nhtt du6i dlly, cae ket qua (15du<;lC
trlnhbay!rang [2][22].Gia su caeham g, M dtt<;lcch911t11baeae yell Cftll

II. Tinh toan ID9ts6 bai toan d9ng life h9C.
1. BiJi roan a{Jng llfc hqc bii'll wa i-chicll. Bi\i loan nfty dtf<;1Cdi)ng df; tJnh
loan khOi plwc I~i dl)ng dulY b cae thng stiu khae nhau dw~i bien khi bi{rt
s6 li~u v~ dong dulY va pIlau b6 Ir11ongv~n We tfen m~t Ihoting (~ic ktrt
(l\\a da d~ng trfmg [3][4]).
Thtl~1loan da dulle kjCmnghi~m IJennuly tfnh, va ~mld6 Sttdl,mg
Hohtoan cho khu V\fCphfa Nam bien DOng,vai cae 86 Ii~u do ('[<,1cd\1OC
clla Phong V~t Iy bi~n, Phan vj~n Dftu kbf phfa Narn.
Cae btrde ltt6'i khOng giao Ihco Irt;leOx, Oy dlUJCIa'y c6 d!nh, CC!D
thea tn,lc Oz dlt9'c Iftythay d6i tily thea df) sau.
. Ax=~y =112000m, 1m ~ ~Zk ~ 100m,
trang d6 ~z/'lftn 111<;11nh~n cae gia tr~: 1m, 1m, 1m, 1m, 1m, Sm, Sm. 5m,
5m, 25m, 25m,,;.25m,50m, 50m, 50m, 5Om, 50m, 100m, 100m, 100m,
100m, 100m., '
BUde Itt61thbi gian t\t = 360CB.
Cae s6 li~u ciia mOi truong du<;1cla'y 0011sau:
- M~t (If) n116'ebi~n p = 1025kg f m3,
- He s6nhdt v== 10-2m2fs,
. z
- V~n 16cg6c quay cua trai d:I'1()) =7,29 x 10-5 /S
-Vi d0 t~i (ti(!mkhao sat <D=0,1745
- 17 -
-Gia t6e tT<,mgwang g = 9,81m 182,
Dl!atxen ibQ s6li~u (eua 2 ngay) v~ v~ t6e M m~L,eho chung La
khoi ph,!e I~i 00 eM dong eMy elm cae tMg san Mn du<'1ithoo LunggiG
mQt.ChUngLoi<Hitinh LoanIan hrql.txong Lnrangh'lP 1.1faG 16peach bb
m~t 1m,2m, 3m, 4m vao gia thu 24 va vao gia thu 48.
2. Edi toan
dqng lifc hqc bi!n va dqi dlldng. Cae sOli~u eua moi Inlang clln
biliLm'innay du'le I~ynhu san [5][6][7]:

phuong phap giai cling nhtl cae ket qua Hnh Loan, chUng tOi nh~n 1My lnl~
hinh va phl1ong pMp dan Mn ket. qua phil hqp ve m~t dinh Huh.
4. Bai tOlln Ian truyen va khutch trIll cua nguon chdt bJn trong nUde £!l{m
ddt. ChUng wi <15Hnh Loancho 6 trl10ng hqp kMc nhau cURdiet! ki~~nbi{',u.
Ke"tqua Hnh Loancho th1I"ymve mr6c Ian truyen dlln vao Mn trong mn~ll, do
h~ s6 thllm K Illy kM be, nen mvc nunc 0 Mil tfong tha'p hon gl1nbien. D6i
v6i nbng d~, chung ta cling e6 ke"tqua tuong tv, khi eho gia tT!tren biCn
tang cfllnleu thi Mn trong mibn gia t1'!nh~n dtl<;>ccOng U\ng <l!lnI~~n.Cae
ke"tgila dllqc t1'lnhbay trong [17J.
.' ' '
- 19 -
TAl LIttUTHAMKHAo
I
[1] N.T.BANG,T.V.LANG,al al., A nonlinear differential equation relating to
the buckling of a nonlinearly elastic bar ~ersed in a fluid, Proceedings
of the HCMC Mathematics Consortium 1st Conference, Vol.1,1993, p.5-
12.
[2] T.l.CU'ONG,H.B.LAN,T.V.LANG,Giro s6 mQt phuC1ngtrlnh phi luy~n
lien kef v6i toan tu Bessel, Proceedings of the 4th Workshop on Applied
Mechanics, 4/l994,TpHCM,1994, fr. XVIII-7.
[3] T.V.LANG,LV.THIEM, Mo hlnh t,!a mQt.chien M Huh loan e0 eM dong
char 1.1cae fang sau dufli bi~n, f/tJi nglzt Cd hqc toi}n q1"5Cfan tlllr lV, Hi)
nQi, 20 - 22/01/1988.
[4] T.V.LANG,LV.THIEM, Mo hloh t,!a mOt ehieu M tlnh loan co che' c\i'H1g
char a cae mug san dtt6i bien, B(IO cao Khoa lu?c stY87.202, TrunR Tam
Tl{(fD&TH, Vi~nKHVN, TpHCM, 1987, 19fr.
[5] T.V.LANG,Mo 111nhdQng l,!e hge bien 3-chieu de tfnh loan dhng cl1<\Y(\
mQt khu v'!c c6 dQsan dang k~, 1'6111tat Cong Trinh Khca Hqc - lltJi Ngh!
Khoa Hqc, PMn vifll Khoa Hqc Vi~tNam, TpHCM, 1988, tf. 105.
[6] T.V.LANG,Mo hlnh dQng l,!e h9C bi~n 3-ehicu de Hnh loan dong eh,\y (~

lz(jpTpHCM, 06/1985.
[15] T.V.LANGva;c~cI~cgia.Nghien c{cudong cMy ngoai bi~n, IMi light
TOIIIIhqc Vi~t Na17lLantluHll, Ha n<)i,07/1985. 1'6m t~Ltr. 133.
[16] T.V.LANGva c~clac gia, Xc! anhlnrbngnhol tTenbili toan dc)ng c:h:\y
khOngon djnh m<)tchi~u,Hqi nght Khoa hqc Lanthu V, TruiJngDIIBK II}
HCM, 06/1990, cr. 42-43.
[17] T.V.LANG,B.M.NGQC,CAnbhng va GMt hcq'J1gmtoc dum o1'\tlrong
mila khO,Bao do klwa hqc stf 83.203, Trung Tam l' oan Hqc (}"/1gDlfl18 WI
Tin Hqc, Vi~n KHVN, TpHCM, 1983,21 cr.
[10] T.V.LANG, N.T.PHONG,51! Ian truyCn va khu~ch h111Gila nguhn (~
nhiCm trong khOng khf, T-I~)iI1ghi Khoa hqc va C6ng I1gh(' l/il1 tilT! VI,
TruilngDHBK Tp HCM, 16-17/02/1995, tr.99.
[19] T.V.LANG,N.T.PHONG,ve bai toan Ian truy"~nva khut!"chtan dla
ngubn 0 nhiem, Proceedings of the 5th Workshop on Appli(~d Mechanics,
4/1995, HCMC, 1995, p. 13.1- 13.7:
- 21 -
[20]T.V.LANG,N.T.PHONG,~ bai toaDIan tmyen va khuech tan cua
ngubn 0 nhiem, Tqp cM Khoa hqc & Gong nghf, BK & SPKT, S6 9,
(1995), Ha nQi.(Bainh~n dang)
[21]T.Y.LANG,
Blii toan u6n thanh dan hOi phi tuy6n dl1(;1cnhung trang 111\)(
cha:'t long, H{3i nghj Co' hqc V «tran hien d{Plg toan qudc Ion thl~'lV, Hi\ n<;\i,
10/1994, T6m t~t, 1r.44-45.
[22]T.V.LANG,N.T.LONG,Phl1cmgpIlar phhn tichull ban ap d~ng VaGhlii
to<1nu6n mQ1thanb dan hbi phi 1uyCn,Proceedings of the 4th Workshop on
Applied Mechanics, 4/1994, Tp HCM, 1994, tr. XIIII1-S.
[23]T.V.LANG,T.LCU'i1NG,H.B.lAN,Gi,\i ;86 mQ1 phl1(1J1g11'111hphi tu)'61
lien kef v6i loan tit Bessel, Tqp eM Khoa hqe & C611gI1gh~,ilK &.SPKT,
S67, (1994), HanQi, tr.13-16. . "
[24] T.V. LANG,at at, Mathematical Modelling of the Hammer Machine,