GIẢI GẦN ĐÚNG PHƯƠNG TRÌNH PHI TUYẾN VÀ PHƯƠNG TRÌNH VI PHÂN TRÊN MÁY TÍNH ĐIỆN TỬ

8/2/2019 GII GN NG PHNG TRNH PHI TUYN V PHNG TRNH VI PHN TRN MY TNH IN T

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I HC THI NGUYN

TRNG I HC SPHM

TRN TH HON

GII GN NG PHNG TRNH PHITUYN

V PHNG TRNH VI PHNTRN MY TNH IN T

LUN VN THC S TON HC

THI NGUYN - 200


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I HC THI NGUYN

TRNG I HC SPHM

TRN TH HON

GII GN NG PHNG TRNH PHI TUYNV PHNG TRNH VI PHN

TRN MY TNH IN T

Chuyn ngnh: Gii tch

M s: 60.46.01

LUN VN THC S TON HC

Ngi hng dn khoa hc:

TS T Duy Phng

THI NGUYN - 2007


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MC LC

Trang

Li ni u..............................................................................................2-3

Chng 1. Gii gn ng phng trnh phi tuyn trn my tnh in

t...............................................4

1. Gii gn ng phng trnh ( ) 0f x .......4

2. Cc phng php tm nghim gn ng ca phng trnh

( ) 0f x ......10

3. Tm nghim gn ng ca phng trnh ( ) 0f x trn my tnh in

t.....24

Chng 2. Gii gn ng nghim ca bi ton Cauchy cho phng trnh vi phn

thng trn my tnh in t..................48

1. Phng php gii gn ng bi ton Cauchy cho phng trnh vi phn

thng......482. Phng php Euler .........52

3. Phng php Runge-Kutta ......57

4. Gii bi ton Cauchy cho phng trnh vi phn trn my tnh in t

.........64

Kt lun..................................................................................................82

Ti liu tham kho...............................................................................83


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LI NI U

Cc bi ton thc t (trong thin vn, o c rung t,) dn n vic cn

phi gii cc phng trnh phi tuyn (phng trnh i s hoc phng trnh vi

phn), tuy nhin, cc phng trnh ny thng phc tp, do ni chung kh c th

gii c (a c v cc phng trnh c bn) bng cc bin i i s. Hn na,

v cc cng thc nghim (ca phng trnh phi tuyn hoc phng trnh vi phn)

thng phc tp, cng knh, nn cho d c cng thc nghim, vic kho st cc

tnh cht nghim qua cng thc cng vn gp phi rt nhiu kh khn. V vy, ngay

t thi Archimedes, cc phng php gii gn ng c xy dng. Nhiu

phng php (phng php Newton-Raphson gii gn ng phng trnh phi tuyn,phng php Euler v phng php Runge-Kutta gii phng trnh vi phn) tr

thnh kinh in v c s dng rng ri trong thc t.

Vi s pht trin ca cng c tin hc, cc phng php gii gn ng li

cng c ngha thc t ln. gii mt phng trnh bng tay trn giy, c khi

phi mt hng ngy vi nhng sai st d xy ra, th vi my tnh in t, thm ch

vi my tnh in t b ti, ch cn vi pht. Tuy nhin, vic thc hin cc tnh ton

ton hc trn my mt cch d dng cng i hi ngi s dng c hiu bit su sc

hn v l thuyt ton hc. Mt khc, nhiu vn l thuyt (s hi t, tc hi t,

chnh xc, phc tp tnh ton,) s c soi sng hn trong thc hnh tnh

ton c th. V vy, vic s dng thnh tho cng c tnh ton l cn thit cho mi

hc sinh, sinh vin. Cng c tnh ton s h tr c lc cho vic tip thu cc kin

thc l thuyt, ging dy l thuyt gn vi thc hnh tnh ton, s gip hc sinh, sinh

vin khng ch tip thu tt hn cc kin thc khoa hc, m cn tip cn tt hn vi

cc phng php v cng c tnh ton hin i.

Ni chung, trong cc trng ph thng v i hc hin nay, vic gn ging

dy l thuyt vi tnh ton thc hnh cn cha c y mnh. iu ny hon ton

khng phi v thiu cng c tnh ton, m c l l v vic ph bin cch s dng cc

cng c tnh ton cn t c quan tm.

Vi mc ch minh ha kh nng s dng my tnh in t trong dy v hc

mn Gii tch s, chng ti chn ti lun vn Gii gn ng phng trnh phi


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tuyn v phng trnh vi phn trn my tnh in t. Lun vn gm hai chng:

Chng 1 trnh by ngn gn cc phng php gii gn ng phng trnh phi

tuyn v c bit, minh ha v so snh cc phng php gii gn ng phng trnh

thng qua cc thao tc thc hnh c th trn my tnh in t khoa hc Casio fx-570ES. Chng 2 trnh by phng php Euler, phng php Euler ci tin v phng

php Runge-Kutta gii phng trnh vi phn thng. Cc phng php ny c so

snh v minh ha qua thc hnh tnh ton trn my tnh Casio fx-570 ESv trn

chng trnhMaple.

C th coi cc qui trnh v chng trnh trong lun vn l cc chng trnh

mu gii bt k phng trnh phi tuyn hoc phng trnh vi phn no (ch cn

khai bo li phng trnh cn gii). iu ny c chng ti thc hin trn rt

nhiu phng trnh c th.

Tc gi xin chn thnh cm n TS. T Duy Phng (Vin Ton hc), ngi

Thy hng dn tc gi hon thnh lun vn ny. Xin c cm n Trng i

hc S phm (i hc Thi Nguyn), ni tc gi hon thnh chng trnh cao

hc di s ging dy nhit tnh ca cc Thy. Xin c cm n Phng Gio dc

Ph Yn (Thi Nguyn), nitc gi cng tc, to mi iu kin thun li tc

gi hon thnh kha hc v lun vn. Cui cng, xin c cm n Gia nh ng

vin, gip v chia x nhng kh khn vi tc gi trong thi gain hc tp.

Thi Nguyn, 20.9.2007

Trn Th Hon


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CHNG I

GII GN NG PHNG TRNH

PHI TUYN TRN MY TNH IN T

1. GII GN NG PHNG TRNH ( ) 0f x

Phng trnh ( ) 0f x thng gp nhiu trong thc t. Tuy nhin, ngoi

mt s lp phng trnh n gin nh phng trnh bc nht, phng trnh bc hai,

phng trnh bc ba v bc bn l cc phng trnh c cng thc nghim biu din

qua cc h s, v mt vi lp phng trnh c gii nh cc k thut ca i s(phn tch ra tha s, t n ph,) a v cc phng trnh bc nht hoc bc

hai, hu ht cc phng trnh phi tuyn l khng gii c chnh xc (khng c

cng thc biu din nghim qua cc h s ca phng trnh), v vy ngi ta

thng tm cch tm nghim gn ng ca phng trnh. V ngay c khi bit cng

thc nghim, do tnh phc tp ca cng thc, gi tr s dng ca cng thc nhiu

khi cng khng cao. Th d, ngay c vi lp phng trnh n gin l phng trnh

a thc bc ba 3 2 0 ax bx cx d , mc d c cng thc Cardano gii,

nhng v cng thc ny cha nhiu cn thc kh cng knh (xem, th d:

Eric W. Weisstein: CRS Concise Encyclopedia of Mathematics, CRS Press, New

York, 1999, mc Cubic Equation, trang 362-365),

nn thc cht chng ta cng ch c th tm c nghim gn ng. Hn na, a s

cc phng trnh, thm ch nhng phng trnh rt n gin v mt hnh thc

nhng li xut pht t cc bi ton thc t, th d, phng trnh cosx x khng ccng thc biu din nghim thng qua cc php ton c bn (cng, tr, nhn, chia,

khai cn, ly tha), ni cch khc, khng gii c hoc rt kh gii bng cc php

bin i i s, nhng c th gii gn ng n chnh xc bt k rt d dng nh

php lp 1 cos n nx x , nht l trn my tnh in t b ti (ch cn bm lin tip

mt phm ).

Nhng phng trnh xut hin trong cc bi ton thc t (th d, khi o

c,) ni chung c thng tin u vo (th hin trn cc h s, trong cng thc) ch


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l gn ng (sai s trong o c, nh gi, tnh ton s b,...). V vy vic tm

nghim chnh xc cng khng c ngha thc t ln, trong khi vi cc phng

php gii gn ng phng trnh, ta thng c cng thc nh gi chnh xc ca

nghim gn ng v c th tm nghim n chnh xc bt k cho trc, nnphng phpgii gn ng phng trnh c ngha rt quan trng trong gii quyt

cc bi ton thc t.

Cc phng php gii chnh xc phng trnh ch mang tnh n l (cho tng

lp phng trnh), cn cc phng php gii gn ng phng trnh mang tnh ph

dng: mt phng php c th dng gii cho nhng lp phng trnh rt rng,

th d, ch i hi hm s l lin tc chng hn, v vy kh nng ng dng ca gii

gn ng l rt cao.

Gii gn ng phng trnh lin quan n nhiu vn quan trng khc ca

ton hc. Th d, theo iu kin cn cc tr (nh l Fermat), im 0x l im cc

tr (a phng) ca hm s ( )y F x th n phi l im dng, tc l

0 0'( ) '( ) 0 y x F x . Nh vy, tm im cc tr, trc tin ta phi gii phng

trnh ' '( ) : ( ) 0 y F x f x tm im dng (im c nghi ng l im cc

tr). Trong thc t tm nghim ti u, ta thng i tm cc im dng (nghi ng

l cc tr) nh gii gn ng phng trnh ' '( ) : ( ) 0 y F x f x .

Bi v mt trong nhng th mnh ca my tnh in t l kh nng lp li

mt cng vic vi tc cao, m gii gn ng phng trnh thc cht l vic thc

hin mt dy cc bc lp, nn nh my tnh m vic gii gn ng phng trnh

tr nn n gin, nhanh chng v thun tin. Khng nhng th, my tnh cn cho

php, thng qua lp trnh, m phng qu trnh thc hin bc lp gii phng trnh,

bi vy n l cng c tt trgip hc sinh v sinh vin tip thu cc kin thc ton

hc ni chung, cc phng php gii gn ng phng trnh ni ring. Do thc

hnh gii gn ng trn my tnh in t c mt ngha nht nh trong ging dy

v hc tp b mn ton trong cc trng ph thng v i hc.

Trong chng ny, gii gn ng phng trnh, chng ta lun gi thit

rng, ( )f x l mt hm xc nh v lin tc trn mt on no ca ng thng


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thc. Nhiu khi iu kin ny l xy dng phng php gii gn ng.

Trong mt s phng php, ta s gi thit rng ( )f x kh vi n cp cn thit (c

o hm cp mt hoc c o hm cp hai).

Nu ( ) 0f x th im x c gi l nghim hoc khng im ca

phng trnh ( ) 0f x . Ta cng gi thit rng cc nghim l c lp, tc l tn ti

mt ln cn ca im x khng cha cc nghim khc ca phng trnh. Khong

ln cn (cha x ) ny c gi l khong cch lica nghim x .

Cc bc gii gn ng phng trnh

Gii gn ng phng trnh ( ) 0f x c tin hnh theo hai bc:

Bc 1. Tm khong cha nghim

Mt phng trnh ni chung c nhiu nghim. Ta cn tm khong cha

nghim, tc l khong ( , )a b trong phng trnh c nghim (c duy nht

nghim), bng mt trong cc tiu chun sau.

nh l 1 (Bolzano-Cauchy)Nu hm ( )f x lin tc trn on ,a b v tha mn

iu kin ( ) ( ) 0f a f b th phng trnh ( ) 0f x c t nht mt nghim trong

khong ( , )a b .

ngha hnh hc ca nh l ny kh r rng: th ca mt hm s lin tc

l mt ng cong lin tc (lin nt), khi chuyn t im ( , ( ))A a f a sang im

( , ( )) B b f b nm hai pha khc nhau ca trc honh, ng cong ny phi ct trc

honh ti t nhtmt im (c th ti nhiu im).

Th d, hm s 3( ) 3 1 y f x x x c ( 2) 3 f ; ( 1) 1 f ; (0) 1 f v

(2) 1f nn phng trnh 3 3 1 0 x x c ba nghim phn bit trong cc

khong ( 3, 1) ; ( 1,0) v (0,2) .


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nh l 2 (H qu ca nh l 1) Gi s ( )f x l mt hm lin tc v n iu

cht trn on ,a b . Khi y nu ( ) ( ) 0 f a f b th phng trnh ( ) 0f x c duy

nht mt nghim trong khong ( , )a b .

ngha hnh hc ca nh l ny l: th ca mt hm s lin tc tng

cht (gim cht) l mt ng cong lin tc (lin nt) lun i ln (i xung). Khi di

chuyn t im ( , ( ))A a f a sang im ( , ( )) B b f b nm hai pha khc nhau ca

trc honh th th phi ct v ch ct trc honh mt ln (Hnh v).

Hai nh l trn ch i hi tnh lin tc m khng i hi tnh kh vi (tn ti

o hm) ca ( )f x . Nu ( )f x c o hm th c th dng tiu chun di y.


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nh l 3 (H qu ca nh l 2) Gi s hm s ( )f x c o hm ( )f x v o

hm ( )f x ca n khng i du (lun dng hoc lun m) trn on ,a b .

Khi y nu ( ) ( ) 0f a f b th phng trnh ( ) 0f x c duy nht mt nghim

trong khong ( , )a b .

T ba nh l trn, ta i n hai phng php tm khong cch li nghim ca

phng trnh ( ) 0f x (khong cha duy nht mt nghim): phng php hnh hc

v phng php gii tch.

Phng php gii tch

Gi s ta phi tm nghim ca phng trnh ( ) 0f x trong khong ( , )a b .

Ta i tnh gi tr ( )f a , ( )f b v cc gi tr ( )if x ca hm s ti mt s im

( , )i x a b , 1,2,...,i n . Nu hm ( )f x n iu cht trn khong 1, i ix x v

iu kin 1( ) ( ) 0 i if x f x c tha mn th 1, i ix x l mt khong cch li

nghim ca phng trnh ( ) 0f x . Nu thng tin v hm ( )f x qu t th ta

thng dng quy trnh chia on thng (chia khong ( , )a b thnh 2, 4, 8,phn) v

th iu kin 1( ) ( ) 0 i if x f x tm khong cch li nghim.

Mt a thc bc n c khng qu n nghim. V vy phng trnh a thc c

khng qu n khong cch li nghim.

Khi hm ( )f x tt (c o hm, c dng c th,...), ta c th kho st

th chia trc s thnh cc khong i du ca o hm (khong ng bin v

nghch bin ca hm s) v xc nh khong cch li nghim.

Phng php hnh hc

Trong trng hp th hm s tng i d v, ta c th v phc th

tm khong cch li nghim hoc gi tr th ca nghim nh l giao im (gn ng)

ca th vi trc honh. Cng c th dng cc my tnh ha (my tnh c kh

nng v hnh nh Casio Algebra fx-2.0 Plushoc Sharp EL-9650) hoc cc phn


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mm tnh ton (Maple, Matlab,) v th. Sau , nh tnh ton, ta tinh

chnh i n khong cch li nghim chnh xc hn.

Bc 2. Gii gn ng phng trnh

C bn phng php c bn gii gn ng phng trnh: phng php chia

i, phng php lp, phng php dy cung v phng php tip tuyn (phng

php Newton-Raphson). Nhm lm c s l thuyt cho cc tnh ton trong 3,

trong 2 chngti s vn tt trnh by ni dung ca cc phng php ny, ch yu

l da vo cc gio trnh Gii tch s [1] - [6].

2. CC PHNG PHP TM NGHIM GN NG CA

PHNG TRNH ( ) 0f x

1. Phng php chia i

Ni dung ca phng php chia i rt n gin: Gi s ( )f x l mt hm

lin tc trn on ,a b v ( ) ( ) 0f a f b . Khi y theo nh l Bolzano-Cauchy,

phng trnh ( ) 0f x c t nht mt nghim trong khong ( , )a b .

Chia i on ,a b v tnh ( )2

a bf .

Nu ( ) 02

a bf th 2

a bx l mt nghim ca phng trnh ( ) 0f x .

Nu ( ) 02

a bf th ( ) ( ) 02

a b f a f hoc ( ) ( ) 0

2

a b f f b nn phng

trnh c t nht mt nghim trong khong ( , )2

a ba hoc ( , )2

a bb .

Gi khong mi (khong nh) cha nghim l 1 1( , )a b .

Li chia i khong 1 1( , )a b v tnh gi tr ti im gia1 1

2

a bx .

Tip tc mi qu trnh ny ta i n:


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Hoc ti bc th n no ta c ( ) 02

n n

a bf , tc l

2

n n

a bx l

nghim, hoc ta c mt dy cc on thng lng nhau [ , ]n na b c cc tnh cht:

1 2 1... ... ... ...n na a a a b b b ,

( ) ( ) 0n nf a f b v 2

n n n

b ab a .

S hi t ca phng php chia i

Dy na l dy n iu tng, b chn trn bi b , dy nb l n iu

gim v b chn di bi a nn c hai dy u c gii hn.

Do2

n n n

b ab a nn lim 0

n n

nb a hay lim lim

n n

n na b x .

Do tnh lin tc ca hm s ( ) y f x ,ly gii hn trong biu thc

( ) ( ) 0n nf a f b ta c2 ( ) lim ( ). ( ) 0

n n

n f x f a f b .

Suy ra ( ) 0f x hayx l mt nghim ca phng trnh ( ) 0f x trong khong

( , )a b .

nh gi sai s

Ti bc th n ta c n na x b v2

n n n

b ab a .

Nu chn nghim gn ng l nx a th2

n n n

b a x x b a ;

Nu chn nghim gn ng l nx b th2

n n n

b a x x b a ;

Nu chn nghim gn ng l2

n n

a bx th ta c nh gi:

12 2

n nn

b a b ax x .


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Nh vy, sau bc th n , nn chn nghim gn ng l2

n nn

a bx c , ta s

c nghim chnh xc hn.

Nu chn2

n nn a bx th 12 2

n nn nb a b ax x . Do vi mi 0

cho trc ( chnh xc 0 cho trc) ta c nx x vi mi

2log

b an

.

Nu ti mi bc n ta u chn2

n nn

a bx

th ta cng c

1 1 2 1( ) ( )

2 2 2n n n n n n n

b a b a b a x x x x x x

.

Do khi tnh ton (trn my tnh b ti vi mn hnh hin th c 10 ch s

chng hn), ta c th dng tnh ton khi 1 1 .... n n n x x x ng n s thp

phn cn thit (th d, ta c th dng tnh ton khi c nghim chnh xc n 10

ch s, tc l 1010 ).

2. Phng php lp

Gi s ( , )a b l khong cch li nghim ca phng trnh ( ) 0f x . Gii

phng trnh ( ) 0f x bng phng php lp gm cc bc sau:

Bc 1. a phng trnh ( ) 0f x v phng trnh tng ng ( ) x g x .

Bc 2. Chn 0 ( , ) x a b lm nghim gn ng u tin.

Bc 3. Thay 0x x vo v phi ca phng trnh ( ) x g x ta c nghim gn

ng th nht 1 0( ) x g x . Li thay 1 0( ) x g x vo v phi ca phng trnh

( ) x g x ta c nghim gn ng th hai 2 1( ) x g x . Lp li qu trnh trn, ta

nhn c dy cc nghim gn ng

1 0( ) x g x , 2 1( ) x g x , 3 2( ) x g x , 4 3( ) x g x ,..., 1( )n n x g x , ...


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Nu dy cc nghim gn ng nx , 1,2,...n hi t, ngha l tn ti

lim

nn

x x th (vi gi thit hm ( )g x l lin tc trn on ,a b ) ta c:

1 1lim lim ( ) (lim ) ( )

n n nn n n

x x g x g x g x .

Chng t x l nghim ng ca phng trnh ( ) x g x (im bt ng ca nh

x g) hay x l nghim ng ca phng trnh ( ) 0f x .

Tnh hi t

C nhiu phng trnh dng ( ) x g x tng ng vi phng trnh

( ) 0

f x . Phi chn hm s ( )g x sao cho dy nx xy dng theo phng phplp l dy hi t v hi t nhanh ti nghim. Ta c tiu chun sau.

nh l 4. Gi s x l nghim ca phng trnh ( ) 0f x v phng trnh

( ) x g x tng ng vi phng trnh ( ) 0f x trn on ,a b . Nu ( )g x v

'( )g x l nhng hm s lin tc trn ,a b sao cho ( ) 1 , g x q x a b th

t mi v tr ban u 0 ( , ) x a b dy nx xy dng theo phng php lp

1( )n n x g x s hi t ti nghim duy nht x trong khong ( , )a b ca phng

trnh ( ) 0f x .

Chng minh.

Gi s 0 ( , ) x a b bt k. V x l nghim ca phng trnh ( ) 0f x trong

khong ( , )a b nn ta c ( ) x g x . Mt khc v 1 0( ) x g x nn

1 0( ) ( ) x x g x g x .

Theo nh l Lagrange tn ti mt im 0,c x x sao cho

1 0 0( ) ( ) '( )( ) x x g x g x g c x x .

Suy ra

1 0 0 0'( )( ) x x g c x x q x x x x .

Chng t 1 ( , ) x a b .

Tng t ta c:


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2 1 x x q x x ; 3 2 x x q x x ;...; 1 n nx x q x x ;...

T cc bt ng thc trn ta suy ra nu 0 ( , ) x a b th ( , )n x a b vi mi n v

21 2 0...

nn n nx x q x x q x x q x x .

Do 1q nn khi n v phi tin ti 0 . Chng t dy nx hi t ti x .

nh gi sai s

nh gi sai s ca nghim gn ng nx (nhn c bng phng php

lp) v nghim chnh xc x ca phng trnh ( ) 0f x ti bc th n ta xt hiu

nx x .

T chng minh trn ta c:

1 1 1 n n n n n n n nx x q x x q x x x x q x x q x x

Vy

1(1 ) n n nq x x q x x

hay

11

n n n

q x x x x

q

Mt khc, p dng cng thc s gia hu hn (cng thc Lagrange) ta c:

1 1 2 1 2( ) ( ) '( )( ) n n n n n n nx x g x g x g c x x

trong

1 2( , ) n n nc x x

Suy ra

1 1 2 1 2'( ) n n n n n n nx x g c x x q x x

T bt ng thc trn, cho n=2,3,4,... ta c:


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2 1 1 0 x x q x x

23 2 2 1 1 0 x x q x x q x x

...

11 1 0

nn n x x q x x .

Thay vo bt ng thc

11

n n n

q x x x x

q

ta c:

1

1 1 0 1 01 1 1

nn

n n n

q q q x x x x q x x x x

q q q

Cng thc trn cho thy phng php lp hi t cng nhanh nu q cng b.

T cng thc trn ta cng suy ra rng, t c xp x (nghim gn ng

sai khc nghim ng khng qu , nx x ), ta phi lm ( )N bc, trong

1 0

(1 )lg

( )lg

q

x xN

q

.

T cng thc 11 1 0

n

n n x x q x x ta c kt lun: nu dy nx hi t

th khi n ln hai nghim gn ng nx v 1nx xp x bng nhau. V vy khi sdng my tnh ta thng dng qu trnh lp khi cc kt qu lin tip 1nx , nx ,

1nx ,... t xp x yu cu (trng nhau ti s ch s thp phn sau du phy cn

thit).

Nhn xt. V ta coi ( , )a b l khong cch li nghim (cha nghim x ) ca

phng trnh ( ) 0f x nn trong nh l 4 ta gi thit s tn ti nghim x . Hn

na, ta i hi ( )g x phi l mt hm kh vi. Di y l mt phin bn ca


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nh l 4 (khng i hi trc tn ti nghim ca phng trnh ( ) 0f x v ch i

hi ( )g x l mt hm lin tc Lipschitz).

nh l 5. Gi s ( )g x l hm s xc nh trn khong ;a b sao cho:

i) ( ) ( ) , ; g x g y q x y x y a b ( ( )g x l Lipschitz trn ;a b ).

ii) Tn ti mt s ; a b sao cho ( ) (1 )( ) g q b a .

Khi y vi mi 0 ; x a b , dy nx xy dng theo phng php lp 1( )n n x g x

s hi t ti im bt ng (tc l ( ) x g x ) duy nht x trong khong ( , )a b ca

nh x g.

3. Phng php dy cung

Gi s ( , )a b l khong cch li nghim. Ta thay cung ca ng cong

( ) y f x trn on [ , ]a b bng dy trng cung y v coi giao im ca dy cung

(ng thng) vi trc honh l nghim xp x ca phng trnh ( ) 0f x .

xy dng dy xp x nx , ta xt hai trng hp:

Trng hp 1. '( ). ''( ) 0 f x f x .

xc nh, ta coi ( ) 0, ( ) 0, f a f b v

'( ) 0, ''( ) 0 f x f x (Hnh 1).

Dy cung AB l ng thng ni hai im Hnh 1

( , ( ))A a f a v ( , ( )) B b f b c phng trnh( )

( ) ( )

y f a x a

f b f a b a.

Honh giao im 1x ca ng thng AB vi trc honh chnh l nghim ca

phng trnh trn khi cho 0y .

Suy ra 1( )

( ) ( )

f a x a

f b f a b ahay 1

( )( )

( ) ( )

f a b ax a

f b f a.

Nghim x by gi nm trong khong 1( , )x b (xem Hnh 1).

a

b

f(a)

f(b)

x1x


18/82

16

Thay khong ( , )a b bng khong 1( , )x b , ta i n nghim1 1

2 1

1

( )( )

( ) ( )

f x b xx x

f b f x.

Tip tc qu trnh trn, ta i n dy nghim xp x:

1

( )( )

( ) ( )

n n

n n

n

f x b xx x

f b f x.

Cng thc trn vn ng trong trng hp

( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x .

Trng hp 2. '( ). ''( ) 0 f x f x .

xc nh, coi

( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 2). Hnh 2

Dy cung AB l ng thng ni hai im ( , ( ))A a f a v

( , ( )) B b f b c phng trnh( )

( ) ( )

y f b x b

f b f a b a.

Honh giao im 1x ca ng thng AB vi trc honh chnh l nghim caphng trnh trn khi cho 0y .

Suy ra 1( )

( ) ( )

f b x b

f b f a b ahay 1

( )( )

( ) ( )

f b b ax b

f b f a.

Nghim x by gi nm trong khong 1( , )a x .

Thay ( , )a b bng khong 1( , )a x , ta i n nghim xp x

1 12 1

1

( )( )

( ) ( )

f x x ax x

f x f a.

Tip tc qu trnh trn, ta i n dy nghim xp x

1

( )( )

( ) ( )

n n

n n

n

f x x ax x

f x f a.


( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x .

a

(b)

X1 b

f(a)

x


19/82

17

Ta c th tng kt thnh mt cng thc nh sau:

1

( )( )

( ) ( )

n n

n n

n

f x x d x x

f x f d ,

trong d b nu ( ). ''( ) 0f b f x , 0 x a ;

d a nu ( ). ''( ) 0 f a f x , 0 x b .

Tnh hi t

Dy xp x lin tip l mt dy tng, b chn trn (trng hp 1):

0 1 1... ... n na x x x x x b

hoc l dy gim , b chn di (trng hp 2):

0 1 1... .... n nb x x x x x a

nn hi t. Hn na, chuyn qua gii hn trong cng thc

1

( )( )

( ) ( )

n n

n n

n

f x x d x x

f x f d ta c

( )( )

( ) ( )

f x x d x x

f x f d .

Suy ra ( ) 0f x hay x l nghim ca phng trnh ( ) 0f x trong khong

( , )a b .

nh gi sai s

Gi s '( )f x khng i du trn ( , )a b v 0 '( ) m f x M vi mi

( , ) x a b .

Ta c cc cng thc nh gi sai s sau y:

( ) nn

f xx x

m

; 1

n n nM m

x x x x

m

.

Chng minh.

p dng nh l gi tr trung bnh Lagrange (cng thc s gia hu hn), ta c

( ) ( ) '( )( ) n nf x f x f c x x

vi ( , ) ( , ) nc x x a b .

V ( ) 0f x v 0 '( ) m f x nn

( ) ( ) '( )( ) n n nf x f x f c x x m x x .


20/82

18

Suy ra( )

nnf x

x xm

.

Nh vy, nh gi chnh xc ca nghim nhn c bng phng php dy

cung, ta c th s dng cng thc max ( ) , [ , ](

nnf x x a bf x

x xm m

.

Ngoi ra, nu bit hai gi tr gn ng lin tip, ta c th nh gi sai s nh sau.

T trn (chng minh s hi t ca phng php dy cung) ta c:

1 11

1

( )( )

( ) ( )

n n

n n

n

f x x d x x

f x f d .

Suy ra 11 11

( ) ( )( ) ( )

n

n n n

n

f x f d f x x x

x d.

V x l nghim ng ca phng trnh ( ) 0f x nn ta c th vit:

11 1

1

( ) ( )( ) ( ) ( )

n

n n n

n

f x f d f x f x x x

x d.

p dng nh l gi tr trung bnh Lagrange, ta c

1 1 1'( )( ) ( ) ( ) n nf c x x f x f x

v

2 1 1'( )( ) ( ) ( ) n nf c x d f x f d ,

trong 1c nm gia x v 1nx , 2c nm gia 1nx v d.

Suy ra

11 1 1 1

1

2 11 2 1

1

( ) ( )'( )( ) ( ) ( ) ( )

'( )( )( ) '( )( ).

( )

nn n n n

n

nn n n n

n

f x f d f c x x f x f x x x

x d f c x d

x x f c x xx d

Vy

1 1 2 1'( )( ) '( )( ) n n n n nf c x x x x f c x x

hay

1 2 1 1

'( )( ) [ '( ) '( )]( ) n n n

f c x x f c f c x x

v


21/82

19

2 11

1

'( ) '( )

'( )

n n n

f c f c x x x x

f c.

Theo gi thit ta c:

2 1'( ) '( ) f c f c M m .

Do

1

n n n

M m x x x x

m.

Nh vy, ta c hai cng thc nh gi sai s

max ( ) , [ , ]( ) nn

f x x a bf xx x

m m

v 1

n n nM m

x x x x

m

.

4. Phng php tip tuyn (Phng php Newton-Raphson)

Gi s ( , )a b l khong cch li nghim. Ta thay cung ca ng cong

( ) y f x trn on [ , ]a b bng tip tuyn ti im ( , ( ))A a f a hoc im

( , ( )) B b f b v coi giao im ca tip tuyn vi trc honh l nghim xp x ca

phng trnh ( ) 0f x

. xy dng dy xp x

nx , ta xt hai trng hp:

Trng hp 1. '( ). ''( ) 0 f x f x .

xc nh, ta coi

( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 3). Hnh 3

Phng trnh tip tuyn vi ng cong ( ) y f x ti im ( , ( ))B b f b c dng:

( ) '( )( )y f b f b x b . Honh giao im 1x ca tip tuyn vi trc honhchnh l nghim ca phng trnh trn khi cho 0y .

Suy ra 0 ( ) '( )( )f b f b x b hay 1( )

'( )

f bx b

f b . Nghim x by gi nm trong

khong 1( , )a x (xem Hnh 3). Thay khong ( , )a b bng khong 1( , )a x , ta i n

nghim xp x 12 1( )

'( )

f xx x

f b

.

Tip tc qu trnh trn, ta i n dy

(a)

a1b

f(b)

a

f(b)

x x1 b

f(a)


22/82

20

nghim xp x: 1( )

'( )

nn n

n

f xx x

f x .

Cng thc trn vn ng trong trng hp Hnh 4

( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 4).

Trng hp 2. '( ). ''( ) 0 f x f x . xc nh, coi

( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x (Hnh 5).

Phng trnh tip tuyn ti im ( , ( ))A a f a c dng:

( ) '( )( )y f a f a x a .

Honh giao im 1x ca tip tuyn

vi trc honh chnh l nghim ca

phng trnh trn khi cho 0y . Hnh 5

Suy ra 0 ( ) '( )( )f a f a x a hay 1( )

'( )

f ax a

f a .

Nghim x by gi nm trong khong 1( , )x b (xem hnh 5).

Thay khong ( , )a b bng khong1

( , )x b , ta i n nghim 12 1

1

( )

'( )

f xx x

f x .

Tip tc qu trnh trn, ta i n dy nghim xp x:

1

( )

'( )

nn n

n

f xx x

f x .


( ) 0, ( ) 0, '( ) 0, ''( ) 0 f a f b f x f x .

Tnh hi tDy cc xp x lin tip l mt dy n iu gim v b chn di (trng

hp 1) hoc n iu tng v b chn trn (trng hp 2) nn tn ti gii hn

lim

nn

x x . D thy rng x l nghim ca phng trnh ( ) 0f x .

f(a)

X1ax b

f(b)


23/82

21

Tht vy, chuyn qua gii hn trong biu thc 1( )

'( )

nn n

n

f xx x

f x ta c:

( )

'( )

f xx x

f x

. Suy ra ( ) 0f x . Do ( , )a b l khongcch li nghim nn x x

chnh l nghim ban u.

nh gi sai s

Gi s 10 '( )m f x v 1''( ) f x M .

Khi y ta c nh gi sai s:

1 1

max ( ) , [ , ]( )nn

f x x a bf x

x x m m

v

21

112

n n n

M

x x x xm .

Chng minh. p dng nh l gi tr trung bnh Lagrange (cng thc s gia hu

hn), ta c

( ) ( ) '( )( ) n nf x f x f c x x

vi ( , ) ( , ) nc x x a b .

V ( ) 0f x v 0 '( )m f x nn

( ) ( ) '( )( ) n n nf x f x f c x x m x x .

Suy ra(

nnf x

x xm

.

Dng khai trin Taylor ca ( )f x ti 1nx :

21 1 1 1

1( ) ( ) '( )( ) ''( )( )

2n n n n n n nf x f x f x x x f c x x ,

trong c nm gia nx v 1nx .

Do 111

( )

'( )

nn n

n

f xx x

f x

nn 1 1 1( ) '( )( ) 0n n n n f x f x x x .

Thay vo ng thc trn ta c:

2

1

1( ) ''( )( )

2n n n f x f c x x .


24/82

22

T cng thc trn v cng thc( )

nnf x

x xm

ta suy ra

21

(( )

2

nn n n

f x M x x x x

m m

.

Nh vy, tc hi t ca phng php tip tuyn l bc hai.

3. TM NGHIM GN NG CA PHNG TRNH

( ) 0f x TRN MY TNH IN T

Nh ta thy trong 2, c bn phng php gii gn ng phng trnh

( ) 0f x trong khong cch li nghim ,a b u dn ti thc hin mt dy lp

1 ( ) n nx x v dy lp ny hi t ti nghim (duy nht ) x ca phng trnh trong

khong ,a b .

Do c thit k sn cho php thc hin cc thao tc tnh ton lin tip,

vic thc hin dy lp trn cc my tnh in t khoa hc (Casio fx-500 MS, Casio

fx-570 MS, Sharp 506 WM, Casio fx-500 ES, Casio fx-570 ES) l kh n gin v

thun tin. Trong ny, chng ti trnh by cch s dng cc loi my ny cho

mc ch gii gn ng phng trnh theo cc phng php trnh by mc trn.

Thc hnh gii gn ng phng trnh trn my tnh in t khoa hc cho php cm

nhn r hn cc vn ca gii tch s (s hi t, tc hi t,) ca tng phng

php.

tin trnh by, chng ti chn my tnh in t khoa hc Casio 570 ES, l

loi my c nhiu u im trong cc tnh nng gii ton v c s dng tng iph bin hin nay trong cc trng ph thng v i hc.

My tnh in t khoa hc Casio 570 ESc mt s phm rt tin dng trong

tnh ton. N c thit k c th tnh ton i s (tnh ton theo cng thc) kt

hp vi nhng nh (9 nh). Thit k ny c bit thch hp cho thc hin dy

lp, do c bit thun tin cho vic gii gn ng phng trnh.

Bi 1. Gii phng trnh i s bc cao 9 10 0 x x .


25/82

23

y tuy ch l mt phng trnh a thc, tuy nhin bc ca n kh cao nn

kh c th gii c bng cc k thut ca i s (t n ph, nhm s hng,)

a v phng trnh bc thp hn.

t 9( ) 10 y f x x x .

Do 8' 9 1 0 y x vi mi x nn hm s ng bin trn ton trc s. Ta d dng

tnh c (1) 8 f v (2) 504 0 f . Do , phng trnh 9 10 0 x x c

duy nht mt nghim trong khong (1;2) . so snh, ta s tm nghim gn ng

ca phng trnh ny theo c bn phng php trnh by trong 2.

Phng php chia i khong cha nghim

a gi tr 1x vo nh A : 1 SHIFT STO A

a gi tr 2x vo nh B : 2 SHIFT STO B

Ch : T nay v sau, cho tin, cc phm s c vit nh l cc s, cn cc

phm ch trn mn hnh c trong cc vung. Th du, phm s 2 ta vn vit l

s 2, cn phm A ta vit l A .

Tnh1 2 3

2 2 2

a bc v a vo nh C :

ALPHA A ALPHA B 2 SHIFT STO C

Khai bo cng thc 9( ) 10 f x x x :

ALPHA X ^x 9 ALPHA X 10 (3.1)

tnh gi tr ca biu thc ti mt im no ta bm phm CALC (Calculate-tnh): CALC

My hi (hin trn mn hnh): ?X

Khai bo3

2c (ang trong C ): ALPHA C ( 15331

512)

Ch : T nay v sau, tin trong trnh by, ta ghi ngay p s ca kt qu tnh

ton hin th trn mn hnh sau phm v trong ngoc. Th d, sau khi khai


26/82

24

bo cng thc (3.1) v bm phm CALC , my hi ?X , ta khai bo ALPHA C

(tc l gi tr ca i s x cn tm gi tr ca hm s 9( ) 10 f x x x chnh

bng gi tr trong C ). Sau khi bm phm , my hin ngay trn mn hnh p

s15331

512, tc l

3 15331( )

2 512f v ta vit (

15331

512).

Nh vy, ta c (1) 8 0 f v3 15331

( ) 02 512

f . Chng t nghim ca phng

trnh nm trong khong 1 13

( , ) ( , ) (1; )2

a b a c . B c gii phng (gi tr c

2x trong nh B khng cn dng na).

Ch : V ti mi bc i ta ch cn nh ba gi tr ,i ia b hoc2

i ii

a bc nn ta

cng ch cn s dng ba nh A , B , C l .

Li chia i khong 1 13

( , ) ( , ) (1; )2

a b a c v gi 1 115

1,252 4

a bc vo

B (va c gii phng):

ALPHA A ALPHA C 2 SHIFT STO B

S dng phm i v dng cng thc (3.1) v dng phm CALC tnh gi

tr ca hm s ti 1 1,25c :

Bm phm CALC (Calculate-tnh): CALC

My hi: ?X

Khai bo 1 115

1,252 4

a bc ( c tnh v lu trong B ):

ALPHA B ( 1.299419403 )

Nh vy, ta c 15

( ) ( ) 1, 2994194034

f c f v 13 15331

( ) ( ) 02 512

f b f .


27/82

25

Ch : tin trnh by, ta thng vit du bng thay cho du gn ng , th

d5

( ) 1,2994194034

f , mc d chnh xc hn, phi vit l:

5( ) 1,2994194034

f (chnh xc n 10 ch s).

Chng t nghim ca phng trnh nm trong khong

2 2 1 1

5 3( , ) ( , ) ( ; )

4 2 a b c b . nh A c gii phng (gi tr c 1a trong nh

A khng cn dng na).

Li chia i khong 2 2 1 1 5 3( , ) ( , ) ( ; )4 2 a b c b , tnh v gi

2 22

5 3114 2 1,375

2 2 8

a bc vo A :

ALPHA B ALPHA C 2 SHIFT STO A


tr ca hm s ti 211

1,3758

c :


My hi: ?X

Khai bo 211

1,3758

c ( c tnh v lu trong A ):

ALPHA A (8.943079315)

Nh vy, ta c 211

( ) ( ) 8,943079315 08

f c f v

2 1

5( ) ( ) ( ) 1, 299419403 0

4 f a f c f . Chng t nghim ca phng trnh

nm trong khong 3 3 2 25 11

( , ) ( , ) ( ; )4 8

a b a c . nh B c gii phng (gi tr

c 2 32

b trong nh B khng cn dng na).


28/82

26

Li chia i khong 3 3 2 25 11

( , ) ( , ) ( ; )4 8

a b a c v gi

3 33

5 1121

4 8 1,31252 2 16

a bc vo B :



tr ca hm s ti 321

1,312516

c :


My hi: ?X

Khai bo 321

1,312516

c ( c tnh v lu trong B ):

ALPHA B (2.870795905)

Nh vy, ta c 321

( ) ( ) 2,870795905 016

f c f v

2 15( ) ( ) ( ) 1, 299419403 04

f a f c f . Chng t nghim ca phng trnh

nm trong khong 4 4 1 35 21

( , ) ( , ) ( ; )4 16

a b c c . nh C c gii phng (gi tr

c11

8c trong nh C khng cndng na).

Li chia i khong4 4 1 3

5 21( , ) ( , ) ( ; )

4 16 a b c c v gi

4 44

5 21414 16 1,28125

2 2 32

a b

c vo C :



tr ca hm s ti 4 41 1,2812532

c :


29/82

27


My hi: ?X

Khai bo 441

1,28125

32

c ( c tnh v lu trong C ):

ALPHA C (0.5860042121)

Nh vy, ta c 441

( ) ( ) 0,586004212 032

f c f v

2 1

5( ) ( ) ( ) 1, 299419403 0

4 f a f c f . Chng t nghim ca phng trnh

nm trong khong 5 5 1 4 5 41( , ) ( , ) ( ; )4 32 a b c c . nh C c gii phng (gi tr

c3

21

16c trong nh C khng cn dng na).

Tip tc qu trnh ny, ta i n bng sau, trong mi dng cho ta cc gi tr nc ,

( )nf c v gi tr ( )nf a hoc ( )nf b tnh c dng trn, tri du vi ( )nf c .

V cc gi tr ( )n

f a hoc ( )n

f b c tnh n 10 ch s dng trn, v ch

cn du ca chng, nn tit kim ch, ta ch vit gi tr khc 0 u tin ca n

sau du phy. Mt khc, nhn vo mi dng, ta cng bit no (khng c) c gii

phng, nhng ch a gi tr mi ca nc vo.

nc trong ( )nf c ( )nf a ( )nf b Khong

a=1; b=2 -8 A 504 B ( ; )a b

1,5c

C 29,94335937 -8 A ( ; )a c

1 1, 25c B -1,299419403 29,9 C 1( ; )c c

2 1,375c A 8,943079315 -1,2 B 1 2( ; )c c

3 1,3125c C 2,870795905 -1,2 B 1 3( ; )c c

4 1, 28125c A 0,5860042121 -1,2 B 1 4( ; )c c

5 1, 265625c C -0,402449001 0,5 A 5 4( ; )c c

6 1, 2734375c

B 0,079843683 -0,4 C 5 6( ; )c c


30/82

28

7 1, 26953125c A 0,164222375 0,07 B 7 6( ; )c c

8 1, 271484375c C 0,042927154 0,07 B 8 6( ; )c c

9 1, 272460938c A 0,018272819 0,04 C 8 9( ; )c c

10 1, 271972656c B 0,012373404 0,01 A 10 9( ; )c c

11 1, 272216797c C 0,0029381332 0,01 B 10 11( ; )c c

12 1, 272094127c A 0,00472052697 0,002 C 12 11( ; )c c

13 1, 272155762c B 0,00089191976 0,002 C 13 11( ; )c c

14 1, 272186279c A 0,0010229262 0,0008 B 13 14( ; )c c

15 1, 272171021c C 0,0000654581 0,0008 B 13 15( ; )c c

16 1, 272163391c A 0,00041324182 0,00006 C 16 15( ; )c c

17 1, 272167206c B 0,00017389435 0,00006 C 17 15( ; )c c

18 1, 272169113c A 0,00005421851 0,00006 C 18 15( ; )c c

19 1, 272170067c B 0,0000056196 0,00005 A 18 19( ; )c c

20 1, 27216959c C 0,00000024847 0,000005 B 20 19( ; )c c

21 1, 272169828c A 0,00000933952 0,000005 B 21 19( ; )c c

22 1, 272169948c C 0,00000185969 0,000005 B 22 19( ; )c c

23 1, 272170007c A 0,00000188 0,000001 C 22 23( ; )c c

24 1, 272169977c B 0,0000000101 0,000001 C 22 24( ; )c c

25 1, 272169963c A 0,00000092473 0,00000001 B 25 24( ; )c c

26 1, 27216997c C 0, 0000004573 0,00000001 B 26 24( ; )c c

27 1, 272169974c A 0,00000022325 0,00000001 B 27 24( ; )c c

28 1, 272169976c C 0,00000010655 0,00000001 B 28 24( ; )c c

29 1, 272169977c A 0,00000004821 0,00000001 B 29 25( ; )c c

Nghim gn ng l 29 1,272169977 x c v ( ) 0,00000004821 f x Li

bnh: Ta thy phng php chia i tuy n gin nhng ni chung hi t chm,

thao tc trn my kh phctp.

Phng php lp


31/82

29

Phng trnh 9 10 0 x x tng ng vi 9 10 ( ) x x x .

V

89

1( )

10

xx

nn9 98 8

1 1( ) 1

9 8 x vi mi 1;2x . Do dy

lp 91 10 n nx x hi t ti nghim 1;2x .

Chn gi tr ban u 0 1,5x : 1.5

Ch : My c thit k mt cch thng minh l gi tr 0 1,5x va c khai

bo, hoc cc kt qu tnh ton hin trn mn hnh, cng s c chuyn ngay vo

Ans (kt qu), s dng vo cc tnh ton tip theo cho tin.

Khai bo 0( )x : ( 10 Ans )^x 1 9

Tnh gi tr ca 0( )x : Bm phm

Tnh cc gi tr tip theo: Bm phm

Sau 6 ln bm phm, ta i n dy nghim xp x:

1,5; 1,268436614; 1,27223043; 1,272168998; 1,272169993; 1,272169977;

1,272169977;

Kt qu nghim gn ng l: 1,272169977 (chnh xc n 10 ch s thp phn).

Phng php dy cung

Xt 9( ) 10 f x x x . Ta c 8'( ) 9 1 0 f x x v 7( ) 72 0 f x x

vi mi x trong khong (1;2) . Theo cng thc dy cung

1

( )( )

( ) ( )

n n

n n

n

f x x d x x

f x f d vi 2d ta c dy lp

9

1 9

( 10)( 2)

10 504

n

n

n n

n n

n

x x xx x

x x, 0 1x .

Khai bo gi tr u 0 1x : 1


32/82

30

Khai bo cng thc9

1 9

( 10)( 2)

10 504

n

n

n n

n n

n

x x xx x

x x:

Ans ( Ans ^x 9 Ans 10 ) ( Ans 2 ) ( Ans ^x 9 Ans

514 )

Tnh cc gi tr 1nx bng cch bm lin tip phm . Ta c dy cc gi tr:

65

64; 1,03069279;

Phi sau khong 300 ln bm phm ta mi ra c kt qu 1,272169976.

Nh vy, phng php dy cung trong bi ny hi t rt chm.

Phng php tip tuyn

Theo cng thc tip tuyn 1( )

'( )

nn n

n

f xx x

f x, 0 2x ta c

9

1 8

10

9 1

n

n

n

n n

x xx x

x

.

Khai bo gi tr ban u: 2

Khai bo cng thc9

1 8

10

9 1

n

n

n

n n

x xx x

x:

Ans ( Ans ^x 9 Ans 10 ) ( 9 Ans ^x 8 1 )

Tnh cc gi tr 1nx bng cch bm lin tip phm . Ta c dy cc gi tr xp

x:

1,781344902; 1,592631378; 1,438653785; 1,331291548; 1,281547657;

1,272435766; 1,272170196; 1,27216977; 1,27216977.

Sau chn ln lp ta i n p s.

Kt lun: C bn phng php (chia i on cha nghim, lp, dy cung v tip

tuyn) u cho nghim gn ng ti 10 ch s thp phn. tm c nghim gn


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ng n 10 ch s thp phn theo phng php chia i cn 30 bc lp, v thao

tc phc tp trong mi bc. tm nghim gn ng n 10 ch s thp phn theo

phng php lp ch cn 6 bc. tm nghim gn ng n 10 ch s thp phn

theo phng php dy cung phi cn n 300 bc lp. tm nghim gn ngn 10 ch s thp phn theo phng php tip tuyn ch cn 9 bc lp. Thao tc

lp theo ba phng php sau rt n gin: sau khikhai bo gi tr ban u v cng

thc lp, ta ch cn lin tip bm phm cho n khi c gi tr khng i (im

bt ng), l nghim gn ng n 10 ch s thp phn.

Bi 2. Gii phng trnh cha cn thc 83 2 5 0 x x .

t 8( ) 3 2 5 y f x x x . Khng cn my ta cng c th d dng tnh

c (0) 5 f v 8 8(5) 3.5 2. 5 5 10 2 5 0 f . Do , phng trnh

83 2 5 0x x c t nht mt nghim trong khong (0;5) . xem dng iu

hnh hc ca th, ta c th nhMaple v th trong khong (0;5) nh lnh

plot (v th):

[> plot(3*x-2*x^(1/8)-5,x=0..5);

Nhn vo th hm s 8( ) 3 2 5 y f x x x ta thy, th ct trc honh ti

mt im trong khong (2;3) . Tuy nhin, chnh xc hn ta cn tnh gi tr ca

hm s ti cc im 2x v 3x .

Ta c th tnh cc gi tr ca hm s cho ti im 2x

nh lnh eval (evaluate -

tnh gi tr ca hm s ti 2x ) v lnh evalf(%) (tnh gi tr ca biu thc trn):


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[> eval(3*x-2*x^(1/8)-5,x=2);

1 2 2( )/1 8

[> evalf(%);

-1.181015466

Cng c th kt hp hai lnh trn vo mt lnh:

> evalf(eval(3*x-2*x^(1/8)-5,x=2));

-1.181015466

Tng t, ta c th tnh cc gi tr ca hm s cho ti im 3x nh lnh eval

v lnh evalf(%):[> eval(3*x-2*x^(1/8)-5,x=3);

4 2 3( )/1 8

[> evalf(%);

1.705594620

Tuy nhin, ta cn c th dng lnh subs (thay th) thay 3x vo biu thc

8( ) 3 2 5 f x x x c gi tr (3)f :

[> subs(x=3,3*x-2*x^(1/8)-5);

4 2 3( )/1 8

[> evalf(%);

1.705594620

C th kt hp hai lnh trn vo mt lnh: [> evalf(subs(x=3,3*x-2*x^(1/8)-5));

1.705594620

Tt nhin, ta cng c th dng my tnh khoa hc tnh cc gi tr trn. Th d, s

dng Casio fx-570 ES tnh gi tr ca hm s 83 2 5 x x ti 2x nh sau:

Khai bo hm s 83 2 5 x x :


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3 ALPHA X 2 ALPHA X x 1 8 5 (3.2)

Tnh gi tr hm s:

Bm phm CALC (Calculate-tnh)My hi: ?X

Khai bo 2x : 2

My tr li (p s hin trn mn hnh), mn hnh c dng:

1

83 2 5

1,181015465

X X .

tnh gi tr ca hm s 83 2 5 x x ti 3x ta khng cn khai bo li cng

thc hm s m ch cn s dng phm CALC :

Bm phm CALC (Calculate-tnh)

My hi: ?X

Khai bo 3x : 3

My tr li (p s hin trn mn hnh):1

83 2 5

1, 705594619

X X .

Vy phng trnh 83 2 5 0 x x c mt nghim duy nht trong khong

( ; ) (2;3)a b .

Phng php chia i khong cha nghim

a gi tr 2x vo nh A : 2 SHIFT STO A a gi tr 3x vo nh B : 3 SHIFT STO B

Tnh2 3 5

2 2 2

a bc v a vo nh C :



tr ca hm s ti 52c :


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Bm phm CALC (Calculate-tnh)

My hi: ?X

Khai bo5

2

c (ang trong C ):

ALPHA C (0.2572932161)

Nh vy, ta c (2) 1,181015465 0 f v5

( ) 0.2572932161 02

f . Chng

t nghim ca phng trnh nm trong khong 1 1( ; ) ( , ) (2;3) a b a c . B c

gii phng (gi tr c 3x trong nh B khng cn dng na).

Li chia i khong 1 1( ; ) ( , ) (2;3) a b a c v gi1 1

1 9 2,252 4 a bc vo

B :



tr ca hm s ti 19

2,254

c :

Bm phm CALC (Calculate-tnh):

My hi: ?X

Khai bo 19

4c (ang trong B ): CALC

ALPHA B ( 0.4633638394 )

Nh vy, ta c1

9( ) ( ) 0,4633638394 0

4 f c f v

1

5( ) ( ) 0, 2572932161 0

2 f b f . Chng t nghim ca phng trnh nm trong

khong 2 2 1 19 5

( ; ) ( , ) ( ; )4 2

a b a c . nh A c gii phng (gi tr c 2a

trong nh A khng cn dng na).


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( ; ) ( , ) ( ; )4 2

a b a c v gi

2 22

9 5194 2

2,3752 2 8

a b

c vo A :

ALPHA B ALPHA C 2 SHIFT STO A


tr ca hm s ti 219

2,3758

c :


My hi: ?X

Khai bo 219

2,3758

c (ang trong A ):

ALPHA A ( 0.103373306 )

Nh vy, ta c 219

( ) ( ) (2,375) 0.103373306 08

f c f f v

1

5( ) ( ) 0.2572932161 02 f b f . Chng t nghim ca phng trnh nm trong

khong 3 3 2 15

( ; ) ( ; ) (2,375; )2

a b c b . nh B c gii phng (gi tr c 19

4c

trong nh B khng cn dng na).

Li chia i khong 3 3 2 15

( ; ) ( ; ) (2,375; )2

a b c b v gi

2 23

19 5 398 2 2,43752 2 16

c bc vo B :



tr ca hm s ti 339

2,437516

c :



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My hi: ?X

Khai bo 339

2,437516

c (ang trong B ):

ALPHA B ( 0,076879549)

Nh vy, ta c 339

( ) ( ) 0, 076879549 016

f c f v

2

19( ) ( ) (2,375) 0.103373306 0

8 f c f f . Chng t nghim ca phng

trnh nm trong khong 4 4 2 319 39

( ; ) ( ; ) ( ; )8 16

a b c c . nh C c gii phng (gi

tr c 52

c trong nh C khng cn dng na).


( ; ) ( ; ) ( ; )8 16

a b c c v gi

4 44

19 39778 16 2,40625

2 2 32

a b

c vo C :



tr ca hm s ti 477

2,4062532

c :


My hi: ?X

Khai bo 477

2,4062532

c (ang trong C

):

ALPHA C ( 0,013267467 )

Nh vy, ta c 477

( ) ( ) 0,013267467 032

f c f v

2

19( ) ( ) (2,375) 0.103373306 0

8 f c f f . Chng t nghim ca phng


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trnh nm trong khong4 4 2 3

19 39( ; ) ( ; ) ( ; )

8 16a b c c . nh C c gii phng (gi

tr c5

2c trong nh C khng cn dng na).

Ta c bng kt qu sau:


a=2; b=3 -1,1 A 1,7 B ( ; )a b

2,5c C 0.2572932161 -1,1 A ( ; )a c

1 2,25c B -0.4633638394 0,2 C 1( ; )c c

2 2,2375c A -0,103373306 0,2 C 2( ; )c c

3 2,4375c B 0,076879549 -0,1 A 2 3( ; )c c

4 2,40625c C -0,013267467 0,07 B 3 4( ; )c c

5 2,421875c A 0,031800956 0,01 C 4 5( ; )c c

6 2,42140625c B 0,00926546534 0,01 C 4 6( ; )c c

7 2,41015625c A 0,0020013218 0,009 B 7 6( ; )c c

8 2,412109375c C 0,00363199172 0,002 A 7 8( ; )c c

9 2, 411132813c B 0,00081531494 0,002 A 7 9( ; )c c

10 2, 410644531c

C 0,00059300843 0,0008 B 10 9( ; )c c

11 2,410888672c

A 0,00011115202 0,0005 C 10 11( ; )c c

12 2,410766602c B 0,00024092851 0,0001 A 12 11( ; )c c

13 2,410827637c C 0,000006488831 0,0001 A 13 11( ; )c c

14 2,410858154c B 0,00002313185 0,000006 C 13 14( ; )c c

15 2,410842896c A 0,0002087824 0,00002 B 15 14( ; )c c

16 2,410850525c C 0,00000112682 0,0002 A 15 16( ; )c c

17 2, 41084671c B 0,0000098757 0,000001 C 17 16( ; )c c

18 2,410848618c A 0,00000437445 0,000001 C 18 17( ; )c c

19 2,410849571c B 0,00000162379 0,000001 C 19 17( ; )c c


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20 2,410850048c A 0,00000024847 0,000001 C 20 17( ; )c c

21 2,410850287c B 0,00000043919 0,0000002 A 20 21( ; )c c

22 2, 410850167c C 0,00000009536 0,0000002 A 20 22( ; )c c

23 2,410850108c B 0,00000007658 0,00000009 C 23 22( ; )c c

24 2,410850138c A 0,0000000094 0,00000007 B 23 24( ; )c c

25 2,410850123c C 0,00000003358 0,000000009 A 25 24( ; )c c

26 2,41085013c B 0,00000001209 0,000000009 A 26 24( ; )c c

27 2, 410850134c C 0,00000000132 0,000000009 A 27 24( ; )c c

28 2,410850136c B

0,00000000404 0,000000001 C 27 28( ; )c c

29 2,410850135c A 0,00000000135 0,000000001 C 27 29( ; )c c

30 2,4108501345c B 0,00000000000 0,00000000132 C 27 30( ; )c c

Nghim gn ng l30 2,4108501345 x c v ( ) 0,00000000000f x (chnh

xc n 10 ch s).

Phng php lp

Phng trnh 83 2 5 0 x x tng ng vi82 5

( )3

xx x .

Dy lp c dng8

1

2 5

3

nn

xx .

Chn gi tr ban u 0 2,5x : 2.5

Khai bo 0( )x : ( 2 Ans x 1 8 5 ) 3


Tnh cc gi tr tip theo: Bm phm

Ta c dy xp x:

2.5; 2.414235595; 2.410980682; 2.410855171; 2.410850329; 2.410850142;

2.410850135; 2.410850134; 2.410850134;

Sau 9 bc ta i n nghim gn ng n 10 ch s.


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Phng php dy cung

Theo cng thc dy cung 1( )( 3)

( ) (3)

n n

n n

n

f x xx x

f x f , 0 2x ta c

8

18

(3 2 5)( 3)

3 2 5 (3)

n n n

n n

n n

x x xx x

x x f , 0 2x .

Khai bo 0 2x : 2

Khai bo cng thc 8( ) 3 2 5 f x x x :

3 ALPHA X 2 ALPHA X x 1 8 5

Tnh gi tr ca (3)f : Bm phm CALC

My hi: X?

Khai bo: 3 (1.705594619)

Gi vo A : SHIFT STO A

Khai bo gi tr ban u: 2

Khai bo cng thc8

18

(3 2 5)( 3)

3 2 5 (3)

n n n

n n

n n

x x xx x x x f

:

Ans ( ( 3 Ans 2 Ans ^x 1 8 5 ) ( Ans 3 ) ( 3 Ans

2 Ans ^x 1 8 5 ALPHA A ) )

Tnh cc gi tr 1nx bng cch bm lin tip phm .Ta c dy cc gi tr:

2,409135779; 2,410843757; 2,410850111; 2,410850134; 2,410850134.

Phng php tip tuyn


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Theo cng thc lp trong phng php tip tuyn 1( )

'( )

nn n

n

f xx x

f x, 0 2x ta c

8

1

78

(3 2 5)

134

n n

n n

n

x xx x

x

.

Khai bo gi tr ban u 0 2x : 2

Khai bo cng thc8

1

78

(3 2 5)

13

4

n nn n

n

x xx x

x

:

Ans ( 3 Ans 2 Ans ^x 1 8 5 ) ( 3 1 ( 4 Ans ^x

7 8 ) )


2,412410874; 2,410850152; 2,410850134; 2,410850134.

Sau ba ln lp ta in p s.

Kt lun: tm c nghim gn ng n 10 ch s thp phn theo phng

php chia i cn 30 bc lp, v thao tc phc tp trong mi bc. tm nghim

gn ng n 10 ch s thp phn theo phng php lp ch cn 9 bc. tm

nghim gn ng n 10 ch s thp phn theo phng php dy cung cn 4 bc

lp. tm nghim gn ng n 10 ch s thp phn theo phng php tip tuyn

ch cn 3 bc lp. Thao tc lp theo ba phng php sau rt n gin: sau khi khai

bo gi tr ban u v cng thc lp, ta ch cn lin tip bm phm cho n khic gi tr khng i (im bt ng), l nghim gn ng n 10 ch s thp

phn.

Bi 3. Gii phng trnh hn hp i s - lng gic 3 cos 0 x x .

t 3( ) cos y f x x x. Ta d dng tnh c (0) 1 f v

(1) 0,4596976941 0 f . Do , phng trnh 3 cos 0 x x c t nht mt


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nghim trong khong (0;1) . Nghim ca phng trnh 3 cos 0 x x c th c

coi nh l giao ca hai th hm s 3y x v cosy x . Ta c th nhMaple

v hai th trong khong (0;1) nh lnh plot (v th) trn cng mt h ta :

[> plot([x^3,cos(x)],x=0..1);

Nhn vo thta thy, nghim ca phng trnh nm trong khong (0,8;0,9).

s dng Casio fx-570 EStnh ton vi radian, trc tin ta phi vo MODE s

dng radian bng cch bm phm: SHIFT MODE 4 .

Phng php chia i

Khai bo hm 3( ) cos y f x x x: ALPHA X ^x 3 cos ALPHA X

Thc hin cc thao tc hon ton tng t nh trong Bi 1 v Bi 2, ta i n bng

kt qu di y.


a=0; b=1 -1 A 0,45969 B ( ; )a b

0,5c C -0,752582561 0,4 B ( ; )c b

10,75c A -0,3098138689 0,4 B 1( ; )c b

2 0,875c C 0,02892501684 -0,3 A 1 2( ; )c c

3 0,8125c B -0,1513086091 0,02 C 3 2( ; )c c


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4 0,84375c A -0,06398823736 0,02 C 4 2( ; )c c

5 0,859375c B -0.01824073474 0,02 C 5 2( ; )c c

6 0,8671875c A 0,05163610306 -0,01 B 6 2( ; )c c

7 0,86328125c C -0,006583038793 0,05 A 7 6( ; )c c

8 0,86523437c B -0,000720852907 0,05 A 8 6( ; )c c

9 0,866210937c C 0,002218591595 -0,0007 B 8 9( ; )c c

10 0,865722656c

A 0,000748172873 -0,0007 B 8 10( ; )c c

11 0,865478515c

C 0,000013485903 -0,0007 B 8 11( ; )c c

12 0,86535644c A -0,00035372701 0,00001 C 12 11( ; )c c

13 0,86541748c B -0,000170132846 0,00001 C 13 11( ; )c c

14 0,86544799c A -0,000078327132 0,00001 C 14 11( ; )c c

15 0,86546325c B -0,000032422234 0,00001 C 15 11( ; )c c

16 0,86547088c A -0,000009469276 0,00001 C 16 11( ; )c c

17 0,8654747c B 0,000002007338 -0,000009 A 16 17( ; )c c

18 0,86547279c C -0,00000373098 0,000002 B 18 17( ; )c c

19 0,86547374c A -0,000000861816

0,000002B

19 17( ; )c c

20 0,86547422c C 0,00000057276 -0,0000008 A 19 20( ; )c c

21 0,86547398c B -0,000000144521 0,0000005 C 20 21( ; )c c

22 0,86547410c A 0,000000214128 -0,0000001 B 21 22( ; )c c

23 0,86547404c C 0,000000034804 -0,0000001 B 22 23( ; )c c

24 0,86547401c A -0,000000054851 0,00000003 C 24 23( ; )c c


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25 0,86547402c B -0,000000010024 0,00000003 C 25 24( ; )c c

26 0,86547403c A 0,00000001239 -0,00000001 B 25 26( ; )c c

27 0,86547403c C 0,00000001183 -0,00000001 B 25 27( ; )c c

28 0,865474031c

A -0,000000004412 0,00000001 C 28 27( ; )c c

29 0,86547403c B -0,000000001615 0,00000001 C 29 27( ; )c c

30 0,86547403c A -0,000000000216 0,00000001 C 30 27( ; )c c

Nghim gn ng l30 0,865474033 x c radian v ( ) 0,000000000216 f x

Phng php lp

Phng trnh 3 cos 0 x x tng ng vi 3 cos ( ) x x x .

Dy lp c dng 31

cos

n n

x x .

Trc tin ta phi vo MODE s dng radian bng cch bm phm:

SHIFT MODE 4

Chn gi tr ban u 04

x

: SHIFT 4

Khai bo ( )nx : SHIFT3 cos Ans )


Tnh cc gi tr tip theo: Bm phm CALC

Ta c dy xp x:

0.8908987181; 0.8566779192; 0.8684331147; 0.8644689718; 0.8658143009;

0.8653587072; 0.8655131056; 0.8654607937; 0.865478519; 0.8654725131;

0.8654745481; 0.8654738586; 0.8654740922; 0.8654740131; 0.8654740399;

0.8654740308; 0.8654740339; 0.8654740328; 0.8654740332; 0.8654740331.


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Nh vy, sau 20 ln bm phm CALC , ta i n nghim xp x 0,8654740331x .

Phng php dy cung

Hm s3

( ) cos y f x x x c2

( ) 3 sin 0 y f x x x v

( ) 6 cos 0 y f x x x vi mi ;14

x

. Theo cng thc dy cung

1

( )( )

( ) ( )n n

n n

n

f x x bx x

f x f b

, 0

4x

ta c

3

1 3

( cos )( 1)

cos (1)

n n nn n

n n

x x xx x

x x f

, 0

4x

.

Khai bo 04

x

: SHIFT 4

Khai bo cng thc

3

1 3

( cos )( 1)

cos (1)n n n

n n

n n

x x xx x

x x f

:

Ans ( Ans SHIFT 3x cos Ans ) ) ( Ans 1 ) ( Ans SHIFT 3x

cos Ans ) ALPHA A )


0.8554102741; 0.8642643557; 0.8653292731; 0.865456721; 0.8654719629;

0.86547376855; 0.8654740035; 0.8654740296; 0.8654740327; 0.8655740331;

0.8655740331; 0.8654740331.

Sau 10 ln bm phm , ta i n nghim xp x 0,8654740331x .

Phng php tip tuyn

Theo cng thc tip tuyn 1( )

'( )n

n n

n

f xx x

f x ta c

3

1 2

cos

3 sin

n nn n

n n

x xx x

x x

.

Ta chn gi tr ban u 04

x

.

Khai bo gi tr ban u: SHIFT 4


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Khai bo cng thc3

1 2

cos

3 sin

n nn n

n n

x xx x

x x

:

Ans ( Ans SHIFT 3x cos Ans ) ) ( 3 Ans 2x sin Ans )


0.8724441024; 0.8655207565; 0.8654740352; 0.8654740331; 0.8654740331;

0.8654740331; 0.8654740331.

Sau bn ln bm phm ta i n p s 0,8654740331x .

Kt lun: C bn phng php u cho p s 0,8654740331x .


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CHNG II

GII GN NG NGHIM CA BI TON

CAUCHY CHO PHNG TRNH VI PHN THNG

TRN MY TNH IN T

1. PHNG PHP GII GN NG BI TON CAUCHY

CA PHNG TRNH VI PHN THNG

S lng phng trnh vi phn gii c bng cu phng (nghim c

biu din thng qua cc php tnh v cc hm c bn) l rt t. Thm ch khi c cng

thc nghim ca phng trnh vi phn, do s phc tp ca cng thc, cha chc ta

c th kho st c cc tnh cht ca nghim. V vy, nghin cu phng

trnh vi phn, ngi ta thng khng gii trc tip phng trnh, m s dng hai

phng php: phng php nh tnh - nghin cu cc tnh cht ca nghim (tn ti

v duy nht, tnh b chn, nghim tun hon, tnh n nh,...) thng qua v phi ca

phng trnh v phng php gii gn ng - tm nghim di dng xp x. Hai

phng php ny h tr v b sung nhau.

gii gn ng phng trnh vi phn, ngi ta thng dng hai phng

php: phng php gii tch - tm nghim xp x di dng mt dy cc hm s

lin tc hi t u ti nghim (l mt hm kh vi tha mn phng trnh vi phn v

iu kin u) trn mt khong ,a b no v phng php s - tm nghim xp

x di dng cc gi tr s ca nghim ti mt s im trn on ,a b v kt qu

c cho di dng bng, nh phng php ng gp khc Euler, phng php

Runge-Kutta, phng php a bc,...

Trong chng ny, chng ti trnh by phng php ng gp khc Euler

v phng php Runge-Kutta gii bi ton Cauchy cho phng trnh vi phn

thng. Nhm minh ha cho kh nng s dng my tnh in t gii phng

trnh vi phn, tr gip cho hc tpca sinh vin khi hc mn hc ny, chng ti th


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hin phng php Euler v phng php Runge-Kutta trn my tnh in t khoa

hc Casio fx-570 ESv trn chng trnhMaple.

1.1. Bi ton Cauchy ca phng trnh vi phn cp mt

Bi ton: Tm nghim ( ) y y x ca phng trnh

( , )y f x y (1.1)

tha mn iu kin ban u

0 0( ) y x y (1.2)

(tc l tm mt hm kh vi ( ) y y x sao cho 0 0( ) y x y v ( ) ( , ( ))y x f x y x

trong mt ln cn no ca 0x ).D dng chng minh c rng, phng trnh vi phn (1.1) tng ng vi

phng trnh tch phn

0

0( ) ( , ( )) x

x

y x y f s y s ds , (1.3)

theo ngha, mi nghim ca phng trnh (1.1) l nghim lin tc ca phng trnh

(1.3) v ngc li (xem, th d, [7], [8]).

Ta c nh l sau y v s tn ti v duy nht nghim ca phng trnh vi

phn (1.1) tha mn iu kin ban u (1.2).

nh l 1 (Picard-Lindelof) Gi s:

1. Hm s ( , ) f x y l lin tc theo c hai bin trong min ng, gii ni D :0 0

0 0

x a x x a

y b y y b

(Do D l mt min ng, gii ni nn t gi thit ny suy ra tn ti mt s dng

M sao cho ( , ) f x y M vi mi ( , ) x y D).

2. Hm hai bin ( , ) f x y tha mn iu kin Lipschitz theo bin y trongD u theo x, tc l tn ti mt hng s dngL sao cho

1 2 1 2( , ) ( , ) f x y f x y L y y

vi mi 1 2( , ) , ( , ) x y D x y D .


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Khi y tn ti duy nht mt nghim ( ) y y x ca phng trnh (1.1) trong khong

0 0 x H x x H , trong min( , )b

H aM

, tha mn iu kin ban u

(1.2).

Chng minh nh l ny c th xem trong [7], [8].

Nhn xt 1. Nu hm ( , ) f x y v( , )

f x y

y l lin tc trn hnh ch nht D th

( , )

f x yL

y vi mi ( , ) x y D . Hn na, theo nh l gi tr trung bnh, vi

mi cp 1( , )x y v 2( , )x y tn ti im*

( , ) x y D sao cho*

1 2 1 2 1 2

( , )( , ) ( , )

f x yf x y f x y y y L y y

y

.

Chng t ( , ) f x y tha mn iu kin Lipschitz.

Nhn xt 2. nh l Picard-Lindelof c tnh cht a phng(tn ti nghim trong

mt ln cn ca 0x l khong 0 0, x H x H ).

Vi mt s lp hm ( , )f x y , ta c th chng minh s tn ti nghim ton ccca

phng trnh (1.1). Th d, s tn ti nghim ton cc ca phng trnh tuyn tnh

' ( ) ( ) y P x y Q x trn ton on ,a b l h qu ca nh l sau.

nh l 2 Gi s ( , ) f x y l hm lin tc theo hai bin v tha mn iu kin

Lipschitz

1 2 1 2( , ) ( , ) f x y f x y L y y

trong min ch nht v hn [ , ] ( , ) a b v 0 0( , )x y l mt im trong ca

min . Khi y bi ton gi tr ban u (1.1)-(1.2) c nghim duy nhttrn on

,a b .

1.2. Bi ton Cauchy ca h phng trnh vi phn cp mt

Cc nh l tn ti nghim ca bi ton Cauchy cho phng trnh vi phn

cp mt c th pht biu tng t cho h n phng trnh vi phn cp mt


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1 1 1 2

2 2 1 2

1 2

( ) ( , , ,... )

( ) ( , , ,... )

.....................

( ) ( , , ,... )

n

n

n n n

y x f x y y y

y x f x y y y

y x f x y y y

(1.4)

H (1.4) c th vit di dng ( , )y f x y , trong 1 2( , ,..., )T

n y y y y

v 1 2( , ,..., )T

n f f f f l nhng vect n chiu. Di y chng ta pht biu mt

dng tng t ca nh l 1 cho h phng trnh vi phn cp mt.

nh l 1 Cho h phng trnh vi phn (1.4), trong hm fxc nh v lin

tc trn tp m D R Rn v thomn iu kin Lipschitz theo y u theo x

trn D :

1 2 1 2( , ) ( , ) f x y f x y K y y vi mi 1 2( , ) , ( , ) x y D x y D .

Khi y vi mi 0 0( , ) x y D tm c mt s 0d sao cho trn khong

0 0, x d x d tn ti v duy nht nghim ca phng trnh vi phn (1.4) tho

mn iu kin 0 0( ) y x y .

Chng minh: Xem [7], [8].

2.PHNG PHP EULER

2.1. Phng php Euler

Khi ( , )f x y l lin tc nhng khng l Lipschitz, nghim ca bi tonCauchy vn tn ti, nhng c th khng duy nht.

Th d 2.1. Xt bi ton Cauchy 3dy

ydx

, (0) 0y .

Bi ton ny c v s nghim: ngoi nghim ( ) 0y x cn c mt h nghim

( ) 0y x khi x c v

3

22( )( )

3

x cy x

khi x c , trong 0c bt k.


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Th d 2.2. Xt bi ton Cauchy 3(sin2 )dy

x ydx

, (0) 0y .

Bi ton ny c ba nghim ( ) 0y x ; 32 2

( ) sin3 3

y x x.

Nhn xt rng cc hm s 3( , ) f x y y v 3( , ) sin 2 . f x y x y khng l

Lipschitz theo y trong ln cn im (0,0).

Cc th d trn cho thy, nu hm ( , )f x y khng l Lipschitz th nh l v tn ti

duy nht nghim (nh l Picard-Lindenlof) khng cn ng. Tuy nhin ta vn c

nh l tn ti nghim sau y:

nh l 1 (Cauchy-Peano) Gi s ( , ) f x y l hm lin tc theo hai bin trong min

ng, gii ni D :

0 0

0 0

x a x x a

y b y y b

(Do D l mt min ng, gii ni nn t gi thit ny suy ra tn ti mt s dng

M sao cho ( , ) f x y M vi mi ( , ) x y D).

Khi y tn ti t nht mt nghim ( )y y x ca phng trnh (1.1) trong khong

HxxHx 00 , trong min( , )b

H aM

, tha mn iu kin ban u (1.2).

Chng minh. Nhm mc ch th hin phng php tnh gn ng nghim, di

y chng ti trnh by chng minh nh l Cauchy-Peano bng phng php da

trn khi nim ng gp khc Euler.

T 0 0( , )x y k hai ng thng vi h s gc l M v M .

Cc ng thng ny c phng trnh l

0 0( )y y M x x

v

0 0( )y y M x x

Chng ct hai ng thng


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song song 0 y y b

v 0 y y bti cc im c honh l

0

bx

Mv

0

bx

M(xem hnh v).

Gi h l di bc (stepsize)ca bin c lp x ( h c th dng hoc m, khi h

dng th nghim c xy dng v bn phi ca im 0x v ngc li, khi h m

th nghim c xy dng v bn tri ca 0x . Di y ta coi 0h , trng hp

0h c th chng minh tng t).

Ta tm gi tr gn ng ca nghim ti cc nt gi tr ca ix vi 1,2,...,i n

( 1

Hn

h):

1 0

2 1 0

0

;

2 ;

.........................

.

n

x x h

x x h x h

x x nh

dc ca ung tip tuyn trong th ca hm s ( ) y f x ti cc gi tr cax c cho bi cng thc : ( ) ( , ( ))y x f x y x .

Chng hn, ti 0x x dc ca ng tip tuyn l

0 0 0 0 0'( ) ( , ( )) ( , ) y x f x y x f x y .

Khi , phng trnh ng tip tuyn vi ng cong y y( x ) i qua im

0 0( , )x y c dng:

0 0 0 0 0 0( )( ) ( , )( )y y y x x x f x y x x

hay

0 0 0 0( , )( )y f x y x x y . (2.1)

Dng ng ny tm gi tr xp x ca nghim y y( x )ti 1x x (k hiu l

1y ), ta c

1 0 0 1 0 0 0 0 0( , )( ) ( , )y f x y x x y hf x y y .


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Tip , dng im 1 1,x y tnh gi tr ca y ti 2x x nh phng trnh

ng tip tuyn

1 1 1 1( , )( )y y f x y x x

hay

1 1 1 1( , )( )y f x y x x y .

Tip tc qu trnh ny, ti nx x ta tnh c:

1 ( , )n n n n y hf x y y . (2.2)

Ni cc im ( , )i ix y vi 1,...,i n , ta c mt ng gp khc ( )hy x , c gi

l ng gp khc Euler (xem hnh v). Vi mi h chn trc, ng gp khcEuler cho ta mt xp x ca ng cong nghim, n trng vi tip tuyn ca cc

ng cong nghim (khc nhau) ti mi im ( , )i ix y . Cng thc hin ca ng

gp khc ny l

0 0( ) h y t x ;

1 1 1 1( ) ( ) ( , ( ))( ) h h k k h k k y t y t f t y t t t , 1 k kt t t , 1,2,...,k n .

Khi h thay i ta c mt h cc ng gp khc Euler. Cho h tin ti 0 ta cmt h cc hm s (cc ng gp khc) gii ni u v lin tc ng bc trn

khong 0 0 x x x H .

Tht vy, th ca mi ng gp khc u nm trong tam gic Tgii hn bi cc

ng thng 0 0( ) y y M x x v 0 x x H (xem hnh v) nn chng b chn

u.

Do

''

'

'

( '') ( ') ( ) '' ' x

h h h

x

y x y x y s ds M x x vi mi h v

0 0', '' x x x x H nn h hm ( )hy x lin tc ng bc.

Theo b Arzel, t ( )hy x c th trch ra mt dy cc hm hi t u ti mt

hm lin tc ( ) y y x trn khong . Hm ( )y y x chnh l nghim ca phng

trnh vi phn (1.1) tha mn iu kin ban u (1.2).

nh l chng minh xong.


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Nhn xt. Phng php ng gp khc Euler ni chung khng cho ta tt c cc

nghim ca phng trnh (21)-(2.2). Th d, mi h ng gp khc Euler p dng

cho bi ton Cauchy 3dy

y

dx

, (0) 0y u hi t ti nghim 0y , v vy ta

khng tm c tt c cc nghim ca phng trnh ny.

Di y chng ta pht biu nh l Cauchy-Peano cho h phng trnh vi

phn (phng trnh vi phn vect).

nh l 1. Gi s

1.Hm : n f D R lin tc trn tp D,

0 0 0( , ) : ,D t x t t t a x x b .

2. M v l nhng s sao cho

( , ) , , min( , )b

f t x M t x D aM

.

Khi y phng trnh vi phn (1.1) c t nht mt nghim tho mn iu kin ban

u (1.2) trn on 0 0,t t .

Phng php Euler tuy cho xp x nghim ca bi ton Cauchy tng i

th, nhng n gin, d tnh ton, v khi h cng nh th xp x nghim cng tt,

iu ny c th hin r rng qua cc tnh ton c th trong cc v d trnh by

trong 4. Vi tc ca my tnh hin nay, ta cng c th s dng phng php

Euler trong cc bi ton thc t, c bit l minh ha phng php s trong

ging dy mn phng trnh vi phn.

Nhn xt rng c nhiu cch i n cng thc (2.2), l c s ca phng php

ng gp khc Euler (xem [8]).

2.1. Phng php Euler ci tin

Phng php Euler c tc hi t chm, n c th ci tin nh dc

trung bnh trong mi khong nhm tng tc chnh xc nh sau.

Trc tin, ta s dng tip tuyn ca ng cong qua im 0 0( , )x y :


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0 0 0 0( , )( ) y f x y x x y

tm gi tr xp x ca nghim ( ) y y x ti 1x x ( v k hiu l*

1y ) . Khi (

vi 1 0h x x ):

*1 0 0 0( , ) . y hf x y y

Nh vy, vi phng trnh ( , ) , y f x y ta tm c dc xp x ca ng tip

tuyn ti 1x x l*

1 1 1'( ) ( , )y x f x y . Gi tr trung bnh ca hai dc 0 0( , ) f x y

v*

1 1( , ) f x y ca tip tuyn ti hai im 0 0( , )x y v*

1 1( , )x y l

*0 0 1 1( , ) ( , )

2

f x y f x yv phng trnh ca ng thng i qua

0 0( , )x y vi dc*

0 0 1 1( , ) ( , )

2

f x y f x yl

*0 0 1 1

0 0

( , ) ( , )( )

2

f x y f x y y x x y.

Khi , ti 1x x gi tr gn ng ca ( )y y x c tnh theo cng thc trn l:

* *

0 0 1 1 0 0 1 11 1 0 0 0( , ) ( , ) ( , ) ( , )( )2 2

f x y f x y f x y f x y y x x y h y .

Tip tc vi cch lm ny, gi tr xp x trong mi bc ca phng php Euler ci

tin ph thuc vo hai tnh ton:

^1 1 1

^1 1

1

. ( , )

( , ) ( , )

2

n

n

n n n

n n n

n n

y h f x y y

f x y f x y y h y

(2.6)

T cng thc trn ta suy ra:

1 1

1 1. ( , ) . ( , . ( , ))

2 2n n n n n n n ny y h f x y h f x y h f x y . (2.7)

S dng cng thc ny, ta tm c nghim xp x tt hn so vi phng php

Euler. iu ny s c thy r trong cc v d 4.

3.PHNG PHP RUNGE - KUTTA


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Mt trong nhng phng php s gii phng trnh vi phn hiu qu, thng

dng v ph bin nht trong cc bi ton k thut l phng php Runge-Kutta. C

th suy ra cc cng thc trong phng php Runge-Kutta t cng thc xp x tch

phn. Trong mc ny, khc vi cc ti liu [1][6], chng ti trnh by phng phpRunge-Kutta xut pht t qui tc cu phng theo [7]. Cch lm ny cho php hiu

cc phng php s gii phng trnh vi phn mt cch nht qun hn (xem [7]).

lm c iu ny, trc tin chng ta nhc li quy tc cu phng c bn

(basic quadrature rules).

3.1. Quy tc cu phng c bn

Ni dung c bn ca quy tc cu phng l: tnh tch phn ( )b

a

f t dt ta

thay ( )f t bi mt a thc ni suy (interpolating polynomial). Tch phn ca hm

( )f t trn on ,a b c xp x bi tch phn ca hm a thc (tnh c chnh

xc).

Gi s ta c s im ni suy khc nhau 1 2, ,..., sc c c trong khong ,a b . a

thc ni suy Lagrange bc nh hn s c dng (xem [1]):

1

( ) ( ) ( )

s

j j

j

t f c L t ,

trong

1,

( )( )

( )

si

j

i i j j i

t cL t

c c.

Khi y

1

( ) ( )

b s

j j

ja

f t dt f c .

Cc trng s j c tnh theo cng thc

( ) . b

j j

a

L t dt

Nu 1s th a thc ni suy 1( ) ( )t f c v ta c:


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1( ) ( ) ( ). b

a

f t dt b a f c

Ta ni chnh xc(precision) ca quy tc cu phng l p nu quy tc ny chnh

xc cho mi a thc bc nh hn p , tc l vi mi a thc ( )kP t bc nh hn p

ta c:

1

( ) ( ).

b s

k j j

ja

P t dt f c

Nu 0( ) b a h th sai s trong quy tc cu phng ca chnh xc p l

10( ).ph

Ta xt mt s trng hp c bit.

Nu chn 1s v 1 c a th ta c

cng thc xp x tch phn bi

din tch hnh ch nht ABCD:

( ) ( ) ( ).

b

af t dt b a f a (3.1)

Nu ( )y x l nghim ca phng trnh vi phn (1.1) - (1.2) th:

( ) ( ) ( , ( )) .

x h

x

y x h y x f s x s ds (3.2)

Kt hp vi cng thc (3.1) ta i n cng thc: ( ) ( ) . ( , ( )). y x h y x h f x y x

T y ta li c cng thc Euler tin bit trong 2:

1 . ( , ).n n n ny y h f x y

Nu chn s=1 v c=b th ta c cng thc xp x tch phn bi din tch hnh ch

nht ABEF:


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( ) ( ) ( ). b

a

f t dt b a f b

T y ta c:

( ) ( ) . ( , ( )) y x h y x h f x h y x h

Suy ra cng thc Euler li:

1 1 1. ( , ). n n n ny y h f x y

Hai phng php Euler tin v li l nhng phng php Runge-Kutta bc nht (c

xp x bc nht).

Nu chn 2s v 1 2, c a c b th 1( ) ( )

t b

L t a b v 2( ) .( )

t a

L t b a

Suy ra

2

1 1

1 ( )( )

( ) ( ) 2 2

bb b

a a a

t b t b b a L t dt dt

a b a b

v

2

2 2

1 ( )

( ) .( ) ( ) 2 2

ab b

a a b

t a t a b a

L t dt dt b a b a

Chng t

( ) [ ( ) ( )]2

b

a

b a f t dt f a f b .

Nh vy nu xp x tch phn ( , ( ))

x h

x

f t y t dt bi cng thc trn (bi din tch

hnh thang ABED)th ta c:

( , ( )) [ ( , ( )) ( , ( ))].2

x h

x

hf t y t dt f x h y x h f x y x

T y ta c cng thc hnh thang:

1 1 1[ ( , ) ( , )].2

n n n n n nh

y y f x y f x y


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Nu chn 1s v 12

a bc th ta c cng thc xp x tch phn bi din tch

hnh ch nht ABMN:

( , ( )) . ( , ( )).2 2

x h

x

h hf t y t dt h f x y x

T y ta c cng thc tnh gi tr

xp x nghim ca phng trnh vi phn:

( ) ( ) . ( , ( ))2 2

h h

y x h y x h f x y x .

T cng thc trn ta c

1 [ ( , ( )].2 2

n n n nh h

y y h f x y x

Cng thc ny c gi lphng php im gia (midpoind method).

Phng php im gia v phng php hnh thang l hai phng php n, chng

c chnh xc 2p .

Nu chn 3s v 1 2 3, ,2

a bc a c c b th, t h b a , ta c:

1 2

( )( )22( ) ( )( ),

2( )( )

2

a bt t b

a b L t t t b

a b ha a b

2 2

( )( ) 4( ) ( )( ),

( )( )2 2

t a t b L t t a t b

a b a b ha b

3 2

( )( )22( ) ( )( ).

2( )( )

2

a bt a ta b

L t t a t a b h

b a b

Suy ra:


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1 1 2 2

3 22

2 2

3

2

2 2( ) ( )( ) ( )( )

2 2

2 2 ( ) ( )[( ) ( )( )] [ ( ) ]

2 3 2 2

2 ( ).

12 6

b b b

a a a

bb

a a

a b a b L t dt t t b dt t b t b dt

h h

a b t b a b t bt b t b dt

h h

b a h

h

v

2 2 2 2

3 2

2

4 4( ) ( )( ) ( )( ( ))

4 ( ) ( ) 4[ ( )] .3 2 6

b b b

a a a

b

a

L t dt t a t b dt t a t a a b dt h h

t a t a ha bh

Do tnh cht i xng (hoc tnh trc tip), ta c

3 16

h

.

T cc tnh ton trn ta i n cng thc Simpson:

( ) [ ( ) 4 ( ) ( )]6 2

b

a

h a b f t dt f a f f b .

Suy ra cng thc xp x nghim ca phng trnh vi phn

( ) ( ) [ ( , ( )) 4 ( , ( )) ( , ( ))]6 2 2

h h h

y x h y x f x y x f x y x f x h y x h

v cng thc sai phn

1 1 1[ ( , ) 4 ( , ( )) ( , )]6 2 2

n n n n n n n nh h hy y f x y f x y x f x y .

y l cng thc n ca phng php Runge- Kutta kinh in cp bn (classical

fourth-order Runge-Kutta method).

2. Dn ti Phng php Runge - Kutta

Tnh 1ny theo cng thc n i hi ti mi bc phi gii xp x mt

phng trnh phi tuyn, iu ny khng n gin v c kh nng lm tng sai s,nn ta c gng xy dng cc cng thc Runge-Kutta hin t cng thc hnh thang


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n, cng thc im gia n v cng thc Runge -Kutta kinh in cp bn n tng

ng nh sau.

Trong cng thc hnh thang n:

1 1 1[ ( , ) ( , )]2

n n n n n nh

y y f x y f x y

ta thay gi tr 1ny v phi bng cng thc Euler tin:

1 ( , ) n n n n y y hf x y .

Khi y ta c cng thc:

1 1 1[ ( , ) ( , )].

2

n n n n n nh

y y f x y f x y

Cng thc ny c gi l phng php hnh thang hin (explicit trapezoidal

method).

Bng cch s dng xp x bc nht ca ( )2

nh

y x theo phng php Euler tin:

1

2

( , )2

n n nn

h y y f x y

v thay vo cng thc ca phng php im gia n

1 [ ( , ( )].2 2

n n n nh h

y y h f x y x

ta nhn cphng php im gia hin (explicit midpoint method):

1 1

2

. ( , ).2

n n nn

h y y h f x y

T phng php Runge-Kutta n cp bn kinh in

1 1 1[ ( , ) 4 ( , ( )) ( , )]6 2 2

n n n n n n n nh h h

y y f x y f x y x f x y

ta c cng thc Runge-Kutta hin bc bn kinh in sau:

1 1 2 3 4( 2 2 ), 0,1,2,...6

n nh

y y k k k k n (3.3)

trong :


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1

12

23

4 3

( , );

( , );2 2

( , );2 2

( , ).

n n

n n

n n

n n

k f x y

h hkk f x y

h hkk f x y

k f x y hk

(3.4)

Nh vy, tnh c 1ny theo cng thc Runge-Kutta hin, ta ch cn tnh cc h

s ik , 1,2,3,4i theo gi tr ca hm s ( , ) f x y ti cc im trc . Vic ny

c th d dng thc hin c trn chng trnhMaple qua cc th d trong 4 di

y.

4. GII BI TON CAUCHY CHO PHNG TRNH VIPHN TRN MY TNH IN T

Do c ci t cc nh, my tnh khoa hc Casio fx-570 ESrt thun tin

cho vic thc hin cc thao tc qu trnh lp. Cng thc tnh xp x nghim theo

phng php Euler (2.2), phng php Euler ci tin (2.7) v v phng php

Runge-Kutta (3.3) cho thy, vic gii gn ng phng trnh vi phn (1.1)-(1.2)

thc cht l thc hin mt qu trnh lp, v vy c th d dng thc hin tnh ton

trn my tnh khoa hc Casio fx-570 EShoc lp trnh trnMaple.

Di y chng ti s trnh by cch gii bi ton Cauchy cho mt phng

trnh vi phn bng phng php Euler, Euler ci tin v phng php Runge-Kutta

vi cc bc ni suy khc nhau trn my tnh khoa hc Casio FX-570 ESv trn

Maple.

Bi 1. S dng phng php Euler, phng php Euler ci tin v phng phpRunge-Kutta vi di bc 0,1h v 0.05h tm xp x nghim ca

phng trnh 2 2dy

x ydx

tha mn iu kin ban u (0) 0y trn on 0 1; .

Phi tm nghim ca phng trnh 2 2 dy

x ydx

vi iu kin ban u

0 00, 0x y .

Vi 0,1h ta c:


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2 21 ( , ) 0,1( )n nn n n n n y hf x y y x y y .

Ta c:

0 0

2 21 00,1( ) (0,1).(0).(0) 0 0 y x y y .

Vi 1 0 0,1 x x h :

1 1

2 2 2 22 10,1 (0,1).(0,1 0 ) 0 0,001 y x y y .

Tip tc nh trn, ta tnh c cc gi tr ny theo cng thc:

2 21 ( , ) 0,1( )n n n n n n n y hf x y y x y y . (4.1)

Thc hin php lp (4.1) trn Casio fx-570 ES:

Khai bo cng thc 2 21 0,1( )n n n n y x y y :

0.1 ( ALPHA X2x + ALPHA Y 2y ) + ALPHA Y

Trong qui trnh ny, ta dng nh X cha gi tr nx v cc nh Y

cha gi tr ca ny .

Dng CALC tnh gi tr ca ny : CALC

My hi: X?

Khai bo0

0x : Bm phm 0 =

My hi: Y?

Khai bo0

0y : Bm phm 0 = (0)

Kt qu trn mn hnh l 0, tc l

2 2 2 21 0 0 00,1( ) 0,1(0 0 ) 0 0 y x y y .

a kt qu 1 0y vo nh Y : SHIFT STO Y

Tr v cng thc (4.1): Bm phm

Quy trnh:

Tnh tip: CALC

My hi: X? Khai bo 1 0 1x . : Bm phm 0.1 =


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My hi: Y? Bm phm = ( 1 0y v c sn trong nh Y nn khng cn

khai bo li).

Kt qu hin trn mn hnh:1

1000

, tc l

2 2 2 2 32 1 1 10,1( ) 0,1(0,1 0 ) 0 0,1 y x y y .

a kt qu vo nh Y : SHIFT STO Y

Tr v cng thc ban u:

Lp li quy trnh vi thay i duy nht l khi my hi X? th ta khai bo cc gi tr

tip theo: 0.2; 0.3; 0.4;...;1.0 ta s c bng gi tr nh trong bng sau.

n 1nx ny n 1nx ny

1 0 0 6 0,5 0,05511234067

2 0,1 1

1000 7 0,6 0,09141607768

3 0,2 35 001 10, 8 0,7 0,1412517676

4 0,3 0,0140026001 9 0,8 0,2072469738

5 0,4 0,03002220738 10 0,9 0,2925421046

Thc hin php lp (4.1) trnMaple 7:

TrongMaple, tm cc gi tr iy theo cng thc lp ta c th s dng mc

nh (option) remember(nh). Mc nh ny caMaplecho php nh cc gi tr c

tnh ny , m khng cn tnh li gi tr 1ny .

Trc tin ta khi ng chng trnhMaplenh lnh restart:

[> restart;

Khai bo hm f:

[>f:=(x,y)->x^2+y^2;

:=f ( ),x y x2 y2

Khai bo bcni suy 0,1h :

[> h:=0.1;

:=h .1

Khai bo cch tnh cc gi tr ca 1n n x x h (vi 0 0x ):


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[> x:=n->n*h;:=x n n h

Khai bo gi tr ban u ca y :

[> y(0):=0;:=( )y 0 0

Khai bo th tc tnh ny theo mc nh remember (nh):

[> y:=proc(n) option remember;

[> y(n-1)+h*f(x(n-1),y(n-1));

[> end;

:=y proc ( ) end procn option ;remember ( )y n 1 h ( )f ,( )x n 1 ( )y n 1

Khai bo lnh seq(sp xp theo dy) sp xp cc gi tr 0 1 9 10, ,..., , y y y y :

[> seq(y(i),i=0..10);

0 0. .001 .0050001 .01400260010.03002220738.05511234067.09141607768, , , , , , , ,

.1412517

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