ti nghin cu khoa hc sinh vin

ti nghin cu khoa hc sinh vin2013

MC LCLI M UChng I: KHI NIM V BIN I LAPLACE1. nh ngha2. iu kin tn ti bin i Laplace3. Bin i Laplace ngc4. Tnh duy nht ca bin i LaplaceChng II: CC TNH CHT1. Bng bin i Laplace c bn2. Tnh tuyn tnh3. Cng thc dch chuyn v bng cng thc dch chuyn4. S cng hng v bng cc cng thc lin quan n s cng hng5. Bin i Laplace ca o hm6. Bin i Laplace ca tch phn7. o hm ca bin i Laplace8. Tch phn ca bin i Laplace9. Bin i Laplace ca hm lin tc tng khc10. Tch chp ca bin i LaplaceChng III: NG DNGA. Gii phng trnh vi phn1. Gii Phng trnh vi phn tuyn tnh cp cao c h s hng2. Gii h phng trnh vi phn tuyn tnh c h s hng3. Gii phng trnh vi phn tuyn tnh c h s l hm a thc bc nh hn hoc bng 14. Gii phng trnh vi phn tuyn tnh cp 2 vi v phi l hm gin on5. Gii bi ton dao ng diu ha trong vt liu khng ti6. Gii phng trnh vi phn c trB. Gii phng trnh tch phn1. Phng trnh voltera loi 12. Phng trnh voltera loi 23. Phng trnh vi-tch phnC. Gii phng trnh o hm ring1. Phng trnh o hm ring cp 12. Phng trnh o hm ring cp 2

Gii thiu tiBin i Laplace ng dng gii cc phng trnh v h phng trnh vi phn tuyn tnht vn Nhiu bi ton thc t, nht l nhng bi ton thuc lnh vc k thut dn n vic phi gii cc phng trnh hoc h phng trnh vi phn, phng trnh o hm ring, phng trnh tch phn, Vic nghin cu cc cch gii nhng bi ton ny l cn thit.Khng c phng php chung gii cho cc phng trnh, h phng trnh vi phn cng nh cc phng trnh o hm ring , phng trnh tch phn. Trong mn hc Gii Tch ta thy, ta c hc v phng trnh vi phn cp 1, cp 2 nhng khng phi tt c cc phng trnh , h phng trnh loi ny ta xt ht v gii c, m ta ch gii c mt s lp nh: phng trnh c bin phn ly, phng trnh ng cp, phng trnh tuyn tnh cp 1, phng trnh cp 2 c th gim cp (khuyt y, y; khuyt x), phng trnh tuyn tnh cp 2 c h s hng, phng trnh tuyn tnh cp 2 c h s hm. Trong ch c phng trnh vi phn tuyn tnh cp 1, phng trnh vi phn ton phn l c cng thc tm nghim ng.V d: Phng trnh vi phn tuyn tnh cp 2 c h s hm dngy + p(x)y + q(x)y = f(x)ch gii c khi bit mt s nghim ring ca phng trnh thun nht l y1. Sau tm nghim y2 c lp tuyn tnh vi y1 bng cng thc Louville. Tip theo dng phng php bin thin hng s Lagrng tm nghim ca phng trnh khng thun nht.Nghin cu bin i tch phn Laplace cho php ta gii c mt lp rng hn cc phng trnh vi phn, h phng trnh vi phn tuyn tnh cp cao (ln hn 2) v nghim tm c dng nghim ng. Khi nhng trng hp gii c bit ch l nhng trng hp ring.Mc ch ca ti nhm gii thiu v bin i Laplace, a ra cc tnh cht ca php bin i Laplace, t a ra ng dng ca Laplace vo vic gii cc phng trnh, h phng trnh vi phn, cc bi ton o hm ring, v mt s phng trnh tch phn. Ni dung ti c sp xp theo 3 chng. Chng I trnh by nhng khi nim ban u v bin i Laplace, nh nh ngha bin i Laplace, bin i Laplace ngc v cc iu kin tn ti bin i Laplace. Chng II trnh by cc tnh cht ca bin i Laplace, a ra bng cc cng thc bin i c bn. Chng III trnh by ng dng ca bin i Laplace vo vic gii bi ton lin quan n phng trnh vi phn , phng trnh o hm ring, phng trnh tch phn. ti c hon thnh trong mt khong thi gian hn ch di s hng dn ca c gio Ph Th Vn Anh. Mc d rt c gng nhng c th ti khng trnh c nhng thiu st. Nhm chng em rt mong nhn c s ng gp xy dng ca cc thy c v bn b._Nhm tc gi_

Li m uChng 1: KHI NIM V BIN I LAPLACE1. nh ngha

nh ngha 1.1: Cho hm f(t) xc nh trn R. Bin i Laplace ca hm f, k hiu , l mt hm c xc nh qua cng thc tch phn sau:

,(1.1)

s, f(t) R

tin s dng ta k hiu: .

Ch : Php bin i Laplace xc nh vi s, f(t)C. Nhng trong phm vi ti ny ta ch cn s dng s, f(t)RDng nh ngha ca bin i Laplace, ta tnh c bin i Laplace ca mt s hm n gin di y:

V d 1.1: Tnh Gii:

= , vi s>0

Khng tn ti khi s0

V d 1.2: Cho , t0. Tnh , aR.Gii:

(s) =

, nu s > a

Phn k khi sa2. iu kin tn ti bin i Laplacenh ngha 1.2: Hm s f(t) c gi l lin tc tng khc trn [a;b] nu nh f(t) lin tc trn mi khong nh, [a;b] c chia thnh hu hn khong nh. f(t) c gii hn hu hn khi t tin ti 2 im u mt ca mi on ny.

Hnh 1.1: th ca hm lin tc tng khc(cc du chm ch ra cc gi tr m hm s gin on)

nh ngha 1.3: Hm f c gi l bc m khi t+nu tn ti cc hng s khng m M, c, T sao cho .

nh L 1.4: Nu hm f lin tc tng khc vi t0 v l bc m khi t+th tn ti Chng minh:

T gi thit f l bc m khi t+

Ta c vi s > c

Cho b+ ta c

F(s) b chn hay tch phn suy rng hi t, vy tn ti bin i Laplace ca hm F(t)

Hn na ta thy khi s+ th . Ta c h qu di y:

H qu 1.5: Nu f(t) tha mn gi thit ca nh l 2 th .3. Php bin i Laplace ngcBin i Laplace ngc gip ta tm li hm gc f(t) t hm nh F(s). Bin i Laplace ngc c nh ngha bi tch phn sau:

, vi cR.Nhng thng thng ta t dng tch phn ny tnh hm gc m dng bng hm gc- hm nh tng ng c sn tm hm gc f(t).Ta nh ngha bin i Laplace ngc nh sau:

nh ngha 1.6: Nu th ta gi f(t) l bin i Laplace ngc ca F(s) v vit .

V d 3.1: Theo v d 1.1, ta c nn c:

V d 3.2: Theo v d 1.2, c nn c 4. Tnh duy nht ca bin i LaplaceNgi ta chng minh c nh l v tnh duy nht ca bin i Laplace nh sau:

nh l 1.7: Gi s rng cc hm f(t), g(t) tha mn gi thit ca nh l 1.4 tn ti F(s), G(s), . Nu f(t)=g(t) ti t m c 2 hm lin tc th F(s)=G(s).

Chng II: CC TNH CHT1. Bng bin i Laplace c bnBng 1: Ta c bng bin i Laplace cc hm c bn:STT

11

2t

3

4

vi

5

6Cos(kt)

7Sin(kt)

8Cosh(kt)

9Sinh(kt)

10u(t-a)

11

12

Trong Chng minh mt s cng thc :

V d 1.1: . Tnh .Gii:

nu s>0.

V d 1.2: Cho . Tnh v Gii:

t u=st ta c

Khi a l s thc a > 1. Khi a=n l s t nhin th

Vy: s > 02. Tnh cht tuyn tnh

nh l 2.1: Cho , l hng s v v , khi

Chng minh:

V d 2.1: Tnh Gii:

Ta c mt s tnh cht ca hm Gamma nh sau

Ta c : , ,

Vy:

Do : S dng bng bin i laplace c bn ta c:

Vy

V d 2.2: Tnh Gii:

s > 23. Cng thc dch chuyn v bng cc cng thc dch chuyn

nh l 2.2:Nu tn ti vi s>c, th tn ti vi s > a+c v c: (2.2)

hay tng ng vi (2.3)

Chng minh:

Cng thc (2.3), hay (2.3) c gi l cng thc dch chuyn, v bin i Laplace ca hm so vi bin i ca Laplace ca hm f(t) dch chuyn i mt lng (s-a).p dng cng thc trn ta tnh mt s hm c bn.V d 3.1:

Vi cc hm khc nh trong Bng 1, ta lm tng t, lp c Bng 2.Bng 2: Cc cng thc dch chuynSTT

1

2

3

4

vi

5

6cos(kt)

7sin(kt)

8cosh(kt)

9sinh(kt)

10u(t-b)

Trong

V d 3.2: Tm php bin i Laplace ngc ca:a)

Gii: Phn tch hm phn thc R(s) thnh cc hm phn thc n gin

Cn bng h s theo ly tha ca s

HocThay s = 0, s = - 2, v s = 4 ta c:

-8A = 1, 12B = 5, 24C = 17

Vy Dng tnh tuyn tnh ca bin i Laplace ngc v bng 1 ta c:

4. Bin i laplace ca o hm

nh l 2.3: Cho hm f(t) lin tc, trn tng khc vi t 0 v l bc m khi

Khi tn ti vi s > c v c:

(2.4)Chng minh:

H qu 2.4: Gi s cc hm s lin tc, trn tng khc vi t 0 v l bc m khi . Khi tn ti vi s > c v c cng thc:

(2.5)Chng minh:Chng minh bng phng php quy npVi n=1, ta c cng thc (2.4), nn (2.5) ng vi n=1.Gi s 2.5 ng vi n=k, tc l:

Ta s chng minh (2.5) ng vi n=k+1. Tht vy ta c:

Vy (2.5) c chng minh Cng thc (2.5) c ng dng ln khi gii cc phng trnh vi phn tuyn tnh cp cao. Di y ta xt mt s v d:V d 4.1 : Gii ptvp

(1)Vi iu kin x(0)=0: x(0) = x(0)=1Gii:

(1)

V d 4.2: Gii h ptvp

Gii:

H pt

5. S cng hng v bng cc cng thc lin quan n s cng hngDng o hm v bng bin i Laplace ta chng minh c cc cng thc sau:STT

1t.sin(kt)

2t.cos(kt)

3t.sinh(kt)

4t.cosh(kt)

5

V d 5.1: S dng php bin i Laplace gii bi ton vi gi tr ban u

Trong . L hng s cho trc .

Tc ng php bin i Laplace vo 2 v ca phng trnh cho, t , ta c:

Trng hp 1:

Phn tch Ly Laplace ngc

Vy

= (vi )

=

Nu th

nn

Trng hp 2: th nghim ca phng trnh i s l:Dng cng thc 5 ca bng 3, ta c:

Xt trng hp 0=1/2 v F0=1 th nghim nhn c l:

th x(t) l ng cong c bin khuych i dn khi t tng nh hnh v di

Hnh 2.1: Nghim cng hng vi v F0=1, cng vi ng bao ca n 6. Bin i Laplace ca tch phn

nh l 2.5: Nu f(t) lin tc tng khc vi t 0 v l bc m khi th

,vi s > c (2.7)

Hay l: (2.8)Chng minh:

+)Do f lin tc tng khc lin tc, trn tng khc vi

t 0. Li do f(t) bc m, nn , vi tn ti M,T,C sao cho:

Do

Vy g(t) l hm bc m khi . Do vy tn ti

+) S dng nh l 2.3 ta c

+) Do g(0) = 0 nn ta c Suy ra cng thc (2.7) c chng minh. Trong mt s trng hp tm hm gc t bin i Laplace ngc cng thc (2.8) gip ta thc hin d dng hn.

V d 6.1: Tm nghch o ca php bin i Laplace ca Gii:Ta c

T v tip tc ta c

7. o hm ca bin i Laplace

nh l 2.6: Gi s f(t) lin tc tng khc vi t 0, () v f(t) bc m khi , khi ta c (2.9)

Hay (2.10)

Tng qut ta c: (2.11)Chng minh:

+) T gi thit hi t tuyt i, u v lin tc, vi

+) Do

Vy cng thc (2.9) c chng minh

+) Ta chng minh (2.11) bng phng php quy np ton hc. Tht vy, n=1 th ta c , nn (2.11) ng vi n=1

Gi s (2.11) ng n=k, tc l c Ta chng minh (2.11) ng vi n=k+1, tht vy,

Vy (2.11) c chng minh

Cng thc (2.11) rt thun li ta tm thm cc bin i Laplace ca cc hm c dng khi bit bin i Laplace ca f(t)

V d 7.1 : Tm Gii:

T (2.11) ta c

V d 7.2: Tm Gii:

t Dng cng thc (2.10), ta c

Vy

8. Tch phn ca bin i Laplace

nh l 2.7: Cho f(t) lin tc tng khc i vi t 0, gi s tn ti ,v gi s f(t) bc m. Khi ta c :

Trong cng thc (2.12), ta thy tch phn ca hm chnh l tch phn ca bin i Laplace ca hm f(t)Chng minh:

+) T gi thit suy ra hi t tuyt i v u, s > c

+) Ta c

+) T i th t tch phn ta c

Nh vy cng thc (2.12) c chng minh

V d 8.1: Tm .Gii:iu cn tm c dng v tri ca (2.12) vi f(t)=sinh(t). Ta kim tra cc gi thit ca nh l 2.7

Ta c . Vy tn ti Theo cng thc (2.12):

Vy

V d 8.2: Tm Gii:

t . Dng cng thc (2.13), ta c:

(theo cng thc 5 bi 3)

9. Bin i Laplace ca hm lin tc tng khca) t vn 1. Cc m hnh ton hc trong h c hc hay h in trng lin quan n cc hm khng lin tc tng ng vi cc lc bn ngoi bt ng o chiu bt hay tt.1. Hm n gin bt, tt l hm bc thang n v ti t=a

C th nh sau

Hnh 2.2: th ca hm n v bc thangb) Php tnh tin trn trc t

nh l2.8: Nu tn ti vi s > c, th ta c

Hay

Hnh 2.3: Tnh tin ca f(t) v pha phi a n vChng minh:

+) Ta c

+) i bin t = + a, ta c

+) Do , nn c

Vy cng thc (2.14) c chng minh

V d 9.1: Tnh

T cng thc (2.15) v cng thc 3 ca bng 1, ta c:

Hnh 2.4: th bin i ngc trong v d 1

V d 9.2: Cho . Tm Gii:

Ta thy hm g(t) chnh l hm khi tnh tin sang phi 3 n v.

Vy coi hm gc l th Theo cng thc (2.14), ta c:

10. Tch chp ca bin i Laplace

t vn : Gi s

Xt

Vn t ra l tm hm h(t) sao cho:a. nh ngha 2.9: Tch chp i vi php bin i Laplace ca hai hm f,g lin tc tng khc c nh ngha nh sau: b. Tnh cht 2.10: Tch chp c tnh giao hon, kt hp hm phn phi v ly thac. nh l 2.11: f(t),g(t) l cc hm lin tc tng khc ,

f(t),g(t) b chn bi khi , cc s M,c khng m

Khi

hayChng minh:

C

do

vi khi u 0Gii:

Tc ng Laplace v t ta nhn c:

y l nghim ng ca h phng trnh tch phn cho.V d 2.2: Xt phng trnh

, x > 0, t > 0. (3.12)Tc ng bin i laplace vo 2 v ca phng trnh (3.9) theo bin (t) ta c:

(3.13)Phng trnh vi phn tuyn tnh cp 1 ny c nghim tng qut l:

V v c 1 nghim b chn th . Vy nghim ca Phng trnh (3.13) l:

Ly Laplace ngc ta c:

3. Phng trnh vi- tch phni khi s dng bin i Laplace cn gip ta gii c mt s phng trnh vi- tch phn.V d gii phng trnh sau tm hm f(t):

(3.14)Vi iu kin f(0)=0Phng trnh cho vit c dng tch chp:

Tc ng Laplace, t F(s)=, ta c:

. y l nghim ng ca phng trnh vi- tch phn.D. Phng trnh o hm ring1. Bi ton gi tr bin ban u cp 1.V d 1.1: Xt phng trnh:

, x >0, t >0(3.15)Vi iu kin bin ban u cho l:u(x,0)=0 vi x >0u(0,t)=0 vi t >0trong hm u(x,t) l hm cn tm.Gii:Dng bin i Laplace tc ng vo 2 v v ly theo bin t, ta c:

t , phng trnh trn tr thnh phng trnh vi phn bc nht vi n

a v phng trnh vi phn tuyn tnh dng , ri p dng cng thc ly nghim tng qut:

ta c:

;(3.16)Vi C l hng s tch phn.

V iu kin u(0,t)=0 (3.8) c 1 nghim b chn th suy ra C=0

Vy Ly bin i Laplace ngc, ta c:

2. Phng trnh o hm ring cp 2V D 2.1: Xt phng trnh truyn nhit na trc nh sau:

, x > t, t > 0 (3.17)Vi iu kin bin v iu kin ban u nh sau:U (x,0) = 0, x > 0U (0,t) = f(t), t > 0(3.18)

Khi

Tc ng bin i Laplace vo phng trnh (3.9), ta c : bin t, v t . Ta c:

(3.19)

y l phng trnh vi phn tuyn tnh cp 2 thun nht vi h s hng , c nghim ca phng trnh c trng l nn nghim tng qut ca phng trnh (3.19) l:

Vi A, B l hng s tch phn.

c 1 nghim b chn th v dng iu kin:

Th c : A=F(s)Vy nghim ca (3.19) vi cc iu kin cho l:

.Ly bin i Laplace ngc:

Ngi ta tnh c nghim c dng:

Nu t th Ta c nghim dng:

V D 2.2: Phng trnh khuych tn trong on hu hn

Xt phng trnh: 0