Bazı tipten kısmi türevli denklemlerin çözümlerinin davranışı

(1)

T.C.

SAKARYA ÜNİVERSİTESİ

FEN BİLİMLERİ ENSTİTÜSÜ

BAZI TİPTEN KISMİ TÜREVLİ DENKLEMLERİN ÇÖZÜMLERİNİN DAVRANIŞI

YÜKSEK LİSANS TEZİ

Sema BAYRAKTAR

Enstitü Anabilim Dalı : MATEMATİK

Enstitü Bilim Dalı : UYGULAMALI MATEMATİK Tez Danışmanı : Doç. Dr. Şevket GÜR

Haziran 2014

(2)

(3)

ii

7(ù(..h5

%X WH]LQ KD]ÕUODQPDVÕ VUHFLQGH GHUV DúDPDVÕQGD \DQÕPGD RODQ VD\JÕGH÷HU KRFDP

'Ro 'U (OPDQ +D]DU¶D KHP DUDúWÕUPD KHP GH \D]ÕP NÕVPÕQGD EDQD KHU WUO

IHGDNkUOÕ÷ÕJ|VWHUHQELOJLVLYHWHFUEHVL\OHEHQLKHU]DPDQ\|QOHQGLUHQYHEDQDKHU

WUO RODQD÷Õ VD÷OD\DQ 6DNDU\D hQLYHUVLWHVL )HQ (GHEL\DW )DNOWHVL 0DWHPDWLN

$QDELOLP 'DOÕ |÷UHWLP \HVL GH÷HUOL 'DQÕúPDQ +RFDP 'Ro 'U ùHYNHW *U¶e VRQVX]WHúHNNUOHULPLVXQDUÕP

(4)

iii

ødø1'(.ø/(5

7(ù(..hR«««««««««««««««««««««««««««.... ii

ødø1'(.ø/(5««««««««««««««««««««««««««... iii 6ø0*(/(59(.,6$/70$/$5/ø67(6ø««««««««««.««« v g=(7 ««««««««««««««««««««««««««««««.... vi

SUMMARY««««««««««««««««««««««««««««.. vii

%g/h0

*ø5øù ... 1

%g/h0

TEMEL TANIM VE KAVRAMLAR«««««««««««««««..««6

%g/h0

'(1ø= 3/$7)250/$5,1,1 'ø1$0øöø1ø 7$1,0/$<$1 '(5(&('(1

/ø1((5 2/0$<$1 %ø5 '$/*$ '(1./(0ø1ø1 8=81 =$0$1

'$95$1,ù,««««««««««««««««««««««««««...««11

*LULúYH3UREOHPLQLIDGHVL««««««««««««««««««11

%g/h0

36(8'23$5$%2/,&'(1./(0/(5ø1dg=h0/(5ø1ø13$7/$0$6,. 28

*LULúYH3UREOHPLQLIDGHVL««««««««««««...««« 28

(5)

iv

%g/h0

/ø1((5 2/0$<$1 '$/*$ '(1./(0ø ødø1 dg=h0/(5ø1 */2%$/

'$95$1,ù,««««««««««««««««««.«««««««««48

*LULúYH3UREOHPLQLIDGHVL«««««««««««««««««... 48

%g/h0

6218d/$59(g1(5ø/(5«««««««««««««««««««.. ⁵⁶

KAYNAKLAR«««««««««««««««««««««««««««57

g=*(d0øù««««««««««««««««««««««««..«««« 60

(6)

v

**6ø0*(/(59(.,6$/70$/$5/ø67(6ø**

' 2 : /DSODFHRSHUDW|U

^:*UDGLDQWRSHUDW|U

: : Rⁿ¶GHG]JQVÕQÕUDVDKLSVÕQÕUOÕE|OJH

^,

u x t : Bilinmeyen fonksiyon

2 1

0

ȍ 1 ȍ

H H H : ^:6REROHYX]D\Õ

^{u v}^, ^: uvdx

:

³

LP

u _: :

P

:

2

(7)

vi

g=(7

$QDKWDU NHOLPHOHU d|]PQ SDWODPDVÕ DVLPSWRWLN GDYUDQÕú GDOJD GHQNOHPL

pseudoparabolic denklem.

%XWH]E|OPGHQROXúPDNWDGÕU

%LULQFL E|OPGH NÕVPL GLIHUDQVL\HO GHQNOHPOHULQ GDYUDQÕúÕQGDQ EDKVHGLOHUHN WH]H

JLULú\DSÕOPÕúWÕU

øNLQFLE|OPGHWH]GHNXOODQÕODQWHPHOWDQÕPYHNDYUDPODUYHULOPLúWLU

hoQF E|OPGH OLQHHU ROPD\DQ ELU GDOJD GHQNOHPLQLQ X]XQ VUHOL GDYUDQÕúÕ

LQFHOHQPLúWLU

'|UGQF E|OPGH SVHXGRSDUDEROLF GHQNOHPOHULQ o|]POHULQLQ SDWODPDVÕ

LQFHOHQPLúWLU

%HúLQFLE|OPGHLVHOLQHHUROPD\DQGDOJDGHQNOHPLLoLQo|]POHULQJOREDOGDYUDQÕúÕ

LQFHOHQPLúWLU

AlWÕQFÕE|OPGHLVHWH]oDOÕúPDVÕQGDQHOGHHGLOHQVRQXoODUEHOLUWLOPLúWLU

(8)

vii

BEHAVIOR OF SOLUTIONS OF SOME TYPES OF PARTIAL DIFFERENTIAL EQUATIONS

SUMMARY

Key Words: Blow-up of Solution, Asymptotic Behavior, Wave Equation This thesis is consists of six chapters.

In the first chapter, it is mentioned about behavior of solution for partial differential equations and there is introduction to the thesis.

In the second chapter, main definitions and concepts used in the thesis are given.

In the third chapter, it is concerned nonlinear wave equation long-time behavior.

In the forth chapter, it is concerned with blow-up of solution to pseudobolic equations.

In the fifth chapter, it is concerned stabilization of the energy and continuous dependence of solution to nonlinear wave equation.

Finally in the sixth chapter, the results are stated gained through the study of thesis.

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Equation Section 1

**%g/h0*ø5øù**

6D\ÕVDO\|QWHPOHULQ\DGDDQDOL]LQDPDFÕPDWHPDWLNVHOSUREOHPOHULQo|]PLoLQ

X\JXQ YH GR÷UX VRQXo YHUHQ WHNQLNOHU JHOLúWLUPHNWLU d|]P LVWHQHQ SUREOHPL

WDQÕPODPDN IRUPOH HWPHN YH o|]P LoLQ NXOODQÕODFDN HQ X\JXQ \|QWHPL VHoPHN

DQDOL]L\DSDQNLúLQLQDPDFÕGÕU$GLGLIHUDQVL\HOGHQNOHPOHUNÕVPLWUHYOLGLIHUDQVL\HO

GHQNOHPOHUVD\ÕVDOLQWHJUDV\RQVHULKHVDSODPDODUÕJLELELUoRNSUREOHPWUYDUGÕU

%L] EX SUREOHP WUOHULQGHQ NÕVPL WUHYOL GLIHUDQVL\HO GHQNOHPOHUL HOH DODFD÷Õ]

%LOLQGL÷L]HUHSUREOHPOHULQo|]PKHU]DPDQYDUROPD\DELOLU%XGXUXPGDo|]P

EXOXQPDVDGDSUREOHPLQ\DNODúÕNo|]PHOGHHGLOHELOLU

%LU RWRPRELOL oDOÕúWÕUGÕ÷ÕQÕ]GD YH\D DPSXO DoWÕ÷ÕQÕ]GD ELUoRN úH\ ROXU %XQODUÕQ

ED]ÕODUÕ VLVWHPLQ EDúODPDVÕQÕQ HúVL] SDUoDVÕGÕU %X JHoLFL úH\OHU VDGHFH EDúODPD

VÕUDVÕQGD ROXU YH GDKD VRQUD VLVWHP NDUDUOÕ GXUXPD JHoHU 6LVWHPOHULQ GDYUDQÕúÕQÕ

EDúODWPDN oRN |QHPOL RODELOLU DPD ED]HQ VLVWHPLQ X]XQ VUHOL GDYUDQÕúÕQÕ \D GD

NDUDUOÕ KDOLQL DUDúWÕUPDN LVWH\HELOLUL] gUQH÷LQ ^{f n}

IRQNVL\RQX J|] |QQH

DOÕQGÕ÷ÕQGD n oRN E\N ROGX÷XQGD IRQNVL\RQXQ |]HOOLNOHULQL DoÕNODPDN LoLQ

GDYUDQÕúÕ LQFHOHQLU ^{f n}

ⁿ²³ⁿ ROGX÷XQGD n oRN E\NVH 3n terimi n ile ² NÕ\DVODQGÕ÷ÕQGD |QHPVL] KDOH JHOLU ^{f n}

fonksiyonunun no f LoLQ n²¶\H

DVLPSWRWLNRODUDNHúGH÷HUROGX÷XV|\OHQLU

'DYUDQÕú LQFHOHPHVL D\QÕ ]DPDQGD ELOJLVD\DU ELOLPLQGHNL DOJRULWPDODUÕQ DQDOL]LQGH

YHUL NPHOHULQH oRN E\N JLULúOHU X\JXODQGÕ÷ÕQGD DOJRULWPDQÕQ SHUIRUPDQVÕQÕ HOH

DOÕUNHQ ND]D DQDOL]LQGH EHOOL ]DPDQ YH PHNkQGDNL oRN VD\ÕGDNL ND]D VD\ÕODUÕQÕ

modelleyerek ND]DQÕQQHGHQLQLWHVSLWHGHUNHQGHNXOODQÕOÕr.

øON GDYUDQÕú LQFHOHPHOHUL \ÕOÕQGD 0LOQH :( WDUDIÕQGDQ DUDOÕ÷Õ VÕQÕUVÕ] RODQ

VRQOX GH÷HU SUREOHPLQLQ GDYUDQÕúÕQÕ LQFHOHPHVL\OH EDúODPÕúWÕU >@ \LQH

(10)

\ÕOÕQGD )RUG :% XoD÷ÕQ X]DN NÕVÕPODUÕQGDNL LQWHJUDO IRQNVL\RQODUÕQ GDYUDQÕúÕQÕ

incelemLúWLU [31].

)LOWUDV\RQ WHRULVL WHUPRGLQDPLNOHU YH KLGURGLQDPLNOHULQ oHúLWOL SUREOHPOHULQGHNL

oDOÕúPDODUGDJ|UQHQOLQHHUROPD\DQSVHXGRSDUDEROLFGHQNOHPOHU>@

^{, ,}

^, ^0,

t t

u ' ' u v u f x u u v! (1.1)

IRUPXQGDGÕU¶LQOLQHHUKDOL¶WH6/6REROHY>@WDUDIÕQGDQoDOÕúÕOPÕúWÕU%X

\]GHQIRUPXQGDNLEXGHQNOHP6REROHYWLSLGHQNOHPRODUDNDGODQGÕUÕOÕU6$

Galpern [13], M ve L OLQHHUHOLSWLNRSHUDW|UOHUROPDN]HUH

Mut Lu f (1.2)

IRUPXQGDNLGHQNOHPLoLQ&DXFK\SUREOHPLoDOÕúPÕúWÕU5(6KRZDOWHU>@ M ve L LNLQFLGHUHFHHOLSWLNRSHUDW|UOHUROPDN]HUHOLQHHUSVHXGRSDUDEROLFGHQNOHPLQL

DUDúWÕUPÕúWÕU 6KRZDOWHU YH 7LQJ >@ \ÕOÕQGDNL 3VHXGRSDUDEROLF SDUWLDO

GLIIHUHQWLDO HTXDWLRQV DGOÕ PDNDOHVLQGH GHQNOHPL LoLQ EDúODQJÕo VÕQÕU GH÷HU

probleminLQ ]D\ÕI o|]PQQ YDUOÕN WHNOLN YH G]JQO÷Q DUDúWÕUPÕúODUGÕU

*HUoHNWHQ>@SVHXGRSDUDEROLFGHQNOHPLQLONPDNDOHVLRODUDNDGODQGÕUÕODELOLU

Lineer olmayan pseudoparabolic denklemlerin ilk makalesi

t

^,

M t u L t u f t u (1.3)

IRUPXQGDNL GLIHUDQVL\HO RSHUDW|U GHQNOHPL LoLQ EDúODQJÕo GH÷HU SUREOHPLQLQ VÕIÕU

o|]PQQ YDUOÕN YH WHNOL÷LQLQ ELOLQGL÷L GXUXPGD 6KRZDOWHU >@ WDUDIÕQGDQ

\D]ÕOPÕúWÕU/LQHHUROPD\DQSVHXGRSDUDEROLFGHQNOHPOHULQJHQLúELUVÕQÕIÕQÕNDSVD\DQ

OLQHHUROPD\DQGLIHUDQVL\HORSHUDW|UGHQNOHPOHULoLQ&DXFK\SUREOHPLQLQWHNOL÷LYH

JOREDO YDUOÕ÷ÕQ VLVWHPDWLN oDOÕúPDVÕ 6KRZDOWHU YH 7LQJ >@¶LQ PDNDOHVLQGH YH

*DMHZVNL*U|JHU=DFKDULDV>@¶LQNLWDEÕQGD\D]ÕOPÕúWÕU

(11)

3

¶LQ|QHPOL|UQHNOHULQGHQELUL

t xx xxt x x 0

u vu u u uu (1.4)

Benjamin-Bona-Mahony-%UJHUV %%0% GHQNOHPLGLU GHQNOHPLQGH f ,

^,

f b u F u IRUPXQGDGÕU:DQJYH<DQJ>@WHNER\XWOX%%0%GHQNOHPL

LoLQSHUL\RGLNEDúODQJÕoVÕQÕUGH÷HUSUREOHPLLOHUHWLOHQ\DUÕJUXEXQVRQOXER\XWOX

JOREDO oHNLFLQLQ YDUOÕ÷ÕQÕ LVSDWODPÕúWÕU dHOHEL .DODQWDURY YH 3RODW >@ denkleminde f , ^f

^b^,^u

^{F u}

^{h x} ROGX÷X GXUXPGD SHUL\RGLN EDúODQJÕo

VÕQÕUGH÷HUSUREOHPLLOHUHWLOHQ\DUÕJUXEXQVWHOoHNLFLVLYHJOREDOoHNLFLVLQLQYDUOÕN

SUREOHPLQLoDOÕúPÕúWÕU

$PLFN %RQD YH 6FKRQEHFN >@ EX GHQNOHP LoLQ &DXFK\ SUREOHPLQLQ ^{L R}²

ve^L^f

^R ¶GH o|]POHULQLQ DVLPSWRWLN GDYUDQÕúÕQÕ DUDúWÕUPÕúWÕU (OGH HGLOHQ EX

VRQXoODU

0, 0

m

t xx xxt x x

u vu u u u u mt

IRUPXQGDNL GHQNOHPOHU LoLQ =KDQJ >@ WDUDIÕQGDQ JHOLúWLULOPLúWLU .DUFK >@ oRN

ER\XWOX %%0% GHQNOHPL LoLQ &DXFK\ SUREOHPLQLQ o|]POHULQLQ DVLPSWRWLN

GDYUDQÕúÕQÕDUDúWÕUPÕúWÕU

Stanislavova, Stefanov ve Wang [22], ^H¹

^R³ ¶WHoRNER\XWOX%%0%GHQNOHPLLoLQ

JOREDOoHNLFLQLQYDUOÕNSUREOHPLQLoDOÕúPÕúWÕU

.RUSXVRYYH6YHVKQLNRY>@DúD÷ÕGDNL%HQMDPLQ-Bona-Mahony-%UJHUVGHQNOHPL

LoLQ EDúODQJÕo VÕQÕU GH÷HU SUREOHPLQLQ o|]PQQ JOREDO \RNOX÷X LoLQ JHUHNOL

NRúXOODUÕEXOPXúWXU

1

3 0

t t x

u ' ' u u uu u

(12)

/LQHHUROPD\DQSVHXGRSDUDEROLFGHQNOHPOHULQo|]POHULQLQSDWODPDVÕQGDLONVRQXo

/HYLQH>@WDUDIÕQGDQHOGHHGLOPLúWLU/HYLQH P ve A OLQHHUSR]LWLIRSHUDW|UOHUYH

F u , H +LOEHUW X]D\ÕQGD SRWDQVL\HO RSHUDW|U ROPDN ]HUH DúD÷ÕGDNL OLQHHU

ROPD\DQGLIHUDQVL\HORSHUDW|UGHQNOHPLoLQ&DXFK\SUREOHPLQLDUDúWÕUPÕúWÕU

t

Pu Au F u

%XVRQXo f ,

0

0, 2

s

f s s k

³

f W Wd t k ! ^(1.5)

ROGX÷X GXUXPGD ut ' ' u ut f u

IRUPXQGDNL GHQNOHPOHU LoLQ EDúODQJÕo

VÕQÕU GH÷HU SUREOHPOHUL YH &DXFK\ SUREOHPL LoLQ o|]POHULQ SDWODPDVÕQÕQ JHUHNOL

NRúXOODUÕQÕYHULU

+DUDX[ YH =XD]XD >@ 1DNDR >@ GHUHFH GDOJD GHQNOHPOHULQLQ o|]POHULQLQ

]D\ÕIODPD KÕ]ÕQÕQ NHVWLULPOHUL YH VÕIÕU o|]PQ DVLPSWRWLN NDUDUOÕOÕ÷Õ WHRUHPOHULQL

HOGHHWPLúOHUGLU'DKD \NVHNGHUHFHOLOLQHHUROPD\DQGDOJDGHQNOHPOHULQLQEHQ]HU

VRQXoODUÕGD0DUFDWL>@WDUDIÕQGDQHOHDOÕQPÕúWÕU.

2 , ȍ

tt t t t

u ' u E u u 'D u x t! (1.6)

, 0 0

, t

, 0 1

, ȍ

u x u x u x u x x (1.7)

GHQNOHPLQLQVRO\DQÕQGDNLVRQWHULP ^{f u}

LOHGH÷LúWLULOHUHN¶GH:HEE>@

WDUDIÕQGDQ oDOÕúÕOPÕúWÕU :HEE YH LoLQ JOREDO JoO o|]PQ YDUOÕ÷ÕQÕ

f u ¶\DJ|UHG|UWNRúXODOWÕQGDn 1, 2,3LoLQLVSDWODPÕúWÕU>@¶GHnt4LoLQJOREDO

JoO o|]PQ YDUOÕ÷Õ ED]Õ YDUVD\ÕPODU DOWÕQGD HOGH HGLOPLúWLU %D]Õ YDUVD\ÕPODU

DOWÕQGDYHLoLQo|]POHULQDVLPSWRWLNGDYUDQÕúÕ5XQ]KDQJYH<DFKHQJ>@

WDUDIÕQGDQoDOÕúÕOPÕúWÕU

(13)

5

E|OPGH9.DODQWDURY YH$.XUWWDUDIÕQGDQ\ÕOÕQGD\D]ÕODQGHUHFHGHQ

OLQHHU ROPD\DQ ELU GDOJD GHQNOHPLQLQ X]XQ ]DPDQ GDYUDQÕúÕ LVLPOL oDOÕúPD >@

E|OPGH00H\YDFÕWDUDIÕQGDQ\ÕOÕQGD\D]ÕODQSVHXGRSDUDEROLFGHQNOHPOHULQ

o|]POHULQLQSDWODPDVÕLVLPOLoDOÕúPD >@GHWD\OÕELUúHNLOGHLQFHOHQHFHNWLU

*|UOG÷ JLEL \ÕOÕQGDQ EX \DQD GHQNOHPOHULQ GDYUDQÕúÕ ]HULQH ELUoRN

DUDúWÕUPD\DSÕOPÕúWÕU*HQLúELUDODQDVDKLSRODQGDYUDQÕúLQFHOHPHVLQLQJQGHQJQH

DUDúWÕUPDODUÕDUWPDNWDLQFHOHPHDODQÕJHQLúOHPHNWHGLr.

(14)

%g/h07(0(/7$1,09(.$95$0/$5

7DQÕP

Rn

: ELUE|OJHYH p SR]LWLIELUUHHOVD\ÕROVXQ : ]HULQGHWDQÕPODQPÕú

^p

u x dx

:

³

f

NRúXOXQXVD÷OD\DQ|OoOHELOLUuIRQNVL\RQODUÕVÕQÕIÕQDLp

: X]D\ÕGHQLU

1d fp LoLQ]HULQGHNLQRUP

1

ȍ ȍ

,

p

p p p p

L p

u _: § u dx· u u dx

¨ ¸

©

³

¹

³

úHNOLQGHWDQÕPODQÕU p LoLQ2 Lp

: +LOEHUWX]D\ÕGÕUYH]HULQGHNLLooDUSÕP

ȍ

, , , 2

u v

³

u x v x dx u vL :

ELoLPLQGHWDQÕPODQÕU

(úLWVL]OLN+|OGHUHúLWVL]OL÷L

1d p q, f ve 1 1

p q 1olsun.

(15)

7

Bu durumda uL Up

^,vL Uq

_LoLQ

p q

L U

U L U

u x v x dxd u v

³

HúLWVL]OL÷LVD÷ODQÕU

(úLWVL]OLN<RXQJHúLWVL]OL÷L

, R 0

p q ! pozitif UHHOVD\ÕROVXQ g\OHNL

1 1

p q 1RODFDNúHNLOGH

, 0

a b t R LoLQ

1

p q

a b

p q

ROPDOÕGÕU

, , 0

a b H

! ve ^q ^p

^p^{1 ,1}

^f^p LoLQDúD÷ÕGDNLHúLWVL]OLNJHoHUOLGLU

2

1 1

2 1 1

, .

2 2

p

ab a b ab a bq

p q

H

H H

H

d d

(úLWVL]OLN&DXFK\-6FKZDU]HúLWVL]OL÷L

, ⁿ

x yR

x y d x y

(úLWVL]OLN3RLQFDUH-)ULHGULFKVHúLWVL]OL÷L Rn

: VÕQÕUOÕELUE|OJHROVXQc : ! c( ) 0 ROPDN]HUH

(16)

1

u Wp

: _LoLQ

¹

* q

Lp c W

u _: d u _:

RODFDNúHNLOGHc c p n( , )!0VDELWLYDUGÕU%XUDGD ^* np

p np ROPDOÕGÕU

(úLWVL]OLN3RLQFDUH¶HúLWVL]OL÷L)

1

0 ȍ

u H

ve C HúLWVL]OLNOHULOJLOLVDELWROPDN]HUH

u dC u

ROPDOÕGÕU

(úLWVL]OLN/DG\]KHQVND\DHúLWVL]OL÷L)

1

0 ȍ

u H

ve C HúLWVL]OLNOHULOJLOLVDELWROPDN]HUH

1 1

2 2

u 4dC u u ROPDOÕGÕU

(úLWVL]OLN6REROHYHúLWVL]OL÷L)

1

0 ȍ

u H

ve qt , 2 C₁ HúLWVL]OLNOHLOJLOLVDELWROPDN]HUH

q 1

u dC ROPDOÕGÕU u

(17)

9

Lemma 2.1. (Lagnese, Haraux) E R: oRartmayan fonksiyon olsun ve bir 0

T ! VDELWLQLQROGX÷XQXIDU]HGHOLPg\OHNL

^,

t

E s ds TE t t R

f d

³

Bu durumda

⁰ ¹ ^T^t^,

E t dE e t t T

ROPDNWDGÕU

Lyapunov $QODPÕQGD.DUDUOÕd|]POHU 7DQÕP

^,

^*

, dx f x y

dt

dy g x y dt

½°°

¾°

°¿

VLVWHPLQLQELUo|]PROVXQ(÷HULVWHQLOGL÷LNDGDUNoNELUH !0VD\ÕVÕQDNDUúÕQ

|\OHELU^{G G H}

^!⁰ bulunabiliyorsa ki;

0 0

x t x t y t y t

G G

.RúXOXQXVD÷OD\DQNH\IL

^{x t}

^,^{y t}

o|]PLoLQ

(18)

, 0

x t x t

y t y t t t

H H

t

VD÷ODQVÕQ %X GXUXPGD

^{x t}

^,^{y t}

^o|]P

^* sisteminin Lyapunov DQODPÕQGD

NDUDUOÕ o|]PGU YH\D VDGHFH NDUDUOÕ o|]PGU GHQLU 7DQÕPGDQ DQODúÕODFD÷Õ

]HUHELUo|]PQNDUDUOÕROPDVÕLoLQKHUKDQJLELUt t0 DQÕQGDEXo|]PH\DNÕQ

RODQ EWQ o|]POHULQ t ! t0 GH÷HUOHULQGH GH EX o|]PH \DNÕQ NDOPDNWD GHYDP

etmelidir.

7DQÕP

^{x t} ^,^{y t}

^,

^* VLVWHPLQLQNDUDUOÕo|]PROVXQ(÷HUNH\IL

^{x t}

^,^{y t}

^o|]P

lim 0

t

x t x t y t y t

of

NRúXOXQX VD÷OÕ\RUVD EX KDOGH

^{x t}

^,^{y t}

DVLPSWRWLN NDUDUOÕGÕU GHQLU %X

WDQÕPODUGDNL o|]P VLVWHPLQ GHQJH QRNWDVÕ VÕIÕU o|]P LVH GHQJH QRNWDVÕ

NDUDUOÕGÕU YH\D DVLPSWRWLN NDUDUOÕGÕU GHQLU $\UÕFD YHULOHQ o|]PQ NDUDUOÕ ROXS

ROPDPDVÕt VD\ÕVÕQDED÷OÕGH÷LOGLUYH|]HORODUDN0 t0 DOÕQDELOLU 0

(19)

Equation Section 3

**%g/h0 '(5(&('(1 /ø1((5 2/0$<$1 %ø5 '$/*$**

'(1./(0ø1ø1 0$5,1( 5,6(5 (48$7,21 8=81=$0$1'$95$1,ù,

*LULúYH3UREOHPLQøIDGHVL

^0, xx

^0, ^, xx

^, ⁰

u t u t u l t u l t (3.1)

VÕQÕUNRúXOODUÕDOWÕQGD

^p ⁰

tt xxxx x x tx t t

mu ku ª¬a x u º¼ Ju bu u ^(3.2)

SUREOHPL HOH DOÕQVÕQ %XUDGD p m k b , , SR]LWLI VD\ÕODU J UHHO VD\Õ YH

¹

> @

^0,

a x C l _¶dir.

6ÕIÕUd|]PQ$VLPSWRWLN.DUDUOÕOÕ÷Õ

Teorem 3.1. Kabul edelim ki

i) , , p m k b SR]LWLIVD\ÕODUJ ELUUHHOVD\Õ

ii) ^a

^C¹

> @

^0,^l ^ve ^x

> @

^0,^l LoLQ c₀ t0 ROGX÷X GXUXPGD a x

t c0 ve

2 2

0 0 0

kc l S d !

úDUWODUÕVD÷ODQVÕQ

(20)

Bu durumda (3.1)-SUREOHPLQLQVÕIÕUo|]P 1

² ²

^1/2

0 l

t uxx dx

u § u ·

¨ ¸

©

³

¹ ^normuna

J|UHJOREDODVLPSWRWLNNDUDUOÕGÕUYHVÕQÕUNRúXOODUÕQÕVD÷OD\DQGHQNOHPLQLQ

KHUo|]P

>

1 / 2

2 0 2

2/ 2

0 0

, 0,1

( , ) ( , )

2 2 , 1, 1,

l l

t xx

p p

p

At p

u x t dx u x t dx

At p

m

t

d

d ®°

t f

³ ³

°¯ ^(3.3)

HúLWVL]OL÷LQL VD÷ODU %XUDGD A , u|_t 0, u_t|_t 0 EDúODQJÕo YHULOHULQH YH l c b p m, 0, , , ,J VD\ÕODUÕQDED÷OÕGÕU

øVSDW 3.1. ( , )u x t , (3.1) - SUREOHPLQLQELUo|]PROVXQ

(3.2) denklemi ile u_t oDUSÕOÕUVD

^p ⁰

tt t xxxx t x x t tx t t t t

mu u ku u ª¬a x u º¼ u Ju u bu u u ^(3.4)

elde edilir.

(3.4) denklemindeki YHULOHUWHNWHNHOHDOÕQGÕ÷ÕQGD

1 2

tt t 2 t

u u u

t w w

t xxxx xxx t tx xxx xxx t tx xx txx xx

u u u u u u u u u u u u

x x x

w w w

¹ ²

xxx t tx xx 2 xx

u u u u u

x t

w w

¹

²

x x t x t tx x x t 2 x

a x u u a x u u u a x u a x u u a x u

x x t

w w w

ª º

¬ ¼ w w w

¹

²

x t 2 x

a x u u a x u

x t

w w

(21)

13

1 2

tx t 2 t

u u u

x w w

elde edilir.

(OGHHGLOHQOHUGHQNOHPLQGH\HULQH\D]ÕOGÕ÷ÕQGD

²

¹ ²

2 ^t ^{xxx t} ^tx ^xx 2 ^xx x ^{x t}

m u k u u u u k u a x u u

t x t

w w w w

w w w w (3.5)

²

² ²

1 0

2 2

p

x t t

a x u u b u

t x

J

w w

ROPDNWDGÕU

(3.5) denklemi

^0,l ]HULQGHLQWHJUHHGLOLUVH

2 2

0 0 0

0 0

| | |

2 2

l l

x l x l x l

t t xxx x tx xx x xx x t x

d m d k

u dx k u u k u u u dx a x u u

dt

³

dt

³

²

² ⁰ ²

0 0

1 | 0

2 2

l l

x l p

x t x t

d a x u dx u b u dx

dt

J

³

olur.

¶GHNLVÕQÕUNRúXOODUÕQDED÷OÕRODUDNGHQNOHPL

2 2 2 2

0 0

1 0.

2 2 2

l l

p

t xx x t

d m k

u u a x u dx b u dx

dt

ª º

« »

¬

³

¼

³

^(3.6)

ROPDNWDGÕU

(22)

(3.2) denklemi u LOHoDUSÕOÕUVD

^p ⁰

tt xxxx x x tx t t

mu u ku uª¬a x u º¼ uJu u bu u u ^(3.7)

denklemi elde edilir.

GHQNOHPLQGHNLYHULOHUWHNWHNHOHDOÕQGÕ÷ÕQGD

²

tt t t

u u u u u

t

w

²

xxxx xxx x xxx xxx xx x xx xxx xx x xx

u u u u u u u u u u u u u u u u

x x x x

w w §¨ w ·¸ w

w w ©w ¹ w

x x

x

²x

a x u u a x u u u a x u a x u u a x u

x x

w w

ª º

¬ ¼ w w

tx t t x

u u u u u u x

w

ifadeleri meydana gelir.

(OGHHGLOHQOHUGHQNOHPLQGH\HULQH\D]ÕOGÕ÷ÕQGD

t t²

xxx xx x

²xx

x

m u u u k u u u u ku a x u u

t x x

w w w

ª º

«w » w w

¬ ¼ (3.8)

x² ^Ȗ

t ^Ȗ t x t t ^p

a x u u u u u buu u x

w

ROPDNWDGÕU

(23)

15

(3.8) denklemi x¶HJ|UHintegre edilirse,

2 2

0 0 0

| | |

l l l

x l x l x l

t t xxx x x xx x xx x x

m d uu dx m u dx k uu k u u k u dx a x uu

dt

³

²

0

0 0 0

Ȗ _

l l l

x l p

x t x x t t t

a x u dx u u J u u dx b uu u dx

³

ifadesi elde edilir.

VÕQÕUNRúXOODUÕQDED÷OÕRODUDNEXGHQNOHP

² ²

²

0 0

, , 0

l l

p

t t xx x x t t t

d m u u m u k u a x u dx u u b uu u dx

dt ª¬ º¼

³

J

³

^(3.9)

ROPDNWDGÕU

(3.9) denklemi, G₁ !0 LOHoDUSÕOÕSGHQNOHPLQHHNOHQLUVH

2 2 2 2 2

1 1 1

0

1 ,

2 2 2

l

t xx x t t xx

d m k

u u a x u dx m u u m u k u

dt ^ª« G ^º»G G

¬

³

¼

²

1 1 1

0 0 0

, 0

l l l

p p

x x t t t t

a x u dx u u b uu u dx b u dx

G G J G

³

^(3.10)

denklemi elde edilir.

² ²

²

1 1

0

1 ,

2 2 2

l

t xx x t

m k

E t u u

³

a x u dxGm u u ^(3.11)

olsun.

(24)

Bu durumda (3.10) denklemi

² ²

²

1 1 1 1 1

0

1 2

1

0 0

,

0

l

t xx x x t

l l

p p

t t

d E t m u k u a x u dx u u

dt

b u u dx b u dx

G G G G J

G

d

³

³ ³

(3.12)

úHNOLQGH\D]ÕODELOLU.ROD\OÕNODJ|VWHULOHELOLUNL

2

2 2

x 2 xx

u l u

dS (3.13)

GÕU%XUDGD

uux x ux²u uxx denklemi x¶HJ|UHLQWHJUHHGLOGLNWHQVRQUD

²

0 0 0

l l l

x x x xx

uu u dx uu dx

³ ³ ³

2 0

0 0

|

l l

l

x x xx

uu

³

u dx

³

uu dx

2

0 0

*

l l

x xx

u dx uu dx

³ ³

elde edilir.

&DXFK\HúLWVL]OL÷LYH3RLQFDUH-)ULHGULFKVHúLWVL]OL÷LQGHQ>@VD\ID

2

2 2

2 0 0

l l

x

u dx l u dx dS

³ ³

^(3.14)

\D]ÕODELOLU

(25)

17

^* HúLWVL]OL÷L\DUGÕPÕ\OD

1 1 1 1

2 2 2 2

2 2 2 2 2

0 0 0 0

0 0 0

l l l l l l l

x xx xx xx xx

u dx uu dx u u dx u dx u dx l u dx u dx

S

§ · § · § · § ·

d d d©¨¨ ¸¸¹ ¨© ¸¹ d ¨© ¹¸ ©¨¨ ¸¸¹

³ ³ ³ ³ ³ ³ ³

ROGX÷XJ|UOU 6RQHúLWVL]OLNWHQ

1 1 2 2

2 2

0 0

l l

x xx

u dx l u dx S

§ ·

d ¨ ¸

¨ ¸ ¨ ¸

©

³

¹ ©

³

¹

elde edilir.

DúD÷ÕGDNLHúLWVL]OLNWHNXOODQÕOÕUVD

² ¹² ² ² ² ¹² ^{2 2} ²

1 1 2

1 1

, 2 2 2 2

x t x t t x t xx

u u u u u G J u u G J l u

G J G J

d d d S (3.15)

ifadesi meydana gelir.

HúLWVL]OL÷LQLNXOODQDUDNGHQNOHPL

¹² ^{2 2} ² ²

²

1 1 2 1 1

0

1 2

1

0 0

1

2 2

0

l

xx t x

l l

p p

t t

l

d E t k u m u a x u dx

dt

b u u dx b u dx

G G J G G

S

G

§ · § ·

¨ ¸ ¨ ¸

© ¹

d

³

³ ³

(3.16)

úHNOLQGH\D]ÕOÕU

2 0

1 2 2

0 2 d

l G S

J (3.17)

(26)

olsun. Bu durumda,

2 2 2 1

1 2 0

2 L Gk G J l

S !

olur. 7HRUHP¶LQLLNRúXOX\DUGÕPÕ\OD

2 0

0 2

d k c l S

2 2 1

1 2 0, 1 0

2 k G J l

G G

S

§ ·

¨ ¸!

2 2 2 2

1 1

2 0, 2

2 2

l l

k G J k G J

S S

! !

2 2 2 2

2 2

0 0

2 2 0 2 2 2

1 1 2

2 2

, ,

0 k k c l k c l

l d l

S S

S J

G J G

S

§ ·

¨ ¸

© ¹

2 0

1 2 2

2 d l G S

J ROPDNWDGÕU

%XGXUXPGDHúLWVL]OL÷LQGHQ

² ² ² ²

²

1 1 1

0

1 2

1

0 0

1 2

l

t t t xx x

l l

p p

t t

d E t m u m u m u L u a x u dx

dt

b u u dx b u dx

G G

G

§ ·

d¨© ¸¹

³

³ ³

(3.18)

0

a x t c ROGX÷X LoLQGHQNOHPLQGHQ

(27)

19

²

2 2 2 2 0 2

0 2

0 0 0 0 0

l l l l l

xx x xx x xx

k u dx a x u dx k u dx c u dx k c l u dx S

§ ·

t t¨ ¸

© ¹

³ ³ ³ ³ ³

^(3.19)

₀ ²

0 l

d

³

u dxxx

ROPDNWDGÕr.

² ²

²

0

1

2 2 2

l

t xx x

m k

E t u u

³

a x u dx

olsun.

%|\OHFH\DUGÕPÕ\OD

² ⁰ ² ^min ^, ⁰ ₁²

2 ^t 2 ^xx 2 2

d d

m m

E t t u u t ® ½¾ u

¯ ¿ (3.20)

ve ^a

^. ^C¹

> @

^0,^l ROGX÷XQGDQ¶GHQ

² ²

²

0

1

2 2 2

l

t xx x

m k

E t d u u

³

a x u dx

2 2 2

0

max

2 2 2

l

t xx x

m k a x

u u u dx

d

³

2 2 2 1 2

2

2 2 2 0

l

t xx xx

l A

m k

u u u dx

d S

³

2 2

2 1 2 1 2

2 max , 2 1

2 ^t 2 2 ^xx 2 2 2

l A l A

m k m k

u u u

S S

§ · ½

d ¨ ¸ d ® ¾

ROPDNWDGÕU

(28)

Burada

> @

1 max0,

x l

A a x

¶GÕU

(3.6) denkleminden

²

0

l p t

d E t b u dx dt

³

d ^(3.21)

elde HGLOLU %|\OHFH ^{E t}

ELU /\DSXQRY IRQNVL\RQGXU YH GHQNOHPLQLQ VÕIÕU

o|]PNDUDUOÕGÕU

HúLWVL]OL÷LQGHQ

²

0 0

t l p

E t E b

³³

ut dxds bulunur.

⁰

E t t ROGX÷XQGDQ

2

0 0 t l 0

p t

u dxds E b

d

³³

^(3.22)

elde edilir.

^ `

1 min 1, , 1

D L kG olsun.

(÷HU

²

0 l

a x u dxx

³

QHJDWLIGH÷LOVH

(29)

21

2 2 2

1 0 l

t xx x

m u L u G

³

a x u dx

^

1

`

² ²

²

0

min 1, ,

l

t xx x

L k G ^§m u k u a x u dx^·

t ¨ ¸

©

³

¹ ^(3.23)

1

2D E t t

ROGX÷XDoÕNWÕU

2 2

1 1

2 min 1, 0 2

2

D d l

k

G G J

S

§ ·½

° °

® ¨ ¸¾

° © ¹°

¯ ¿ olsun.

(÷HU

²

0 l

a x u dxx

³

negatifse

2 2 2 2 2 2

1 1 0

0 0

l l

t xx x t xx x

m u L u G

³

a x u dxtm u L u Gc

³

u dx

2

2 2 1 0 2

2 0 l

t xx xx

m u L u G c l u dx

t S

³

2 2 2 2

2 1 1 0 2

1 2 2 2

t xx

l c l

m u Gk G J G u

S S

§ ·

¨ ¸

© ¹

2 2 2 2

2 0 1 2

1 2 2

t 2 xx

c l l

m u G k G J u

S S

ª § · º

« ¨ ¸ »

© ¹

¬ ¼

2 2 2

2 1 2

1 0 2

t 2 xx

d l

m u G G J u

S

ª º

« »

¬ ¼

(30)

2 1 2 2 2

1 0 2

t 2 xx

l u d

m u G G J

S

§ ·

¨ ¸

© ¹

2 2

1 1

0 2

min 1,

2 ^t ^xx

d l m u k u

k

G G J

S

§ ·½

° °

t ® ¨ ¸¾

° © ¹°

¯ ¿

2 2 2

2 2

0

2

l

t xx x

D §m u k u a x u dx· D E t

t ¨ ¸

©

³

¹

ifadesi elde edilir. %|\OHFH

2 2 2

1 3

0

2

l

t xx x

m u L u G

³

a x u dxt D E t ^(3.24)

GÕU%XUDGD^D₃ ^min

^

^{D D}₁^, ₂

`

¶GLU

HúLWVL]OL÷L\DUGÕPÕ\OD¶GHQ

² ² ² ²

1 1

1

2 ^t ^t ^t ^xx

d E t m u m u m u L u

dt d^§¨©G ^·¸¹

1

² 1 ¹ ²

0 0

0

l l l

p p

x t t

a x u dx b u u dx b u dx

G G

³

ROPDNWDGÕU%XUDGDQ

²

¹ ²

1 3 1

0 0

2

l l

p p

t t t

d E t B u D E t b u u dx b u dx

dt d G

³

^(3.25)

HúLWVL]OL÷LHOGHHGLOLU%XUDGD 1

1

B G m 2 m ¶dir.

(31)

23

GHQNOHPLQLQWUHYLQHJDWLIWLU %X ^{E t}

¶QLQDUWPD\DQROGX÷XQX J|VWHULU %X

\]GHQHúLWVL]OL÷LQLQ t¶\HJ|UHLQWHJUHHGLOPHVL\OH

²

3 3 1 1

0 0 0

1 2

1

0 0 0 0

2 2 0

t t l

t

t l t l

p p

t t

D tE t D E s ds E E t B u dxds

bG u u dxds b u u dxds

d dª¬ º¼

³ ³³

³³ ³³

(3.26)

ROPDNWDGÕU

¶GDNLHúLWVL]OLNWHQ

² ¹

3 1 1 1

0 0 0 0

2 0

t l t l

p

t t

D tE t d¬ªE E t º¼B

³³

u dxds b G

³³

u u dxds ^(3.27)

Poincare-)ULHGULFKVHúLWVL]OL÷LYH¶WHNLHúLWVL]OLN\DUGÕPÕ\OD

4

2 2

4 xx

u l u

dS (3.28)

2 2 2 4

2 2 2 2

2 x 2 2 xx 4 xx

l l l l

u u u u

S S S S

§ ·

d d d

¨ ¸

© ¹

elde edilir. %XHúLWVL]OLNNXOODQÕODUDNE t1

tahmin edilebilir.

² ²

²

1 1

0

1 ,

2 2 2

l

t xx x t

m k

E t u u

³

a x u dxGm u u

2 2 0 2

1 0

2 2 2 ,

l

t xx x t

m k c

u u u dx G m u u

t

³

(32)

2 2 0 2 2

2 1 ,

2 ^t 2 ^xx 2 ^xx ^t

c

m k l

u u u Gm u u

t S

2 0 2 2 1 1 2

2

1 2

2 2

2 ^t 2 ^xx ^t

c l m m

m u k u G u G u

S

§ ·

t ¨ ¸

© ¹

2 0 2 1 2 1 2

2 ^t 2 ^xx 2 2 ^t

d m m

m u u G u G u

t

1

² 0 ¹ 4 ⁴ ²

1 1

2 ^t 2 ^xx

m ml

u d G u

G S

§ ·

t ¨ ¸

ROPDNWDGÕU

%|\OHFH 1 2 2² ⁰ ⁰ 4⁴

min 2 d ,1,d

l ml

S S

G J

½

® ¾

¯ ¿ ^LoLQ

² ²

1 1 t xx

E t td u u (3.30)

VD÷ODQÕU%XUDGD 1 1 0 ¹ 4 ⁴

1min ,

2

d m Gm d Gml S

½

® ¾

¯ ¿^¶GLU

%|\OHFH

1 0 1 1 0

E E t dE (3.31)

dir. ^u

^0,^t ⁰ ROGX÷XQGDQ

0

, ,

x

u x t

³

ux s t ds^¶GLU

> @

^0,

x l

_LoLQ

(33)

25

1 2 2

0 0 0

, , , ,

x l l

x x x

u x t u s t ds u x t ds l§ u x t dx·

d d d ¨ ¸

© ¹

³ ³ ³

1 1 1

2 2 2

2 2

2

0 0 0 0

0

, 1 1

l l l l l

x x x x

u x t dx u dx dx u dx l u dx

§ § · § · § · ·

¨ d¨ ¸ ¨ ¸ d ¨ ¸ ¸

¨ ¸

© ¹

³ ³ ³ ³ ³

olur. Bu durumda

> @

1 2 2

0,

0

max ,

l

x l u x t l ux dx

§ ·

d ¨ ¸

©

³

¹ ^(3.32)

HOGHHGLOLU%|\OHFH ¶GHQqt1LoLQ

_{> @}

1 1

2 2

0, 0 0

1 1

1

, ^qmax , ^q 2 ,

l q l

q

x l x

u x t dx l u x t l u x t dx

§ · § ·

¨ ¸ ¨ ¸

§ · § ·

d d

¨ ¸ ¨ ¸

¨ ¸ © ¹

©

³

¹

³

ROPDNWDGÕU%uradan

² ²

²

0 2

0

, ,

l l

q q

x

q

u x t dx l § u x t dx·

d ¨ ¸

© ¹

³ ³

^(3.33)

olur.

+|OGHU HúLWVL]OL÷L YH HúLWVL]OL÷L NXOODQÕODUDN ²

0 0 t l

B

³³

u dxdst ^ve 1

1 0 0 t l

p

bG

³³

u ut dxds terimleri hesaplanabilir.

Bazı tipten kısmi türevli denklemlerin çözümlerinin davranışı

FEN BİLİMLERİ ENSTİTÜSÜ

BAZI TİPTEN KISMİ TÜREVLİ DENKLEMLERİN ÇÖZÜMLERİNİN DAVRANIŞI

YÜKSEK LİSANS TEZİ

7(ù(..h5

ødø1'(.ø/(5

6ø0*(/(59(.,6$/70$/$5/ø67(6ø

³

g=(7

BEHAVIOR OF SOLUTIONS OF SOME TYPES OF PARTIAL DIFFERENTIAL EQUATIONS

SUMMARY

Equation Section 1

%g/h0*ø5øù

³

%g/h07(0(/7$1,09(.$95$0/$5

³

³

³

³

³

³

Equation Section 3

%g/h0   '(5(&('(1 /ø1((5 2/0$<$1 %ø5 '$/*$

'(1./(0ø1ø1 0$5,1( 5,6(5 (48$7,21 8=81=$0$1'$95$1,ù,

> @

> @

> @

³

>

³ ³

³

³

³

³

³

³

³

**6ø0*(/(59(.,6$/70$/$5/ø67(6ø**

**%g/h0*ø5øù**

%g/h07(0(/7$1,09(.$95$0/$5

**%g/h0 '(5(&('(1 /ø1((5 2/0$<$1 %ø5 '$/*$**