Doğrusal dalga denkleminin çözümlerinin düzgün kararlılığı

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$]L]H087/8¶\DVRQVX]WHúHNNUOHULPLVXQDUÕP

<NVHNOLVDQVoDOÕúPDODUÕPVÕUDVÕQGDKLoELUNRQXGDEHQGHQ\DUGÕPODUÕQÕHVLUJHPH\HQ

ablam +DVHQH 087/8 *(1d.$/¶D YH HQLúWHP %HUNDQW *(1d.$/¶D VRQVX]

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Į : Alfa

ȕ : Beta

Ȗ : Gama

İ : Epsilon

Ș : Eta

ȟ : Ksi

ĭ : Phi

Ȍ : Psi

ȍ : Omega

Ȟ : Nu

ǻ : Laplasyen

׏ : 1DEODRSHUDW|U

ܴ_ା : ሾͲǡ λሻ

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Anahtar kelimeler: 'R÷UXVDOøQWHJUDO(úLWVL]OL÷L'R÷UXVDO Dalga Denklemi.

%X WH]  E|OPGHQ ROXúPDNWDGÕU %LULQFL E|OPGH WH]GH NXOODQÕODQ QRWDV\RQODU YH

WHPHO HúLWVL]OLNOHU YHULOPLúWLU $\UÕFD GR÷UXVDO LQWHJUDO HúLWVL]OL÷L YHULOPLú YH

LVSDWODQPÕúWÕU øNLQFL E|OPGH GR÷UXVDO dalga denkleminin o|]POHULQLQ G]JQ

NDUDUOÕOÕ÷Õ LQFHOHQPLúWLU hoQF E|OPGH \DUÕ GR÷UXVDO dalga denkleminin o|]POHULQLQ G]JQ NDUDUOÕOÕ÷Õ LQFHOHQPLúWLU '|UGQF E|OPGH LVH WH]

oDOÕúPDVÕQGDQHOGHHGLOHQVRQXoODUEHOLUWLOPLúWLU

UNIFORM STABILIZATION OF THE SOLUTION TO LINEAR

WAVE EQUATION

SUMMARY

KeyWords: Linear Integral Inequality, Linear Wave Equation

This thesis consists of four chapters. In the first chapter, notations and main inequalities used in the thesis are given. Linear integral inequality is given and proved. In the second chapter, uniform stabilization of the solution to linear wave equation is examined. In the third chapter, uniform stabilization of the solution to quasilinear wave equation is examined. Finally in the fourth chapter, the results are stated gained through the study of thesis.

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1.1. Notasyonlar

%XE|OPGHLúDUHWOHUYHVHPEROOHUWDQÕWÕODFDNWÕU

ܴ^௡QER\XWOXgNOLWX]D\ÕGÕU

ݔ ൌ ሺݔ_ଵǡ ݔ_ଶǡ ǥ ǡ ݔ_௡) ܴ^௡ GHELUQRNWDGÕU

ȳ, ܴ^௡GHVÕQÕUOÕELUE|OJHGLU.

߲ȳ, ȳ E|OJHVLQLQG]JQVÕQÕUÕGÕU

ݑ௫ ve ݑ௧ sembolleri, ^డ௨

డ௫ ve ^డ௨

డ௧ NÕVPLWUHYOHULGLU

ܮ௣ሺߗሻǡ ሺ݌ ൒ ͳሻȳ ]HULQGH|OoOHELOLUIRQNVL\RQODUÕLoHUHQ%DQDFKX]D\ÕGÕUYH

ԡݑԡ_௣ǡఆ ൌ ቀ׬ ȁݑȁ_ఆ ^௣݀ݔቁ^{ଵ ௣Τ} (1.1)

sonlu norma sahiptir.

Genel olarak ܮ_ଶሺߗሻ deki norm ԡήԡ úHNOLQGHJ|VWHULOPLúWLUݑ ve ߥ QLQVNDOHUoDUSÕPÕ

ሺݑǡ ߥሻ_ఆ ൌ ׬ ݑߥ݀ݔ_ఆ (1.2)

úHNOLQGHJ|VWHULOLU

ԡݑԡଶ ൌ ቀ׬ ݑ_ఆ ^ଶ݀ݔቁ^{ଵ ଶΤ} (1.3)

ԡ׏ݑԡଶ ൌ ቀ׬ ȁ׏ݑȁ_ఆ ^ଶ݀ݔቁ^{ଵ ଶΤ} (1.4)

*UHHQ)RUPO.ÕVPLLQWHJUDV\RQXQJHQHOOHúWLULOPLúKDOLRODQ*UHHQIRUPO ^డ௚

డఔ

,

GÕúQRUPDOLQHJ|UHWUHYLJ|VWHUPHN]HUH

׬ ݂ο݃݀ݔ ൌ െ ׬ ׏݂ ή ׏݃݀ݔ ൅ ׬ ݂_{ఆ ఆ డఆ} ^߲݃_߲ߥ݀ݔ (1.5)

úHNOLQGHGLU

1.2. 7HPHO(úLWVL]OLNOHU

1) &DXFK\(úLWVL]OL÷L:

ܽǡ ܾ VDELWUHHOVD\ÕODUÕLoLQ

ܾܽ ൑ ^௔_ଶ^మ

൅

dir.

ʹሻߝ- <RXQJ(úLWVL]OL÷L

ܽǡ ܾǡ ߝ SR]LWLIUHHOVD\ÕODUYH݌ǡ ݍ ൐ ͳ LoLQ ^ଵ_௣

൅

ܾܽ ൑^ߝ݌_݌^ܽ݌

൅

úHNOLQGHGLU

+|OGHU(úLWVL]OL÷L

ͳ ൑ ݌ǡ ݍ ൑ λ ve ^ଵ_௣

൅

׬ ȁݑߥȁ݀ݔ ൑ ԡݑԡఆ ௣ԡߥԡ௤ (1.8)

GLUg]HORODUDN ݌ ൌ ݍ ൌ ʹ DOÕQÕUVD Cauchy-6FKZDU]HúLWVL]OL÷LHOGHHGLOLU

4) PoincareȂ )ULHGULFKV(úLWVL]OL÷L

׬ ݑఆ ^ଶ݀ݔ ൑ ܿଵሺߗሻ ׬ ȁ׏ݑȁ^ଶ݀ݔǡݑ א ܪ௢ଵ

ఆ ሺߗሻ (1.9)

ȳ, ܴ^௡X]D\ÕQGDVÕQÕUOÕELUE|OJHGLUve ܿଵሺߗሻ sabiti ȳE|OJHVLQHED÷OÕGÕU

5) *URQZDOO(úLWVL]OL÷L7UHY)RUPX

ߟሺήሻǡሾͲǡ ܶሿ GHQHJDWLIROPD\DQVUHNOLELUIRQNVL\RQROVXQ $\UÕFDKHPHQKHPHQher ݐ LoLQ

ߟ^ƍሺݐሻ ൑ ߔሺݐሻߟሺݐሻ ൅ ߖሺݐሻ (1.10)

HúLWVL]OL÷L VD÷ODQVÕQ. Burada Ȱሺݐሻ ve Ȳሺݐሻ negatif olmayan ve ሾͲǡ ܶሿ de integre edilebilHQIRQNVL\RQODUGÕU%XUDGDQKHU0൑ ݐ ൑ ܶ LoLQ

ߟሺݐሻ ൑ ݁^{׬ ఃሺ௦ሻௗ௦బ೟} ቂߟሺͲሻ ൅ ׬ ߖሺݏሻ݀ݏ_଴^௧ ቃ (1.11)

dir.

*URQZDOO(úLWVL]OL÷LøQWHJUDO)RUPX

ߦሺݐሻǡ ሾͲǡ ܶሿ GH QHJDWLI ROPD\DQ LQWHJUDOOHQHELOLU ELU IRQNVL\RQ ROVXQ $\UÕFD KHPHQ

hemen her ݐ ve ܿ_ଵǡ ܿ_ଶ ൒ Ͳ LoLQ

ߦሺݐሻ ൑ ܿ_ଵ׬ ߦሺݏሻ݀ݏ ൅ ܿ௧ ଶ

଴ (1.12)

HúLWVL]OL÷LVD÷ODQVÕQ%XGXUXPGDKHPHQKHPHQKHU Ͳ ൑ ݐ ൑ ܶ LoLQ

ߦሺݐሻ ൑ ܿ_ଶሺͳ ൅ ܿ_ଵݐ݁^௖భ௧ሻ (1.13)

dir.

1.3. 'R÷UXVDO øQWHJUDO(úLWVL]OL÷L

Lemma1.1.

ܧǣ ܴ_ା ՜ ܴ_ା fonksiyonu artmayan bir fonksiyon olsun.

׬ ܧሺݏሻ݀ݏ ൑ ܶܧሺݐሻǡ׊ݐ א ܴ’ ା

௧ (1.14)

oODFDNELoLPGHELU ܶ ൐ Ͳ var ise bu durumda

ܧሺݐሻ ൑ ܧሺͲሻ݁^ଵି೅೟ǡ׊ݐ ൒ ܶ (1.15)

dir.

øVSDW

݂ሺݔሻ ൌ ݁^௫Ȁ்׬ ܧሺݏሻ݀ݏǡ_௫^’ ݔ א ܴା

úHNOLQGH ELU IRQNVL\RQ WDQÕPODQVÕQ f fonksiyonu VUHNOLGLU D\UÕFD (1.14) HúLWsizOL÷LQGHQ f fonksiyonu arWPD\DQGÕU%XUDGDQ IIRQNVL\RQXQGDWUHYDOÕQDUDN

݂^ƍሺݔሻ ൌ_்^ଵ݁^௫Ȁ்׬ ܧሺݏሻ݀ݏ ൅ ݁_௫^{’ ௫Ȁ் ௗ}_ௗ௫൫׬ ܧሺݏሻ݀ݏ_௫^’ ൯ (1.16)

elde edilir. (1.16) HúLWOL÷LQGH /HLEQL]IRUPOQGHQ

ௗ

ௗ௫൫׬ ܧሺݏሻ݀ݏ_௫^’ ൯ ൌ ܧሺ’ሻ^ௗ’_ௗ௫െ ܧሺݔሻ^ௗ௫_ௗ௫ൌ െܧሺݔሻ

bulunur. െܧሺݔሻ HúLWOLNWH \HULQH\D]ÕOÕUVD

݂^ƍሺݔሻ ൌ_்^ଵ݁^௫Ȁ்׬ ܧሺݏሻ݀ݏ െ ݁_௫^{’ ௫Ȁ்}ܧሺݔሻ

úHNOLQLDOÕU $\UÕFDHúLWOLN ^ଵ_்݁^௫Ȁ் RUWDNoDUSDQSDUDQWH]LQHDOÕQÕUVD

݂^ƍሺݔሻ ൌ_்^ଵ݁^௫Ȁ்൫׬ ܧሺݏሻ݀ݏ െ ܶܧሺݔሻ_௫^’ ൯

elde edilir. (1.14) HúLWsizOL÷LQGHQ

ଵ

்൐ Ͳǡ ݁^{௫ ்Τ} ൐ Ͳ ve ׬ ܧሺݏሻ݀ݏ െ ܶܧሺݔሻ ൑ Ͳ_௫^ஶ ROGX÷XELOLQGL÷LQGHQ

݂^ᇱሺݔሻ ൌ^ଵ_்݁^௫Ȁ்൫׬ ܧሺݏሻ݀ݏ െ ܶܧሺݔሻ_௫^ஶ ൯ ൑ Ͳ

bulunur. HúLWsizOL÷LQGHQ

݂ሺݔሻ ൑ ݂ሺͲሻ ൌ ׬ ܧሺݏሻ݀ݏ ൑ ܶܧሺͲሻǡ׊ݔ א ܴஶ ା

௢

GÕr. Buradan

݂ሺݔሻ ൌ ݁^௫Ȁ்׬ ܧሺݏሻ݀ݏ ൑ ܶܧሺͲሻ_௫^ஶ

HOGHHGLOLUhVWWHNLHúLWVL]OL÷LQKHULNLWDUDIÕ ݁^ି௫Ȁ் LOHoDUSÕOÕUVD

׬ ܧሺݏሻ݀ݏ ൑ ܶܧሺͲሻ݁’ ^ି௫Ȁ்ǡ׊ݔ א ܴା

௫ (1.17)

bulunur. ܧ SR]LWLIYHDUWPD\DQROGX÷XQGDQGROD\Õ

׬ ܧሺݏሻ݀ݏ ൌ ׬_௫^’ _௫^௫ା்ܧሺݏሻ݀ݏ ൅ ׬ ܧሺݏሻ݀ݏ ൒ ׬_௫ା்^’ _௫^௫ା்ܧሺݏሻ݀ݏ

GLUhVWWHNLHúLWVL]OL÷LQVD÷WDUDIÕQGDNLLIDGH\HLQWHJUDOOHULoLQRUWDODPDGH÷HUWHRUHPL

X\JXODQÕUVD

׬_௫^௫ା்ܧሺݏሻ݀ݏ ൌ ሺݔ ൅ ܶ െ ݔሻܧሺݐሻ ൌ ܶܧሺݐሻݔ ൏ ݐ ൏ ݔ ൅ ܶ

׬_௫^௫ା்ܧሺݏሻ݀ݏ ൌ ܶܧሺݐሻ ൒ ܶܧሺݔ ൅ ܶሻ

ROXUHúLWVL]OL÷LQGHQ

ܶܧሺݔ ൅ ܶሻ ൑ ׬ ܧሺݏሻ݀ݏ ൑ ܶܧሺͲሻ݁_௫^{’ ି௫Ȁ்}

ROGX÷XQGDQGROD\Õ

ܶܧሺݔ ൅ ܶሻ ൑ ܶܧሺͲሻ݁^ି௫Ȁ்

GLUhVWWHNLHúLWVL]OL÷LQKHULNLWDUDIÕ ͳ ܶΤ LOHoDUSÕOÕUVD

ܧሺݔ ൅ ܶሻ ൑ ܧሺͲሻ݁^ି௫Ȁ்ǡ׊ݔ א ܴ_ା

bulunur. ݔ ൅ ܶ ൌ ݐ ֜ ݔ ൌ ݐ െ ܶ G|QúP\DSÕOÕUVD

ܧሺݐሻ ൑ ܧሺͲሻ݁^{ି೟ష೅೅}

elde edilir. hVWWHNLHúLWVL]OLNWHNUDU\D]ÕOÕUVD

ܧሺݐሻ ൑ ܧሺͲሻ݁^ଵି೅೟ǡ׊ݐ ൒ ܶ

bulunur YHLVSDWWDPDPODQÕU

1.4..DUDUOÕOÕN

ௗ௫

ௗ௧ ൌ ݂൫ݐǡ ݔሺݐሻ൯ (1.18)

GHQNOHPLQLQo|]POHULQLQNDUDUOÕOÕ÷ÕQՁoNDWHJRULGHLQFHOHQHELOLU

1) Lyapunov $QODPÕQGD.DUDUOÕOÕN:

׊ߝ ൐ Ͳ ve ݐ଴ א ܴ YHULOGL÷LQGHGHQNOHPLQLQo|]PRODQݔሺݐሻ LoLQH÷HU

ȁݔሺݐ଴ሻ െ ݔҧሺݐ଴ሻȁ ൏ ߜ iken ȁݔሺݐሻ െ ݔҧሺݐሻȁ ൏ ߝ׊ݐ ൒ ݐ଴

úDUWÕQÕVD÷OD\DQELU ߜ ൌ ߜሺߝǡ ݐ_଴ሻ ൐ Ͳ bulunabilirse ݔҧሺݐሻ o|]PQHNDUDUOÕGÕUGHQLU

$VLPSWRWLN.DUDUOÕOÕN:

(1.18) in o|]P RODQݔҧሺݐሻ Lyapunov anlamÕQGD NDUDUOÕ YH׊ݐ_଴ א ܴ YHULOGL÷LQGH

ݔሺݐሻ o|]PLoLQH÷HU

ȁݔሺݐ଴ሻ െ ݔҧሺݐ଴ሻȁ ൏ ߜ iken Ž‹௧՜ஶȁݔሺݐሻ െ ݔҧሺݐሻȁ ൌ Ͳ

úDUWÕQÕVD÷OD\DQELUߜ ൌ ߜሺݐ_଴ሻ ൐ Ͳ bulunabilirse ݔҧሺݐሻ o|]PQHDVLPSWRWLNNDUDUOÕGÕU

denir.

']JQ.DUDUOÕOÕN:

׊ߝ ൐ Ͳ YHULOGL÷LQGHGHQNOHPLQLQo|]PRODQݔሺݐሻ LoLQH÷HU ED]Õ ݐ_଴ א ܴ LoLQ

ȁݔሺݐ଴ሻ െ ݔҧሺݐ଴ሻȁ ൏ ߜ iken ׊ݐ ൒ ݐ଴ LoLQȁݔሺݐሻ െ ݔҧሺݐሻȁ ൏ ߝ

úDUWÕQÕVD÷OD\DQߜ ൌ ߜሺߝሻ ൐ Ͳ bulunabilirse ݔҧሺݐሻ o|]PQHG]JQNDUDUOÕGÕUGHQLU

'R÷UXVDOYH<DUÕ'R÷UXVDO'LIHUHQVL\HO'HQNOHP

'R÷UXVDO'LIHUHQVL\HO'HQNOHP

(÷HUELU diferensiyel denklem bilinmeyen fonksiyon ve bilinmeyen fonksiyonun var RODQ WUHYOHULQH J|UH ELULQFL GHUHFHGHQGR÷UXVDO LVH GLIHUHQVL\HO GHQNOHPH

GR÷UXVDOGÕUGHQLU

<DUÕ'R÷UXVDO'LIHUHQVL\HO'HQNOHP

%LUNÕVPLGLIHUHQVL\HOGHQNOHPGHQNOHPGHJ|UOHQ HQ\NVHNPHUWHEHGHQWUHYOHUH

J|UHGR÷UXVDOLVHGHQNOHPH\DUÕGR÷UXVDOGHQNOHPGHQLU

%g/h0'2ö586$/ DALGA '(1./(0ø1ø1

dg=h0/(5ø1ø1'h=*h1.$5$5/,/,ö,

%X E|OPGH GR÷UXVDO dalga denkleminin o|]POHULQLQ G]JQ NDUDUOÕOÕ÷Õ

incelenecektir. $úD÷ÕGDNLGDOJDGHQNOHPLHOHDOÕQVÕQ

ݑ௧௧െ οݑ ൅ ݍݑ ൌ Ͳሺݔǡ ݐሻ א ሺߗ ൈ ܴାሻ

߲_ఔݑ ൅ ܽݑ ൅ ݈ݑ_௧ ൌ Ͳሺݔǡ ݐሻ א ሺī_ଵൈ ܴ_ାሻ (2.3) ݑሺͲሻ ൌ ݑ଴ ve ݑ௧ሺͲሻ ൌ ݑଵݔ א ߲ȳ (2.4)

ve gǣ ܴ ՜ ܴ IRQNVL\RQXD]DOPD\DQVUHNOLELUIRQNVL\RQROVXQ.

gሺͲሻ ൌ Ͳ ve

݉ሺݔሻ ൌ ݔ െ ݔ_଴, ݔ א ܴ^௡

ܴ ൌ ܴሺݔ_଴ሻ ൌ ݏݑ݌ሼȁݔ െ ݔ଴ȁǣ ݔ א ȳሽ ī_ା ൌ ሼݔ א īǣ ݉ሺݔሻ ή ߥሺݔሻ ൐ Ͳሽ

݀ī_୫ൌ ሺ ή Ȟሻ†ī

ROPDN]HUHDúD÷ÕGDNLúDUWODUÕQVD÷ODQGÕ÷ÕQÕNDEXOHGHOLP

ݍ א ܮ^’ሺȍሻǡܽǡ ݈ א ܥ^ଵሺī_ଵሻ

ī_଴ת ī_ଵ ൌ ׎ (2.6)

ī_଴ ് ׎ veya ݍ ء Ͳ veya ܽ ء Ͳ (2.7)

݊ ൒ ͵ (2.8)

Sistemin enerjisi

ܧሺݐሻ ൌ_ଶ^ଵ׬ ሺݑ_ȍ ௧ଶ൅ ȁ׏ݑȁ^ଶ൅ ݍݑ^ଶሻ݀ݔ ൅_ଶ^ଵ׬ ܽݑ_īభ ^ଶ݀ī (2.9)

úHNOLQGHWDQÕPODQÕU

Lemma2.1.

Verilen ሺݑ_௢ǡ ݑ_ଵሻ א ܦሺܣሻ NH\ILROPDN]HUH (2.1) ± (2.4) SUREOHPLQLQo|]P

ܧሺܵሻ െ ܧሺܶሻ ൌ ׬ ׬ ݈ݑ_୻ ௧ଶ݀Ȟ†–ǡͲ ൑ ܵ ൏ ܶ ൏ λ

భ

்

ௌ (2.10)

HQHUMLHúLWOL÷LQLVD÷ODU

øVSDW

(2.1) denklemi ݑ_௧LOHoDUSÕOÕUYH ȍ E|OJHVLQGHintegre edilirse

׬ ݑ_ஐ ௧ݑ_௧௧݀ݔ െ ׬ ݑ_ஐ ௧οݑ݀ݔ ൅ ׬ ݑ_ஐ ௧ݍݑ݀ݔ ൌ Ͳ (2.11)

elde edilir. (2.11) HúLWOL÷LQLQVROWDUDIÕQGDNLELULQFLWHULPG]HQOHQLUVH

׬ ݑஐ ௧ݑ௧௧݀ݔ ൌ^ଵ_ଶௗ௧^ௗ ׬ ݑ௧ଶ

ஐ ݀ݔ (2.12)

bulunur. (2.11) HúLWOL÷LQLQVROWDUDIÕQGDNLLNLQFLWHULPH *UHHQ)RUPOX\JXODQÕUVD

െ ׬ ݑஐ ௧οݑ݀ݔ ൌ െ ቀെ ׬ ׏ݑ௧׏ݑ݀ݔ ൅ ׬ ݑ௧డ௨

୻ డఔ

ஐ ݀Ȟቁ

ൌ ׬ ׏ݑ௧׏ݑ݀ݔ െ ׬ ݑ௧డ௨

୻ డఔ

ஐ ݀Ȟ

ൌ ׬ ׏ݑஐ ௧׏ݑ݀ݔ െ ׬ ݑ୻_బ ௧ డ௨

డఔ݀Ȟ െ ׬ ݑ௧డ௨

୻_భ డఔ݀Ȟ

\D]ÕODELOLU (2.2) GHQGROD\Õ ׬ ݑ୻_బ ௧ డ௨

డఔ݀Ȟ ൌ Ͳ GÕUO halde

െ ׬ ݑஐ ௧ȟݑ݀ݔ ൌ ׬ ׏ݑஐ ௧׏ݑ݀ݔ െ ׬ ݑ୻_భ ௧ డ௨ డఔ݀Ȟ

ൌ ^ଵ_ଶௗ௧^ௗ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ െ ׬ ݑ୻_భ ௧ డ௨

డఔ݀Ȟ (2.13)

elde edilir. (2.11) HúLWOL÷LQLQVRO WDUDIÕQGDNLoQFWHULPG]HQOHQLUVH

׬ ݑ_ஐ ௧ݍݑ݀ݔ ൌ^௤_ଶௗ௧^ௗ ׬ ݑ_ஐ ^ଶ݀ݔ (2.14)

bulunur. (2.12), (2.13), (2.14) ifadeleri, (2.11) de \HUOHULQH\D]ÕOÕUVD

ଵ ଶ

ௗ

ௗ௧׬ ݑ௧ଶ

ஐ ݀ݔ ൅^ଵ_ଶௗ௧^ௗ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ െ ׬ ݑ௧డ௨

୻భ డఔ݀Ȟ ൅^௤_ଶௗ௧^ௗ ׬ ݑ_ஐ ^ଶ݀ݔ ൌ Ͳ

elde edilir. hVWWHNLHúLWOLNG]HQOHQLUVH

ଵ ଶ

ௗ

ௗ௧׬ ݑ௧ଶ

ஐ ݀ݔ ൅^ଵ_ଶௗ௧^ௗ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅^௤_ଶௗ௧^ௗ ׬ ݑ_ஐ ^ଶ݀ݔ ൌ ׬ ݑ୻_భ ௧ డ௨ డఔ݀Ȟ

olur. hVWWHNLHúLWOL÷LQKHULNLWDUDIÕQD_ௗ௧^ௗ ቀ^ଵ_ଶ׬ ܽݑ_୻భ ^ଶ݀Ȟቁ eklenirse

ଵ ଶ

ௗ

ௗ௧׬ ݑ௧ଶ

ஐ ݀ݔ ൅^ଵ_ଶௗ௧^ௗ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅^௤_ଶௗ௧^ௗ ׬ ݑ_ஐ ^ଶ݀ݔ ൅_ௗ௧^ௗ ቀ^ଵ_ଶ׬ ܽݑ_୻ ^ଶ݀Ȟ

భ ቁ

ൌ _ௗ௧^ௗ ቀ^ଵ_ଶ׬ ܽݑ_୻ ^ଶ݀Ȟ

భ ቁ ൅ ׬ ݑ୻_భ ௧ డ௨ డఔ݀Ȟ HOGHHGLOLUhVWWHNLHúLWOLNG]HQOHQLUVH

ௗ

ௗ௧ቂ^ଵ_ଶ׬ ݑ௧ଶ

ஐ ݀ݔ ൅^ଵ_ଶ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅^ଵ_ଶ׬ ݍݑ_ஐ ^ଶ݀ݔ ൅^ଵ_ଶ׬ ܽݑ_୻భ ^ଶ݀Ȟቃ

ൌ _ௗ௧^ௗ ቀ^ଵ_ଶ׬ ܽݑ_୻భ ^ଶ݀Ȟቁ ൅ ׬ ݑ୻_భ ௧ డ௨

డఔ݀Ȟ (2.15)

bulunur. (2.15) GHo|]POHULQHQHUMLVLRODUDN

ܧሺݐሻ ൌ^ଵ_ଶ׬ ݑ௧ଶ

ஐ ݀ݔ ൅^ଵ_ଶ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅^ଵ_ଶ׬ ݍݑ_ஐ ^ଶ݀ݔ ൅^ଵ_ଶ׬ ܽݑ_୻భ ^ଶ݀Ȟ (2.16) úHNOLQGHELU ܧሺݐሻ IRQNVL\RQXWDQÕPODQÕUVD (2.15) ifadesi

ௗ

ௗ௧ܧሺݐሻ ൌ_ௗ௧^ௗ ቀ^ଵ_ଶ׬ ܽݑ_୻భ ^ଶ݀Ȟቁ ൅ ׬ ݑ௧డ௨

୻_భ డఔ݀Ȟ (2.17)

KDOLQLDOÕU(2.3) de

߲_ఔݑ ൅ ܽݑ ൅ ݈ݑ_௧ ൌ Ͳ

veya

డ௨

డఔ ൌ െܽݑ െ ݈ݑ_௧

ROGX÷XQGDQGROD\Õ (2.17) GH\HULQH\D]ÕOÕUVD

ௗ

ௗ௧ܧሺݐሻ ൌ_ௗ௧^ௗ ቀ^ଵ_ଶ׬ ܽݑ_୻భ ^ଶ݀Ȟቁ ൅ ׬ ݑ_୻భ ௧ሺെܽݑ െ ݈ݑ௧ሻ݀Ȟ

olur. hstteki HúLWOLNG]HQOHQLUVH

ܧ^ᇱሺݐሻ ൌ ׬ ܽݑݑ୻_భ ௧݀Ȟ െ ׬ ܽݑݑ୻_భ ௧݀Ȟ െ ׬ ݈ݑ௧ଶ

୻_భ ݀Ȟ

ܧ^ᇱሺݐሻ ൌ െ ׬ ݈ݑ௧ଶ

୻_భ ݀Ȟ (2.18)

elde edilir.

ܧ^ᇱሺݐሻ ൌ െ ׬ ݈ݑ௧ଶ

୻_భ ݀Ȟ ൑ Ͳ (2.19)

ROGX÷XQGDQ GROD\Õ ܧ artmayan bir fonksiyondur. (2.18) HúLWOL÷LQLQ KHU LNL WDUDIÕ

Ͳ ൑ ܵ ൏ ܶDUDOÕ÷ÕQGD integre edilirse

׬ ܧ^ᇱሺݐሻ݀ݐ ൌ െ ׬ ׬ ݈ݑ௧ଶ

୻_భ

்

ௌ

்

ௌ ݀Ȟ†– (2.20)

bulunur. (2.20) LQWHJUDOOHULoLQRUWDODPDGH÷HUWHRUHPLX\JXODQÕUVD

ܧሺܶሻ െ ܧሺܵሻ ൌ െ ׬ ׬ ݈ݑ௧ଶ

୻_భ

்

ௌ ݀Ȟ†–

ve

ܧሺܵሻ െ ܧሺܶሻ ൌ ׬ ׬ ݈ݑ௧ଶ

୻_భ

்

ௌ ݀Ȟ݀ݐ (2.21)

elde edilir.

ܧሺܵሻ െ ܧሺܶሻ ൌ ׬ ׬ ݈ݑ௧ଶ

୻_భ

்

ௌ ݀Ȟ†– ൒ Ͳ

GÕUO halde

ܧሺܵሻ െ ܧሺܶሻ ൒ Ͳ

ve

ܧሺܵሻ ൒ ܧሺܶሻ

úeklinde \D]ÕODELOLUYHLVSDWWDPDPODQÕU

Teorem 2.1.

ሺʹǤͷሻ െ ሺʹǤͺሻ

Ȟ଴ ]HULQGH ݉ ή ߥ ൑ Ͳ ve Ȟଵ ]HULQGH݉ ή ߥ ൒ Ͳǡ (2.22)

ܳ_ଵ ൏ ͳǡ (2.23)

VD÷ODQVÕQ

݈ ൌ ሺ݉ ή ߥሻ ܴΤ ve ܽ ൌ ሺ݊ െ ͳሻሺ݉ ή ߥሻ ሺʹܴΤ ^ଶሻ (2.24) VHoLOVLQ+HU ሺݑ_଴ǡ ݑ_ଵሻ א ܪ_୻^ଵ_బሺȳሻ ൈ ܮ^ଶሺȳሻ LoLQ (2.1) ± (2.4) problemi

ܧሺݐሻ ൑ ܧሺͲሻ݁^{ଵିሺభషೂభሻ೟మೃ} ǡ׊ݐ א ܴା (2.25)

NHVWLULPLQLVD÷ODU

Lemma2.2.

Verilen ሺݑ଴ǡ ݑ_ଵሻ א ܦሺܣሻ ve Ͳ ൑ ܵ ൏ ܶ ൏ λ keyfi, ሺʹǤͳሻ െ ሺʹǤͶሻ probleminin o|]P

ʹ ׬ ܧሺݐሻ݀ݐ ൌ ቂ׬ ݑ_ஐ ௧ܯݑ݀ݔቃ

்

் ௌ

ௌ െ ׬ ׬ ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ݀ݔ݀ݐ

൅ ׬ ׬ ቚ^డ௨_డఔቚ^ଶ݀Ȟ_௠݀ݐ ൅ ׬ ׬ ݑ௧ଶ

୻_భ

்

ௌ

୻_బ

்

ௌ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ_௧൅ ܾݑሻܯݑ݀Ȟ_௠݀ݐ (2.26) HúLWOL÷LQLVD÷ODU

øVSDW

(2.1) denklemi ܯݑ LOHoDUSÕOÕS ȳ E|OJHVLQGHLQWHJUHHGLOLUVH

ௗ

ௗ௧׬ ݑஐ ௧ܯݑ݀ݔ െ ׬ ݑஐ ௧ܯݑ௧݀ݔ െ ׬ ܯݑοݑ݀ݔ ൅ ׬ ݍݑܯݑ݀ݔ ൌ Ͳ_{ஐ ஐ} (2.27)

elde edilir. (2.27) HúLWOL÷Lnin sol WDUDIÕQGDNLെ ׬ ݑஐ ௧ܯݑ௧݀ݔ ifadesinde

ܯݑ ൌ ʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑ

úeklinde DOÕQÕUYHLIDGHGH\HULQH\D]ÕOÕUVD

െ ׬ ݑ_ஐ ௧ሺʹ݉ ή ׏ݑ௧൅ ሺ݊ െ ͳሻݑ_௧ሻ݀ݔ (2.28)

olur. (2.27) HúLWOL÷LQLQVROWDUDIÕQGDNLoQF LIDGH\H*UHHQIRUPOX\JXODQÕUVD

െ ׬ ܯݑοݑ݀ݔ ൌ െ ቀെ ׬ ׏ݑ ή ׏ሺܯݑሻ݀ݔ ൅ ׬ ܯݑ_{ஐ ஐ ୻} ^డ௨_డఔ݀Ȟቁ

ൌ ׬ ׏ݑ ή ׏ሺܯݑሻ݀ݔ െ ׬ ܯݑ_{ஐ ୻} ^డ௨_డఔ݀Ȟ

ൌ ׬ ሺ݊ ൅ ͳሻȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔെ ׬ ܯݑ_୻ ^డ௨_డఔ݀Ȟ

elde edilir. Ȟ VÕQÕUÕ Ȟ_଴ ve Ȟ_ଵGHQROXúWX÷XLoLQ

ൌ ׬ ሺ݊ ൅ ͳሻ_ஐ ȁ׏ݑȁ^ଶ݀ݔ ൅ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔ െ ׬ ܯݑ_୻బ ^డ௨_డఔ݀Ȟ െ ׬ ܯݑ_୻భ ^డ௨_డఔ݀Ȟ(2.29)

úeklinde \D]ÕODELOLU (2.27) HúLWOL÷LQLQVROWDUDIÕQGDNL ׬ ݍݑܯݑ݀ݔ_ஐ ifadesinde

ܯݑ ൌ ʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑ

yD]ÕOÕUVD ifade

׬ ݍݑሺʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑሻ݀ݔ_ஐ (2.30)

KDOLQLDOÕU. (2.28), (2.29) ve (2.30) ifadeleri ሺʹǤʹ͹ሻ de \HUOHULQH\D]ÕOÕUVD

ௗ

ௗ௧׬ ݑஐ ௧ܯݑ݀ݔ െ ׬ ݑஐ ௧ሺʹ݉ ή ׏ݑ௧൅ ሺ݊ െ ͳሻݑ_௧ሻ݀ݔ ൅ ׬ ሺ݊ ൅ ͳሻ_ஐ ȁ׏ݑȁ^ଶ݀ݔ

൅ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔെ ׬ ܯݑ_୻బ ^డ௨_డఔ݀Ȟ െ ׬ ܯݑ_୻భ ^డ௨_డఔ݀Ȟ

൅ ׬ ݍݑሺʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑሻ݀ݔ ൌ Ͳ_ஐ

HOGHHGLOLUhVWWHNLHúLWOLNWH ܯݑ ൌ ʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑ ifadesi ile (2.3) den

డ௨

డఔ൅ ܽݑ ൅ ݈ݑ_௧ ൌ Ͳ ֜ ^డ௨_డఔ ൌ െܽݑ െ ݈ݑ_௧ úHNOLQGHEXOXQDQ ifade yerine \D]ÕOÕUVD

ௗ

ௗ௧׬ ݑఆ ௧ܯݑ݀ݔ െ ׬ ݑఆ ௧ʹ݉ ή ׏ݑ_௧݀ݔ ൅ ሺͳ െ ݊ሻ ׬ ݑ௧ଶ

ఆ ݀ݔ ൅ ሺ݊ ൅ ͳሻ ׬ ȁ׏ݑȁ_ఆ ^ଶ݀ݔ

൅ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔ െ ׬ ሺʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑሻ_୻బ ^డ௨_డఔ݀Ȟ െ ׬ ܯݑሺെܽݑ െ ݈ݑ_୻భ ௧ሻ݀Ȟ

൅ ׬ ݍݑʹ݉ ή ׏ݑ݀ݔ ൅ ሺ݊ െ ͳሻ ׬ ݍݑ_{ஐ ஐ} ^ଶ݀ݔ ൌ Ͳ

bulunur. (2.2) den

׬ ሺ݊ െ ͳሻݑ_୻బ ^డ௨_డఔ݀Ȟ ൌ Ͳ

oOGX÷XQGDQ GROD\ÕEXGXUXPVWWHNLHúLWOLNWH\HULQH\D]ÕOÕUVD

ௗ

ௗ௧׬ ݑஐ ௧ܯݑ݀ݔ െ ׬ ݑஐ ௧ʹ݉ ή ׏ݑ_௧݀ݔ ൅ ሺͳ െ ݊ሻ ׬ ݑ௧ଶ

ஐ ݀ݔ ൅ ሺ݊ ൅ ͳሻ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ

൅ ׬ ݉ ή ׏ȁ׏ݑȁ_ఆ ^ଶ݀ݔ െ ׬ ሺʹ݉ ή ׏ݑሻ_௰బ ^డ௨_డఔ݀߁ െ ׬ ܯݑሺെܽݑ െ ݈ݑ_௰భ ௧ሻ݀߁

൅ ׬ ݍݑʹ݉ ή ׏ݑ݀ݔ ൅ ሺ݊ െ ͳሻ ׬ ݍݑ_{ఆ ఆ} ^ଶ݀ݔ ൌ Ͳ

elde edilir. hVWWHNLHúLWOLNWHLIDGHOHUHúLWOL÷LQNDUúÕWDUDIÕQDJHoLULOLUYHG]HQOHQLUVH

Ͳ ൌ െ_ௗ௧^ௗ ׬ ݑஐ ௧ܯݑ݀ݔ ൅ ׬ ݑஐ ௧ʹ݉ ή ׏ݑ௧݀ݔ െ ሺͳ െ ݊ሻ ׬ ݑ௧ଶ ஐ ݀ݔ

െሺ݊ ൅ ͳሻ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔെ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ሺʹ݉ ή ׏ݑሻ_୻బ ^డ௨_డఔ݀Ȟ െ ׬ ܽݑܯݑ݀Ȟ_୻భ

െ ׬ ݈ݑ_୻భ ௧ܯݑ݀Ȟെ ׬ ݍݑʹ݉ ή ׏ݑ݀ݔ െ ሺ݊ െ ͳሻ ׬ ݍݑ_{ஐ ஐ} ^ଶ݀ݔ

úeklinde \D]DELOLUL]hVWWHNLHúLWOL÷LQKHULNLWDUDIÕQD (2.16) dan ʹܧሺݐሻ eklenirse

ʹܧሺݐሻ ൌ ׬ ݑ௧ଶ

ஐ ݀ݔ ൅ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ݍݑ_ஐ ^ଶ݀ݔ ൅׬ ܽݑ_୻భ ^ଶ݀Ȟ െ_ௗ௧^ௗ ׬ ݑஐ ௧ܯݑ݀ݔ

൅ ׬ ݑஐ ௧ʹ݉ ή ׏ݑ௧݀ݔ െ ሺͳ െ ݊ሻ ׬ ݑ௧ଶ

ஐ ݀ݔ െ ሺ݊ ൅ ͳሻ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ െ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔ

൅ ׬ ሺʹ݉ ή ׏ݑሻ_୻బ ^డ௨_డఔ݀Ȟ െ ׬ ܽݑܯݑ݀Ȟ െ ׬ ݈ݑ୻భ ୻భ ௧ܯݑ݀Ȟ െ ׬ ݍݑʹ݉ ή ׏ݑ݀ݔ_ஐ

െሺ݊ െ ͳሻ ׬ ݍݑ_ஐ ^ଶ݀ݔ

bulunur. (2.24) den ݀Ȟ_௠ ൌ ሺ݉ ή ߥሻ݀Ȟ ROGX÷XQGDQYH ݇ ൌ_ோ^ଵ, ܾ ൌ^௡ିଵ_ଶோమ DOÕQÕUVD VWteki HúLWOL÷L

ʹܧሺݐሻ ൌ ׬ ݑ௧ଶ

ஐ ݀ݔ ൅ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ݍݑ_ஐ ^ଶ݀ݔ ൅ ׬ ܾݑ୻_భ ^ଶ݀Ȟ௠െ_ௗ௧^ௗ ׬ ݑ_ஐ ௧ܯݑ݀ݔ

൅ ׬ ݑ௧ʹ݉ ή ׏ݑ_௧݀ݔ െ ሺͳ െ ݊ሻ ׬ ݑ௧ଶ ஐ

ஐ ݀ݔ െ ሺ݊ ൅ ͳሻ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ െ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔ

൅ ׬ ሺʹ݉ ή ׏ݑሻ_୻బ ^డ௨_డఔ݀Ȟ െ ׬ ሺ݇ݑ_୻భ ௧൅ ܾݑሻܯݑ݀Ȟ௠െ ׬ ʹݍݑ݉ ή ׏ݑ݀ݔ_ஐ

െሺ݊ െ ͳሻ ׬ ݍݑ_ஐ ^ଶ݀ݔ

úHNOLQGH\D]DELOLUL]hVWWHNLHúLWOLNG]HQOHQLUVH

ʹܧሺݐሻ ൌ ׬ ݑ௧ଶ

ஐ ݀ݔ ൅ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ݍݑ_ஐ ^ଶ݀ݔ ൅ ׬ ܾݑ_୻భ ^ଶ݀Ȟ௠െ_ௗ௧^ௗ ׬ ݑ_ஐ ௧ܯݑ݀ݔ

൅ ׬ ݑஐ ௧ʹ݉ ή ׏ݑ_௧݀ݔ െ ׬ ݑ௧ଶ

ஐ ݀ݔ ൅ ׬ ݊ݑ௧ଶ

ஐ ݀ݔ െ ׬ ݊ȁ׏ݑȁ_ஐ ^ଶ݀ݔ െ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ

െ ׬ ݉ ή ׏ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ሺʹ݉ ή ׏ݑሻ_୻బ ^డ௨_డఔ݀Ȟ െ ׬ ሺ݇ݑ_୻భ ௧൅ ܾݑሻܯݑ݀Ȟ_௠

െ ׬ ʹݍݑ݉ ή ׏ݑ݀ݔ_ஐ െ ׬ ݊ݍݑ_ஐ ^ଶ݀ݔ ൅ ׬ ݍݑ_ஐ ^ଶ݀ݔ

HOGHHGLOLUhVWWHNLHúLWOLNG]HQOHQLUVH

ʹܧሺݐሻ ൌ ׬ ݑ௧ଶ

ஐ ݀ݔ ൅ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ݍݑ_ஐ ^ଶ݀ݔ ൅׬ ܾݑ_୻భ ^ଶ݀Ȟ௠െ_ௗ௧^ௗ ׬ ݑ_ஐ ௧ܯݑ݀ݔ

െ ׬ ݊ݑ_ஐ ௧ଶ݀ݔ൅ ׬ ݑ௧ଶ

୻_భ ݀Ȟ௠൅ ׬ ݊ݑ௧ଶ

ஐ ݀ݔ െ ׬ ݑ௧ଶ

ஐ ݀ݔ െ ׬ ݊ȁ׏ݑȁ_ஐ ^ଶ݀ݔ

െ ׬ ȁ׏ݑȁ_ஐ ^ଶ݀ݔ ൅ ׬ ݊ȁ׏ݑȁ_ஐ ^ଶ݀ݔെ ׬ ȁ׏ݑȁ_୻భ ^ଶ݀Ȟ௠൅ ׬ ሺʹ݉ ή ׏ݑሻ_୻బ ^డ௨_డఔ݀Ȟ

െ ׬ ሺ݇ݑ_୻భ ௧൅ ܾݑሻܯݑ݀Ȟ_௠െ ׬ ʹݍݑ݉ ή ׏ݑ݀ݔ_ஐ െ ׬ ݊ݍݑ_ஐ ^ଶ݀ݔ ൅ ׬ ݍݑ_ஐ ^ଶ݀ݔ

bulunur. hVWWHNLHúLWOLNG]HQOHQLUVH

ʹܧሺݐሻ ൌ ׬ ݑ௧ଶ

୻భ ݀Ȟ_௠െ ׬ ȁ׏ݑȁ_୻భ ^ଶ݀Ȟ_௠൅ ׬ ሺܾݑ_୻భ ^ଶെ ሺ݇ݑ_௧൅ ܾݑሻܯݑሻ݀Ȟ_௠

െ_ௗ௧^ௗ ׬ ݑஐ ௧ܯݑ݀ݔ െ ׬ ʹݍݑ݉ ή ׏ݑ݀ݔ ൅ ሺʹ െ ݊ሻ ׬ ݍݑ_{ஐ ஐ} ^ଶ݀ݔ ൅ ׬ ሺʹ݉ ή ׏ݑሻ_୻బ ^డ௨_డఔ݀Ȟ

\D]ÕODELOLU hVWWHNLHúLWOLNG]HQOHQLUVH

ʹܧሺݐሻ ൌ െ_ௗ௧^ௗ ׬ ݑ_ஐ ௧ܯݑ݀ݔ െ ׬ ൫ሺ݊ െ ʹሻݍݑ_ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ൯݀ݔ ൅ ׬ ʹ݉ ή ׏ݑ_୻బ ^డ௨_డఔ݀Ȟ

൅ ׬ ሺݑ୻_భ ௧ଶെ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሻ݀Ȟ௠

elde edilir. hVWWHNLHúLWOLN ሺܵǡ ܶሻ DUDOÕ÷ÕQGD integre edilirse

ʹ ׬ ܧሺݐሻ݀ݐ ൌ ׬ ݑ_ஐ ௧ܯݑ݀ݔቚ

்

் ௌ

ௌ െ ׬ ׬ ൫ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ ൅ ʹݍݑ݉ ή ׏ݑ൯݀ݔ݀ݐ (2.31)

൅ ׬ ׬ ʹ݉ ή ׏ݑ_ௌ^் _୻బ ^డ௨_డఔ݀Ȟ݀ݐ ൅ ׬ ׬ ሺݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሻ݀Ȟ௠݀ݐ

bulunarak HúLWOL÷LVD÷ODQÕU%|\OHFH LVSDWWDPDPODQÕU

øVSDW7HRUHP

Teorem LQLVSDWÕ/HPPD/HPPD¶QLQVRQXoODUÕNXOODQÕODUDN\DSÕOÕU

ฬ׬ ݑ_ஐ ௧ܯݑ݀ݔቚ

்

ௌฬ ൑ ܴܧȁ_்^ௌ ൌ ܴܧሺܵሻ െ ܴܧሺܶሻ ൑ ܴܧሺܵሻ ൅ ܴܧሺܶሻ

dLU2KDOGHD\QÕ]DPDQGD

ฬ׬ ݑ_ஐ ௧ܯݑ݀ݔቚ

்

ௌฬ ൑ ʹܴܧሺܵሻ ൅ ʹܴܧሺܶሻ

GLUYHEX\]GHQ (2.31) HúLWOL÷Lni

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൑ ʹܴܧሺܵሻ ൅ ʹܴܧሺܶሻ െ ׬ ׬ ൫ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ൯݀ݔ݀ݐ

൅ ׬ ׬ ʹ݉ ή ׏ݑ_୻బ ^డ௨_డఔ݀Ȟ݀ݐ ൅ ׬ ׬ ሺݑ௧ଶ

୻_భ

்

ௌ

்

ௌ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሻ݀Ȟ௠݀ݐ

HúLWVL]OL÷L VHNOLQGH \Dzabiliriz. ׏ݑ ൌ ߥ^డ௨_డఔ ve ݀Ȟ_௠ ൌ ሺ݉ ή ߥሻ݀Ȟ ROGX÷XQGDQ GROD\Õ

VWWHNLHúLWVL]OL÷LQVD÷WDUDIÕQGDNLG|UGQFWHULP LoLQGHNLintegral

׬ ʹ݉ ή ׏ݑ_୻బ ^డ௨_డఔ݀Ȟ ൌ ׬ ʹ݉ ή ߥ_{୻బ డఔ}^డ௨డ௨_డఔ݀Ȟ ൌʹ ׬ ቚ୻_బ ^డ௨_డఔቚ^ଶ݀Ȟ௠

úHNOLQGHEXOXQXUYH VWWHNLHúLWVL]OLN G]HQOHQLUVH

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൑ ʹܴܧሺܵሻ ൅ ʹܴܧሺܶሻ െ ׬ ׬ ൫ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ൯݀ݔ݀ݐ

൅ ׬ ʹ ׬ ቚ_୻బ ^డ௨_డఔቚ^ଶ݀Ȟ௠݀ݐ ൅ ׬ ׬ ሺݑ௧ଶ

୻_భ

்

ௌ

்

ௌ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሻ݀Ȟ௠݀ݐ (2.32) KDOLQLDOÕU

׬ ʹ݉ ή ߥ_{୻బ డఔ}^డ௨డ௨_డఔ݀Ȟ ൌ ʹ ׬ ቚ_୻బ ^డ௨_డఔቚ^ଶ݀Ȟ_௠

eúLWOL÷LQGH ݉ ή ߥ ൏ Ͳ ve ^డ௨_డఔ^డ௨_డఔ ൐ Ͳ ROGX÷XQGDQGROD\Õ

ʹ ׬ ቚ_୻బ ^డ௨_డఔቚ^ଶ݀Ȟ௠ ൏ Ͳ

olur.  HúLWVL]OL÷LQGHQ VWWHNL LIDGH\L oÕNDUWÕUVDN  HúLWVL]OL÷LQLQ VD÷ WDUDIÕ

GDKDE\PúROXUYH (2.32) HúLWVL]OL÷LWHNUDUG]HQOHQLUVH

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൅ ׬ ׬ ൫ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ൯݀ݔ݀ݐ൑ ʹܴܧሺܵሻ ൅ ʹܴܧሺܶሻ

൅ ׬ ׬ ሾݑ_ௌ^் _୻భ ௧ଶെ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ (2.33)

elde edilir. (2HúLWVL]OL÷LQLQVD÷WDUDIÕQdaki oQFWHULPGH

ܯݑ ൌ ʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑ

LIDGHVL\HULQH\D]ÕOÕUVD

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ

ൌ ׬ ׬ ሾݑ_ௌ^் _୻భ ௧ଶെ ȁ׏ݑȁ^ଶ ൅ ܾݑ^ଶെ ݇ݑ௧ܯݑ െ ܾݑܯݑሿ݀Ȟ௠݀ݐ

ൌ ׬ ׬ ሾݑ_ௌ^் _୻భ ௧ଶെ ȁ׏ݑȁ^ଶ ൅ ܾݑ^ଶെ ݇ݑ௧ሺʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑሻ

െܾݑሺʹ݉ ή ׏ݑ ൅ ሺ݊ െ ͳሻݑሻሿ݀Ȟ௠݀ݐ

elde edilir. Buradan

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ

ൌ ׬ ׬ ሾݑ_ௌ^் _୻భ ௧ଶെ ȁ׏ݑȁ^ଶ ൅ ܾݑ^ଶെ ʹሺ݇ݑ௧൅ ܾݑሻ݉ ή ׏ݑ

൅ሺͳ െ ݊ሻሺ݇ݑݑ_௧൅ ܾݑ^ଶሻሿ݀Ȟ௠݀ݐ (2.34)

HúLWOL÷L bulunur. (2.34) QVD÷WDUDIÕQGDNLLQWHJUDOLoLQGHNL

ʹሺ݇ݑ_௧൅ ܾݑሻ݉ ή ׏ݑ

ifadesine &DXFK\HúLWVL]OL÷LX\JXODQÕUVD

ȁʹሺ݇ݑ௧൅ ܾݑሻ݉ ή ׏ݑȁ ൑ ʹȁܴሺ݇ݑ_௧൅ ܾݑሻȁȁ׏ݑȁ

ȁʹሺ݇ݑ௧൅ ܾݑሻ݉ ή ׏ݑȁ ൑^{൫ξଶȁ׏௨ȁ൯}_ଶ ^మ൅^{ቀξଶோሺ௞௨೟ା௕௨ሻቁ}

మ

ଶ

ȁʹሺ݇ݑ௧൅ ܾݑሻ݉ ή ׏ݑȁ ൑ ȁ׏ݑȁ^ଶ൅ ܴ^ଶሺ݇ݑ௧൅ ܾݑሻ^ଶ (2.35)

HúLWVL]OL÷LHOGHHGLOLU (2.35) HúLWVL]OL÷L (2.34) HúLWVL]OL÷LQGH \HULQH\D]ÕOÕUVD

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ ൑ ׬ ׬ ሾݑ_ௌ^் _୻భ ௧ଶെ ȁ׏ݑȁ^ଶ ൅ ܾݑ^ଶ

൅ȁ׏ݑȁ^ଶ൅ ܴ^ଶሺ݇ݑ௧൅ ܾݑሻ^ଶ൅ ሺͳ െ ݊ሻሺ݇ݑݑ௧൅ ܾݑ^ଶሻሿ݀Ȟ௠݀ݐ

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ ൑ ׬ ׬ ሾݑ_ௌ^் _୻భ ௧ଶ൅ ܾݑ^ଶ

൅ܴ^ଶሺ݇ݑ௧൅ ܾݑሻ^ଶ൅ ሺͳ െ ݊ሻሺ݇ݑݑ௧൅ ܾݑ^ଶሻሿ݀Ȟ௠݀ݐ

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ ൑ ׬ ׬ ሾݑ_ௌ^் _୻భ ௧ଶ൅ ܾݑ^ଶ

൅ܴ^ଶሺ݇^ଶݑ௧ଶ൅ ʹܾ݇ݑݑ௧൅ ܾ^ଶݑ^ଶሻ ൅ ሺͳ െ ݊ሻሺ݇ݑݑ௧൅ ܾݑ^ଶሻሿ݀Ȟ௠݀ݐ

׬ ׬ ሾݑ௧ଶ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ ൑ ׬ ׬ ሾݑ௧ଶ൅ ܾݑ^ଶ൅ ܴ^ଶ݇^ଶݑ௧ଶ

୻_భ

்

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୻_భ

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൅ʹܴ^ଶܾ݇ݑݑ௧൅ ܴ^ଶܾ^ଶݑ^ଶ൅ ሺͳ െ ݊ሻ݇ݑݑ௧൅ ሺͳ െ ݊ሻܾݑ^ଶሿ݀Ȟ௠݀ݐ

HúLWVL]OL÷LHOGHHGLOLUhVWWHNLHúLWVL]OLNG]HQOHQLUVH

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ

൑ ׬ ׬ ሾݑ௧ଶ

୻_భ

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ௌ ൅ ሺʹ െ ݊ ൅ ܴ^ଶܾሻܾݑ^ଶ൅ ܴ^ଶ݇^ଶݑ௧ଶ൅ ሺͳ െ ݊ ൅ ʹܴ^ଶܾሻ݇ݑݑ௧ሿ݀Ȟ௠݀ݐ

HúLWVL]OL÷LEXOXQXUhVWWHNLHúLWVL]OLNWH ܾ ൌ^௡ିଵ_ଶோమ ve ݇ ൌ^ଵ_ோLIDGHOHUL\HUOHULQH\D]ÕOÕUVD

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ൑

׬ ׬ ቂݑ_ௌ^் _୻భ ௧ଶ൅ ቀʹ െ ݊ ൅ ܴ^{ଶ ௡ିଵ}_ଶோమቁ ܾݑ^ଶ൅ ܴ^{ଶ ଵ}_ோమݑ௧ଶ൅ ቀͳ െ ݊ ൅ ʹ^௡ିଵ_ଶோమܴ^ଶቁ ݇ݑݑ௧ቃ ݀Ȟ௠݀ݐ

ROXUhVWWHNLHúLWVL]OLNG]HQOHQLUVH

׬ ׬ ሾݑ_ௌ^் _୻భ ଶ௧ െ ȁ׏ݑȁ^ଶ൅ ܾݑ^ଶെ ሺ݇ݑ௧൅ ܾݑሻܯݑሿ݀Ȟ௠݀ݐ

൑ ׬ ׬ ቂʹݑ_ௌ^் _୻భ ଶ௧ ൅ ቀ^ଷି௡_ଶ ቁ ܾݑ^ଶቃ ݀Ȟ௠݀ݐ (2.36)

HúLWVL]OL÷LHOGHHGLOLU (2.36), (2.33) de \HULQH\D]ÕOÕUVD

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൅ ׬ ׬ ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ݀ݔ݀ݐ൑ ʹܴܧሺܵሻ ൅ ʹܴܧሺܶሻ

൅ ׬ ׬ ቂʹݑ_୻ ௧ଶ൅ ቀ^ଷି௡_ଶ ቁ ܾݑ^ଶቃ ݀Ȟ_௠݀ݐ

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olur. hVWWHNLHúLWVL]OLNG]HQOHQLUVH

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൅ ׬ ׬ ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ݀ݔ݀ݐ൑ ʹܴܧሺܵሻ ൅ ʹܴܧሺܶሻ

൅ʹ ׬ ׬ ݑ_ௌ^் _୻భ ௧ଶ݀Ȟ_௠݀ݐ ൅^ଷି௡_ଶ ׬ ׬ ܾݑ_ௌ^் _୻భ ^ଶ݀Ȟ_௠݀ݐ (2.37)

HúLWVL]OL÷LEXOXQXU HúLWOL÷LQGH (2.24) ifadesindeki ݈ ൌ ሺ݉ ή ߥሻ ܴΤ ifadesi yerine

\D]ÕOÕUVD

ܧሺܵሻ െ ܧሺܶሻ ൌ ׬ ׬ ݈ݑ௧ଶ

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ௌ ݀Ȟ݀ݐ

ܧሺܵሻ െ ܧሺܶሻ ൌ_ோ^ଵ׬ ׬ ሺ݉ ή ߥሻݑ_ௌ^் _୻భ ௧ଶ݀Ȟ݀ݐ

eúLWOL÷LHOGHHGLOLU ሺ݉ ή ߥሻ݀Ȟ ൌ ݀Ȟ_௠ROGX÷XQGDQ GROD\Õ VWWHNLHúLWOLN

ܧሺܵሻ െ ܧሺܶሻ ൌ_ோ^ଵ׬ ׬ ݑ_ௌ^் _୻భ ௧ଶ݀Ȟ௠݀ݐ

úHNOLQGH\D]ÕODELOLUhVWWHNLHúLWOLNG]HQOHQLUVH

ܴሾܧሺܵሻ െ ܧሺܶሻሿ ൌ ׬ ׬ ݑ_ௌ^் _୻భ ௧ଶ݀Ȟ௠݀ݐ (2.38)

bulunur. (2.38), (2.37HúLWVL]OL÷LQGH\HULQH\D]ÕOÕUVD

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൅ ׬ ׬ ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ݀ݔ݀ݐ൑ ʹܴܧሺܵሻ ൅ ʹܴܧሺܶሻ

൅ʹܴሾܧሺܵሻ െ ܧሺܶሻሿ ൅^ଷି௡_ଶ ׬ ׬ ܾݑ_୻ ^ଶ݀Ȟ_௠݀ݐ

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HúLWVL]OL÷LHOGHHGLOLUhVWWHNLHúLWVL]OLNG]HQOHQLUVH

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൅ ׬ ׬ ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ݀ݔ݀ݐ൑ Ͷܴܧሺܵሻ

൅^ଷି௡_ଶ ׬ ׬ ܾݑ_ௌ^் _୻భ ^ଶ݀Ȟ_௠݀ݐ

HúLWVL]OL÷LHOGHHGLOLUhVWWHNLHúLWVL]OL÷LQVD÷WDUDIÕQGDNLLNLQFLWHULP (2.8) ve (2.22),

ܾ ൐ Ͳve ݑ^ଶ ൐ Ͳ ROGX÷XQGDQ GROD\Õ QHJDWLI ROXU YH HúLWVL]OLNWHQ DWÕODELOLU %|\OHFH

VWWHNLHúLWVL]OLN

ʹ ׬ ܧሺݐሻ݀ݐ_ௌ^் ൅ ׬ ׬ ሺ݊ െ ʹሻݍݑ_ௌ^் _ஐ ^ଶ൅ ʹݍݑ݉ ή ׏ݑ݀ݔ݀ݐ൑ Ͷܴܧሺܵሻ

úHNOLQGH\D]ÕODELOLU (2.23) den VWWHNLHúLWVL]OLN

ʹሺͳ െ ܳଵሻ ׬ ܧሺݐሻ݀ݐ ൑ Ͷܴܧሺܵሻ_ௌ^்

úHNOLQGHEXOXQXU hVWWHNLHúLWVL]OLNG]HQOHQLUVH

׬ ܧሺݐሻ݀ݐ ൑ ቀ_ଵିொ^ଶோ

భቁ ܧሺܵሻ

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HúLWVL]OL÷LHOGHHGLOLUܶ ื λ LoLQ

ܧሺݐሻ ൑ ܧሺͲሻ݁^{ଵିሺభషೂభሻ೟మೃ} ǡ׊ݐ א ܴା

EXOXQXUYHE|\OHFHLVSDWWDPDPODQÕU

%g/h0<$5,'2ö586$/ '$/*$'(1./(0ø1ø1

dg=h0/(5ø1ø1'h=*h1.$5$5/,/,ö,

%X E|OPGH \DUÕ GR÷UXVDO GDOJD GHQNOHPLQLQ o|]POHULQLQ G]JQ NDUDUOÕOÕ÷Õ

incelenecektir.

$úD÷ÕGDNLGDOJDGHQNOHPLLoLQEDúODQJÕoVÕQÕUGH÷HU SUREOHPLHOHDOÕQVÕQ

ݑ௧௧െ οݑ ൌ െߙݑ௧൅ ׏Ȱሺݔሻ ή ׏ݑ െ ߚ פ ݑ פ^௣ିଶݑݔ߳ߗǡݐ ൐ Ͳ (3.1) ݑሺݔǡ ݐሻ ൌ Ͳݔ א ߲ߗǡݐ ൐ Ͳ (3.2) ݑሺݔǡ Ͳሻ ൌ ݑ_଴ሺݔሻǡ ݑ௧ሺݔǡ Ͳሻ ൌ ݑ_ଵ(x) ݔ א ߗǡ (3.3)

Burada ߙǡ ߚ ൐ Ͳve ݌ ൐ ʹ dir. ȳ, ܴ^௡GHVÕQÕUOÕELUE|OJHGLU߲ߗ iseܥ^ଶ VÕQÕIÕQGDQROXS

ȳE|OJHVLQLQVÕQÕUÕGÕU

HúLWOL÷LQLQKHULNLWDUDIÕ ݁^஍ሺ௫ሻݑ௧LOHoDUSÕOÕU ve ȳ E|OJHVLQGHLQWHJUHHGLOLUVH

׬ ݁ఆ ^஍ሺ௫ሻݑ௧ݑ௧௧݀ݔ െ ׬ ݁_ఆ ^஍ሺ௫ሻݑ௧οݑ݀ݔ ൌ െ ׬ ݁ఆ ^஍ሺ௫ሻݑ௧ߙݑ௧݀ݔ

൅ ׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧׏Ȱሺݔሻ ή׏ݑ݀ݔ െ ׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧ߚȁݑȁ^௣ିଶݑ݀ݔ (3.4)

elde edilir. (3.4) HúLWOL÷LQLQVROWDUDIÕQGDNLELULQFLWHULPG]HQOHQLUVH

׬ ݁_ఆ ^஍ሺ௫ሻݑ௧ݑ௧௧݀ݔ ൌ^ଵ_ଶௗ௧^ௗ ׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧^ଶ݀ݔ (3.5)

bulunur. (3.4) HúLWOL÷LQLQVROWDUDIÕQGDNLLNLQFLWHULPH*UHHQIRUPOX\JXODQÕUVD

െ ׬ ݁_ఆ ^஍ሺ௫ሻݑ௧οݑ݀ݔ ൌ െ ቀെ ׬ ׏൫݁_ఆ ^஍ሺ௫ሻݑ௧൯ ή ׏ݑ݀ݔ ൅ ׬ ݁_డఆ ^஍ሺ௫ሻݑ௧డ௨ డఔ݀ݔቁ

ൌ ׬ ׏൫݁_ఆ ^஍ሺ௫ሻݑ_௧൯ή ׏ݑ݀ݔ െ ׬ ݁_డఆ ^஍ሺ௫ሻݑ_௧^డ௨_డఔ݀ݔ

\D]ÕODELOLU (3.3) VÕQÕUNRúXOXQGDQGROD\Õ ׬ ݁^஍ሺ௫ሻݑ௧డ௨

డఆ డఔ݀ݔ ൌ Ͳ GÕU2KDOGH

െ ׬ ݁_ఆ ^஍ሺ௫ሻݑ௧οݑ݀ݔ ൌ ׬ ׏൫݁_ఆ ^஍ሺ௫ሻݑ௧൯ ή ׏ݑ݀ݔ

ൌ ׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧׏Ȱሺݔሻ ή ׏ݑ݀ݔ ൅ ׬ ݁_ఆ ^஍ሺ௫ሻ׏ݑ_௧ή ׏ݑ݀ݔ

ൌ ׬ ݁_ఆ ^஍ሺ௫ሻݑ௧׏Ȱሺݔሻ ή ׏ݑ݀ݔ ൅^ଵ_ଶௗ௧^ௗ ׬ ݁_ఆ ^஍ሺ௫ሻȁ׏ݑȁ^ଶ݀ݔ(3.6)

bulunur. (3.4) HúLWOL÷LQLQVD÷WDUDIÕQGDNLELULFLWHULPG]HQOHQLUVH

െ ׬ ݁_ఆ ^஍ሺ௫ሻݑ௧ߙݑ௧݀ݔ ൌ െߙ ׬ ݁^஍ሺ௫ሻݑ௧ଶ

ఆ ݀ݔ (3.7)

HOGHHGLOLUHúLWOL÷LQLQVD÷WDUDIÕQGDNLLNLQFLWHULPRODQ

׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧׏Ȱሺݔሻ ή ׏ݑ݀ݔ (3.8)

D\QHQ\D]ÕOÕUHúLWOL÷LQLQVD÷WDUDIÕQGDNLoQFWHULPG]HQOHQLUVH

െ ׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧ߚȁݑȁ^௣ିଶݑ݀ݔ ൌ െ^ఉ_௣ௗ௧^ௗ ׬ ݁_ఆ ^஍ሺ௫ሻݑ^௣݀ݔ (3.9)

bulunur. (3.5), (3.6), (3.7), (3.8) ve (3.9) ifadeleri, (3.4) GH\HUOHULQH\D]ÕOÕUVD

ଵ ଶ

ௗ

ௗ௧׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧^ଶ݀ݔ ൅ ׬ ݁_ఆ ^஍ሺ௫ሻݑ_௧׏Ȱሺݔሻ ή ׏ݑ ൅ ׬ ݁_ఆ ^஍ሺ௫ሻ׏—_୲ή׏ݑ݀ݔ

ൌ െߙ ׬ ݁^஍ሺ௫ሻݑ௧ଶ݀ݔ ൅ ׬ ݁^஍ሺ௫ሻݑ௧׏Ȱሺݔሻ ή ׏ݑ݀ݔ െ^{ఉ ௗ} ׬ ݁^஍ሺ௫ሻݑ^௣݀ݔ