Kısmi türevli diferensiyel denklemlerin diferansiyel transform metodu ile çözümü ve diğer yöntemlerle karşılaştırılması

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Mehmet Mustafa TOKER

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ÖNSÖZ

.ÕVPL GLIHUDQVL\HO GHQNOHPOHU |]HOOLNOH ELULQFL YH LNLQFL PHUWHEHGHQ OLQHHU ROPD\DQGLIHUDQVL\HOGHQNOHPOHUWHRULNYHSUDWLNEDNÕPÕQGDQE\N|QHPWDúÕPDNWDYH bütün fen ve mühendislikELOLPGDOODUÕQGDoRNJHQLúX\JXODPD\HULEXOPDNWDGÕU

%X oDOÕúPDGD NÕVPL GLIHUDQVL\HO GHQNOHPOHULQ \DNODúÕN o|]POHULQGHQ WUDQVIRUP$GRPLDQYHVRQOXIDUNPHWRWODUÕLQFHOHQPLúWLU/LQHHUYH\DOLQHHUROPD\DQ NÕVPLGLIHUDQVL\HOGHQNOHPOHULQ\DNODúÕNo|]münde diferansiyel transform yönteminin, GL÷HU \DNODúÕN \|QWHPOHU RODQ $GRPLDQ D\UÕúWÕUPD PHWRGX LOH VRQOX IDUNODU \|QWHPOHULQLQ NDUúÕODúWÕUPDODUÕ \DSÕOPÕúWÕU 7UDQVIRUP \|QWHPL NXOODQÕODUDN NÕVPL GLIHUDQVL\HO GHQNOHPOHU FHELUVHO GHQNOHPOHUH G|QúWUOHELlir ve elde edilen cebirsel GHQNOHPOHUGHED]ÕEDVLWLúOHPOHUOHNROD\OÕNODVLVWHPDWLNELUúHNLOGHo|]OHELOLU

7H]NRQXVXQXQVHoLPL\UWOPHVLNRQXVXQGDNL\DUGÕPODUÕYH\DNÕQLOJLVLQGHQ GROD\ÕVD\ÕQKRFDP<UG'Ro'U$\GÕQ.851$=¶DWHúHNNUOHULPLVXQDUÕP

Mehmet Mustafa TOKER Konya,2008

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IV

ÖZET

Diferansiyel transfoUP\|QWHPLLOHNDUPDúÕNYH\NVHNPHUWHEHGHQNÕVPLWUHYOL GLIHUDQVL\HO GHQNOHPOHU YH GL÷HU PKHQGLVOLN SUREOHPOHULQLQ o|]PQ HOGH HWPHN PPNQGU 'LIHUDQVL\HO GHQNOHPOHUGHNL WUDQVIRUP \DNODúÕPÕNXOODQÕPNROD\OÕ÷ÕYH oDEXN VRQXFD J|WUHQ \|QWHP ROGX÷X LoLQ ¶OÕ \ÕOODUGDQ EHULGLU E\N LOJL X\DQGÕUPÕúWÕU *HOLúHQ ELOJLVD\DU SURJUDPODPDVÕQD X\JXQ ROPDVÕ EX \|QWHPOHUH KHU JHoHQJQEXLOJL\LDUWÕUPDNWDGÕU%X\|QWHPOHUNXOODQÕODUDNNÕVPLWUHYOLGLIHUDQVL\HO GHQNOHPOHUFHELUVHOGHQNOHPOHUHG|QúWUOHbilir ve elde edilen cebirsel denklemler de ED]ÕEDVLWLúOHPOHUOHNROD\OÕNODVLVWHPDWLNELUúHNLOGHo|]OHELOLU

Anahtar Kelimeler: Lineer Diferansiyel Denklemler, Lineer Olmayan

Diferansiyel DenNOHPOHU .ÕVPL 'LIHUDQVL\HO 'HQNOHPOHU 'LIHUDQVL\HO 7UDQVIRUP <|QWHPL$GRPLDQ$\UÕúÕP<|QWHPL6RQOX)DUNODU

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V

ABSTRACT

It is possible to get the solutions of partial differential equations and the solutions of other engineering problems via differantial transform method. Since 1990s the solutions of differantial equations by the transform method have awakened great interest as they are practical and easy to use. Its being reasonable to developing computer technology arises the interest through it, day by day. By using this method, partial differential equations can be converted to algebraic equations and those algebraic equations can be solved easily with some simple transformations.

This study, which is about the transform solutions of partial differential equations and its comparison with other methods, has six chapters. Chapter one gives basic derfinitions of the subject. The differantial transform method is described in chapter two. In the third chapter, Adomian decomposition method is given. Chapter four gives information about finite diffrence method. In chapter five, some applications with differential transform, Adomian decomposition and finite difference mehods in the solutions of partial differential equations are presented comperatively. In the sixth chapter A brief conlusion has been drawn for this study.

KEY WORDS : Linear Differential Equations, Nonlinear Differential Equations,

Partial Differential Equations, Differential Transform Method, Adomian decomposition method, finite difference method.

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1 ødø1'(.ø/(5 *ø5øù««««««««««««««««««««««««««««««« 1. TEMEL KAVRAMLAR ……….……… 3 1.1 Denklemler ve Çözümler ……….………… 3 1.2 Fonksiyonlar ve Denklemler ……… 3 'LIHUDQVL\HO'HQNOHPOHULQ6ÕQÕIODQGÕUÕOPDVÕ««……….4

1.4 Diferansiyel Denklemlerin ÇÕkÕú Yerleri ve UygulamalarÕ ...……….….5

1.5 Diferansiyel Denklemlerde Çözüm ……….. 6

'ø)(5$16ø<(/75$16)250<g17(0ø«««««««««««««« 2.1 Bir Boyutlu Diferansiyel Transform Yöntemi………... 7

øNL%R\XWOX'LIHUDQVL\HO7UDQVIRUP<|QWHPL ..………..… 13

2.3 N-Boyutlu Diferansiyel Transform ……….… 18

$'20ø$1$<5,ù,00(72'8«««««««««««««««««« 3.1 Lineer Olmayan Polinomlar ………..……….………..….. 25

3.2 Lineer Olmayan Türevler ……..………. 26

3.3 Trigonometrik Lineer Olmama Durumu ……….…….. 27

3.4 Hiperbolik Lineer Olmama Durumu ……….. 28

3.5 Üstel Lineer Olmama Durumu ………...…… 30

3.6 Logaritmik Lineer Olmama Durumu ……….……… 31

4. SONLU)$5.<$./$ù,0/$5,«««««««««««««««««« 4.1 Hiperbolik Denklemler……….……….……... 33 4.2 Parabolik Denklemler……….………..…… 35 4.3 Eliptik Denklemler ……….………....………. 37 5. UYGULAMALAR……….……… 39 *(1(/6218d/$59(g1(5ø/(5««««««««««««««««« 7. KAYNAKLAR………..………..…….. 57

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Amaç ve Kapsam

)HQYHPKHQGLVOLNELOLPGDOODUÕQGDRUWD\DoÕNDQNÕVPLGLIHUDQVL\HOGHQNOHPOHU LoLQ NXOODQÕODQ oRN oHúLWOL \|QWHP EXOXQPDVÕQD UD÷PHQ o|]PGH NDUúÕODúÕODQ ED]Õ ]RUOXNODU ELOLP DGDPODUÕQÕ GDKD NXOODQÕúOÕ YH NROD\ \|QWHPOHU EXOPD\D VHYN HWPLúWLU gUQH÷LQ PKHQGLVOLN SUREOHPOHULQGH /DSODFH YH )RXULHU G|QúPOHUL JLEL LQWHJUDO G|QúPOHU \D\JÕQ RODUDN NXOODQÕOPDNWDGÕU $QFDN JHOLúHQ ELOJLVD\DU oD÷ÕQGD GLIHUDQVL\HO WUDQVIRUP \|QWHPL $GRPLDQ D\UÕúÕP PHWRGX YH VRQOX IDUNODU \|QWHPOHUL GL÷HU /DSODFH YH )RXULHU G|QúP \|QWHPOHUL LOH NDUúÕODúWÕUÕOGÕ÷ÕQGD \DNODúÕN \|QWHPOHULQGDKDSUDWLNYHELOJLVD\DUSURJUDPODPDVÕQDX\JXQGur.

Diferansiyel transform yöntemi, diferansiyel denklemlerin çözümünde NXOODQÕODQ QPHULN PHWRWWXU .ÕVPL GLIHUDQVL\HO GHQNOHPOHULQ \DNODúÕN o|]PQGH NXOODQÕODQGLIHUDQVL\HOWUDQVIRUPPHWRGXGL÷HU\DNODúÕNPHWRWODUDJ|UHOLQHHUYHOLQHHU olmayan problemOHULQ o|]PQQ \DQÕ VÕUD VUHNOL ROPD\DQ VÕQÕU úDUWODUÕQD VDKLS SUREOHPOHULQo|]PQGHGHoDOÕúWÕ÷ÕJ|UOHELOLU%XoDOÕúPDGDJHQHOOLWHUDWUWDUDPDVÕ \DSÕODUDN\|QWHPOHULQNDUúÕODúWÕUÕOPDVÕ]HULQGHGXUXOPXúWXU

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3

1. TEMEL KAVRAMLAR

1.1 Denklemler ve Çözümler

'LIHUDQVL\HO GHQNOHPOHU NRQXVX PRGHUQ PDWHPDWL÷LQ JHQLú YH oRN |QHPOL ELU NÕVPÕQÕROXúWXUXU'LIHUDQVL\HOYHLQWHJUDOKHVDEÕQ\]\ÕODNDGDULQHQJHoPLúLQGHQ EHUL EX GDO JHQLú WHRULN DUDúWÕUPDODUD YH LQVDQÕQ EX GQ\DGDNL KD\DWÕQÕ NROD\ODúWÕUDQ |QHPOLX\JXODPDODUDVDKQHROPXúWXU

1.2 Fonksiyonlar ve Denklemler

0KHQGLVOHUYHELOLPLQGL÷HUDODQODUÕQGDX]PDQODUX]XQ]DPDQGÕUSUREOHPOHULQL PDWHPDWLN\DUGÕPÕ\ODGDKDL\LDQODWDELOHFHNOHULQLYHGDKDNROayca inceleyebileceklerini DQODPÕúEXOXQX\RUODU$QFDNEXQXQLoLQLQFHOHQHQROD\ÕQL\LFHWDQÕQPÕúYHGDYUDQÕúÕQÕ \|QHWHQ \DVDODUÕQ WDP RODUDN LIDGH HGLOPLú ROPDVÕ JHUHNL\RU %WQ EX DQODWÕPODUGD denklemler ve fonksiyonlar bulunur[ Ross,2004 ].

Bir f IRQNVL\RQXQXQ WDQÕP E|OJHVLQGHNL KHU x LoLQ DOGÕ÷Õ GH÷HU f(x)ile gösterilir. Burada x’HED÷ÕPVÕ]GH÷LúNHQf(x)¶HGHED÷ÕPOÕGH÷LúNHQGHQLU

ùLPGL ELU \D GD GDKD ID]OD ELOLQPH\HQOL IRQNVL\RQX YH RQODUÕQ WUHYOHULQL EXOXQGXUDQGLIHUDQVL\HOGHQNOHPOHULHOHDODFD÷Õ] e x dx dy ₌ x₊_sin _(1.1) y′′−2y′+y=cosx (1.2) t u y u x u ∂ ∂ = ∂ ∂ + ∂ ∂ 2 2 2 2 (1.3) 3 2 _{+ ydy}2 ₌0 dx x (1.4)

øIDGHOHULQGH IRQNVL\RQODUÕQ GH÷HULQL J|VWHUHQ GH÷LúNHQOHUH ED÷ÕPOÕ GH÷LúNHQ denir. Bu durumda (1.1) ve (1.2)’de yED÷ÕPOÕxED÷ÕPVÕ]GH÷LúNHQ¶WHu ED÷ÕPOÕ

t,x,ve y ED÷ÕPVÕ] GH÷LúNHQGLU ¶WH LVH x ya da y¶GHQ ELUL ED÷OÕ GH÷LúNHQ olarak

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4 'LIHUDQVL\HOGHQNOHPOHULQ6ÕQÕIODQGÕUÕOPDVÕ

7DQÕP%LU\DGDGDKDoRNGH÷LúNHQLQELU\DGD daha çok VHUEHVWGH÷LúNHQHJ|UH türevlerini ya da diferansiyellerini bulunduran denklemlere diferansiyel denklem denir.

Örnek1.1$úD÷ÕGDNLOHULGLIHUDQVL\HOGHQNOHP|UQH÷LRODUDNYHUHELOLUL] 0 2 2 2 =       + dx dy xy dx y d (1.5) x t dt y d dt x d sin 3 5 ₂ 2 4 2 = + + (1.6) x v t v s v = + ∂ ∂ + ∂ ∂ ₃ _(1.7) ₂ 0 2 2 2 2 2 = ∂ ∂ + ∂ ∂ + ∂ ∂ z u y u x u (1.8) 'L÷HUWDUDIWDQWHPHOWDQÕPDX\VDODUGD ax ax ae e dx d = ) ( , dx du v dx dv u uv dx d + = ) ( (1.9) 'LIHUDQVL\HO GHQNOHPOHULQL VÕQÕIODQGÕUPDN LoLQ oHúLWOL |]HOOLNOHU NXOODQÕODELOLU

Adi vH NÕVPÕ GLIHUDQVL\HO GHQNOHPOHU ED÷ÕPVÕ] GH÷LúNHQOHULQLQ VD\ÕVÕQD

EXOXQGXUGXNODUÕWUHYOHULQoHúLGLQHJ|UHVÕQÕIODQGÕUÕODELOLU

7DQÕP 'LIHUDQVL\HO GHQNOHPGH EXOXQDQ WUHYOHU ELU \D GD GDKD oRN ED÷OÕ GH÷LúNHQLQ ELU WHN VHUEHVW GH÷LúNHQH J|UH DGL Würev ise, bu diferansiyel denklem adi

diferansiyel denklem olarak isimlendirilir.

Buna göre (1.5) ve (1.6)’daki örnekler adi diferansiyel denklemdir.

7DQÕP%LU\DGDGDKDoRNED÷OÕGH÷LúNHQLQHQD]LNLVHUEHVWGH÷LúNHQHJ|UHNÕVPÕ türevlerini bulunduran diferansiyel denkleme NÕVPÕGLIHUDQVL\HOGHQNOHP denir.

YH ¶GHNL GHQNOHPOHU NÕVPL GLIHUDQVL\HO GHQNOHPOHUGLU [S.L. Ross.,2004 ].

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5 7DQÕP : x ve yED÷ÕPVÕ]GH÷LúNHQOHULQLQELUIRQNVL\RQXz = f( yx, ) olsun. x ve y ED÷ÕPVÕ]ROGXNODUÕLoLQy sabit iken xGH÷LúHELOLUx sabit iken yGH÷LúHELOLUx ve yD\QÕDQGDGH÷LúHELOLUøONLNLGXUXPGDz,WHNELUGH÷LúNHQLQIRQNVL\RQXJLELHWNLOHQLU YHELOLQHQNXUDOODUDJ|UHGLIHUDQVL\HOOHQHELOLUùD\Ht y sabit iken x GH÷LúL\RUVDz, x’in bir fonksiyonu olur. Bu durumda x’e göre türev;

x y x f y x x f x z y x f x x ∆ − ∆ + ∂ ∂ = → ∆ ) , ( ) , ( lim ) , ( 0

úHNOLQGHROXSEXLIDGH\Hz= f( yx, )’nin x¶HJ|UHELULQFLONÕVPLWUHYLGHQLU(÷HUx sabit iken y GH÷LúWLULOL\RUVDz, y’nin bir fonksiyonudur. Bu durumda y’ye göre türev; y y x f y y x f y z y x f y y ∆ − ∆ + ∂ ∂ = → ∆ ) , ( ) , ( lim ) , ( 0

úHNOLQGH ROXS EX LIDGH\H z= f( yx, )fonksiyonunun y’\H J|UH ELULQFLO NÕVPL WUHYL denir>&2ù.8N, 2003].

'LIHUDQVL\HOGHQNOHPOHUEDVDPDNODUÕQDJ|UHGHVÕQÕIODQGÕUÕOÕUODU

7DQÕP Bir diferansiyel denklemin mertebesi, denklemde bulunan en yüksek mertebesidir.

Örneklerde (1.7) birinci,(1.5) ve (1.8) ikinci, (1.6) da üçüncü basamaktan denklemlerdir.

%DúND |QHPOL ELU VÕQÕIODQGÕUPD úHNOL GH ED÷OÕ GH÷LúNHQOH WUHYOHULQLQ GHQNOHP LoLQGH\HUDOÕúODUÕQDJ|UHGLU

7DQÕP'HQNOHPGHEXOXQDQED÷OÕGH÷LúNHQOHUNPHVLQLQELUHOHPDQÕQÕ\DGDRQXQ bir türevini bulundurDQKHUWHULPEXGH÷LúNHQHYHWUHYOHULQHJ|UHELULQFLGHUHFHGHQ LVH EX GLIHUDQVL\HO GHQNOHP EX ED÷OÕ GH÷LúNHQ NPHVLQH J|UH lineer¶GLU %D÷OÕ GH÷LúNHQH J|UH OLQHHU ROPD\DQ GHQNOHPH EX ED÷OÕ GH÷LúNHQH J|UH lineer olmayan

diferansiyel denklem denir.

(1.6), (1.7), (1.8)denklemleri lineer, (1.5) ise lineer olmayan bir diferansiyel denklemdir.

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1.4 Diferansiyel Denklemlerin ÇÕkÕú Yerleri ve UygulamalarÕ

Diferansiyel denklemleri çeúitli yönlerine göre sÕnÕflandÕrdÕktan sonra, bu gibi denklemlerin nereden ve nasÕl çÕktÕ÷Õndan kÕsaca bahsedelim. Bu sayede diferansiyel denklemlerin teori ve yöntemlerinin ne kadar geniú bir uygulama alanÕQVDKLSROGX÷XQX DQODUÕ]'LIHUDQVL\HOGHQNOHPOHUIHQYHPKHQGLVOLNELOLPOHULQLQoRNoHúLWOLGDOODUÕQGD pek çok sayÕda problemle ilintili olarak ortaya çÕNDUODUgUQH÷LQ

(1) Füze, roket, uydu ve gezegen hareketlerinin belirlenmesi, (2) Elektrik devrelerinde yük ya da akÕmÕQEXOXQPDVÕ (3) Çubukta ve levhalarda ÕsÕ yayÕlmasÕ problemi, (4) Telin ya da levhanÕn titreúimleri,

(5) Radyoaktif cismin bozunmasÕ veya bir canlÕ toplulu÷unun nüfus artÕúÕ problemi, (6) Kimyasal reaksiyonlarÕn incelenmesi,

(7) Belli geometrik özelliklere sahip e÷ULOHULQEXOXQPDVÕ

1.5 Diferansiyel Denklemlerde Çözüm

Bir x GH÷LúNHQLQH ED÷OÕ FHELUVHO \D GD WUDQVDQGDQW ELU GHQNOHPLQ o|]P EX GHQNOHPL VD÷OD\DQ ELU VD\ÕGÕU +kOEXNL GLIHUDQVL\HO GHQNOHPOHULQ o|]POHUL VD\ÕODU GH÷LOEXGHQNOHPOHULVD÷OD\DQIRQNVL\RQODUGÕU

Örnek: _{′′ k}₊ 2 ₌0

y diferansiyHO GHQNOHPLQL J|] |QQH DODOÕP f(x)=sinkx ve kx

k

y′′=− 2sin RODFD÷ÕQGDQ GHQNOHPL KHU x LoLQ VD÷ODU $\QÕ úHNLOGHg(x)=coskx

IRQNVL\RQXGDD\QÕGHQNOHPLQ(−∞,∞)DUDOÕ÷ÕQGDo|]PGU>S.L. Ross,2004].

/LQHHU ROPD\DQ GLIHUDQVL\HO GHQNOHPOHULQ NHVLQ YH\D \DNODúÕN VRQXoODUÕQÕ EXOPDN LoLQ oHúLWOL PHWRWODU X\JXODQDELOLU /LQHHU ROPD\DQ GLIHUDQVL\HO GHQNOHPOHULQ \DNODúÕN VRQXoODUÕQÕ EXOPDGD X\JXODQDQ PHWRWODUGDQ ELUL GH diferansiyel transform metoduGXUùLPGLEXPHWRWKDNNÕQGDELOJLYHUHOLP

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7

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%LUoRNPKHQGLVOLNX\JXODPDODUÕQGDOLQHHUYHOLQHHUROPD\DQNÕVPLGLIHUDQVL\HO GHQNOHPOHUOH NDUúÕODúÕUÕ] %X GHQNOHPOHULQ o|]POHULQL EXOPDN LoLQ oHúLWOL PHtotlar NHVLQ YH \DNODúÕN PHWRWODU X\JXODQDELOLU %LUoRN GXUXPGD EX GHQNOHPOHUL DQDOLWLN RODUDNo|]PHNoRNNDUÕúÕNRODELOLUøQWHJUDOG|QúPOHU/DSODFHYH)RXULHUG|QúPOHUL JLEL \|QWHPOHU NXOODQÕODELOLU )DNDW EXQODUÕQ \DQÕQGD \DNODúÕN \|QWHPOHU NXOODQÕODUDN SUREOHP EDVLWOHúWLULOHUHN NHVLQ o|]PH \DNÕQ VRQXoODU HOGH HGLOHELOLU %X \DNODúÕN yöntemlerden biri olan diferansiyel transform yöntemi diferansiyel denklemi aritmetik GHQNOHPHG|QúWUHUHNGDKDEDVLWYHVLVWHPDWLNo|]PVD÷ODU

2.1 Bir Boyutlu Diferansiyel Transform Yöntemi

%X\|QWHPWHNGH÷LúNHQLoHUGL÷LQGHQDGLGLIHUDQVL\HOGHQNOHPOHULQo|]POHUL

LoLQNXOODQÕOÕU

7DQÕP[Chen,1996]

7HNELOHúHQOLw(x) fonksiyonunun diferansiyel transform fonksiyonu W(k) olmak

üzere, w(x)’nin tek boyutlu diferansiyel transformu

0 ) ( ! 1 ) ( =       = x k k x w dx d k k W (2.1) úHNOLQGHGLU 7DQÕP[Chen,1996]

W(k) transform fonksiyonunun tersi; diferansiyel ters transform fonksiyonu,

0 ( ) ( ) k k w x W k x ∞ = =

∑

(2.2) ELoLPLQGHWDQÕPODQÕUYHHúLWOLNOHULGLNNDWHDOÕQDUDNDúD÷ÕGDNLHúLWOL÷L elde edilir.

∑

∞ =  ₌     = 0 ₀ ) ( ! 1 ) ( k k x k k x x w dx d k x w (2.3)

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8 YHGHQNOHPOHULQLNXOODQDUDNWHPHOPDWHPDWLNLúOHPOHU\DUGÕPÕ\ODWHN ER\XWOXGLIHUDQVL\HOG|QúPLoLQDúD÷ÕGDNLWHRUHPOHULYHUHELOLUL] Teorem 2.1.1 [Chen,1996] 7HNELOHúHQOLZ[X[YHY[IRQNVL\RQODUÕQÕDODOÕP(÷HU w(x)=u(x)±v(x) LVHVÕUDVÕ\OD:N8NYH9NYHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUP fRQNVL\RQODUÕROPDN]HUH W(k)=U(k)±V(k) (2.4) HúLWOL÷LVD÷ODQÕU Teorem 2.1.2 [Chen,1996]

7HNELOHúHQOLw(x) ve u(xIRQNVL\RQODUÕQÕDODOÕPc∈RROPDN]HUHH÷HU w(x)=cu(x) LVHVÕUDVÕ\ODW(k) ve U(kYHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUPIRQNVL\RQODUÕ olmak üzere, W(k)=cU(k) (2.5) HúLWOL÷LVD÷ODQÕU Teorem 2.1.3 [Chen,1996] 7HNELOHúHQOLw(x) ve u(xIRQNVL\RQODUÕQÕDODOÕP(÷HU _m m dx x u d x w( )= ( ) LVHVÕUDVÕ\ODW(k) ve U(kYHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUPIRQNVL\RQODUÕ olmak üzere, . ( ) ! )! ( ) ( U k m k m k k W = + + (2.6) HúLWOL÷LVD÷ODQÕU

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Teorem 2.1.4 [Chen,1996]

7HNELOHúHQOLw(x) ve u(xIRQNVL\RQODUÕQÕDODOÕPm∈NROPDN]HUHH÷HU _m m dx x u d x w( )= ( )

LVHVÕUDVÕ\ODW(k) ve U(k) verLOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUPIRQNVL\RQODUÕ olmak üzere, . ( ) ! )! ( ) ( ) )....( 2 )( 1 ( ) ( U k m k m k m k U m k k k k W = + + + + = + + (2.7) HúLWOL÷LVD÷ODQÕU Teorem 2.1.5 [Chen,1996]

7HNELOHúHQOLw(x), u(x) ve v(x)IRQNVL\RQODUÕQÕDODOÕPr∈NROPDN]HUHH÷HU

w(x)=u(x)v(x)

LVHVÕUDVÕ\ODW(k), U(k) ve V(k)YHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUP IRQNVL\RQODUÕROPDN]HUH

∑

= − = k r r k V r U k W 0 ) ( ). ( ) ( (2.8) HúLWOL÷LVD÷ODQÕU Teorem 2.1.6 [Chen,1996]

7HNELOHúHQOLw(x) IRQNVL\RQXQXDODOÕPm∈N ROPDN]HUHH÷HU

m

x x w( )=

LVHVÕUDVÕ\ODW(k), U(k) ve V(k)YHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUP fonksiyonu _  = = − = halde aksi m k için m k k W 0 1 ) ( ) ( δ (2.9) HúLWOL÷LVD÷ODQÕU

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10

Teorem 2.1.7 [Ayaz,1996]

7HNELOHúHQOLw(x), u(x) ve v(x)IRQNVL\RQODUÕQÕDODOÕP ( ) ( ) ₂ ( ) 2 x v dx d x u x w =

LVHVÕUDVÕ\ODW(k), U(k) ve V(k)YHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUP IRQNVL\RQODUÕROPDN]HUH

∑

= + − + − + − = k r r k V r U r k r k k W 0 ) 2 ( ). ( ) 1 )( 2 ( ) ( (2.10) úHNOLQGH\D]ÕODELOLU Teorem 2.1.8[Ayaz,1996]

7HNELOHúHQOLw(x), u(x) v(x) ve s(x) fonkVL\RQODUÕQÕDODOÕPr∈N olmak üzere H÷HU

w(x)=u(x)v(x)s(x)

LVHVÕUDVÕ\ODW(k), U(k) ve V(k)YHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUP IRQNVL\RQODUÕROPDN]HUH ) ( ) ( ). ( ) ( ) ( ) ( ) ( 0 0 t r k s t V r U k S k V k U k W k r r k t − − = ⊗ ⊗ =

∑∑

= − = (2.11) HúLWOL÷LVD÷ODQGÕ÷ÕJ|UOU Teorem 2.1.9 [Abel-Halim,2004]

7HNELOHúHQOLw(x) IRQNVL\RQXQXDODOÕPλ∈NROPDN]HUHH÷HU

x

e x w( )= λ

LVHVÕUDVÕ\ODW(k) verilen fonksiyonun diferansiyel transform fonksiyonu olmak üzere, ! ) ( k k W k λ = (2.12) HúLWOL÷LVD÷ODQGÕ÷ÕJ|UOU Teorem 2.1.10 [Abel-Halim,2004]

7HNELOHúHQOLw(x) IRQNVL\RQXQXDODOÕPa,b∈RROPDN]HUHH÷HU

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11 LVHVÕUDVÕ\ODW(k) verilen fonksiyonun diferansiyel transform fonksiyonu olmak üzere, _      ₊ = k b k a k W k 2 sin ! ) ( π (2.13) HúLWOL÷LVD÷ODQÕU Teorem 2.1.11 [Abel-Halim,2004]

7HNELOHúHQOLw(x) IRQNVL\RQXQXDODOÕPa,b∈R olmak ]HUHH÷HU

w(x)=cos(ax+b)

LVHVÕUDVÕ\ODW(k) verilen fonksiyonun diferansiyel transform fonksiyonu olmak üzere,       ₊ = k b k a k W k 2 cos ! ) ( π (2.14) HúLWOL÷LVD÷ODQÕU Teorem 2.1.12 >$ULNR÷OX@ 7HNELOHúHQOLw(x) ve u(xIRQNVL\RQODUÕQÕDODOÕP(÷HU =

∫

x x u t dt x w 0 ) ( ) (

ise W(k) ve U(kYHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUPIRQNVL\RQODUÕ olmak üzere, k k U k W( )= ( −1) (2.15) HúLWOL÷LVD÷ODQÕU Teorem 2.1.13 >$ULNR÷OX@

7HNELOHúHQOLw(x), u(x) ve v(x)IRQNVL\RQODUÕQÕDODOÕP(÷HU =

∫

x x u t dt x v x w 0 ) ( ) ( ) (

ise W(k), U(k) ve V(x) YHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUPIRQNVL\RQODUÕ olmak üzere,

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12 k k U k v k W( )= ( )⊗ ( −1) (2.16) HúLWOL÷LVD÷ODQÕU Teorem 2.1.14 >$ULNR÷OX@

7HNELOHúHQOLw(x), u(x) ve v(x)IRQNVL\RQODUÕQÕDODOÕP(÷HU =

∫

x x u t v x dt x w 0 ) ( ) ( ) (

ise W(k), U(k) ve V(x) YHULOHQIRQNVL\RQODUÕQGLIHUDQVL\HOWUDQVIRUPIRQNVL\RQODUÕ olmak üzere, k k V k U k W( )= ( −1)⊗ ( −1) (2.17) HúLWOL÷LVD÷ODQÕU

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13 Bir boyutlu Diferansiyel transform ile verilen teoremleri bir tablo içinde gösterelim. Tablo-1 Bir boyutlu diferansiyel trasform

Fonksiyon formu Transform formu

) ( ) ( ) (x u x v x w = ± W(k)=U(k)±V(k) ) ( . ) (x u x w =α W(k)=α.U(k) m m dx x u d x w( )= ( ) . ( ) ! )! ( ) ( U k m k m k k W = + + ) ( ). ( ) (x u x v x w =

∑

= − = k r r k V r U k W 0 ) ( ). ( ) ( m x x w( )=    = = − == halde aksi m k için m k k W 0 1 ) ( ) ( δ ) ( ) ( ) ( ₂ 2 x v dx d x u x w =

∑

= + − + − + − = k r r k V r U r k r k k W 0 ) 2 ( ). ( ) 1 )( 2 ( ) ( ) ( ) ( ) ( ) (x u x v x s x w = ) ( ) ( ). ( ) ( 0 0 t r k s t V r U k W k r r k t − − =

∑∑

= − = x e x w( )= λ ! ) ( k k W k λ = ) sin( ) (x ax b w = +       ₊ = k b k a k W k 2 sin ! ) ( π ) cos( ) (x ax b w = +       ₊ = k b k a k W k 2 cos ! ) ( π

∫

= x x u t dt x w 0 ) ( ) ( k k U k W( )= ( −1)

∫

= x x u t dt x v x w 0 ) ( ) ( ) ( k k U k v k W( )= ( )⊗ ( −1)

∫

= x x u t v x dt x w 0 ) ( ) ( ) ( k k V k U k W( )= ( −1)⊗ ( −1) øNL%R\XWOX'LIHUDQVL\HO7UDQVIRUP<|QWHPL 7DQÕP.[Zhou,1986]

øNLELOHúHQOLw(x,y) fonksiyonunun transform fonksiyonu W(k,h) olmak üzere, Z[\¶QLQLNLER\XWOXGLIHUDQVL\HOG|QúP ) 0 , 0 ( ) , ( ! ! 1 ) , ( _      = + w x y dy dx d h k h k W _k _h h k (2.18) RODUDNWDQÕPODQÕU

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14 7DQÕP.[Zhou,1986] W(k,h) fonksiyonun ters dönüúPLVH

∑∑

∞ = ∞ = = 0 0 ) , ( ) , ( k h k h y x h k W y x w (2.19) úHNOLQGHGLUGHQNOHPLGHQNOHPGH\HULQH\D]GÕ÷ÕPÕ]GD

∑∑

∞ = ∞ = +       = 0 0 ₍₀_,₀₎ ) , ( ! ! 1 ) , ( k h k h h k h k y x y x w dy dx d h k y x w (2.20) elde edilir. Teorem 2.2.1.[Zhou,1986]

øNLELOHúHQOLw(x,y), u(x,y) ve v(x,y)IRQNVL\RQODUÕQÕDODOÕP(÷HU w(x,y)=u(x,y)±v(x,y) LVHVÕUDVÕ\ODW(k,h),U(x) ve V(k,h)YHULOHQIRQNVL\RQODUÕQWUDQVIRUPIRQNVL\RQODUÕROPDN üzere W(k,h)=U(k,h)±V(k,h) (2.21) HúLWOL÷LVD÷ODQÕU Teorem 2.2.2.[Zhou,1986] øNLELOHúHQOLZ[\YHX[\IRQNVL\RQODUÕQÕDODOÕPc∈RROPDN]HUHH÷HU w(x,y)=cu(x,y)

LVH VÕUDVÕ\OD W(k,h) ve U(k,h) YHULOHQ IRQNVL\RQODUÕQ WUDQVIRUP IRQNVL\RQODUÕ ROPDN üzere

W(k,h)=cU(k,h) (2.22) HúLWOL÷LVD÷ODQÕU

Teorem 2.2.3.[Zhou,1986]

øNLELOHúHQOLw(x,y) ve u(x,y) IRQNVL\RQODUÕQÕDODOÕP(÷HU x y x u y x w ∂ ∂ = ( , ) ) , (

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15 W(k,h)=(k+1)U(k+1,h) (2.23) HúLWOL÷LVD÷ODQÕU

Teorem 2.2.4.[Ayaz,2003]

øNLELOHúHQOLw(x,y) ve u(x,y) IRQNVL\RQODUÕQÕDODOÕP(÷HU y y x u y x w ∂ ∂ = ( , ) ) , (

W(k,h)=(h+1)U(k,h+1) (2.24) HúLWOL÷LVD÷ODQÕU

Teorem 2.2.5.[Ayaz,2003]

øNLELOHúHQOLw(x,y) ve u(x,y) IRQNVL\RQODUÕQÕDODOÕPr,s∈NROPDN]HUHH÷HU

_r _s s r dy x y x u y x w ∂ ∂ = + ( , ) ) , (

isH VÕUDVÕ\OD W(k,h) ve U(k,h) YHULOHQ IRQNVL\RQODUÕQ WUDQVIRUP IRQNVL\RQODUÕ ROPDN üzere W(k,h)=(h+1)(k+2)...(k+r)(h+1)(h+2)....(h+s)U(k+r,h+s) ( , ) ! )! ( . ! )! ( ) , ( U k r h s s s h k r k h k W = + + + + (2.25) HúLWOL÷LVD÷ODQÕU Teorem 2.2.6.[Ayaz,2003]

øNLELOHúHQOLw(x,y), u(x,y)ve v(x,y) IRQNVL\RQODUÕQÕDODOÕP(÷HU

w(x,y)=u(x,y).v(x,y)

LVH VÕUDVÕ\OD W(k,h),U(k,h) ve V(k,h) YHULOHQ IRQNVL\RQODUÕQ WUDQVIRUP IRQNVL\RQODUÕ olmak üzere

∑∑

= = − − = ⊗ = r r h h s r k V s h r U h k V h k U h k W 0 0 ) , ( ). , ( ) , ( ) , ( ) , ( (2.26) HúLWOL÷LVD÷ODQÕU

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16

Teorem 2.2.7.[Ayaz,2003]

øNLELOHúHQOLw(x,y) IRQNVL\RQXQXDODOÕPm∈ZROPDN]HUHH÷HU m n y x y x w( , )=

ise W(k,h) verilen fRQNVL\RQXQWUDQVIRUPIRQNVL\RQODUÕROPDN]HUH _  = = = − − = halde aksi n h ve m k için n h m k h k W 0 1 ) , ( ) , ( δ (2.27) HúLWOL÷LVD÷ODQÕU Teorem 2.2.8.[Ayaz,2003]

øNLELOHúHQOLw(x,y), u(x,y)ve v(x,y) IRQNVL\RQODUÕQÕDODOÕP(÷HU ₂ 2 ₍ _, ₎ ). , ( ) , ( x y x v y x u y x w ∂ ∂ =

∑∑

= = − − + − + − = r r h h s r k V s h r U r k r k h k W 0 0 ) , ( ). , ( ) 1 )( 2 ( ) , ( (2.28) HúLWOL÷LVD÷ODQÕU Teorem 2.2.9.[Ayaz,2003]

øNLELOHúHQOLw(x,y), u(x,y)ve v(x,y) IRQNVL\RQODUÕQÕDODOÕP(÷HU x y x v x y x u y x w ∂ ∂ ∂ ∂ = ( , ). ( , ) ) , (

∑∑

= = + − − + + − + = r r h h s r k V s h r U r k r h k W 0 0 ) , 1 ( ). , 1 ( ) 1 )( 1 ( ) , ( (2.29) HúLWOL÷LVD÷ODQÕU Teorem 2.2.10.[Ayaz,2003]

øNLELOHúHQOLw(x,y), u(x,y)ve v(x,y) IRQNVL\RQODUÕQÕDODOÕP(÷HU

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17 y y x v y y x u y x w ∂ ∂ ∂ ∂ = ( , ). ( , ) ) , (

∑∑

= = + − + − + − + = r r h h s r k V s h r U s k s h k W 0 0 ) 1 , ( ). 1 , ( ) 1 )( 1 ( ) , ( (2.30) HúLWOL÷LVD÷ODQÕU Teorem 2.2.11.[Ayaz,2003]

øNLELOHúHQOLw(x,y), u(x,y)ve v(x,y) IRQNVL\RQODUÕQÕDODOÕP(÷HU y y x v x y x u y x w ∂ ∂ ∂ ∂ = ( , ). ( , ) ) , (

∑∑

= = + − + − + − + − = r r h h s h r V s r k U s k r k h k W 0 0 ) 1 , ( ). , 1 ( ) 1 )( 1 ( ) , ( (2.31) HúLWOL÷LVD÷ODQÕU.

Bu yüzden iki boyutlu diferansiyel transformdan elde edilen temel matematiksel LúOHPOHUWDEOR-2 deki gibidir.

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18 Tablo-øNLER\XWOXGLIHUDQVL\HOWUDQVIRUP

Fonksiyon formu Transform formu

) , ( ) , ( ) , (x y u x y v x y w = ± W(k,h)=U(k,h)±V(k,h) ) , ( . ) , (x y u x y w =α W(k,h)=α.U(k,h) x y x u y x w ∂ ∂ = ( , ) ) , ( W(k,h)=(k+1).U(k+1,h) y y x u y x w ∂ ∂ = ( , ) ) , ( W(k,h)=(h+1).U(k,h+1) s r s r dy x y x u y x w ∂ ∂ = + ( , ) ) , ( ( , ) ! )! ( . ! )! ( ) , ( U k r h s s s h k r k h k W = + + + + ) , ( ). , ( ) , (x y u x y v x y w =

∑∑

= = − − = ⊗ = r r h h s r k V s h r U h k V h k U h k W 0 0 ) , ( ). , ( ) , ( ) , ( ) , ( n m y x y x w( , )= _  = = = − − = halde aksi n h ve m k için n h m k h k W 0 1 ) , ( ) , ( δ 2 2 ₍ _, ₎ ). , ( ) , ( x y x v y x u y x w ∂ ∂ =

∑∑

= = − − + − + − = r r h h s r k V s h r U r k r k h k W 0 0 ) , ( ). , ( ) 1 )( 2 ( ) , ( x y x v x y x u y x w ∂ ∂ ∂ ∂ = ( , ). ( , ) ) , (

∑∑

= = + − − + + − + = r r h h s r k V s h r U r k r h k W 0 0 ) , 1 ( ). , 1 ( ) 1 )( 1 ( ) , ( y y x v y y x u y x w ∂ ∂ ∂ ∂ = ( , ). ( , ) ) , (

∑∑

= = + − + − + − + = r r h h s r k V s h r U s k s h k W 0 0 ) 1 , ( ). 1 , ( ) 1 )( 1 ( ) , ( y y x v x y x u y x w ∂ ∂ ∂ ∂ = ( , ). ( , ) ) , (

∑∑

= = + − + − + − + − = r r h h s h r V s r k U s k r k h k W 0 0 ) 1 , ( ). , 1 ( ) 1 )( 1 ( ) , (

ùLPGL Q GH÷LúNHQOL NÕVPL GHQNOHPOHU LoLQ Q-boyutlu diferansiyel transform \|QWHPLQLYHLúOHPOHULQLWDQÕWDOÕP

2.3 N-Boyutlu Diferansiyel Transform:

7DQÕP[Kurnaz,2005]

x=(x1,x2...,xn)QGH÷LúNHnli bir vektör ve k =(k1,k2...,kn) ise pozitif

WDPVD\ÕYHNW|UROVXQ%|\OHFHQ-ER\XWOXGLIHUDQVL\HOG|QúPDúD÷ÕGDNLJLELROXU ) 0 ,.... 0 , 0 ( 2 1 2 1 .... 2 1 2 1 ( , ..., ) ... ! !... ! 1 ) ,...., , ( 2 1 2 1         ∂ ∂ ∂ ∂ = + + + k n n k k k k k n n w x x x x x x k k k k k k W n n (2.32)

Burada w(x)=w(x₁,x₂...,xn)orijinal fonksiyondur ve ) ..., , ( ) (k W k₁ k₂ k_n W = G|QúWUOPúIRQNVL\RQGXU

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19 7DQÕP[Kurnaz,2005] W(k)¶QÕQGLIHUDQVL\HOWHUVWUDQVIRUPXDúD÷ÕGDNLJLELEXOXQXU

∑∑ ∑

∏

= ∞ ∞ ∞ = n i k i k k k n n i n x k k k W x x x w 1 2 1 2 1 1 2 ) ,.... , ( ... ) ..., , ( (2.33) úHNOLQGHGLUYHGHnklemlerinden; =

∑∑ ∑

∞ ∞ ∞ 1 2 ! !.... ! 1 ... ) ..., , ( 2 1 2 1 k k k n n n k k k x x x w x

∏

= + + +       ∂ ∂ ∂ n i k i n k k k k k k i n n x x x x w x x x d 1 ) 0 ,.... 0 , 0 ( 2 1 2 1 .... ) ..., , ( ... 2 1 2 1 (2.34) HOGHHGLOLUùLPGLED]ÕPDWHPDWLNVHOLúOHPOHULWHRUHPOHUOHLQFHOH\HOLP TEOREM 2.3.1[Kurnaz,2005]: QELOHúenli w(x₁,x₂...,xn), u(x1,x2...,xn)ve u(x1,x2...,xn) IRQNVL\RQODUÕQÕDODOÕP(÷HU ) ..., , ( ) ..., , ( ) ..., , (x₁ x₂ x_n u x₁ x₂ x_n v x₁ x₂ x_n w = ± LVHVÕUDVÕ\ODW(x1,x2...,xn),U(x1,x2...,xn) ve V(x1,x2...,xn)verilen IRQNVL\RQODUÕQWUDQVIRUPIRQNVL\RQODUÕROPDN]HUH W(x₁,x₂...,xn)=U(x₁,x₂...,xn)±V(x₁,x₂...,xn) (2.35) TEOREM 2.3.2 [Kurnaz,2005]: QELOHúHQOLw(x₁,x₂...,x_n)veu(x₁,x₂...,x_n)IRQNVL\RQODUÕQÕDODOÕP(÷HU w(x₁,x₂...,x_n)=α.u(x₁,x₂...,x_n)

LVH VÕUDVÕ\ODW(x₁,x₂...,x_n) ve U(x₁,x₂...,x_n) YHULOHQ IRQNVL\RQODUÕQ WUDQVIRUP

IRQNVL\RQODUÕROPDN]HUH

W(x₁,x₂...,x_n)=α.U(x₁,x₂...,x_n) ( :α sabit ) (2.36) HúLWOL÷LVD÷ODQÕU

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20 TEOREM 2.3.3 [Kurnaz,2005]: QELOHúHQOLw(x1,x2...,xn)veu(x1,x2...,xn)IRQNVL\RQODUÕQÕDODOÕP(÷HU i n i n i dx x x x u x x x w( ,... ..., ) ( 1,... ..., ) 1 ∂ =

LVH VÕUDVÕ\ODW(x₁,x₂...,x_n) ve U(x₁,x₂...,x_n) YHULOHQ IRQNVL\RQODUÕQ WUDQVIRUP

IRQNVL\RQODUÕROPDN]HUH W(k₁,...k_i...,k_n)=(k_i +1)U(k₁,...k_i +1,...,k_n), i=1...n (2.37) TEOREM 2.3.4 [Kurnaz,2005]: QELOHúHQOLw(x₁,x₂...,x_n)veu(x₁,x₂...,x_n)IRQNVL\RQODUÕQÕDODOÕP(÷HU n n r n r n r r n x x x x x u x x w ∂ ∂ ∂ = + + .... ) ..., , ( ) , ,... ( 1 1 1 2 1 ... 1

LVH VÕUDVÕ\ODW(x₁,x₂...,xn) ve U(x1,x2...,xn) YHULOHQ IRQNVL\RQODUÕQ WUDQVIRUP

IRQNVL\RQODUÕROPDN]HUH ( ,...,..., ) ! !... )! )!....( ( ) , ,... , ( ₁ 1 1 1 2 1 n n n n n n n U k r k r k k r k r k k k k W = + + + + (2.38) TEOREM 2.3.5[Kurnaz,2005]: QELOHúHQOLw(x₁,x₂...,xn), u(x1,x2...,xn)ve u(x1,x2...,xn) IRQNVL\RQODUÕQÕDODOÕP(÷HU w(x₁,x₂...,x_n)=u(x₁,x₂...,x_n).v(x₁,x₂...,x_n) ise w fonksiyonun diferansiyel transformunu;

W(k1,k2,....,kn)=U(k1,k2,....,kn)⊗V(k1,k2,....,kn)

∑ ∑ ∑

= = = − − − = 1 1 2 2 0 0 0 2 1 1 2 2 1, ,...., ). ( , ...., ) ( .... k a k a k a n n n n n a a a k V a k a k a U (2.39) úHNOLQGH\D]DUÕ]

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21                  = = + − × − − − + − + − + ≤ < = + − × − − − + − + ≤ ≤ < + − × − − + − × − − + − =

∑ ∑

= = = = = = 1 ) ,... , ( ) ,... ,... , ( )! ( )! ( ! )! ( ... 1 ) ,... ,... , ( ) ,... ,... , ( ! )! ( )! ( )! ( ... 1 ) ,... ,... ,... ( ) ,... ),... ...( ( ! )! ( )! ( )! ( ... 2 2 1 1 2 2 1 1 0 0 1 1 2 1 1 1 1 1 2 2 1 1 2 2 1 1 0 0 2 1 1 1 2 1 1 1 1 0 0 2 1 1 1 1 1 1 1 j i için a a r a k V a k a k a k r a U a k r a k a r a n j i için a r a a a k V a k a k a k r a U a r a a r a n j i için a r a a a k V a k a k r a k a U a r a a k r a k n n n j j k a k a n j n n j j k a k a j j n j i n n j j i i k a k a j j i i i i n n n n n n TEOREM 2.3.6[Kurnaz,2005]: QELOHúHQOLw(x₁,x₂...,x_n), u(x₁,x₂...,x_n)ve u(x₁,x₂...,x_n) IRQNVL\RQODUÕQÕDODOÕP(÷HU ) / ) ..., , ( )( / ) ..., , ( ( ) ..., , ( 1 1 2 2 2 1 2 1 2 1 r j n r r i n r n u x x x x v x x x x x x x w = ∂ ∂ ∂ ∂

ise w fonksiyonun diferansiyel transformunu; ) ,...., , (k₁ k₂ kn W (2.40) úHNOLQGHJ|VWHULOLU TEOREM 2.3.7[Kurnaz,2005]: QELOHúHQOLw(x₁,x₂...,x_n)IRQNVL\RQXQXDODOÕP(÷HU mn n m m n x x x x x x w( , ..., ) 1, 2..., 2 1 2 1 =

ise w fonksiyonun diferansiyel transformu;

∏

= − = − − − = n i i i n n n k m k m k m k m k k k W 1 2 2 1 1 2 1, ,...., ) ( , ,...., ) ( ) ( δ δ    = = hallerde diger m k eger i i 0 1 (2.41) úHNOLQGHJ|VWHULOLU

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22

$'20ø$1$<5,ù,00(72'8

.ÕVPLGLIHUDQVL\HOGHQNOHPOHULQ\DNODúÕNo|]POHULQLQHOGHHWPHQLQELU\ROXGD $GRPLDQ D\UÕúÕP PHWRGXGXU %X PHWRW \DUGÕPÕ\OD SROLQRPODUÕ KHVDSODQDQ ve ELOLQPH\HQ IRQNVL\RQXQ D\UÕúWÕUÕOPDVÕQD GD\DQDQ $GRPLDQ D\UÕúÕP PHWRGX \DNODúÕN ¶OL \ÕOODUGD *$GRPLDQ WDUDIÕQGDQ EXOXQGX %X PHWRWOD OLQHHU YH OLQHHU ROPD\DQ NÕVPLGLIHUDQVL\HOGHQNOHPOHULEHOLUOLVÕQÕUúDUWODUÕDOWÕQGDNODVLN\|QWHPOHUHJ|UHGDKD EDVLWYHGDKDNDUPDúÕNGHQNOHPOHUHX\JXODQDELOHQELU\|QWHPGLU

%X PHWRGXQ HWNLQOL÷L EDúODQJÕoWD \DOQÕ] |UQHNOHUH GHQHQPLú YH \DNÕQVDNOÕ÷Õ RUWD\D NR\DFDN KLoELU DoÕNODPD \DSÕOPDPÕúWÕU 'DKD VRQUD &KHUUXDXOW \DSWÕ÷Õ oDOÕúPDODUGD D\UÕúÕP PHWRGXQ \DNÕQVDNOÕ÷ÕQÕ NDQÕWODGÕ $GRPLDQ D\UÕúÕP PHWRGXQXQ X\JXODPDVÕVÕUDVÕQGDSUREOHPGHKHUKDQJLELUGH÷LúLNOLN\DSPDLKWL\DFÕGX\PD]EXGD SUREOHPLQ DQD \DSÕVÕQÕ GH÷LúWLUPH] *HQHOOLNOH YHULOHQ ELU SUREOHPLQ o|]PQ oRN oDEXN \DNÕQVD\DQ VHUL IRUPXQGD YH EHOLUOL VÕQÕU úDUWODUÕ DOWÕQGD HOGH HGLOLU %X PHWRW GL÷HU ELUNDo PHWRWOD RUWDN |]HOOLNOHUL YDUGÕU IDNDW JHQLú oDSOÕ DUDúWÕUPD \DSÕOGÕ÷ÕQGD RQODUGDQD\UÕOGÕ÷ÕJ|UOHFHNWLU

$GRPLDQ D\UÕúÕP PHWRGX ]RU o|]OHELOHQ SUREOHPOHUL NROD\OÕNOD çözebilmekteGLU$GRPDLQD\UÕúÕPPHWRGXELUVHULo|]P\|QWHPLGLU%LUoRNOLQHHUYH OLQHHU ROPD\DQ GLIHUDQVL\HO GHQNOHPOHUH EDúDUÕ\OD X\JXODQDELOPHNWHGLU %X PHWRGX incelenecek olursak, F hem lineer hem de lineer olmayan terimleri içeren genel bir lineer olmayan adi diferansiyel operatör olmak üzere,

Fu x( )=g x( ) (3.1) GHQNOHPLQLHOHDODOÕPGHQNOHPLQGH1GLIHUDQVL\HOGHQNOeminde lineer olmayan WHULPL5OLQHHURSHUDW|UGHQNDODQNÕVPÕYH/YHULOHQGLIHUDQVL\HOGHQNOHPLQHQ yüksek mertebeden türevini göstermek üzere,

Lu Ru Nu+ + = (3.2) g úHNOLQGH\D]ÕOVÕQLOLQHHUELURSHUDW|UGU9HWHUVLGHPHYFXWWXUHúLWOL÷L Lu= −g Ru−Nu (3.3) úHNOLQGH\D]ÕODELOLU(úLWOL÷LQKHULNLWDUDIÕQD 1 L− RSHUDW|UVROWDUDIÕQGDQX\JXODQÕUVD 1 1 1 1 L Lu− =L g− −L Ru− −L Nu− (3.4)

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23

HúLWOL÷LHOGHHGLOLU

L Operatörünün ikinci mertebeden ve tersi mevcut olan lineer bir operatör ROGX÷XNDEXOHGHOLPHúLWOL÷LQGH\DSÕODQLúOHPOHUVRQXFXQGD

₍₀₎ ₍₀₎ 1 1

u=u +tu′ −L Ru− −L Nu− (3.5)

o|]P IRQNVL\RQX EXOXQXU HúLWOL÷LQGHNL Nu lineer olmayan bir terim ve

0 n n

Nu=

∑

∞₌ A biçiminde ifade edilebilir. Buradaki A_nSROLQRPODUÕ|]HOSROLQRPODUGÕU

%X SROLQRPODU ]HULQGHQ NRQXúXODFDNWÕU HúLWOL÷LQGH NL u D\UÕúWÕUÕOPÕú ELU VHUL çözüm fonksiyonudur. Bu seri çözüm fonksiyonunun birinci terimi olan u₀, verilen

EDúODQJÕoGH÷HULYHGHQNOHPLQVD÷WDUDIIRQNVL\RQXQXQLQWHJUDOLDOÕQPDN]HUH

1

0

u = + −a bt L g−

úHNOLQGHEXOXQXU

Daha sonra seri çözümün birinci terimi olan u₀ terimini kullanaraku u ₁, ,₂

WHULPOHULHOGHHGLOLU$\UÕúWÕUÕOPÕúVHULo|]PIRQNVL\RQX ₀ 0 ( , ) ( , ) n u x t u x t ∞ = =

∑

(3.6) ELoLPLQGH\D]ÕODELOLUYHVHULQLQ\DNÕQVDNROGX÷XJ|UOU%XVHULo|]PQNXOODQDUDN HúLWOL÷LQLWHNUDU\D]ÕOÕUVD 1 1 0 0 0 0 n n n n n n u u L R u L A ∞ ∞ ∞ − − = = = = − −

∑

(3.7) biçiminde ki genel seri formu elde edilir. Bununla beraber, (3.7) belirgin biçimde

1 1 1 0 0 u = −L Ru− −L A− 1 1 2 1 1 u = −L Ru− −L A− (3.8) 1 1 1 n n n u ₊ = −L Ru− −L A− , n≥0

úHNOLQGH \D]ÕODELOLU %XUDGDNL An SROLQRPODUÕ OLQHHU ROPD\DQ KHU ELU WHULP LoLQ

(29)

24

0, ,1 2

u u u ¶\HED÷OÕYHEHQ]HUúHNLOGHHúLWOL÷LQGHNLEWQ A_n Adomian polinomlar

elde edilebilir. An Adomian polinomu,

0 ( )0 A = f u , 1 1 0 0 ( ) d A u f u du   =    

,

2 2 1 2 2 0 2 0 0 0 ( ) ( ) 2! u d d A u f u f u du du      = _ _ +_ __ _      

, (3.9)

3 2 3 1 3 3 0 1 2 2 0 3 0 0 0 0 ( ) ( ) ( ) 3! u d d d A u f u u u f u f u du du du        = _ _ + _ _ +_ __ _         úHNOLQGH\D]ÕODELOLU>$GRPLDQ@$\UÕúWÕUÕOPÕúSROLQRPODUÕQJHQHOGXUXPX 1 ₁ 1 , 0 ! n n n n k k d A u n n dλ λ _λ ∞ = ₌    = _ Φ_ __ >   

∑

 (3.10) úHNOLQGH JHQHOOHúWLULOHUHN $GRPLDQ 6HQJ YH DUNDGDúODUÕ WDUDIÕQGDQ

OLWHUDWUHND]DQGÕUÕOPÕúWÕU%X$GRPLDQSROLQRPODUÕQÕHOGHHWPHNLoLQELUDOWHUQDWLILVH >:D]ZD]@WDUDIÕQGDQJHOLúWLULOPLúWLU%D]ÕSUREOHPOHULQVD\ÕVDOo|]POHULQLQGDKD KDVVDVROPDVÕLVWHQLOGL÷LGXUXPODUGDD\UÕúÕPVHULVLLoLQoRNVD\ÕGDWHULPLQKHVDSODQPDVÕ JHUHNHELOLU%XGXUXPODUGDJHQHOIRUPOQQNXOODQÕOPDVÕLVWHQLOGL÷LNDGDUoRN VD\ÕGDD\UÕúWÕUPDVHUVLQLQWHULPOHULQLQKHVDSODQPDVÕQGDNROD\OÕNODUVD÷ODPDNWDGÕU $\UÕúÕPPHWRGXNXOODQDUDN ( , )u x t NDSDOÕo|]PIRQNVL\RQXYHEXIRQNVL\RQD DLWVD\ÕVDOo|]PHOGHHWPHNLoLQ 1 1 ( , ) ( , ); n n n n x t u x t − = Φ =

∑

n≥0 (3.11) olmak üzere, lim _n ( , ) x→∞Φ =u x t (3.12) LIDGHVLQGHLQGLUJHPHED÷ÕQWÕVÕLQFHOHQHUHNNROD\FDKHVDSODQDELOLU%XQDHNRODUDN úHNOLQGHNLD\UÕúÕPVHULo|]PJHQHOOLNOHIL]LNVHOSUREOHPOHUGHoRNKÕ]OÕRODUDN

(30)

25 \DNÕQVDNOÕN RUWD\D oÕNDUPDNWDGÕU %LU oRN D\UÕúÕP VHULVLQLQ \DNÕQVDNOÕ÷ÕQÕ DUDúWÕUDQ >5HSDFL@ D\UÕúÕP VHULVLQLQ \DNÕQVDNOÕ÷ÕQÕ WHRULN RODUDN OLWHUDWUH ND]DQGÕUPÕúWÕU $úD÷ÕGDNL $GRPLDQ D\UÕúÕP SROLQRPODUÕ LOH LOJLOL NRQXODU >:D]ZD]@ YH >:D]ZD]@ND\QDNODUÕQGDQ\DUDUODQÕODUDN|]HWOHQPLúWLU

3.1 Lineer Olmayan Polinomlar

₍ ₎ 2 u u f = Öncelikle,

∑

∞ = = 0 n n u u (3.13) D\UÕúWÕUPDVÕQÕNXUDUÕ] (3.13)’ ₍ ₎ 2 u u f = ’de yazarak, 2 2 1 0 ...) ( ) (u = u +u +u + f (3.14) RODUDNEXOXUX]YDLIDGHQLQVD÷WDUDIÕQÕDoÕNRODUDN\D]DUVDN ( ) 2 2 2 2 ₀ ₃ 2 ₁ ₂ ... 1 2 0 1 0 2 0 + + + + + + =u u u u u u u u uu u f (3.15)

HOGH HGLOLUL YH EX DoÕGDQ EX DoÕOÕP WP WHULPOHULQu0ELOHúHQOHULQLQ LQGLVOHUL WRSODPÕ

D\QÕRODFDNúHNLOGHJUXSODQGÕUÕODUDN\HQLGHQG]HQOHQHELOLU<DQL , ... 2 2 2 2 ) ( 3 2 1 0 2 1 3 0 2 1 2 0 1 0 2 0 + + + + + + = A A A A u u u u u u u u u u u f (3.16) úHNOLQGH\D]ÕODELOLU%|\OHFH ₍ ₎ 2 u u f = için AdRPLDQSROLQRPODUÕ 2 0 0 u A = A₁ =2 uu₀ ₁ 2 1 2 0 2 2u u u A = + A₃ =2u₀u₃ +2u₁u₂ 2 2 3 1 4 0 4 2u u 2uu u A = + + (3.17)

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26

3.2 Lineer Olmayan Türevler f(u)=uu_x (3.13)’deki ifadeden,

∑

∞ = = 0 n n x u x u (3.18)

kurulur ve (3.18¶\Õ f(u) denkleminde yerine yazarsak f(u)=(u₀ +u₁ +u₂ +u₃ +...) ×(u₀x +u₁x +u₂x +u₃x +...) (3.19) HOGHHGLOLUYHEXoDUSPD\DSÕODUDN f(u)=u0u0_x +u0xu1_x +u0u1x +u0_x u2 +u1_xu1 +u2_xu0 +u0_xu3 u₁ u₂ u₂ u₁ u₃ u₀ u₀ u₄ u₀u₄ u₁ u₃ x x x x x x + + + + + + +u₁u₃_x +u₂u₂_x + (3.20)

bulunur. Tüm terimleri u_nELOHúHQOHULWRSODPÕD\QÕRODFDNúHNLOGHWRSODUVDN

2 1 0 0 2 1 1 2 0 1 0 1 0 0 0 ) ( A A x x A u u u u u u u u u u u u u f x x x x x + + + + + = 3 0 3 1 2 2 1 3 0 A u u u u u u u u x x x x + + + + + ++ + + + + + 4 0 4 1 3 2 2 3 1 4 0 A u u u u u u u u u u x x x x x (3.21)

sonuç olarak Adomian polinomlaUÕ x u u A₀ = ₀ ₀ A u _xu u u _x x 0 1 1 0 1 = + A₂ u₁ u₁ u₂ u₀ u₀ u₃ u₁ u₂ u₂ u₁ x x x x x + + + + = A₄ u₀ u₄ u₁ u₃ u₂ u₂ u₃ u₁ u₄ u₀ x x x x x ++ + + + = (3.22) úHNOLQGHHOGe edilir.

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27

3.3 Tirigonometrik Lineer Olmama Durumu

f(u)=sinu øONRODUDNA₀ = f(u₀)¶ÕGL÷HUWHULPOHUGHQD\ÕUPDOÕ\Õ]gQFH

∑

∞ = = 0 n n u u (3.23)

D\UÕúWÕUPDVÕQÕNXUXS f(u)=sinu’da yazarsak,

f(u)=sin

[

u₀ +(u₁+u₂ +u₃ +u₄ +)

]

(3.24) elde ederiz ve

sin

(

θ +φ

)

=sinθcosφ+cosθsinφ 7ULJRQRPHWULN|]GHúOL÷LQLNXOODQDUDN

f(u)=sinu0cos(u1 +u2 +u3 +u4 +)

+cosu₀sin(u₁+u₂ +u₃+u₄ +) (3.25)

olarak bulunur. f(u0)=sinu0¶Õ GL÷HU oDUSDQODUÕQGDQ D\ÕUÕS cos(u1+ u2 +) ve

) sin(u₁+ u₂ + LoLQ7D\ORUDoÕOÕPÕNXOODUDN _      ₋ ₊ ₊ ₊ ₊ ₊ ₋ = 1 2 4 2 2 1 0 ( ) ! 4 1 ) ( ! 2 1 1 sin ) (u u u u u u f 3 0 1 2 1 2 1 cos ( ) ( ) 3! u  u u u u  + _ + + − + + + _   (3.26) bulunur. Böylece _      ₋ ₊ ₊ ₊ = ( 2 ) ! 2 1 1 sin ) ( 2 ₁ ₂ 1 0 u uu u u f 3 0 1 2 1 1 cos ( ) 3! u  u u u  + _ + + − + _  

HOGH HGLOLU YH EX DoÕOÕP WP WHULPOHULQ LQGLVOHUL WRSODPÕ D\QÕ RODFDN úHNLOGH JUXSODQGÕUÕOPDVÕ\OD

(33)

28 2 1 0 ) sin ! 2 1 cos ( cos sin ) ( 0 2 1 0 2 0 1 0 A A A u u u u u u u u f = + + − 3 ) cos ! 3 1 sin cos ( 0 3 1 0 2 1 0 3 A u u u u u u u − + + + + + + − + 4 ) sin ! 4 1 cos ! 2 1 sin ) ! 2 1 ( cos ( 4 ₀ 1 0 2 2 1 0 3 1 2 2 0 4 A u u u u u u u u u u u (3.27)

elde ederiz. Böylece f(u)=sinuLoLQ$GRPLDQSROLQRPODUÕ A0 =sin u0 A₁ =u₁cosu₀ 0 2 1 0 2 2 sin ! 2 1 cosu u u u A = − 0 3 1 0 2 1 0 3 3 cos ! 3 1 sin cosu uu u u u u A = − + 4 ₀ 1 0 2 2 1 0 3 1 2 2 0 4 4 sin ! 4 1 cos ! 2 1 sin ) ! 2 1 ( cosu u uu u u u u u u u A = − + + + (3.28) úHNOLQGHHOGHHGLOLU

3.4 Hiperbolik Lineer Olmama Durumu

f(u₀)=sinhu önce,

∑

∞ = = 0 n n u u D\UÕúWÕUPDVÕQÕNXUXS f(u)=sinhu’da yazarsak,

f(u)=sinh

[

u0 +(u1+u2 +u3 +u4 +)

]

(3.29) olur ve

sinh

(

θ +φ

)

=sinhθcoshφ+coshθsinhφ KLSHUEROLN|]GHúOL÷LQLNXOODQDUDk (3.29) denklemi,

(34)

29 f(u)=sinhu₀cosh(u₁+u₂ +u₃+u₄ +)

+coshu0sinh(u1 +u2 +u3 +u4 +) (3.30)

úHNOLQGH\D]DELOLUL] f(u0)=sinhu0¶ÕGL÷HUoDUSDQODUÕQGDQD\ÕUÕScosh(u1 + u2 +) ve

) sinh(u₁+ u₂ + LoLQ7D\ORUDoÕOÕPÕNXOODQDUDN _      ₋ ₊ ₊ ₊ ₊ ₊ ₋ = 1 2 4 2 2 1 0 ( ) ! 4 1 ) ( ! 2 1 1 sinh ) (u u u u u u f _      ₊ ₊ ₋ ₊ ₊ 3₊ 2 1 2 1 0 ( ) ! 3 1 ) ( coshu u u u u _      ₋ ₊ ₊ ₊ = ( 2 ) ! 2 1 1 sinh 2 ₁ ₂ 1 0 u u u u       ₊ ₊ ₋ 3 ₊ 1 2 1 0 ! 3 1 ) ( coshu u u u HOGHHGLOLU7PWHULPOHULQLQGLVOHULWRSODPÕD\QÕRODFDNúHNLOGHJUXSODQGÕUÕUVDN 2 1 0 ) sinh ! 2 1 cosh ( cosh sinh ) ( 2 ₀ 1 0 2 0 1 0 A A A u u u u u u u u f = + + − 3 ) cosh ! 3 1 sinh cosh ( 3 ₀ 1 0 2 1 0 3 A u u u u u u u − + + + + + + − + 4 ) sinh ! 4 1 cosh ! 2 1 sinh ) ! 2 1 ( cosh ( 0 4 1 0 2 2 1 0 3 1 2 2 0 4 A u u u u u u u u u u u (3.31)

elde ederiz. Böylece f(u)=sinhu içiQ$GRPLDQSROLQRPODUÕ

A₀ =sin u₀ A1 =u1cosu0 0 2 1 0 2 2 sin ! 2 1 cosu u u u A = − 3 ₀ 1 0 2 1 0 3 3 cos ! 3 1 sin cosu uu u u u u A = − +

(35)

30 0 4 1 0 2 2 1 0 3 1 2 2 0 4 4 sin ! 4 1 cos ! 2 1 sin ) ! 2 1 ( cosu u uu u u u u u u u A = − + + + (3.32)

olarak elde edilir.

3.5 Üstel Lineer Olmama Durumu

u e u f( )=

∑

∞ = = 0 n n u u D\UÕúWÕUPDVÕVWHOIRQNVL\RQGD\HULQH\D]ÕOÕUVD 3 2 1 0 ) ( u u u u e u f = + + + (3.33)

veya denk olarak,

_f(_u)=_eu0_eu1+u2+u3 (3.34) EXOXQXU%XUDGDNLLONoDUSDQD\QHQ\D]ÕOÕSGL÷HUoDUSDQLoLQ7D\ORUDoÕOÕPÕNXOODQDUDN = × + ₁+ ₂ + ₃ + + ( ₁+ ₂ + ₃ +)2 + ! 2 1 ) ( 1 ( ) (_u _e 0 _u _u _u _u _u _u f u (3.35) HOGHHGLOLUYHWPWHULPOHULQGLVOHUWRSODPÕRODFDNúHNLOGHJUXSODQGÕUÕOÕUVD , , 3 0 2 0 1 0 0 0 ) ! 3 1 ( ) ! 2 1 ( ) ( 3 1 2 1 3 2 1 2 1 A u A u A u A u e u u u u e u u e u e u f = + + + + + + + + + + + + + 4 0 ) ! 4 1 ! 2 1 ! 2 1 ( 4 1 2 2 1 2 2 3 1 4 A u e u u u u u u u (3.36) bulunur ve böylece, 0 0 u e A = 0 1 1 u e u A = ) 0 ! 2 1 ( 2 1 2 2 u e u u A = + ) 0 ! 3 1 ( 3 1 2 1 3 3 u e u u u u A = + +

(36)

31 ) 0 ! 4 1 ! 2 1 ! 2 1 ( 4 1 2 2 1 2 2 3 1 4 4 u e u u u u u u u A = + + + ++ (3.37) elde edilir.

3.6 Logaritmik Lineer Olmama Durumu

f(u)=lnu, u>0

∑

∞ = = 0 n n u u

D\UÕúWÕUPDVÕQÕ f(u)=lnu’da yazarsak,

) ln( ) (u = u0 +u1 +u2 +u3 + f (3.38) olur ve (3.38) denklemi ( ) ln( (1 )) 0 3 0 2 0 1 0 + + + + = u u u u u u u u f (3.39) úHNOLQGH\D]ÕODELOLU β α αβ) ln ln ln( = + |]GHúOL÷LNXOODQÕODUDNGHQNOHPL ( ) ln ln(1 ) 0 3 0 2 0 1 0 + + + + + = u u u u u u u u f (3.40) RODUDN\D]ÕOÕU f(u0)=lnu0¶ÕD\ÕUÕSNDODQWHULPLoLQ7D\ORUDoÕOÕPÕQÕNXOODQDUDN 2 0 3 0 2 0 1 0 3 0 2 0 1 0 2 1 ln ) ( _      + + + −          + + + = u u u u u u u u u u u u u u f     +       + + + −       + + + + 4 0 3 0 2 0 1 3 0 3 0 2 0 1 4 1 3 1 u u u u u u u u u u u u (3.41)

elde edilir ve önceki gibi hareket ederek (3.41) denklemi , , 3 2 1 0 3 0 3 1 0 2 1 0 3 2 0 2 1 0 2 0 1 0 2 1 2 1 ln ) ( A A A A u u u u u u u u u u u u u u u f = + + − + + −

(37)

32 4 4 0 4 1 2 0 2 2 1 2 0 3 1 2 0 2 2 0 4 4 1 2 1 A u u u u u u u u u u u u + − + + − + (3.42) úHNOLQGH\D]ÕODELOLU%XQDJ|UH$GRPLDQSROLQRPODUÕ A₀ =ln u₀, 0 1 1 u u A = , ₂ 0 2 1 0 2 2 2 1 u u u u A = − , ₃ 0 3 1 0 2 1 0 3 3 2 1 u u u u u u u A = + − , ₄ 0 4 1 2 0 2 2 1 2 0 3 1 2 0 2 2 0 4 4 4 1 2 1 u u u u u u u u u u u u A = − + + − , (3.43) úHNOLQGHROXU

(38)

33

621/8)$5.<$./$ù,0/$5,

%X E|OPGH ELUoRN WHPHO ND\QDNWD \HU DODQ ELU IRQNVL\RQXQ NÕVPL WUHYOHULQ LIDGH HGHQ VRQOX IDUN \DNODúÕPODUÕ LQFHOHQHFHNWLU>2WXUDQo@ øNLQFL PHUWHEHGHQ NÕVPLGLIHUDQVL\HOGHQNOHPOHULQJHQHOJösterimi, ₂ 0 2 2 2 2 = + ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ f y u C y x u B x u A úHNOLQGHROXSEXUDGDf fonksiyonu y u x u u y x ∂ ∂ ∂ ∂ _, , ,

, nin bir fonksiyonudur. Bu denklemi oVÕQÕIDD\ÕUDELOLUL] 2 _{− AC}4 _<0 B ise eliptik 2 _{− AC}4 ₌0 B ise parabolik 2 _{− AC}4 _>0 B ise hiperbolik 4.1 Hiperbolik Denklemler +LSHUEROLNNÕVPLGLIHUDQVL\HOGHQNOHPOHULQELU|UQH÷L ( , ) 2 ( , ) t x u c t x utt = xx 0< x<a ve 0<t <b (4.1) YHVÕQÕUúDUWODUÕ u(0,t)=0 ve u(a,t)=0 0≤t≤b u(x,0)= f(x) 0≤x≤a (4.2) u_t(x,0)= g(x) 0<x<a úHNOLQGHYHULOHQGDOJDGHQNOHPLGLU%XGHQNOHPLQDQDOLWLNo|]P)RXULHUVHULOHULLOH EXOXQDELOPHVLQHUD÷PHQúLPGLVD\ÕVDOo|]PQLQFHOH\HFH÷L]

∆x=h,∆t =k olmak üzere R =

{

(x,t):0≤x≤a,0≤t≤b

}

bölgesini

) 1 ( ) 1 (n− × m− ER\XWOXGLNG|UWJHQVHOE|OJHOHUHD\ÕUDOÕPt= t₁ =0ROGX÷XQGD ) ( ) , (xi t1 f xi

u = çözümleri mevcuttur. Biz

{ }

u_ij i=1 ,2, ,n ve j=1 ,2 ,m

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u_tt( tx, ) ve u_xx( tx, )NÕVPLWUHYOHULQHPHUNH]LIDUNIRUPOOHULLOH\DNODúÕP

Kısmi türevli diferensiyel denklemlerin diferansiyel transform metodu ile çözümü ve diğer yöntemlerle karşılaştırılması

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ÖZET

ABSTRACT

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1. TEMEL KAVRAMLAR

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