Kesirli kablo denkleminin uyumlu türev operatörü ile yaklaşık analitik çözümlerinin bulunması

Matematik

-2018 KONYA

Burcu YAŞKIRAN tarafından hazırlanan "KESİRLİ KABLO DENKLEMİNİN

UYUMLU TÜREV OPERATÖRÜ İLE YAKLAŞIK ANALİTİK ÇÖZÜMLERİNİN

BULUNMASI" adlı tez çalışması 11/05/2018 tarihinde aşağıdaki jüri tarafından oy

birliği ile Necmettin Erbakan Üniversitesi Fen Bilimleri Enstitüsü Matematik Anabilim

Dalı'nda YÜKSEK LİSANS TEZİ olarak kabul edilmiştir.

Jüri Üyeleri

Başkan

Prof. Dr. Necati ÖZDEMİR

Danışman

Dr. Öğr. Üyesi Mehmet YAVUZ

Üye

Dr. Öğr. Üyesi Nihat AKGÜNEŞ

Yukarıdaki sonucu onaylarım.

Prof.Dr. Mehmet KARALI

FBE Müdürü

Bu tezdeki bütün bilgilerin etik davranış ve akademik kurallar çerçevesinde elde

edildiğini ve tez yazım kurallarına uygun olarak hazırlanan bu çalışmada bana ait

olmayan her türlü ifade ve bilginin kaynağına eksiksiz atıf yapıldığını bildiririm.

DECLARATION PAGE

I hereby declare that all information in this document has been obtained and

presented in accordance with academic rules and ethical conduct. I also declare that, as

required by these rules and conduct, I have fully cited and referenced all material and

results that are not original to this work.

Burcu YAŞKIRAN

11/05/2018

iv Necmettin Erbakan Matematik Dr. Mehmet YAVUZ 2018, 66 Sayfa Bu tezde zaman-kesirli operat (HAM), homotopi -uygul

Anahtar Kelimeler: , Homotopi analiz

metodu, Kesirli kablo

v ABSTRACT MS THESIS

APPROXIMATE ANALYTICAL SOLUTIONS OF FRACTIONAL CABLE EQUATION WITH CONFORMABLE DERIVATIVE OPERATOR

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCE OF UNIVERSITY

THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS

Advisor: Asst. Prof. Mehmet YAVUZ 2018, 66 Pages

Jury

Asst. Prof. Dr. Mehmet YAVUZ Asst. Prof. Dr.

In this thesis, time-fractional one dimensional cable equation has been considered. Conformable derivative operator (CDO) has been used as a fractional operator in cable equation. Adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), homotopy perturbation method (HPM), modified homotopy perturbation method (MHPM) and reduced differential transform method (RDTM) have been emphasized in the solution of the conformable fractional cable equation (CFCE). The main aim of this study is to redefine the approximate-analytical methods that are mentioned above with CDO and to find the approximate-analytical solutions of CFCE with these suggested methods. Furhermore, since the conformable derivative operator had been defined in 2014, there are a little bit studies in this area. Therefore, a new application of CDO has been brought to the literature with this thesis.

Keywords: Adomian decomposition method, Conformable derivative operator, Fractional cable equation, Homotopy analysis method, Homotopy perturbation method, Modified homotopy perturbation method, Reduced differential transform method, Variational iteration method.

vi

ni esirgemeyen

Mehmet YAVUZ sonsuz

zaman esirgemeyen

Burcu

vii ... iv ABSTRACT ... v ... vi ... vii ... ix TA ... x ... xi ... 1 1.1. Kaynak A ... 2 ... 2 r ... 4 ... 5

2. TEMEL TANIM VE TEOREMLER ... 8

... 8 2.2. Temel Fonksiyonlar ... 10 2.2.1. Gamma Fonksiyonu ... 10 2.2.2. Beta Fonksiyonu ... 11 ... 12 2.3.1. Riemann- ... 12 ... 13 2.3.3. Uyumlu ... 14 ... 19 ... 22 4.1. ... 22 ... 24

Kesirli Diferansiyel Denklemlere ... 26

viii

... 28

4.2. Va ... 29

4.2.1. Kesirli Diferansiyel Denklemlere ... 30

4.2.2. Uyum ... 32

4.3. Homotopi Analiz Metodu ... 35

4.3.1. en Deformasyon Denklemi ... 36

... 38

4.3.3. Homotopi Analiz Metodunun Kesirli Diferansiyel Denklemlere ... 39

40 syon Metodu ... 42

nun Kesirli Diferansiyel Denklemlere ... 43

4.4.2. Uyumlu ... 44

... 46

4.4.4. Modi nun Kesirli Diferansiyel Denklemlere ... 48 ... 49 ... 51 4.5.1. ... 51 4.5.2. ... 52 4.5.3. ... 52

4.5.4. nun Kesirli Diferansiyel Denklemlere ... 55 4.5.5. ... 56 ... 59 KAYNAKLAR ... 61 ... 66

ix ... 19 ... 20 1. 0.30 ve 0.70 29 0.30 ve 0.70 ... 34 ... 35 0.35 ve 0.65 ... 46 0.90 ve 1 ... 46 0.35 ve 0.65 ... 50 Kablo denkleminin 0.90 ... 51 Kablo denkleminin 1 ... 58

x TABLO

Tablo 2.1. ... 10

Tablo 4.1. 24

Tablo 4.2. e edilen mutlak hata

, , k u x t u x t ... 34 Tablo 4.3. t 0.1 ... 42 Tablo 4.4. t ... 53 Tablo 4.5. t diferansiyel ... 54 Tablo 4.6. ... 56

xi Simgeler z : Gamma fonksiyonu , B z w : a D : Riemann-Liouville kesirli a J : Riemann-Liouville kesirli *a D : Caputo kesirli t T : Uyumlu t I : Uyumlu integral n A : L : Lineer terimler

N : Lineer olmayan terimler

R : geri : : n u : p : Homotopi parametresi m D : m. mertebeden homotopi : Alfa : alfaya : Ksi UTO : Uyumlu

UKKD : Uyumlu Kesirli Kablo Denklemi

AAY :

:

HAM : Homotopi Analiz Metodu

HPM : Homoto

MHPM : Modifiye Metodu

DDM :

bu

matematiksel ifadelerden yani denklemlerden .

problemlerin modellenme

. Bu sebeple klasik a

mertebeden reel veya komple naliz ortaya

Kesirli analizde en uygun

ta nda en iyi

elektromanyetik teori, fizik ve kontrol teorisi,

esirli a

n incelenmesi bu sistemlerin matematiksel modellemeler .

matematiksel olarak sistemlerin modellenmesi

zaman-kesirli bir boyutlu kablo denklemi Kablo denklemi

K ise bu denklemde uyumlu t o

(UTO) Uyumlu anan uyumlu kesirli

kablo denklemi (UKKD) nin -analitik

metotlardan; A AAY), varyasyonel iterasyon metodu ( homotopi analiz metodu (HAM), homotopi p n metodu (HPM), modifiye

homotopi p n metodu (MHPM) ve indirgenmi etodu

( de U

Bu tezde b kesirli analiz ve kablo denkleminin

olan

U -analitik

U .

el geometrik olarak

fonksiyona ait .

ve son de ise U -analitik

lar ile ilgili Kesirli a e n 1 2 n n d dx (Nishimoto, 1991). daha d (Weilbeer, 2005). 1 ile kesirli

son Euler bu konuya ilk defa

lan Gam (Weilbeer, 2005).

Langrange

ve daha sonra (Lagrange, 1775).

Laplace, . Bu kitapta Laplace bir r (Weilbeer, 2005).

(Weilbeer, 2005). in Gamma

fonksiyonunu kullanarak k in elde

Riemann- (Ross, 1975).

(Fourier, 1822). 1

(Abel, 1823). Samko vd. de lerdir (Samko, Kilbas ve Marichev, 1993).

de analizle ilgili ilk

(Liouville, 1837). Liouville i (Ross, 1975; Weilbeer, 2005) lar 1 f x x i fonksiyonlar (Fowler, 1975).

Riemann, Taylor serisinin gen

ve fonksiyonun . . Fakat Riemann bu

nschaftlicher Nachlass- Bernhard

(Ross, 1975).

Liouville

-(Hilfer, 2000; Kilbas, Srivastava ve Trujillo, 2006; Oldham ve Spanier, 1974).

1867-1868 fark yakla

nda Riemann-unu ispatla (Kilbas vd., 2006; Podlubny, 1999; Samko vd., 1993).

Khalil vd. tar yeni bi uyumlu (UTO) (Khalil, Al Horani, Yousef ve Sababheh, 2014).

U nu ve

UTO kesir mertebeli diferansiyel denklemlerin

.

UTO

Abdeljawad, uyumlu ve uyumlu kesirli

(Abdeljawad, 2015).

Batarfi vd. uyumlu ine

(Batarfi, Losada, Nieto ve Shammakh, 2015).

2016 vd. silindirik bir pl uyumlu

( )

radyal s ne

U yu ( ).

Acan vd. U yu varyasyonel iterasyon metoduna (Acan, Oturanc ve Keskin, 2017), daha sonra

metodu

- bu (Acan ve Baleanu, 2017).

U yu kullanarak kesin

( ).

Yavuz, U Adomian ayr

kesirli diferansiyel denklemlerin lerini (Yavuz, 2018).

Ilei vd. kesirli Bernoulli ve Riccati de UTO

(Ilei, Biazar ve Ayati, 2017).

1.1.2.

Kesirli k k az olup, 2009

vd. (Murillo ve Yuste, 2011) ve 2012 de ise fark metodu (Hu ve Zhang, 2012) ile

to ya da Riemann-Liouville olarak k UTO u U ve varyasyonel ite -( ).

1.1.3. Kesirli Kablo Denk

AAY ilk defa lineer ve lineer olmayan

difer

(Adomian, 1988).

denklemlerin s (Abbaoui ve Cherruault, 1995; Adomian, 1990).

Wazwaz vd.

yeni bir algoritma (Wazwaz, A.-M., 2000; Wazwaz, A.-M. ve El-Sayed, 2001).

Duan vd. kesirli diferansiyel denklem ele al lard (Duan, Rach, Baleanu ve Wazwaz, 2012).

problemle genel (Inokuti,

Sekine ve Mura, 1978).

ilk kez ileri

(He, 1997) ve 1998 de kesirli difera ilk kez (He, 1998).

2006 da Odibat vd. kesirli mertebeye sahip diferansiyel denklemlerin yakla -lard (Odibat ve Momani, 2006).

Bu metot daha sonra

(Soltanian, Karbassi ve Hosseini, 2009).

diferansiyel denk

AAY (Wazwaz, A. M., 2009).

HAM ilk olarak 1992 de rak lineer

(Liao, 1992).

(Tan, Xu ve Liao, 2007), (Abbasbandy, 2007), Zakharov-Kuznatsov denklemi (Molliq, Noorani, Hashim ve Ahmad, 2009) vb.

Zurigat vd. kesirli mertebeye sahip cebirsel diferansiyel denklemlere (Zurigat, Momani ve Alawneh, 2010).

(HPM) ve modifiye homotopi metodu (MHPM)

He, homotopi

ve lineer olmayan problemlerin problemleri lineer (He, 1999a). Daha sonra ba

(He, 2000, 2003, 2004, 2005). 2007 de Odibat

(Odibat, 2007).

2008 de Abdulaziz vd. linee yi

(Abdulaziz, Hashim ve Momani, 2008).

mevcuttur (Odibat ve Momani, 2008; Yavuz, 2018; 2019).

DDM ilk olarak Zhou n elektrik devre analizindeki

problemlerin de (Zhou, 1986).

Chen vd. ise ilk kez DDM den hareketle diferansiyel denklemlerin

yi lar

Ayaz iki boyutlu DDM den (Ayaz, 2003),

Kurnaz vd. ise N-boyutlu lard ( ).

Keskin vd. l diferansiyel

( ).

ise Keskin vd. lineer olmayan fonksi

(Keskin ve Oturanc, 2009).

Acan vd. k lere

2. TEMEL TANIM VE TEOREMLER 2.1. . f x , I 0 0 0 0 lim x x f x f x f x x x

limiti varsa, f x x0 I nok lenebilirdir (diferansiyellenebilirdir)

denir ve bu limite f fonksiyonunun x₀ (Thomas, Finney,

Weir ve Giordano, 2003).

.2. nin mertebesi tam

rasyon (Podlubny, 1998).

.3. bir ya

kesirli denkleme diferansiyel denklem denir (Podlubny, 1998).

.4. Diferansiyel denklem, b

kesirli denkleme kesirli adi diferansiyel denklem denir (Podlubny, 1998). _{D y t}1 2 _{5y t}2 _{3 0 denklemi bir kesirli adi}

diferansiyel denklemdir.

.5. Diferansiyel denklem, b bir ya ini denkleme kesirli denir (Podlubny, 1998). 2 2 , , , 0, , 0 1 t u x t D u x t t x x denklemi

Diferansiyel denklemlerde bilinmeyen

bu denir (Debnath, 2011).

2.7. esine

denklemin mertebesi ve e n derecesine (kuvvetine) ise denklemin derecesi denir (Podlubny, 1999).

2.8. Bir diferansiyel denklemde,

b birinci mertebeden

denkleme lineer diferansiyel denklem denir. Lineer bir diferansiyel denklemin genel formu P x0 0

1 1 0 n n n n P x D y x P x D y x P x y x Q x (2.1) , 0,1, , j D j n

(2.1) denkleminin lineerlik lineer olmayan

diferansiyel denklemler denir (Debnath, 2011).

3 3 2 x

x D y x y x e ve _{D y x}2 _{D y x y x 0 denklemleri lineer iken,}

2 3 2

D y x y x ve _{y x D y x}3 2 _{D y x}3 5 _{x denklemleri lineer olmayan} 3

diferansiyel denklemlerdir.

2.9. (2.1) denkleminde Q x 0 ise bu denkleme homojen diferansiyel denklem denir. Q x 0 ise bu denkleme homojen olmayan diferansiyel denklem denir (Debnath, 2011).

. f x fonksiyonunun x ₀

yani k 0,1, 2, f k x mevcutsa bu fonksiyon ₀

2 0 0 0 0 0 0 0 0 2! ! k k n f x f x f x f x f x x x x x x x n (2.2) (2.2) f x fonksiyonunun x ₀

Taylor serisi ve x₀ 0 i urin serisi denir (Thomas vd., 2003). 2.2. Temel Fonksiyonlar 2.2.1. Gamma Fonksiyonu 2.2.1. Gamma fonksiyonu z / 0, 1, 2, 1 0 , Re 0 1 / , Re 0, 0, 1, t z e t dt z z z z z z (2.3) (Weilbeer, 2005).

Teorem 2.2.1. Gamma fonksiyonu

1. z / 0, 1, 2, 1 z z z z ! 2. z z z 1 ! 3. z / 0,1, 2, 1 z z z 4. Re z 0 lim ! 1 2 z n n n z z z z z n 5. z 1 sin z z z ve z z zsin z sahiptir (Weilbeer, 2005). Gamma fonksiyonuna ait b (Podlubny, 1999). Tablo 2.1. 3 2 4 3 1 2 2 0

1 2 1 1 3 2 1 2 2 1 5 2 3 4 3 2 2.2.2. Beta Fonksiyonu .2.2. z w, , z w B z w z w (2.4) (Weilbeer, 2005).

Teorem 2.2.2. Beta (Weilbeer, 2005).

1. Re z , Re w 0 (2.4) denklemi 1 1 1 0 , z 1 w B z w t t dt 2. Beta fonksiyonu a) B z w, B w z , b) B z w, B z 1,w B z w, 1 (Weilbeer, 2005).

2.3. ler

Riemann-Liouville, Caputo ve uyumlu

n bahsedi esirli kablo denkleminin

. 2.3.1. Riemann- i 2.3.1.1. 0 . mertebeden Riemann-Liouville (R-L) 1 1 ( ) , x a a J f x x t f t dt t a 0 0 a J f x f x dir (Ahmad, 2015). ,

f x a , 0 olmak -Liouville integral

(Das, 2011) : 1. J J f x_{a a} J_a f x J J f x _{a a} 2. J_a c f x_{1 1} c f x_{2 2} c J f x_{1 a 1} c J f x_{2 a 2} , c c₁, ₂ Teorem 2.3.1.1. f x x a 1 , , 0, 1 1 a J f x x a x a (Diethelm, 2010). 2.3.1.2. 0, a , x a ve m 1 m -Liouville 1 1 x m m m m m a a a m m a f x d D f x D D f x D J f x dt dt m x t

olar (Ahmad, 2015; Das, 2011; Diethelm, 2010).

R-L ine ait

,

1. D J f x_{a a} f x , 2. 1 1 1 0 , k m k a a a k x J D f x f x D f a k 3. D c f x_{a 1 1} c f x_{2 2} c D f x_{1 a 1} c D f x c c_{2 a 2} , ,_{1 2} , 4. 1 1 1 0 , k n k a a a a k x D D f x D f x D f a k 1 , 1 , , . m m n n m n 5. 0,b kli f x fonksiyonu ve D0 0 0, 0,1, 2, , 1 k f k n , 0 0 0 0 0 , 1 , . n n n D D f x D f x D D f x m m n

(Das, 2011; Oldham ve Spanier, 1974).

Teorem 2.3.1.2. f x x a ve x a, 0, L L, max 1, 1 olmak

1 1 a D f x x a R-L 1 f t 1 1 1 0 1 a D (Samko vd., 1993). 2

Caputo kesirli integrali R-L k J _a

2.3.2.1. m ,m 1 m i D _a 1 1 ( ) ( ) , m x m m a a a m a f t D f x J D f x dt x a m _{x t}

revine ait , a b ar f , f1, f2 Da 1. D J f xa a ( ) f x( ) 2. ( ) 1 0 , 0 ! k m k a a k f a J D f x f x x a k 3. Da c f x1 1 c f x2 2 c D f x1 a 1 c D f x2 a 2 , c c1, 2 4. 0,b f x fonksiyonu ve D 0 0 0, , 1, , k f k m m n 0 0 0 0 0 , 1 , n n n D D f x D f x D D f x m m n 5. _{f C a b , ,}k , _{0 , l k k l, , ve , l 1,l} a a a D D f x D f x (Ahmad, 2015). R-Teorem 2.3.2.1. 0, m 1 m m, , 1 0 1 k m k a a a k x D f x D f x D f a k d (Caputo, 1969). Teorem 2.3.2.2. m 1 m m, ve ,k 1. D ka 0, 2. 1 , ve ya da ve 1 1 0, 0,1, 2, , 1 a x m m D x m r (Ahmad, 2015). 2.3.3. Uyumlu i

2.3.3.1. f : 0, bir fonksiyon olsun. t 0 ve 0,1 f fonksiyonunun . mertebeden uyumlu

1 0 lim t f t t f t T f t (Khalil vd., 2014).

Teorem 2.3.3.1. 0,1 f ve ,g t 0 da diferansiyellenebilir olsun. O halde 1. a b, Tt af bg aTt f bTt g , 2. Tt fg fTt g gTt f , 3. / t ₂ t , t gT f fT g T f g g 4. f t diferansiyellenebilirse 1 _, t d T f t t f t dt 5. k k k , t T t kt 6. f t k T_t f t 0 (Khalil vd., 2014). . 1. 1 1 0 lim t af t t bg t t af t bg t T af bg 1 1 0 1 1 0 0 lim lim lim t t a f t t f t b g t t g t a f t t f t b g t t g t aT f bT g 2. 1 1 0 lim t f t t g t t f t g t T fg 1 1 1 1 0 lim f t t g t t f t g t t f t g t t f t g t 1 1 1 0 0 lim f t t f t g t t lim g t t g t f t 1 0 lim t t T f t g t t f t T g t

y g fonksiyonu t de 1 0

lim g t t g t olup ispat

3. 1 1 0 / lim t f t t _{f t} g t g t t T f g 1 1 1 0 lim f t t g t f t g t f t g t f t g t t g t t g t 1 1 1 0 lim f t t f t g t g t t g t f t g t t g t 1 1 1 0 lim f t t f t g t t g t g t f t g t t g t 1 0 lim t t gT f fT g g t t g t g fonksiyonu t 1 0 lim g t t g t dir. 4. Tan 2.3.3.1 de _{h t}1 1 1 1 0 0 lim lim t _{h h} f t h f t f t h f t d T f t t t f t h dt ht elde edilir. 5. 4 1 0 lim k k k t _h t h t T t ht limitindeki t h k 1 2 2 1 0 0 1 2 lim k k k k r r k k h k k k k k t t h t h t h h t r k t h 1 2 2 1 0 2 lim k k k k r r k k h k k k t kt h t h t h h t r k t h

1 2 1 1 1 1 0 2 lim k k k r r k h k k k h kt t h t h h r k t h 1 1 2 1 1 1 0 lim 2 k k k r r k h k k k t kt t h t h h r k

elde edilir. Buradan

k k t T t kt olur. 6. 1 t d T f t t f t dt den faydalanarak t f t k 1 1 _{.0 0} t d T k t k t dt olur.

Tan 2.3.3.2. n n, 1 f, t 0 n kez diferansiyellenebilir

olsun. f fonksiyonunun . mertebeden uyumlu

1 1 0 lim t f t t f t T f t

(Anderson ve Ulness, 2015; Atangana, Baleanu ve Alsaedi, 2015; Batarfi vd., 2015; Khalil vd., 2014). Burada , dan

.

Lemma 2.3.3.1. ,f t de n kez diferansiyellenebilir olsun. t 0 ve n n, 1

t

T f t t f t

(Khalil vd., 2014).

Tan 2.3.3.3. 0 1, t a ve f fonksiyon mertebeli uyumlu integrali 1 t t t a f I f t T f t d

olarak (Khalil vd., 2014).

Teorem 2.3.3.2. f bir fonksiyon ve 0 1 olsun. Bu durumda t a 1. T I_{t t} f t f t 2. I T_{t t} f t f t f a olur (Khalil vd., 2014). f 1. 1 t t t d T I f t t I f t dt 1 1 1 1 t a f d t d dt f t t t f t 2. 1 t t t d I T f t I t f t dt 1 1 t a t a d _f d _d f d f t f a

Teorem 2.3.3.3. : 0,f bir fonksiyon, _f n _{t n n, 1 olsun.}

Bu durumda, t a 1. T It t f t f t 2. 0 ! k k n t t k f a t a I T f t f t k olur (Khalil vd., 2014).

3.

nsan

mize ve o sistemle

deneysel

dendrit, bir tane olan Dendritler,

aksona .

1.

. Mi herhangi bir bozulur.

merkezi sinir

sisteminden gelen emirleri ise ezi sinir sistemine

ileten sinirlere duyu sinirleri,

denir. Duyu sinirleri ve motor si sinirlere de ara

sinirler denir. lerin

Sinapslar, bilgi iletiminden sorumludur.

devrelerine benzeyen bir ilet ( ).

2.

(Rall, 1957). Burada

Rall, bir-boyutlu kablo modelinde n elektrik smi

diferansiyel denkle r.

n s

kesirli kablo denklemi

(Langlands, Henry ve Wearne, 2005),

modellemede (Reynolds, 2005) kesirli kablo denklemini

Genel haliyle kesirli kablo denklemi (KKD):

1 2 2 1 2 1 0 2 0 0 , , , , , t t u x t u x t T K T u x t f x t t x (3.1) ,0 ( ), 0 , u x g x x L (3.2) 0, ( ), L, ( ), 0 , u t t u t t t T (3.3)

(3.2) (3.3) (3.1) denklemiyle verilir (Liu

vd., 2009). Burada 0 ₁, ₂ 1, K 0 ve 2 0 sabitler ve 1 1 0Tt u x t de , 1 1 mertebeli uyumlu (3.1) denkleminde 1 2 ve denkleme 1 0Tt

zaman-kesirli bir boyutlu kablo denklemi

2 1 0 2 0 , , , , , 0 1, t t u x t T u x t u x t T f x t x (3.4)

veya Lemma 2.3.3.1 den

2 1 1 1 0 2 , , , _t , , 0 1, u x t u x t t u x t t T f x t t x t (3.5) . Burada b ,0 0, 0 1, u x x (3.6) 0, 0, 1, 0, 0 u t u t t T (3.7)

olarak f fonksiyonu ise

1 2 , 2sin 1 2 t f x t x t (3.5) denkleminin kesin 2 ( , ) sin u x t t x dir (Liu vd., 2009).

4. MATERYAL VE

uyumlu kesirli kablo

denklemi

-arya omotopi analiz metodu

(HAM), h HPM) ve i

k olup kesirli kablo denklemi (KKD) nin bu . 4.1. ( , ) ( , ) F u x t g x t (4.1) ( , ) u x t bilinmeyen fonksiyon ve ( , )g x t bir fonksiyon olup F

L

R lineer oper , N ise .

(4.1) denklemini Lu Ru Nu g (4.2) verebiliriz. (4.2) denklemi 1 L 1 1 1 1 L Lu L g L Ru L Nu (4.3) ( , )u x t bilinmeyen fonksiyonu 0 ( , ) _n , n u x t u x t (4.4)

ve lineer olmayan terimlerini de

0 n n Nu A (4.5) . u ve Nu 0 n n n u u (4.6) 0 0 n n n n n i f u Nu N u A (4.7)

olarak elde edilir. An genel hali 0 ₀ 1 , 0 ! n k n n k k d A N u n n d (4.8) r (Adomian, 1988). Burada parametredir.

(4.8) denkleminde lineer olmayan terim Nu yani f u , u u₀

2 3

0 0 0 0 0 0 0

1 1

2! 3!

f u f u f u u u f u u u f u u u (4.9)

elde edilir. Burada

0 1 2 3 u u u u u (4.10) (4.10) denklemi (4.9) r ve indis 0 0 1 1 0 2 2 2 0 1 0 3 3 3 0 1 2 0 1 0 2 2 4 4 4 0 1 1 2 0 1 2 0 1 0 1 2! 1 _{( )} 3! 1 1 1 ( ) 2! 2! 4! iv A f u A u f u A u f u u f u A u f u u u f u u f u A u f u u u u f u u u f u u f u (4.11)

bulunur (Adomian, 1994; Seng, Abbaoui ve Cherruault, 1996). (4.4) ve (4.5) denklemlerini (4.2) de , 1 1 1 0 0 0 ( ,0) n n n n n n u u x L g L R u L A (4.12)

elde ederiz. Burada

0 n n u ile 1 0 1 1 1 0 0 1 1 1 ( ,0) , k 0 k k k u u x L g u L Ru L A u L Ru L A (4.13) 0 n n u seri (4.1)

0 , lim n _k , n k u x t u x t (4.14) Fakat uygulamada 0 n n u zordur. Bu nede 1 0 n k n i u u (4.15) veya 1 0 2 0 1 3 0 1 2 1 0 1 2 , 0 n n u u u u u u u u u u u u u u n (4.16) eklinde bulunur. 4.1.1. Adomian Poli

hesaplamada daha kul (Wazwaz, A.-M., 2000).

n

A

4.1. Tablo 4.1. Fonksiyon Adomian P 2 ( ) F u u 2 0 0 1 0 1 2 2 1 0 2 3 1 2 0 3 2 4 2 1 3 0 4 , 2 , 2 , 2 2 , 2 2 , A u A u u A u u u A u u u u A u u u u u Polinom tipinde n ( ) n F u u 0 0 ( 1) 1 0 1 ( 2) 2 ( 1) 2 0 1 0 2 ( 3) 3 ( 2) ( 1) 3 0 1 0 1 2 0 3 , , 1 ( 1) , 2 1 ( 1)( 2) ( 1) , 6 n n n n n n n A u A nu u A n n u u nu u A n n n u u n n u u u nu u

Polinom tipinde n ( ) n F u u 0 0 ( 1) 1 0 1 ( 2) 2 ( 1) 2 0 1 0 2 ( 3) 3 ( 2) ( 1) 3 0 1 0 1 2 0 3 , , 1 _{( 1) ,} 2 1 ( 1)( 2) ( 1) , 6 n n n n n n n A u A nu u A n n u u nu u A n n n u u n n u u u nu u ( ) t F u uu 0 0 0 1 0 1 0 1 2 0 2 1 1 2 0 3 0 3 1 2 2 1 3 0 , , , , t t t t t t t t t t A u u A u u u u A u u u u u u A u u u u u u u u ( ) sin F u u 0 0 1 1 0 2 2 2 0 1 0 3 3 3 0 1 2 0 1 0 sin , cos , 1 cos sin , 2! 1

cos sin cos ,

3! A u A u u A u u u u A u u u u u u u ( ) cos F u u 0 0 1 1 0 2 2 2 0 1 0 3 3 3 0 1 2 0 1 0 cos , sin , 1 sin cos , 2! 1

sin cos sin ,

3! A u A u u A u u u u A u u u u u u u ( ) sinh F u u 0 0 1 1 0 2 2 2 0 1 0 3 3 3 0 1 2 0 1 0 sinh , cosh , 1 cosh sinh , 2! 1

cosh sinh cosh ,

3! A u A u u A u u u u A u u u u u u u ( ) cosh F u u 0 0 1 1 0 2 2 2 0 1 0 3 3 3 0 1 2 0 1 0 cosh , sinh , 1 sinh cosh , 2! 1

sinh cosh sinh ,

3! A u A u u A u u u u A u u u u u u u

( ) u F u e 0 0 0 0 0 1 1 2 2 2 1 3 3 3 1 2 1 , , 1 , 2! 1 , 3! u u u u A e A u e A u u e A u u u u e ( ) u F u e 0 0 0 0 0 1 1 2 2 2 1 3 3 3 1 2 1 , , 1 , 2! 1 , 3! u u u u A e A u e A u u e A u u u u e ( ) ln , >0 F u u u 0 0 1 1 0 2 2 1 2 2 0 0 3 3 1 2 1 3 2 3 0 0 0 ln , , 1 , 2 1 , 3 A u u A u u u A u u u u u u A u u u

4.1.2. Kesirli Diferansiyel Denklemlere

1 t

T t

t

t ifadesi . mertebeden uyumlu f lineer

olmayan bir fonksiyon ve v

, , , , , 1 ,

t x xx

T u x t f u u u v x t m m m N (4.17)

lineer olmayan kesirli (4.17) denklemini

lineer ve lineer olmayan

, , , , t T u x t L u x t N u x t v x t (4.18) veya , , , , u x t t L u x t N u x t v x t t (4.19)

denklemi elde edilir (Acan ve Baleanu, 2017). Burada L lineer, N lineer olmayan (4.19) denkleminin T_t uyumlu 1 0 1 . t t I d yi uygularsak, , , , , t t t t t I T u x t I v x t I L u x t I N u x t (4.20)

elde ederiz. (4.20) denklemini

1 0 ,0 , , , , ! k k m t t t k k u x t u x t I v x t I L u x t I N u x t k t (4.21) olur. Ado u x t, 0 , n , n u x t u x t (4.22)

olarak ve (4.18) denklemindeki lineer

0 1 0 , , , , n n n N u A u u u (4.23)

ifadesi ile verilir. Burada A ler Adomian n

0 ₀ 1 ! n n i n n i i d A N u n d (Adomian, 1988). (4.22) ve (4.23) (4.21) konulursa 1 0 0 0 0 ,0 ! k k m n n k t t n t n n k n n u x t u I v I L u I A k t (4.24) bulunur. 0 n n

u serisinin iterasyon terimleri

0 1 0 0 1 ,0 , , , 0 t t t n t n t n u u x I v u I Lu I A u I Lu I A n (4.25) Bu durumda (4.18) -0 , k , k n n u x t u x t

veya

, lim_k k ,

u x t u x t

minde bulunur.

4.1.3. Uyumlu Kesirli Kablo Denkleminin

kesirli kablo denkleminin A (AAY) ile

. (3.5) denkleminin I_t uygularsak, 2 1 2 ( , ) , ,0 t ( , ) t ( , ) u x t u x t u x I t f x t I u x t x (4.26) elde edilir. (4.22) denklemi (4.26) denkleminde yerine

2 1 2 0 0 ( , ) ( , ) ,0 ( , ) ( , ) n t t n n n u x t u x t u x I t f x t I u x t x elde edilir. 0 ( , ) n n

u x t serisinin iterasyon terimleri, (3.6)

2 2 2 1 0 2 2 2 2 2 2 2 0 1 2 0 2 3 2 2 2 2 3 2 2 1 2 2 1 1 , 0 ( , ) 2sin , 2 (3 ) 1 1 2sin , 2( 2) (2 2) (3 ) 1 1 2sin 2( 2)(2 2) (2 t t t t t u u x I t f x t x t t u u I u x x t t u u I u x x 2)(3 2) (3 ) , 2 1 1 2 n n t n u u I u x 2 2 1 2 ( 1) 2 1 2( 1) sin 2( 2)(2 2) ( 2) 1 . (2 2)(3 2) (( 1) 2) (3 ) n n n n _n t x n t n .

2 2 2 2 2 2 2 3 2 2 2 3 2 1 1 2 1 , lim , sin 1 (3 ) ( 2) (2 2) (3 ) 1 2 1 ( 2)(2 2) (2 2)(3 2) (3 ) k k t t t u x t u x t t x t t

(3.5) denkleminin AAY ile 1 kesin

2 , sin u x t t x elde edilir ( ). 1. 0.30 ve 0.70 ablo denkleminin 4.2.

Bu metotta, L lineer, N lineer olmayan , ( , )u x t bilinmeyen fonksiyon ve ( , )g x t homoj

( , )

Lu Nu g x t (4.27)

diferansiyel denklemi

Varyasyonel iterasyon metoduna

1 0 , , , , , t n n n n u x t u x t Lu x Nu x g x d (4.28)

formundaki varyasyon fonksiyonu kurulur. Burada (Inokuti vd.,

1978), u _n (Finlayson, 1972) olup u_n 0 .

(4.28) onksiyonele

(0) 0

n

1 0 0 , , , , , = , , , =0 t n n n n t n n u x t u x t Lu x Nu x g x d u x t Lu x g x d (4.29) elde edilir. -0 u olarak 0 n u_n , lim _n , n u x t u x t

olarak elde edilir (He, 1999b).

4.2.1. nun Kesirli Diferansiyel Denklemlere

, , , , , 1 ,

t x xx

T u x t f u u u v x t m m m N (4.30)

. Burada f lineer olmayan bir fonksiyon, u bilinmeyen fonksiyon, v homoj ve T ise . t

mertebeden uyumlu (4.30) ,0 , 0 1 , 0, , 0 u x h x u x t x t (4.31) ve ,0 ,0 , , 1 2 , 0, , 0 u x u x h x k x t u x t x t (4.32)

(4.30) denklemi Lemma 2.3.3.1 den ,

, ,_{x xx} , u x t

t f u u u v x t

t (4.33)

Varyasyonel ite (4.33) denklemi

1 0 , , , , , , m t n n n m n n x n xx u x u x t u x t f u u u v x d (4.34)

edilir (Acan vd., 2017). Burada , 0 n u olur. (4.34) denklemine 1 0 , , , , m t n n n m u x u x t u x t v x d (4.35)

denklemi elde edilir.

(4.35) denkleminde m 1 u kat

1 0,

0

elde edilir ve 1 olarak (4.34)

1 1 0 ( , ) ( , ) ( , ) , , ( , ) t n n n n n x n xx u x u x t u x t f u u u v x d (4.36) 0 ,0 u x h x olur. 2 m u 1 0, 0

t olarak bulunur ve (4.34) denkleminde

1 1 0 , , , , , , t n n n n n x n xx u x u x t u x t t f u u u v x d (4.37) 0 ,0 u x h x tk x olur. (4.35) denkleminde u ,

1 0, 0

m m

elde edilir ve buradan L

1 1, 1 , 2 1 , 1 ( 1)! m m m t m t m m (Wu, 2011). (4.34) -analitik , lim _n , n u x t u x t (Momani ve Odibat, 2006). 4.2.2. Uyumlu

Bu uyumlu kesirli kablo denkleminin varyasyonel iterasyon metodu ( ile . (3.5) denklemine uygularsak 0 1 1 ve 2 1 1 1 1 2 0 , , , , , , t n n n n n u x u x u x t u x t u x T f x d x

iter (3.6) kullanarak iterasyon

de 0 2 0 0 1 1 1 0 2 0 0 3 3 2 ,0 0, , , , , 2sin 1 , 3 3 (2 ) t u u x u x u x u u u x f x d x t t x

2 1 1 1 1 2 1 2 1 0 2 3 2 4 3 4 2 2 2 4 2 4 , , , , 1 1 4sin 2sin 3 3 (2 ) 4 2 (4 ) (2 ) 1 1 2sin , (4 )(3 ) 4 3 (2 ) t _{u x u x} u u u x f x d x t t t t x x t t x 2 2 2 1 1 3 2 2 2 0 2 3 2 4 3 4 2 2 2 4 2 4 5 3 , , , , 1 1 6sin 6sin 3 3 (2 ) 4 2 (4 ) (2 ) 1 1 6sin (4 )(3 ) 4 3 (2 ) 2sin 5 t _{u x u x} u u u x f x d x t t t t x x t t x t x 2 ₁ 5 2 3 (5 2 ) (2 ) t 2 2 5 2 2 5 2 2 5 2 2 5 2 3 2 5 2 5 1 1 2sin (5 2 )(3 ) 3(5 ) (2 ) 1 1 2sin (5 2 )(4 2 ) (5 )(4 ) (2 ) 1 1 2sin , (5 )(4 )(3 ) 5 4 3 (2 ) t t x t t x t t x

olarak bu iterasyon devam ettirilerek seri hesaplanabilir. (3.5) denkleminin , lim _n , n u x t u x t eklinde bulunur ve 1 2 2 2 2 2 2 6 6 6 6 6 ( , ) sin 1 1 1 2 3 (3) 2 3 (3) 3 2 6 1 4 3 (3) n t t t u x t t x t

2

, sin

u x t t x

( ).

2. 0.30 ve 0.70 ablo denkleminin

Tablo 4.2. Kablo denkleminin AAY ilen mutlak hata u x t_k , u x t,

x t 0.35 0.75 0.95

AAY M AAY M AAY M

3 10 0.10 0.00046588 0.00000281 0.00000076 0.00000042 0.00000004 0.00000000 0.50 13.5023030 13.7112263 0.00847201 0.00038284 0.00206843 0.00143082 0.70 322.724387 323.473669 0.05919425 0.00774314 0.02011917 0.01250838 5 10 0.10 0.00000466 0.00000003 0.00000001 0.00000000 0.00000000 0.00000000 0.50 0.13502325 0.13711249 0.00008472 0.00000383 0.00002068 0.00001431 0.70 3.22724918 3.23474201 0.00059194 0.00007743 0.00020119 0.00012508 6 10 0.10 0.00000047 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.50 0.01350233 0.01371125 0.00000847 0.00000038 0.00000207 0.00000143 0.70 0.32272492 0.32347420 0.00005919 0.00000774 0.00002012 0.00001251 8 10 0.10 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.50 0.00013502 0.00013711 0.00000008 0.00000001 0.00000002 0.00000001 0.70 0.00322725 0.00323474 0.00000059 0.00000008 0.00000020 0.00000013

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 0,0 5,0x10-3 1,0x10-2 1,5x10-2 2,0x10-2 x AAY VIM KESIN AAY VIM 3. kablo denkleminin AA

4.3. Homotopi Analiz Metodu

Homotopi analiz metodu lineer ve lineer olmayan diferansiyel denklemlerin seri -analitik bir Homotopi analiz metodu t .3.1. X ve Y topolojik uzaylar f X: Y , :g X Y x X ,0 x f x ve x,1 g x :X 0,1 Y f ve g ye homotopiktir

denir ve f g f ve g da homotopidir denir.

Yani;

, 1

x p p f x pg x

f x ve g x p parametresine

4.3.2. , p homotopi parametresinin bir fonksiyonu olsun. m 0 bir 0 1 ! m m m p p D m p (4.38) m

D m. mertebeden homotopi veya deformasyon

(Liao, 2009).

.3.3. u 0 bir lineer olmayan denklem, ; Maclaurin serisi

0 m m m

u p (4.39)

olan p 0,1 homotopi parametresinin bir fonksiyonu olsun. ,p 0, p 0,1

denklemler ailesine u 0 i denir.

1 p ise bu denklem 1 0 k p k u u (4.40)

o u 0 denklemine denktir. (4.39) serisine homotopi

serisi ve (4.40) serisine u 0 u lerim u

denklemlere m. derece deformasyon denklemleri denir (Liao, 2009).

4.3.1.

lineer u x t bilinmeyen fonksiyon, , x konum t

, 0

u x t (4.41)

lineer olmayan genel bir diferansiyel denklemi ele ala u x t bir 0 ,

0 H x t( , ) 0 ve L ise

, 0 , 0

0 0 ( , ; ); , , , , , 1 ( , ; ) , , ( , ; ) x t p u x t H x t p p L x t p u x t p H x t x t p (4.43)

homotopisi kurulabilir. H x t( , ), metot

0 , u x t , L , ni ve H x t( , ) (4.43) 0 , ; ; , , , , , 0 x t p u x t H x t p ve s lemi 0 1 p L x t p, ; u x t, p H x t, x t p , ; (4.44)

elde edilir. x t p , sadece , ; u x t , ( , )₀ , H x t , L ve 0,1 p (4.44) p 0 0 , ;0 , 0 L x t u x t (4.45) (4.42) (4.45) de uygularsak 0 , ;0 , x t u x t (4.46) elde edilir. (4.44) denkleminde 0 ve ( , ) 0H x t p 1 iken , ;1 0 x t (4.47)

denklemi elde edilir. Bu denklem ve (4.41) denkleminden

, ;1 ,

x t u x t (4.48)

denklemi elde edilir. (4.46) ve (4.48) p

artarken x t p , , ; u x t den ₀ , u x t kesin , e

homotopide deformasyon denir. m. derece defo

0 0 , ; , m m m p x t p u x t p (4.49) , ; x t p p

0 1 , , ; , ;0 ! m m m u x t x t p x t p m (4.50)

elde edilir. Burada

0 , ; 1 , ! m m m m p x t p u x t D m p (4.51) olarak Dm , m. (4.46) denklemi ve (4.51) denklemi (4.50) denkleminde 0 1 , ; , , m m m x t p u x t u x t p (4.52)

olarak bulunur. (4.52) denklemi .

0 , u x t , H x t( , ), L ve 1. Her p 0,1 x t p , (4.44) , ; 2. m 1, 2, 0 , m u x t 3. x t p, ; (4.52) kuvvet serisi p 1 (4.48) ve (4.52) den 0 1 , , _m , m u x t u x t u x t (4.53)

olarak bulunur (Liao, 2003).

4.3.2. 0 , , 1 , , 2 , , , , n n u u x t u x t u x t u x t .3.2 ye , m u x t denklemi (4.44) lir. (4.44) denkleminde p m !

m p 0 m. mertebeden deformasyon denklemi olarak

1 1

, , , ,

m m m m m

L u x t u x t H x t R u x t (4.54)

0, 1, 1, 1. m m m (4.55) ve 1 1 1 0 , ; 1 , 1 ! m m m m p x t p R u x t m p (4.56) (4.56) denkleminde (4.52) 1 1 1 0 ₀ 1 , , 1 ! m n m m m n n _p R u x t u x t p m p

denklemi elde edilir.

Verilen herhangi bir lineer olmayan (4.54) deformasyon denklemi, R u_{m m 1} terimi ve L

(4.54) u_m1 e ba (4.56) 1 , , 2 , , u x t u x t u x t nin , m. 0 , m k k u x t (4.57)

serisi ile bulunur (Liao, 2003).

4.3.3. Homotopi Analiz Metodunun Kesirli Diferansiyel Denklemlere

F lineer olmayan kesirli u x t bilinmeyen fonksiyon ,

, 0

F u x t (4.58)

genel bir kesirli diferansiyel denklemi

0 1 p L x t p, ; u x t, p H x t F, x t p , ; (4.59) 0,1 p homotopi parametresini, i, ( , ) H x t yar ve L L T t , ; _t , ; , 1 L x t p T x t p n n

0

, ;0 , , , ;1 ,

x t u x t x t u x t (4.60)

olup p x t p de , ; u x t den 0 , u x t kesin ,

0 1 , ; , , m m m x t p u x t u x t p (4.61) 0 1 , ! m m m p u x t m p (4.62) dir. p 1 de (4.61) . Bu durumda 0 1 , , _m , m u x t u x t u x t

elde edilir. u _n u_n u x t u x t₀ , , ₁ , , ,u x t _n , m. dereceden deformasyon denklemi

1 1

, , , ,

t m m m m m

T u x t u x t H x t R u x t (4.63)

olarak elde edilir. Burada

1 1 1 0 , ; 1 , 1 ! m m m m p F x t p R u x t m p (4.63) I _t 1 ( ) 1 1 1 0 , , ,0 , , ! k n k m m m m m t m m k t u x t u x t u x I H x t R u x t k (4.64)

elde edilir (Dehghan, Manafian ve Saadatmandi, 2010).

4.3.4. Uyumlu

uyumlu kesirli kablo denkleminin homotopi analiz metodu (HAM) (4.63) denkleminde H x t, 1 (4.63) denkleminde H x t y, 1 1 , , , , 0 t m m m m m T u x t u x t R u x t (4.65) elde edilir. (3.5) denklem

2 1 1 1 1 1 1 2 1 , ( , ) , m m ( , ) 1 ( , ) m m m m t u x t u x t R u x t t u x t t T f x t t x t olup denklemi 2 1 1 1 1 1 2 1 , ( , ) , m m ( , ) 1 ( , ) m m m m u x t u x t R u x t t u x t t f x t t x (4.66) elde edilir (4.65) I t 1 1 1 0 1 , , , t m m m m m u x t u x t R u x d (4.67)

olur. (4.55) ve (4.66) ifadeleri (4.67) den , (3.5) denkleminin

(3.6) ; 0 2 2 2 1 , 0, , 2 sin 1 , 2 3 u x t t t u x t x 2 2 2 2 2 ₂ 2 2 2 2 2 , 2 1 sin 1 2 3 2 sin 1 1 2 2 2 2 3 t t u x t x t t x 2 2 2 ₂ 3 , 2 1 sin 1 2 3 t t u x t x 2 ₂ 2 2 2 2 2 2 2 3 2 2 3 3 2 2 4 1 sin 1 1 2 2 2 2 3 2 sin 1 1 2 2 2 2 3 2 2 2 3 t t x t t x

eklinde elde edilir (3.5) denkleminin

0 1 2 3 2 2 2 2 2 2 2 2 2 , , , , , 2 1 2 1 1 sin 1 3 3 1 2 1 2 2 2 3 u x t u x t u x t u x t u x t t t t x t t 1 _{u x t, t}2_{sin x dir.}

Tablo 4.3. t 0.1 ablo denkleminin x 0.01 0.05 0.5 1 0.1 0.40 0.0003524602 0.0015827153 0.0042553848 0.0199840077 0.60 0.0002245001 0.0010373006 0.0038109095 0.0060632691 0.80 0.0001582580 0.0007427406 0.0033620446 0.0037064371 0.2 0.40 0.0006704190 0.0030105034 0.0080942229 0.0380118415 0.60 0.0004270246 0.0019730629 0.0072487806 0.0115330231 0.80 0.0003010247 0.0014127766 0.0063949888 0.0070500624 0.3 0.40 0.0009227527 0.0041436025 0.0111407420 0.0523188114 0.60 0.0005877490 0.0027156881 0.0099770906 0.0158738445 0.80 0.0004143249 0.0019445202 0.0088019470 0.0097035784 0.4 0.40 0.0010847608 0.0048710968 0.0130967277 0.0615044515 0.60 0.0006909403 0.0031924829 0.0117287734 0.0186608234 0.80 0.0004870681 0.0022859206 0.0103473093 0.0114072406 0.5 0.40 0.0011405850 0.0051217743 0.0137707145 0.0646696074 0.60 0.0007264977 0.0033567751 0.0123323622 0.0196211509 0.80 0.0005121337 0.0024035592 0.0108798049 0.0119942825 0.6 0.40 0.0010847608 0.0048710968 0.0130967277 0.0615044515 0.60 0.0006909403 0.0031924829 0.0117287734 0.0186608234 0.80 0.0004870681 0.0022859206 0.0103473093 0.0114072406 0.7 0.40 0.0009227527 0.0041436025 0.0111407420 0.0523188114 0.60 0.0005877490 0.0027156881 0.0099770906 0.0158738445 0.80 0.0004143249 0.0019445202 0.0088019470 0.0097035784 0.8 0.40 0.0006704190 0.0030105034 0.0080942229 0.0380118415 0.60 0.0004270246 0.0019730629 0.0072487806 0.0115330231 0.80 0.0003010247 0.0014127766 0.0063949888 0.0070500624 0.9 0.40 0.0003524602 0.0015827153 0.0042553848 0.0199840076 0.60 0.0002245001 0.0010373006 0.0038109095 0.0060632691 0.80 0.0001582580 0.0007427406 0.0033620446 0.0037064371 1.0 0.40 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.60 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.80 0.0000000000 0.0000000000 0.0000000000 0.0000000000 4

Bu metotta L lineer , N lineer olmayan f r( ) homojenl

,

Lu Nu f r r (4.68)

, / 0,

olan genel lineer olmayan diferansiyel denklemi p 0,1

parametresi ve v r p, : 0,1 (4.68) denklemine homotopi , 1 _o 0 v p p L v L u p L v N v f r (4.70) veya 0 , _o 0 v p L v L u pL u p N v f r (4.71) . Burada u_o (4.68) (4.71) denkleminde p 0 ve p 1 ,0 _o 0 v L v L u ve ,1 0 v L v N v f r

elde edilir. (4.71) p nin kuvvet serisi olarak

2 3 0 1 2 3 v v pv p v p v ifade edilebilir. (4.68) 0 1 2 1 ₀ , lim _n p _n u x t v v v v v (He, 1999a).

4.4.1. nun Kesirli Diferansiyel Denklemlere

Lineer olmayan

, , , , , , , 0,

t x xx x xx

T u x t L u u u N u u u v x t t (4.72)

Burada L N lineer

olmayan v ve T_t, m 1 m, mertebeli uyumlu

,0 , 0,1, , 1. k k u x f x k m olmak (4.72) h (1 p T u x t) t , p T u x tt , L u u u, ,x xx N u u u, ,x xx v x t, 0, (4.73)

veya

, , , , , , 0, 0,1

t x xx x xx

T u x t p L u u u N u u u v x t p (4.74)

elde edilir.

Homotopi parametresi p 0 (4.74) hali

0

t

T u (4.74) denkleminde p 1 (4.72) denklemini verir.

Bu nedenle (4.74) p 2 3 0 1 2 3 u u pu p u p u (4.75) (4.75) denklemi (4.74) denkleminde ye p 0 0 0 1 1 0 0 1 2 2 1 0 1 2 : 0, ,0 , : , , ,0 0, : , , ,0 0, k t k k t k t p T u u x f x p T u L u N u v x t u x p T u L u N u u u x (4.76)

elde edilir ve (4.76) deki denklemlerin her birine I t sa 1 0 0 1 0 0 2 1 0 1 0 , ! , , , , k m k k t t t t t t u u k u I L u I N u I v x t u I L u I N u u

elde edilir. (4.72) denkleminin

-0

, _n ,

n

u x t u x t

(Abdulaziz vd., 2008).

4.4.2. Uyumlu Kesirli Kablo Denkleminin Homotopi

uyumlu kesirli kablo denklemini h

(HPM) . (3.6) (3.5) denklemine HPM

0 0 0 2 1 0 1 1 2 0 1 2 2 1 2 2 1 2 2 3 2 3 2 2 3 : 0, ,0 0, : , , ,0 0, : , ,0 0, : , , 0 0, t t t t t t p T u u x u p T u u T T f x t u x x u p T u u u x x u p T u u u x x (4.77)

eklinde elde edilir. (4.77) I_t u u u₀, ₁, ₂ ve u₃ ni 0 2 2 2 1 , 0, , 2sin 1 , 2 (3 ) u x t t t u x t x 2 ₂ 2 2 2 2 2 , 2sin 1 1 , 2( 2) (2 2) (3 ) t t u x t x 2 2 3 2 2 3 2 2 3 , 2sin 1 1 , 2 2 2 2 3 2 2 2 3 t t u x t x

b nin geri kalan terimleri

(3.5) denklemi 0 1 2 3 2 2 2 ₂ 2 2 2 2 2 2 2 3 2 2 3 2 2 , , , , , 2sin 1 1 1 2 (3 ) 2( 2) (2 2) (3 ) 1 1 2 2 2 2 3 2 2 2 3 u x t u x t u x t u x t u x t t t t t x t t (3.5) denkleminin HPM ile 1 2 , sin u x t t x (3.5) 1 0, 1, 2, u u u

4. 0.35 ve 0.65 ablo denkleminin HPM 4.5. 0.90 ve 1 ablo denkleminin HPM Modifiye (4.68) denkleminde f fonksiyonu, 0 ( ) n n f r f r par . f fonksiyonu 0 1 ( ) ( ) f r f r f r

1 0 ( , )v p 1 p L v L u_o p L v N v f r f r (4.78) veya 0 1 0 ( , )v p L v L u_o p L u N v f r f r (4.79) (4.79) denkleminde p 0 ve p 1 0 ( ,0)v L v L u_o f r ve 1 0 ( ,1)v L v N v f r f r

elde edilir. f r₀ 0 ve f r₁ f r olarak s

(Odibat, 2007). f fonksiyonu 0 ( ) n n f r f r ise (4.68) denklemin homotopisi 0 ( , ) 1 n o n n v p p L v L u p L v N v p f r (4.80) veya 0 0 ( , ) n o n n v p L v L u p L u N v p f r (4.81) (Odibat, 2007).

f fonksiyonu ikiden daha fazla terimden (4.80) veya (4.81) denkleminde f terimi 0 u , 0 f terimi 1 u , 1 f terimi 2 u 2 hesaplama

Verilen

Bu metotla (4.78)-(4.81) ve

4.4.4. nun Kesirli Diferansiyel Denklemlere (4.72) denkleminde v fonksiyonu 0 , _n( , ) n v x t v x t bir fonksiyon olsun. O zaman (4.72) denkleminin homotopisi

0 (1 ) , , , , , , n ( , ) t t x xx x xx n n p T u x t p T u x t L u u u N u u u p v x t (4.82) veya 0 , , , , , n ( , ), 0,1 t x xx x xx n n T u x t p L u u u N u u u p v x t p (4.83) (4.83) denkleminin p 2 3 0 1 2 3 u u pu p u p u (4.83) denkleminde yerine 0 0 0 0 1 1 0 0 1 1 2 2 1 0 1 2 2 3 3 2 0 1 2 3 3 : , , ,0 , : , , ,0 0, : , , , ,0 0, : , , , , , 0 0, k t k k t k t k t p T u v x t u x f x p T u L u N u v x t u x p T u L u N u u v x t u x p T u L u N u u u v x t u x (4.84) elde edilir ve (4.84) I t uy 1 0 0 0 1 0 0 1 2 1 0 1 2 0 , , ! , , , , , k m k t k t t t t t t t u u I v x t k u I L u I N u I v x t u I L u I N u u I v x t 3 t 2 t 0, ,1 2 t 3 , , u I L u I N u u u I v x t bulunur (Abdulaziz vd., 2008). 0 , , v x t v x t ve v x t₁ , 0 0 , 0 v x t ve v x t₁ , v x t , de ise siyo ve HPM deki u u u ve 0, 1, 2 u3,

4.4.5. Uyumlu Metodu i

uyumlu kesirli kablo denklemini modifiye h

(3.5) denkleminde 1 _, t T f x t fonksiyonu 1 1 1 1 1 1 2 2 2 2 , , , 2sin 1 2 2sin 1 2 t t T f x t t T f x t t t f x t t t x t t x t olur. Burada 1 _{, ,} t T f x t g x t olsun. g fonksiyonu, 2 0 , 2 sin g x t t x ve 2 2 1 , 2 1 sin 2 t g x t x olarak MHPM uygularsa, (3.6) (4.83) 0 0 0 0 2 1 0 1 2 0 1 1 2 2 1 2 2 1 2 : , , ,0 0, : , , ,0 0, : , ,0 0, t t t p T u g x t u x u p T u u g x t u x x u p T u u u x x (4.85) (4.85) I _t 0, 1, 2 u u u ve u ₃ 2 0 2 2 2 2 1 , sin , , 1 sin 2 1 sin , 2 (3 ) u x t t x t t u x t x x 2 2 2 2 2 2 2 2 2 , 1 _{(2 2)( 2)}sin 2 1 _{(2 2) (3 )}sin , t t u x t x x

3 3 2 3 2 2 3 2 3 1 1 , sin 2 sin , 3 2 2 2 2 3 2 2 2 3 t t u x t x _x

olarak elde edilir.

0 1 2 3 2 2 2 2 2 2 2 2 2 , , , , , 1 2 1 1 2 1 sin 1 2 (3 ) (2 2)( 2) (2 2) (3 ) u x t u x t u x t u x t u x t t t t t t x

seri toplam da 1 da (3.5) denkleminin kesin

2

( , ) sin

u x t t x

elde edilir.

7. Kablo denkleminin 0.90 MHPM .

4

ndirgen

d etodu (DDM) Daha sonra i

diferan ansiyel denklemlere

. Uyumlu kesirli kablo denkleminin

t olarak t boyunca hesaplanacak

4.5.1. Metodu v x fonksiyonu V k olur. V x 0 1 ( ) ! k k x d V k v x k dx (4.86) 0 k k v x V k x (4.87) (4.86) ve (4.87) 0 ₀ 1 _{( )} ! k k k k _x d v x v x x k dx

elde edilir (Zhou, 1986).

4.5.2. Metodu

Benzer iki de i kenli v x y fonksiyonunun diferans, fonksiyonu V k h , v x y, 0 0 1 , ( , ) ! ! k h k h x y V k h v x y k h x y (4.88) 0 0 , , k h k h v x y V k h x y (4.89) (4.88) ve (4.89) 0 0 0 0 1 , ( , ) ! ! k h k h k h x k h y v x y v x y x y k h x y

elde edilir (Chen ve Ho, 1999).

4.5.3.

0 0

, , k h

k h

u x t U k h x t

Buradan u x t fonksiyonun diferansiyel d n,

0 0 1 , , ! ! k h k h x t U k h u x t k h x t

olarak elde edilir.

, , 0 0 k h x t k h k h u U x t 2 2 2 2 0,0, 1,0 , 2,0 , , 0,1, 1,1 , 1,2 , , 0,2 , 1,2 , U U x U x U t U xt U xt U t U xt

0 ,0 0 k k k t U x , 1 ,1 0 k k k t U x , 2 ,2 0 , k k k t U x

gibi t nin kuvvetlerine g re d

, 0 h x t h h u U x t (Keskin ve Oturanc, 2009). 4.5.3.1. ki de i kenli u x t , u , U k h u x t, t boyunca 0 , h h h u x t U x t (4.90) (Keskin, 2010). 4.5.3.2. ki de i kenli u x t , , U k h u x t, t diferansiy 0 1 , ! h h h t U x u x t h t (4.91) eklindedir (Keskin, 2010).

4.5.3.3. U x indirgenmi diferansiyel d n_h m fonksiyonunun tersi

0 , h h h u x t U x t (4.92) (4.91) ve (4.92) 0 ₀ 1 , , ! h h h h _t u x t u x t t h t (4.93)

elde edilir (Keskin, 2010).

Tablo 4.4. t Fonksiyon , u x t 0 1 , ! h h h t t U x u x t h t

, , , u x t av x t bw x t U xh aV xh bW xh , , , u x t v x t w x t 0 h h r h r r U x V x W x , , s s u x t v x t t s h s h U x V x x , , r r u x t v x t t ! ! h h r h r U x V x h , m n u x t x t , 1, 0, m h h n U x x h n h n h n 4.5.3.4. u x t, Nu x t , , Nu x t t el 0 0 0 1 1 , ! ! h h h h h h h h t t N x Nu x t Nu U x t h t h t (4.94) (Keskin, 2010).

Tablo 4.5. t boyunca ind

Fonksiyon , m , Nu x t u x t 0 0 m N x U x 1 1 0m 1 N x mU x U x 2 2 1 2 0 1 0 2 1 1 2 m m N x m m U x U x mU x U x , cos , Nu x t u x t 0 cos 0 N x U x 1 sin 0 1 N x U x U x 2 2 0 2 0 1 1 sin cos 2! N x U x U x U x U x , , u x t Nu x t e 0 0 U x N x e 0 1 1 U x N x U x e 0 2 2 2 1 1 2 U x N x U x U x e

4.5.4. nun Kesirli Diferansiyel Denklemlere

T 4.5.4.1. x ve t u x t fonksiy,

diferansiyellenebilir bir fonksiyondur. u x t, t boyunca hesaplanacak z m n indirgenmi kesirli diferansiyel d n m ,

0 ( ) 1 ( ) ! h h h t _{t t} U x T u h

tan (Acan ve Baleanu, 2017). Burada U xh( ) d

( )h _{, ,}

t t t t

h kez

T u x t T T T u x t ise . 0 1 mertebeden h kez

diferansiyellenebilen uyumlu

T 4.5.4.2. U x nin t boyunca indirgenmi diferansiyel d nh( ) m fonksiyonunun tersi 0 ( ) 0 0 0 0 1 , ! h _h h h h t _{t t} h h u x t U x t t T u t t h (Acan ve Baleanu, 2017). 0 1 _, h ! 0,1, 2, , 1 0 h h h t t u x t if h _n t U x for h if h

dur (Acan ve Baleanu, 2017). Burada n, uyumlu diferansiyel denkleminin mertebesidir.

Lineer olmayan uyumlu kesirli diferansiyel denklem

, , , , , t T u x t Lu x t Nu x t v x t (4.95) ,0 u x f x (4.96) (4.95) 1 1 _{h h h h} h U x LU x NU x V x (4.97) (4.96) ise 0 ( ) h ( ) U x F x (4.98)

olur. (4.98) denklemi (4.97) denkleminde h 0,1, 2,3, ,n

basit iteratif hesaplamalar kullanarak U xh( ) fonksiyonu elde edilir. Daha sonra

0 ( ) n h _h U x (4.95) denkleminin 0 ( , ) ( ) h , n h h u x t U x t

eklindedir ve burada n (4.95) denkleminin

( , ) lim ( , )_n n u x t u x t elde edilir. 4.5.4.2 uyumlu 4.6 da (Acan ve Baleanu, 2017). Tablo 4.6. Fonksiyon , u x t 0 ( ) 1 ! h h h t _{t t} U x T u h , , , u x t av x t bw x t Uh x aVh x bWh x , , , u x t v x t w x t 0 h h r h r r U x V x W x , t , u x t T v x t Uh x h 1Vh1 x 0 , m n u x t x t t 1, , 0, m h n if h n n U x x h h n if h 4.5.5. Uyumlu

uyumlu kesirli kablo denklemini i

metodu ( (3.5) denklemine denkleminin

4.5.4.1 den 2 1 1 2 1 h , h h U x h U x U x t f x t x (4.99)

, f x t 2 1 2 2 1 2 2 2sin 1 2 h h h U x h U x U x x h x h (4.100) elde edilir. (3.6) de 0 0 U x (4.101) (4.101) denklemi (4.100) denkleminde h uygun 1 2 0, sin , U x U x x 2 2 3 2 2 2 2 4 1 2 1 sin sin , 2 3 1 2 1 sin sin , 2 2 2 2 2 3 U x x x U x x x ( ) h U x Daha sonra 0 ( ) n h _h U x (3.5) denkleminin 0 2 2 2 2 2 , 1 2 1

sin sin sin

2 3 h n h h u x t U x t t x t x t x (4.102)

elde edilir. (4.102) denkleminde 1 (3.5) olan _{u x t( , ) t}2_{sin x elde edilir.}

5

ada, zaman-kesirli bir boyutlu kesirli kablo denkleminin

-sirli da uyumlu

Uyumlu kesirli kablo denklemi (U

- aryasyonel

iterasyon metodu, homotopi analiz metodu, homotopi pe modifiye h

.

1. U inin 0.30 ve 0.70

4.1 ve 4.2 Daha

sonra Tablo 4.2 de ,x t ve hata tablosu

tur. te ise U kesin

AA .

2. U elde edilen - nde x

0.1 t iken 3. U -analitik fonksiyonunda 4. HAM da 1 5. U -fonksiyonunda 6. U gra 7. fonksiyonunun AAY

8. 1 nin U 9. U k olar 10. kesirli kablo denkleminde uyumlu yakl -elde edilebilir.

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: : T.C. : Konya- 04.06.1989 Telefon : 05556685512 Faks : e-mail : burcu.yaskiran89@gmail.com Derece

Lise : Cumhuriyet Anadolu Lisesi, Konya 2007

: 2013 : Doktora : Kurum 2015-2016 _kulu YAYINLAR

, 2017, Approximate-analytical solutions of cable equation using conformable fractional operator, New Trends in Mathematical Sciences, 5 (4), 209-219.