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MJEN 1694-7398 4 1 2016ISSN Volume Issue Year

Authors Article Title PagesAvıt Asanov Sedat Yanık

Solving linear Fredholm-Stieltjes integral equations of the second kind by using the generalized Simpson’s rule 1-11

Dağıstan Şimşek Mustafa Eröz

T _ x n-3 x n+1 _ j

Solutions of the rational difference equations n n-1 n - 212- 20

Tuğba Aydemir Ömer Faruk Gözükızıl

Exact travelling wave solutions of the benjamin-bona-mahony-burgers type (bbmb) nonlinear pseudoparabolic equations by using the (G’/G) expansion method 21- 37

Dağıstan Şimşek Mustafa Eröz

Burak Oğul

r _ 1 y n 1 _ \ 1 X n 1 n+1 | , f ; y n+1 1 , fI X n-1 X n J I y n - 1 y n J

Maksimumlu fark denklem sisteminin çözümleri

38-51

Şamil Akçağıl Ömer Faruk Gözükızıl

The Tanh-Coth method for two system of Sobolev type equations in mathematical physics 52-62

Ahmet Hulusi Dinçoğlu Zafer Gönülalan Determination of Escherichia coli O157:H7 in chicken meats sold in sanliurfa region 63-68

Aibek Bodoshov ^eprH^HKTYY aK coprrory Tee öyypnaKTap, K 6np $H3HKa^HK KacnerrepH *aHa aMHHKHCflOTa HK KypaM e3reHe^YKTepY 69-77

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MANAS Journal of EngineeringVolume 4 (Issue 1) (2016) Pages 1-11

Solving linear Fredholm-Stieltjes integral equations of the second kind by using the generalized Simpson’s rule

Avıt A SA N O V Kırgızistan Türkiye Manas Üniversitesi, Bişkek, Kırgızistan

[email protected] Sedat YANIK

Kırgızistan Türkiye Manas Üniversitesi, Bişkek, Kırgızistan [email protected]

Received: 04.06.2015; Accepted: 09.07.2015

Abstract: In this paper, the generalized Simpson's rule (GSR) is applied to solve linear Fredholm-Stieltjes integral equations o f the second kind (LFS IESK ). A numerical example is presented to illustrate the method by using Maple. In some cases depending on the number o f subintervals “n ” , the results are calculated and compared. The graph o f these results is plotted. An algorithm o f this application is given by using Maple.

Keywords: Approximate Solutions, Linear Fredholm-Stieltjes Integral Equations, Simpson's Rule.

İkinci tür lineer Fredholm-Stieltjes integral denklemlerinin genelleştirilmişSimpson kuralı ile çözümü

Öz: Bu çalışmada, ikinci tür lineer Fredholm-Stieltjes integral denklemlerinin çözümü içingenelleştirilmiş Simpson kuralı uygulanmıştır. Metodu göstermek için Maple programı kullanılarak sayısal bir örnek sunulmuştur. "n "nin alt aralıklarına göre bazı durumlarda sonuçlar hesaplanmış ve karşılaştırılm ıştır. Bu sonuçların grafiği çizilm iştir.Maple kullanılarak oluşturulmuş bu uygulamanın algoritması verilmiştir.

Anahtar Kelimeler:_____ Yaklaşık Çözümler, Lineer Fredholm-Stieltjes İntegral Denklemleri, Simpson Kuralı.

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ASANOV and YANIK, Solving linear fredholm-stieltjes integral equations o f the second kind by using the generalizedSimpson’s rule

INTRODUCTION

The theory of integral equation with its applications plays an important role in applied mathematics. Integral equations are used as mathematical models for many and varied physical situations and they also occur as reformulations of other mathematical problems [7]. For many integral equations, it is necessary to use approximation methods. As an example, most of the geophysical problems connected with electromagnetic and seismic wave propagation can only be solved approximately. Among the integral equations, linear Fredholm integral equations of second kind is one of the most popular types of integral equations [7] [13]. Many approximation methods can be used to solve linear Fredholm integral equations of second kind. However, only a few of them are useful to solve LFSIESK. The generalized Simpson's rule is one of the most suitable method with its pretty close result to solve LFSIESK.

Solving LFSIESK by Using the GSR

Given LFSIESKb

u(x) = — J K (x, y) u(y) dg(y) + f (x), x e [a, b] (2.1)

where K (x,y) e C [a ,b] , g (y) is the continuous function on the closed interval [a, b] which can be

written as a difference of two strictly increasing functions (p(y), y (y ) on the closed interval [a, b ], f (x) given function and u(x) is the unknown function to be determined. The parameter A is a known quantity. Now, instead of g (y) in (2.1), if the difference p(y) ~ y (y ) is substituted, then it becomes of the form

b b b

u (x) =i(x ) = /¡J K (x , v ) u (y ) d g (v ) + /(x ) = A j K {x , v) u {v ) dcp{v) - 1 J K {x , v) n {v ) dy/{v) + f {x ) (2.2)

The integrals in (2.2) can be calculated separately as follows by using the generalized Simpson's rule w h i c h i s ca l l ed here shortly as GSR [1],

b /LI = — J K ( x , y ) u ( y ) d P ( y ) « “ X [ K ( x = x2, ) u ( x2, ) + 4 K ( x = x 2*, + 1) u ( x 2*, + 1) + K ( x = x2, +2) u ( x 2, + 2 ) ] [ P ( x 2, + 2) “ P ^ ] =

6 1=0a ‘ 0

= - { [ K ( x , x0) u ( x0) + 4 K ( x , x*) u ( x ’ ) + K ( x , x2) u ( x2) ] [ p ( x2) “ P ( x0) ] + ■■■ +6

+ [ K ( x , x2n_2 ) u ( x 2„_2 ) + 4 K ( x = x2*n-1 ) u ( 1 ) + K ( x = ) u (x 2n ) ] [ P ( ^ n ) “ P ( x 2„-2 ) ] }

P ( x 2, ) + P ( X 2, - 2 )where x* = p2i-1 t and

b — n-111 = — J K ( x ’ y ) u ( y ) d V ( y ) “ _ X [ K ( x ’ x 2i ) u ( x 2i ) + 4 K ( x = x2*+1) u ( x 2*+1) + K ( x = x 2,+2) u ( x 2i + 2 ) x 2, + 2 ) “ ^ ( ] =

6 ,=0a ‘ 0

= - { [ K ( x = x0) u ( x 0) + 4 K ( x = x1** ) u ( x 1** ) + K ( x = x2) u ( x 2) ] [ ^ (x 2 ) “ W (x 0 ) ] + ■■■ +6

+ [ K ( x , x2n-2 ) u ( x 2n-2 ) + 4 K ( x = x2"n-1) u ( x 2n-1 ) + K ( x = x2n ) u ( x 2n ) ] [ ^ ( x 2n ) “ ^ ( ^ -2 ) ] }

I

B aM JEN M A N A S Journal of Engineering, Volume 4 (Issue 1) © 2016 www.journals.manas.edu.kg2



where x 2,-1 = V V ( X 2i ) + V ( * 2 i- 2 )

Therefore, the integral equation (2.2) becomes— n-1

U (x) = _ X { [ K (X, x2,) u(x2, ) + 4K (X, x2,+1 ) u (x2,+1 ) + K (X, x2,+2 ) u (x2, + 2 ) ] [ P (X2.+2 ) - P ( x2, ) ] -6 , = 0

- [ K ( x, x2, ) u( x2, ) + 4 K ( x, x2,+1) u( x2,+1) + K ( X x2, + 2) u( x2,+2) ] [W( x2, + 2) - W( x2, ) ] } + f ( x)

where u (x) is the approximate solution of (2.2).Now, in (2.3), if we use the following substitution:

A , ( x ) = 1 K ( X , x 2 i - 2 ) [ P ( x 2 i ) - P ( x 2 i - 2 ) ]6

4 * r i

B , ( x ) = 7 K ( x , x 2 , - 1 ) [ P ( x 2 , ) - P ( x 2 , - 2 ) ]6

C , ( x ) = 1 K ( X , x 2 i ) [ P ( x 2 i ) - P ( x 2 i - 2 ) ]6

D , ( X ) = 7 K ( X X 2 i - 2 ) [ W ( X 2 i ) - W ( X 2 i - 2 ) ]6

E , ( X ) = 4 K ( X , X 2 2- 1 ) [ W ( X 2 , ) - W ( X 2 i - 2 ) ]6

F , ( X ) = 1 K ( X , X 2 i ) [ W ( X 2 i ) - V ( X 2 i - 2 ) ]6

then the equation (2.3) can be written as

u (x) = X [A (x)u(x2,_2) + B (x)u(XL , ) + C (x ) u (x2j) - D (x)u(^ 2) - E ( x ) u ( ) - F (x)u(x _) ] + f (x)

for i = 1,2,3, ,■■, n ,

From ( 2.5 ) , we have

U ( x ) = — j ''X ( j ( j ] ) - j j ( j ] ) ) M ( X 2 , - 2 ) + X 5 , . ( x ) U ( X 2*i - ) j E i ( j ( X 2 i - 1 ) + X ( C i ( x ) - j ( x ) ) u ( X 2 i )

= 1

H ow eve r, some of the terms of the equation (2.6) can be written as

—X ( 4 (x ) - D (x )) u (x2,-2 ) = — ( A 1(x ) - D 1(x )) u (x0) + —Xn ( A i +1(x ) - A+1(x )) u (X2i)¿=1 ¿=1

n-1^ X ( C i (X) - F (X )) u(X2i ) = ^ X ( C i (X ) - F (X )) u (X2i ) + ( C n (x ) - F n (x )) u (X2n )

i l ¿=1

Taking into account (2.7) and (2.8), from (6), we get

(2.3)

(2.4)

(2.5)

+ f ( x )

(2.6)

(2.7)

(2.8)

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T(x ) = — ( A 1 ( x ) - D 1 ( x ) ) u ( x 0 ) + —X ( A ,+1 (X ) - D ,+1 (X ) + C , (X ) - F , ( X ) ) u (X 2i ) +

i=1 (2.9)

+ — ( C n (X ) - F n ( X ) ) u (X 2n ) + —X 5 ( X ) u ( X *V 1 ) - —X E ( X ) u ( X T -1 ) + f (X )ni l i l

In (2.9), ifP (x ) = A1 (X) - D 1 (x ) R (X ) = A i+1 (X ) - D ,+1(X) + C i (X) - F (XX «S'(X ) = C n (X ) - F n (x ) (2.10)

then (2.9) becomesn-1 n n

u (x) = —P (x )u (X0) + —X R (x)u(X2i) + —S (x )u (X 2n) + —X (x)u(x*_ 1) - —X E (x )u (x*"_ 1) + f (x)i=1 i=1 i=1

(2.11)

Substituting x2 . , x^ j and x2tl for j = 0 ,1 ,_ ,n and & = 1 ,2 ,_ ,n into (2.11), we get the following system of linear equations

n-1 n

u ( X 2 j ) = —P ( X 2 j ) u ( X 0 ) + —X R i (X 2 j M X2i ) + —S ( X 2 j ) u ( X 2n ) + —X B i (X 2 j ) u ( X *i-1 ) -i=1 i =1

- —X E i (X 2 j M X"-1 ) + f (X 2 j )

u ( -1 ) = —P ( *2;-1 )u(X0 ) + R i (X2&-1 )u(X2i ) + —S (XL -1 )u (X2n ) + — B i (XL -1 V (4-1 ) -i=1 i=1

n- —X E (x*. _ > ( x*i_ , )+ f ( x*. _,)

i=1

u ( X*.- ) = —P (X * ;- )u(X0 ) + — X R (X*;-1 u (X2, ) + — S ( X*“ - )u (X „ ) + — f i B , (X *;- )» (X*,- ) -i=1 i=1

n- ** \ y- ** \ r f ** \

- —X E i (X2;-1 M X2,-1 ) + f (X2;-1 ) t ! (2.12)

Fr om ( 2.1 2 ), we get the following system of linear equations,n-1 n

u ( x 0 ) = — P ( X 0 ) u ( X 0 ) + — X R ( X 0 ) u ( X2i ) + — S ( X 0 ) u ( X 2n ) + — X 5 ( ^ ( X ! i -1 ) -

- —X E, (X0)u (x22_.)+f (x„)

i =1

i=1

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n-1

u (X 2n ) = — P ( X 2n ) u ( X 0 ) + —X R (X 2n ) u ( X 2i ) + — S (X 2n M X2n ) + —X B (X 2n ) u ( 4 - 1 ) 'i=1 i = 1

n

- —X E (X 2n ) u ( 4 - 1 ) + f (X 2n )i=1

U ( x 12) = — P ( x * ) u ( x 0 ) + —X R i (X *) u ( X 2i ) + — S (X * ) u ( X 2n ) + —] ^ B i (X * ) u ( 4 - 1 ) -i=1 i=1

n

- — ' X E i (X * ) u ( 4 - 1 ) + f ( x 12)

u (X *n-1 ) = —P ( X *n-1 ) u ( X 0 ) + —X R i (X *n-1 M X2i ) + —S ( 4 - 1 ) u ( X 2n ) + —X B ( 4 - 1 M X*i-1 ) -i=1 i = 1

n

- —X E (X *n-1 ) u ( 4 - 1 ) + f (X *n-1 ) (2.13)

If the system of linear equations (2.13) is converted into matrix form, then( \

f P ( X y ) ••• R „ - l ( X 2 j ) S ( X 2 j ) B A X V ) ~ E A X 2 j ) ]

I - —

,«(x2*L),

(2.14)for j = 0,1,_, n, i = 1,2,_, n, . = 1,2,_, n

Now, the system of linear equation (2.14) ( I - —A )• U = F has a unique solution U = ( I - —A )-1 • F if and only if det(I - —A ) ^ 0.

Now, let p(x) e C a [a , b ], v (x ) e C p [a , b ] where 0 < a < 1, 0 < ^< 1.Then in the equations

A i (X) = 1 K ( X, X2i-2 ) [ P (X2i ) - P (X2i-2 ) ] , B i (X ) = 4 K ( X, X2i-1 ) [ P (X2i ) - P (X2i-2 ) ] ,6 6

C i (X ) = 1 K ( X, X2i ) [ P (X2i ) - P (X2i-2) ] , D i (X ) = 1 K ( X X2i-2 ) [W (X2i ) - W(X2i-2 ) ] ,6 6 4 1

E i (x ) = T K ( X, X2i-1 ) [ W(X2i ) - W(X2i-2 ) ] , F i (x ) = 7 K ( X, X2i ) [W (X2i ) - W(X2i-2 ) ] for I = 1, 2, ■■■,6 6the terms approaches 0 as x2 - x2,_2 approaches 0 , at least as fast as |x - x |a,^ approaches 0

So, A (x) ^ 0, B (x) ^ 0, C (x) ^ 0, D (x) ^ 0 , E (x) ^ 0, F (x) ^ 0 for all/ = 1,2,3,■■■,n . [18], [19].

i =1

i =1

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i * 0 « (ASo, the coefficient matrix will be of the form A ; and we can conclude that det(I - A) ^ 0

Therefore, the system (2.14) has a unique solution, namely U = ( l - A )-1 • F .Thus, if the solution of the system of linear equations (2.14) is substituted back into the (2.12), then the general solution is defined as

n-1 n nU (x) = A P (x ).U 0 + A O 0 ( 0 0 0 0 ( 0 0 O 0 ( X)U„+i O 0 (x )U 2„+; + f (x) (2.15)

i=1 i=1 i =1

Numerical Example

Let us consider the following LFSIESK

uCx) = j (ı + x2s)u(s) d (ln (ı + yfs ))------- + xyfx . (3.1)

x 'i x

Here K (x, s) = 1 + x2 s ,p (x ) = ln (1 + V x ) , w (x ) = 0,—= 1 and f (x) = --------+ xyfx .v ’ 4 6

b - a 1 - 0Let us take n = 4 , then h = ----=---- = 0J25 and x.. = a + 2ih = 0J25 • 2i for i = 1,2,3,4.

2n 8If it is calculated, then it can be obtained as x0 = 0, x2 = 0 25, x4 = 0 5, x6 = 0 75, x8 = 10 . Then, if the GSR is used to integrate (3.1),

1

I = Í(— + x 2 s ) u(s) d (ln I1 + * ))■

! - X [ K CX , X 2i ) U CX2i ) + 4 K C X , X 2i+— ) U C X 21 +1 ) + K C X , X 2i+2 ) U C X 2i+2 ) ] [ P ( X2,J - P ( X 2, ) ]Ó i = 0

(3.2)

where x ^ = p 1 X2j ) +p(X2i-2 )

if calculated, then it can be obtained as x* = 0.0505, x* = 0.3602, x* = 0.6159, x* = 0.8683 . So the equation (3.2)

1 -3u (X) = 7 X [ [ ( [i ) [ ([i ) + 4 K ( [ i+1 ) u (2*i+1 ) + K ( X, X2i + 2 ) [ (X2i + 2) ] [P (X2i + 2 ) - P (]i ) ] + f ( ]

6 i.0

Here if

A (x) = 1 K (x, x2,_2 ) [p(X2i ) - p(X2i_2)] , B i (x) = 4 K (X x*i_1 ) [p(X2i ) - p(X2i-2 )] ,6 6

C (x) = — K (x,x2.) [p(x2.) -p(x2. 2)] fori = 1,2,3,4 , then (3.3) becomes 6

becomes

(3.3)

0

0

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4

r ( x ) = 0 [ 4 ( x ) u ( X 2i- 2 - + - i ( - C - i - J + C — x 2i ) ] + / ( X ) . (3.4)

Then if the valuesx0,x*,x2,x*,x4,x*,x6,x*,x8 are substituted into the equation (3.4), then the following system is obtained and solution is found by using Maple as follows

' 0 .9 3 2 4 - 0 .2 7 0 3 - 0 .0 8 9 1 - 0 .0 8 6 2 - 0 .0 3 6 4 - 0 .0 5 9 3 - 0 .0 2 6 4 - 0 .0 4 6 2 - 0 .0 1 1 6 ^ ' u ( x o > '0 . 0 0 0 0 ^ ^ u ( x o >

^ 0 .3 0 6 8 N

- 0 .0 6 7 6 0 .7 2 9 7 - 0 .0 8 9 2 - 0 .0 8 6 3 - 0 .0 3 6 4 - 0 .0 5 9 4 - 0 .0 2 6 4 - 0 .0 4 6 3 - 0 .0 1 1 6 u ( x 0 .0 0 2 3 u ( x

0 .3 0 9 4

- 0 .0 6 7 6 - 0 .2 7 1 2 0 .9 0 9 5 - 0 .0 8 8 2 - 0 .0 3 7 5 - 0 .0 6 1 6 - 0 .0 2 7 6 - 0 .0 4 8 7 - 0 .0 1 2 3 u ( x 0 .0 6 7 7 u ( x 20 .3 8 2 1

- 0 .0 6 7 6 - 0 .2 7 2 1 - 0 .0 9 2 0 0 .9 0 9 7 - 0 .0 3 8 8 - 0 .0 6 4 1 - 0 .0 2 9 0 - 0 .0 5 1 4 - 0 .0 1 3 1 u ( x 0 .1 2 3 7 u ( x 30 .4 4 6 3

- 0 .0 6 7 6 - 0 .2 7 2 4 - 0 .0 9 4 7 - 0 .0 9 4 0 0 .9 5 9 1 - 0 .0 6 8 5 - 0 .0 3 1 3 - 0 .0 5 6 3 - 0 .0 1 4 5 u ( x = 0 .2 0 7 7 u ( x 4 =0 .5 4 4 9

- 0 .0 6 7 6 - 0 .2 7 5 5 - 0 .0 9 7 6 - 0 .0 9 8 0 - 0 .0 4 3 3 0 .9 2 6 8 - 0 .0 3 3 9 - 0 .0 6 1 5 - 0 .0 1 5 9 u ( x 0 .2 8 5 9 u ( X0 .6 3 8 8

- 0 .0 6 7 6 - 0 .2 7 8 0 - 0 .1 0 1 7 - 0 .1 0 3 7 - 0 .0 4 6 6 - 0 .0 7 9 9 0 .9 6 2 5 - 0 .0 6 8 8 - 0 .0 1 8 1 u ( x 0 .3 8 3 9 u ( x 6

0 .7 5 9 1

- 0 .0 6 7 6 - 0 .2 8 0 6 - 0 .1 0 5 9 - 0 .1 0 9 6 - 0 .0 5 0 1 - 0 .0 8 6 9 - 0 .0 4 1 3 0 .9 2 3 5 - 0 .0 2 0 3 u ( x 0 .4 7 5 9 u ( X

V 1 .0 1 1 8 J

V - 0 .0 6 7 6 - 0 .2 8 4 0 - 0 .1 1 1 4 - 0 .1 1 7 3 - 0 .0 5 4 6 - 0 .0 9 5 9 - 0 .0 4 6 2 - 0 .0 8 6 4 0 .9 7 6 9 J V u ( x8J V 0 .5 8 3 3 J V u ( x8) J

(3.5)

=1

Then, this solution is substituted back into (3.4) and simplified by Maple to get

U (x) = 0.306805528886602 - 0.128312149449382 • x2 - 1 • x + x\fx (3.6)6

which is pretty close to the exact solution u (x) = 0.3058111302- 0.1289085929• x2 - — • x + xyfx6

As the number of subintervals "n" increased, the accuracy in the approximate solution increases and the error decreases. The following Table 1 shows how the approximate solution approaches the exact solution as the number of subintervals "n" increases.

Table 1. Comparison determinant of the coefficient matrix in (16) and the approximate solution, as nincreases

«I det(A)l u (x)l u(x)

4 0.2619621 0.3068055 - 0.1283121 • x2 - 1 • x + xyfx 0.3058111 - 0.1289085 • x2 - - • x + x yp6 6

16 0.2621706 0.3058447 - 0.1288960 • x2 - 1 • x + xyfx 0.3058111 - 0.1289085 • x2 - 1 • x + x>/x6 6

64 0.2621742 0.3058122 - 0.1289082 • x2 - 1 • x + xyfx 0.3058111 - 0.1289085 • x2 - 1 • x + x^x6 6

256 0.2621743 0.3058111 - 0.1289085 • x2 - 1 • x + xyfx 0.3058111 - 0.1289085 • x2 - - • x + x yp6 6

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In the following Figure 3-1, we have Maple plot the solutions of the Table 1. It can be observed that as the number of subintervals " n " increases, the graph of the solutions are accumulating around the exact solution which is close enough to the solution of n = 256.

Figure 1: Comparison graphs of the approximate solutions as n increases

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Algorithm for Solving LFSIESK By Using The GSR In Maple

restartM'ith{ Linear A Igebra) : mth(plots) :

InputsK - = U j ) - 1 + j c 2 s :

f ~ - y + x sqrt(x) :

p •■= *ln( 1 + sqrt(x)) : t ■= x—>0:

= 4:= 0.0: = 1.0 :

lambda == 1 :

U(x)Simpson_General_Soher •■= p r o c (£ p, t , f a b, lambda *= 1, n ■■= 100) :: function,

local imp, h == — — >; x : : list. Id, Temp, i,j, C, F, Pr,u \ list, l£

im p ■■= x—* (solve(x= p(y), >’) ) ; for ¡from 0 b y 2 to 2 «do

Xj •■= a + i h ; end d o ;for ; from 1 ton do

Xj.,-1 := imp | P jX l d + P jX l i- j ) )

end do:

g ■■= (V. 0 - P t o w ( i = I, - K ( y . x . _ ,) ■ ( p ( x . + ,) - p (x f _ ,)), i ::: even, j

K{y>xi - & { P { xi ) - P ( xi - 2) ) ’ i ^ 2 n + l> J K(y'xl - i ) (P{xi - l ) ~P(xi - l ) ) 'J-K( V,AT, _ , ) • (p(xl_ , ) - p(xt _ , ) ) + - i ■ _ , ) ( / > ( * , + , ) - />(.r( _ , ) ) ] ;

A := Matrix( 1-n + I);fo r/from 1 to 2 « 4- 1 dofor / from 1 to 2 « + 1 AoA(j, /) := g(x [j — 1], /); end do end do;C == IdentityMcarix(2n+ 1) — A:F ■= Malrix( 2 n + 1,1); for ifrom 1 to2 n -1-1

d o F ( u ) = = / ( ^ _ j ) ;end do;Pr ■■= Midtipfy(Matrixlnverse( C),F): for i from l t o 2 n + 1 do

ut ■■= Pr[i, 1): end do;U ■•= y —*add(lambda g(y, i) u[/]3 i= 1 .2- n + 1) + /(> ');U

end proc:

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CONCLUSION

This paper deals with the operative algorithms for solving LFSIESK. Indeed, it enables the algorithms that implement the approximation method of the generalized Simpson's rule and its modification using Maple. A numerical example is given and the approximate results are compared with respect to the number of subintervals "n". Also the graph of the example is plotted with respect to increasing "n". Eventually, this shows that the algorithm yield acceptable results.

REFERENCES:[1] A. Asanov, M. H. Chelik ve M. Sezer, «Approximating the Stieltjes Integral by Using the Generalized

Simpson's Rule,» Com. in Diff. and D ifference Eq., cilt 1, no. 3, pp. 1-11, 2012.[2] L.M. Delves , J. Walsh, Numerical Solution of Integral Equations, London: Oxford University Press,

1974.[3] P.Cerone , S.S.Dragomir , «Approximation of the Stieltjes Integral and Applications in Numerical

Integration,» Application o f M athem atics, pp. 37-47, 2006.[4] F. G. Dressel, «A note on Fredholm-Stieltjes Integral Equations,» Bull. Am er. M at. Soc., cilt 44, no. 6,

pp. 434-437, 1938.[5] A. Chakrabarti , S.C. Martha, «Approximate Solutions of Fredholm Integral Equations of The Second

Kind,» Applied M athem atics and Com putation, no. 211, p. 459-466, 2009.[6] A. T. Lonseth, «Approximate Solutions of Fredholm-Type Integral Equations,» Bull. Am er. M ath. Soc.,

cilt 60, no. 5, pp. 415-430, 1954.[7] K. E. Atkinson, The Numerical Solution Of Integral Equations Of The Second Kind, Cambridge:

Cambridge University Press, 1997.[8] M. Munteanu, «Quadrature Formulas for The Generalized Riemann-Stieltjes Integral,» Bull. Braz.

M ath. Soc., cilt 38, no. 1, pp. 39-50, 2007.[9] S.S. Dragomir, C. Bu§e, M. V. Boldea, L. Braescu, «A Generalization of The Trapezoidal Rule for The

Riemann-Stieltjes Integral and Applications,» Nonlinear Analysis Forum, cilt 6, no. 2, p. 337-351,2001.

[10] L.A. Lusternik, V.J. Sobolev , Elements of Functional Analysis, Delhi: Hindustan Publishing Corporation, 1974.

[11] V. Culjak , J. Pecaric , L.E. Persson , «A note on Simpson Type Numerical Integration,» Soochow Jou rnal o f M athem atics, cilt 29, no. 2, pp. 191-200, 2003.

[12] S. J. Majeed, «Modified Midpoint Method For Solving System of Linear Fredholm Integral Equations of The Second Kind,» Am erican Jou rn a l o f Applied M athem atics, cilt 2, no. 5, pp. 155-161, 2014. Press, 1971.

[13] V.D.Watsworth, Approximate Integration Methods Applied to Wave Propagation, Cambridge, 1958.

[14] A. M. Wazwaz, Linear and Nonlinear Integral equations: Methods and Applications, New York: Springer, 2011.

[15] R. P. Kanwal, Linear Integral Equations: Theory and Technique, New York: Academic Press, 1971.

[16] F. Mirzaee, S. Piroozfar, «Numerical Solution of Linear Fredholm Integral Equations Via Modified Simpson's Quadrature Rule,» J. King Saud University (Science), no. 23, p. 7-10, 2011.

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[17] S. Rahbar, E.Hashemizadeh, «A Computational Approach to The Fredholm Integral Equation of The Second Kind,» Proceeding of the World Congress on Engineering, London, 2008.

[18] J.Engelbrecht, I.Fedotov , T.Fedotova, A.Harding , «Error Bounds for Quadrature Methods Involving Lower Order Derivatives,» In ternational Jou rn a l o f M athem atica l Education in Science and Technology, cilt 34, no. 6, 2003.

[19] A.D.Gadjiev , A.Aral , «The Estimates of Approximation by Using a New Type of Weighted Modulus of Continuity,» Computers and M athem atics w ith Applications, cilt 54, pp. 127-135, 2007.

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x

Solutions of The Rational Difference Equationsxn+1

n-31 ^ xnxn-1 xn-2

Dağıstan Ş İM ŞEKKyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan; Selcuk University, Konya, Turkey

dagistan. [email protected] Mustafa ERÖZ

Sakarya University, Sakarya, Turkey [email protected]


Abstract:

Keywords:

In this paper the solutions o f the following difference equation is examined,

xn+l = ------— -----, n=0,1,2,...1 + xnxn-1 xn-2 (1)

where the in itial conditions are positive real numbers.

Difference Equation, Period Four Solution

x _ xn-3 xn+1

1 + XnXn-iXn-2 Rasyonel Fark Denkleminin Çözümleri

Öz: Bu çalışmada aşağıdaki fark denkleminin çözümleri incelenmiştir,

x „ ixn+\ =~-----— -----, n=0,1,2,...

1 + xnxn-1 xn-2

Burada başlangıç şartları pozitif reel sayılardır.

Anahtar Kelimeler:_____ Fark Denklemi, Dört Periyotlu Çözüm____________________________

ch M JEN M A N A S Journal of Engineering, Volume 4 (Issue 1) © 2016 www.journals.manas.edu.kg




ŞİMŞEK and ERÖZ, Solutions o f The Rational Difference Equations

INTRODUCTION

Recently there has been a lot of interest in studying the periodic nature of non-linear difference equations. For some recent results concerning among other problems, the periodic nature of scalar nonlinear difference equations see, [1-25].Cinar, studied the following problems with positive initial values

r - Xn~lr n+1 1 + axnxn_x

xn_1xn+1 - _ , ,-1 + axnxn-1

_ axn_1xn+1 1 + brnr n-1

for n=0,1,..., in [2,3,4], respectively.

In [18] Stevic assumed that ¡ = 1 and solved the following problem

r n-1 rrn+l = i---- for n = 0,1,2, ...

1 + r nwhere r _ r 0 e (0,<»). Also, this results was generalized to the equation of the following form:

rn-1 „rn+1 = Z 7 for n = 0,1,2, ...

g (rn )where r-1, r0 e (0, ro).

Simsek et. al., studied the following problems with positive initial values

n+1 _ 1 ,1 + xn_1

x - Xn_5xn+1 - 1 + r n-1 rn-3

for n=0,1,..., in [19,20,21] respectively.

In this paper we investigated the folloving nonlinear difference equation

r n+1 = "----— -----> n=0,1,2,... (1)1 + r nr n-1 r n- 2

where x_3 ,x_2 ,x_1,x0 e (0,œ).

xn_3

xn_5x„,, —n_2

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m a in r e s u l t

Let x be the unique positive equilibrum of Eq. (1), then clearly

- x - -4 - -4 -x =--x + x = x x = 0 ^ x = 01 + xxx

We can obtain x = 0.

Theorem 1. Consider the difference equation (1). The n t h e following statements are true.

a) The sequences (x4n_3), (x4n_2 ), (x4n_i), and (x4n) are decreasing and there exist p.q.r.s > 0 such that

lim x4n_3 = p, lim x4n_2 = q, lim x4n_x = r and lim x4n = s.n——to n——to n——to n—to

b) (p, q, r, s, p, q, r, s,...) is a solution of equation (1) of period four.c) p.q.r.s = 0.d) If there exist n0 e N such that xnxn_1xn_2 > xn+1xnxn_1 for all n > n0, then

e) The following formulas hold:

x4n+1 = x—3

lim xn = 0 .

xr,x ıx 9 n 4ji — 0 1 2 s n- 1

1 + x0 x— 1x— 2 j =0 ¿=i 1 + xi—2 xi—1xi(

x4n+2 = x— 2

x4n+3 = x— 1

x4n+4 = x0

x>x ix o n 4j +11— 0 —1 —3 s n 1 A

1 + x0 x— 1x—2 j =0 i=1 1 + xi—2 xi—1xi

x0 x— 2 x—3 n 4j+2 11— 0 2 3 s n

1 + x0 x— 1x— 2 j =0 i=1 1 + xi—2 xi—1x

1x ix 9 n 4j +31-, 1 2 3 s n ----------1 + x0 x— 1x—2 j =0 i=1 1 + xi—2 xi—1xi

f) If x*4W_|_1 —— p ^ 0, %4n+2 —— q ^ 0 and x4n+3 —— r ^ 0 then ^4^+1 —— 0 as

Proof. a) Firstly, we consider the equation (1). From this equation we obtainxn+1(1 + xnxn_1xn_2) = xn_3 .

If xn,xn_x, xn_2 e (0, +to) , then (1 + xnxn_1xn_2) e (1, +to). Since xn+1 < xn_3, n e N , we obtain that lim x4n_3 = p, lim x4n_ 2 = q, lim x4n_1 = r and lim x4n = s.n—to n—to n—to n—tob) (p, q, r, s, p, q, r, s,...) is a solution of equation (1) of period four.c) In view of the equation (1), we obtain

x4n-3x4n+1 =1 + x4nx4n—1x4n—2

Taking limit as n — to on both sides of the above equality, we get

lim x4n+1 = lim —— X4n_n—to

Thenp

n 1 + x4nx4n—1x4n—2

P = 1 + s.r.q• p + p.qr.s = p ^ p.q.r.s = 0 .

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d) If there exist n0 e N such that xnxn_1xn_2 > xn+1xnxn_1 for all n > n0, then p < q < r < 5 < p . Since p.q.r.s = 0 we obtain the result.e) Subracting xn_3 from the left and right-hand sides of equation (1) we obtain

1Xn+1 xn-3 , ( xn xn-4)

1 + xnxn -1 xn-2and the following formula

n-1n > 1 for \ xn -x n-4 = (x1 -x-3) n 1

i=1 1 + xi_ 2 x _1 xiholds. Replacing n by 4j in (2) and summing from j = 0 to j = n we obtain

n 4jx4n+1 -x-3 = (x1 -x-3) 2 n

1j=0 i_11 + X-2Xi-1 Xi

(n = 0,1,2,...).

Also, replacing n by 4j+1 in (2) and summing from j = 0 to j = n we obtain1n 4 j +1

X4n+2 -x-2 = (X1 -X-3) 2 n “j_0 ¿=1 1 + xi-2xi-1 xi(n = 0,1,2,...).

Also, replacing n by 4j+2 in (2) and summing from j = 0 to j = n we obtain

X4n+3'n 4 j +2

-X- 1 = (X1 -X-3) 2 n 1(n = 0,1,2,...).

j =0 i=i 1 + 2 xi _1 xi Also, replacing n by 4j+2 in (2) and summing from j = 0 to j = n we obtain

1n 4 j +3X4n+4 -x0 = (X1 -X-3) 2 nj=0 i=1 1 + X-2xi-1 xi

(n = 0,1,2,...).

From the formulas above, we obtain(

X4n+1 = x- 3XnX iX 9 n 4j

1— 0 1 2 2 n-1

1 + X0 x- 1x-2 j =0 i=11 + xi-2 xi-1xi(

x4n+2 = x- 2

fX4n+3 = x-1

(

XnX 1X 0 n 4 j +11— 0 -1 -3 2 n 1

1 + x0 x- 1x-2 j =0 i=1 1 + xt-2xi-1xi

1^ X 9X n n 4j+21--- 0 2 3 2 n ---------1 + x0 x- 1X-2 j =0 i=1 1 + Xt-2Xt

X4n+4 = X0X ,X n 4j+3

1— -1 -2 -3 2 n 11 + X0 X-1X-2 j =0 i=1 1 + Xi-2 Xi-1Xi

f) Suppose that p = q = r = 5 = 0. By e) we have

lim X4n+1 = lim X-3 n n

1Xr\X iX 9 n 4j1— 0 -1 -2 2 n -

1 + X0 X-1X-2 j=0 i=1 1 + Xt-2Xt-1Xi

P = X- 3XnX iX 9 “ 4j

1— 0 1 2 2 n-1

1 + X0 X-1X-2 j =0 i=1 1 + Xt-2Xt-1Xi

11 + XnX iX 9 “ 4jp = 0 ^ — 0 -1 -2 = 2 n

X0X-1X- 2 =0 i=11 + Xi- 2 Xi -1XiSimilarly,

q = X- 2X\X 1X 0 “ 4j +1

1- . 0-1-3 2 n1

1 + X0 X-1X-2 j =0 i=1 1 + Xt-2Xi-1Xi~ 4 j +1 11 + XnX 1X 9q = 0 --- Ox- ! ^ 2 = 2 n

X0X-1X-3 j =0 i=i 1 + X- 2 X- 1X1

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

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Similarly,

x-1 ! x0x-2x-3 œ 4++2 11 + x0 x_ 1x-2 j —0 ¿—1 1 + xX-2 xX-1xi

1 + x0x >x 2 œ 4j +2 1 r - 0 --- 0 1 2 - x n

x0 x-2x-3 j-0 i-1 1 + x-2xi-1x,Similarly,

S - x0 1----1 2 3 x nœ 4j +3 1

1 + x0 x- 1x-2 j -0 i- 1 1 + xi-2xi-1xi

s - 0 1 + x0 x- 1x- 2 4 j +3 1- x nx-x 2x-3 j -0 i-1 1 + x - 2x i- ^

From the equations (11) and (12),

1 + xnx 1 x n “ 4j 1 1 + j x ++ 9 “ 4+ +1 10 1 2 - x n ---------- >— 0 1 2 - x n

x0x- 1x-2 + o X l l + +_2+_ 1i x0X_ + 3 j +0 X 1 1 + +_2+_pthus, x._3 > x^ .From the equations (12) and (13),

1 + xnx_,x_9 “ 4+ 1 1 1 + %)x_-ix_n x ++ 2 1— 0 1 2 - x n ---------- >— 0 1 2 - x n ----------

x0x- 1x-3 +0 İxx 1 + +x-2 +x- 1+ x01—2x-3 J+ 0 X 1 1 + +x-2 +x- 1xithus, x_2 > x- 1 .From the equations (13) and (14),

(13)

(14)

(15)

(16)

1 + x0x ix 2 “ 4++2 1 1 + x0x i+ 2 “ 4++3 10 1 2 - x n ---------- >— 0 1 2 - x n

x0 x-2x-3 j+0 X 1 1 + 2 ^ x_ 1x_21 3 j+0 X 1 1 + xX_21 _ 1xi(17)

thus, x_ i > x0 .From here we obtain x_3 > x_2 > x_ l > x0. We arrive at a contradiction which completes the proof of theorem.

œ

EXAMPLES

Example 1: If the initial conditions are selected as follows:

x[-3]=2;x[-2]=3;x[-1]=4;x[0]=5;

The following solutions are obtained:

x(n)={ 0.0327869, 1.81188, 3.08397, 4.22581, 0.0013321, 1.78096, 3.05336, 4.19542, 0.000055937, 1.77969, 3.05208, 4.19414, 2.35212x10-6, 1.77963, 3.05203, 4.19409, 9.89108x10-8, 1.77963, 3.05203, 4.19409, 4.15939x10"9, 1.77963, 3.05203, 4.19409, 1.7491x10'10, 1.77963, 3.05203, 4.19409, 7.35532x10" 12, 1.77963, 3.05203, 4.19409, 3.09306x10"13,1.77963,3.05203, 4.19409,...}

The graph of the solutions is given below.

B aM JEN16



ÇlMÇEK and EROZ, Solutions o f The Rational Difference Equations

x [ i l l


x[-3]=5;x[-2]=4;x[-1]=3;x[0]=2;The following solutions are obtained:

x(n)={0.2, 1.81818, 1.73684, 1.22581, 0.0410596, 1.67202, 1.60202, 1.10435, 0.0103735, 1.64189, 1.57245, 1.07554, 0.00274663, 1.63429, 1.56489, 1.06804, 0.000736065, 1.63229, 1.56289, 1.06604, 0.000197891, 1.63175,1.56235, 1.0655, 0.0000532489, 1.6316, 1.5622, 1.06536, 0.0000143316, 1.63156,1.56217, 1.06532,3.85751x10-6, 1.63155, 1.56216, 1.06531, ... }


B aM JEN17



ÇlMÇEK and EROZ, Solutions o f The Rational Difference Equations

x j n [


x[-3]=2;x[-2]=0.1;x[-1]=0.01;x[0]=0.001;

The following solutions are obtained:

x(n)={2, 0.099998, 0.009998, 0.000998004, 2, 0.099996, 0.00999601, 0.000996013, 1.99999, 0.099994, 0.00999401, 0.000994027, 1.99999, 0.099992, 0.00999203, 0.000992044, 1.99999, 0.09999,0.00999005, 0.000990066, 1.99999, 0.0999881, 0.0099861, 0.000986123, 1.99998, 0.0999841, 0.00998216, 0.000982198, ...}


0.00998807,0.00998413,

0.000988093,0.000984159,

1.99999,1.99998,

0.0999861,0.0999822,

cfaM JEN18


n


ÇlMÇEK and ERÖZ, Solutions o f The Rational Difference Equations

x j n f

REFERENCES

[1] A.M. Amleh, E.A. Grove, G. Ladas and D.A. Georgiou, " On the recursive sequence

Math. Anal. Appl., 233, no. 2, 790-798, 1999.V

[2] C. Cinar, " On the positive solutions of the difference equation xn+1 =

Comp., 158 (3), 809-812, 2004.

[3] C. Cinar, " On the positive solutions of the difference equation xn+1 = xn_ 1_ j + axnxn _ 1

[4]

[5]

[6]

[7]

Comp., 158 (3), 793-797, 2004.

C. Cinar, " On the positive solutions of the difference equation xn+1 =1 + bxnxn_ 1

J.

Appl. Math.

Appl. Math.

Appl. Math.

Comp., 156 (3), 587-590, 2004.E. M. Elabbasy, H. El-Metwally and E. M. Elsayed,"On the difference equation

bx„, ", Advances in Difference Equation, Volume 2006,Article ID 82579, 1-10,xn+1 = axn ------"cx „-d x „_ i 2006.E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, "Qualitative behavior of higher order difference equation", Soochow Journal of Mathematics, 33(4), 861-873, 2007.E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, "Global attractivity and periodic character of a fractional difference equation of order three", Yokohama Mathematical Journal, 53, 89-100, 2007.

n

xn _1 "x..,, = a + x

axn 1

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xn- k[8] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed,"On the difference equation xn+1 =-^+ rnf=o xn~ı

", J. Conc. Appl. Math., 5(2), 101-113, 2007.[9] E. M. Elabbasy and E. M. Elsayed, "On the Global Attractivity of Difference Equation of Higher

Order", Carpathian Journal of Mathematics, 24 (2), 45-53, 2008.10] E. M. Elsayed, "On the Solution of Recursive Sequence of Order Two", Fasciculi Mathematici, 40,

5-13, 2008.11] E. M. Elsayed, "Dynamics of a Recursive Sequence of Higher Order", Communications on Applied

Nonlinear Analysis, 16 (2), 37-50, 2009.12] E. M. Elsayed, "Solution and atractivity for a rational recursive sequence", Discrete Dynamics in

Nature and Society, Volume 2011, Article ID 982309, 17 pages, 2011.13] E. M. Elsayed, "On the solution of some difference equation", Europan Journal of Pure and Applied

Mathematics, 4 (3), 287-303, 2011.14] E. M. Elsayed, "On the Dynamics of a higher order rational recursive sequence", Communications

in Mathematical Analysis, 12 (1), 117-133, 2012.15] E. M. Elsayed, "Solution of rational difference system of order two", Mathematical and Computer

Modelling, 55, 378-384, 2012.16] C. H. Gibbons, M. R. S. Kulenovic and G. Ladas, "On the recursive sequence x „+1 = a + Xn—1 ", Math.

Z + xnSci. Res. Hot-Line, 4, no. 2, 1-11, 2000.

17] M.R.S. Kulenovic, G. Ladas and W.S. Sizer, "On the recursive sequence x „+1 = a x " + ^Xn—1 , Math. Sci.^ n + &c n-1

Res. Hot-Line, Vol. 2, No. 5, 1-16, 1998.

18] S. Stevic, "On the recursive sequence xn+1 = Xn-1 ", Taiwanese J. Math., Vol.6, No. 3, 405-414,g (xn)

2002.19] D. Şimşek, C. Çınar and İ. Yalçınkaya, "On the recursive sequence x „+1 = Xn-3 ", Int. J. Contemp.

1 + xn-1Math. Sci., 1, no. 9-12, 475-480, 2006.

20] D. Şimşek, C. Çınar, R. Karataş and İ. Yalçınkaya, "On the recursive sequence x „ +1 = Xn-5 ", Int. J.1 + x n -2

Pure Appl. Math., 27, no. 4, 501-507, 2006.

21] D. Şimşek, C. Çınar, R. Karataş and İ. Yalçınkaya, "On the recursive sequence xn+1 =1 + xn—1 xn-3

Int. J. Pure Appl. Math., 28, no.1, 117-124, 2006.22] D. Şimşek, C. Çınar and İ. Yalçınkaya, "On The Recursive Sequence x(n+1) = x[n-(5k+9)] /

1+x(n-4)x(n-9) ... x[n-(5k+4)] ", Taiwanese Journal of Mathematics, Vol. 12, No.5, 1087-1098, 2008.

23] D. Şimşek and A. Doğan , "On A Class of Recursive Sequence", Manas Journal of Engineering, Vol. 2, No.1, 16-22, 2014.

[24] I. Yalcinkaya, B. D. Iricanin and C. Cinar, "On a max-type difference equation", Discrete Dynamics in Nature and Society, Volume 2007, Article ID 47264, 10 pages, doi: 1155/2007/47264, 2007.

25] H. D. Voulov, " Periodic solutions to a difference equation with maximum", Proc. Am. Math. Soc., 131 (7), 2155-2160, 2002.

B aM JEN20



MANAS Journal of Engineering

Volume 4 (Issue 1) (2016) Pages 21-37

Exact Travelling Wave Solutions Of The Benjamin-Bona-Mahony- Burgers Type (BBMB) Nonlinear Pseudoparabolic Equations By

Using The (G’/G) Expansion Method

Tuğba A Y D E M İR Sakarya University, Applied Mathematics Department, Sakarya, Turkey.

[email protected] Ömer Faruk GÖZÜKIZIL

Sakarya University, Applied Mathematics Department, Sakarya, Turkey. farukg@sakarya. edu.tr


Abstract: In this paper, we consider some nonlinear pseudoparabolic Benjamin-Bona-Mahony-Burger(BBMB) equations by using the (G ’/G) expansion method with the aid o f computer algebraic system Maple. These equations are o f a class o f nonlinear pseudoparabolic or Sobolev-type equationsut -A u .-a A u = f (x,u ,Vu) ,

1 v ' ,a is a fixed positive constant, arising from mathematical physics.The method is straightforward and concise, and it be also applied to other nonlinear pseudoparabolic equations.The (G ’/G) expansion method; Travelling wave solution; Nonlinear pseudoparabolic equation;

Keywords: Benjamin-Bona-Mahony-Burger-type(BBMB) equation.

(G’/G) Açılım Metoduyla Lineer Olmayan Pseudoparabolik Tipte Benjamin-Bona-Mahony-Burgers Denkleminin Yürüyen Tam

Dalga Çözümleri

Öz: Bu makalede Maple yardım ıyla (G ’/G) açılım metodu kullanarak lineer olmayan pseudoparabolikBenjamin-Bona-Mahony-Burger(BBMB) denklemlerini ele aldık. Bu denklemler matematiksel fizikte ortaya çıkan, a pozitif bir sabit pozitif bir katsayı olmak üzereu. -A u .-a A u = f (x, u, Vu) ,, , , , , , ,, ,

1 1 __________________ v ' şeklinde lineer olmayan pseudoparabolik veya Sobolev-tıpı sınıfındandır. Metod açık anlaşılır ve öz bir metod ve aynı zamanda diğer lineer olmayan pseudoparabolik denklemlere de uygulanabilir bir metoddur.

a ut it v ı ■ The (G ’/G) açılım metodu; Yürüyen dalga çözümü; Lineer olmayan pseudoparabolik na ar e ime er._____ denklem; Benjamin-Bona-Mahony-Burger-type(BBMB) denklem.______________________

M JEN M A N A S Journal of Engineering, Volume 4 (Issue 1) © 2016 www.journals.manas.edu.kg




AYDEMİR and GÖZÜKIZIL, Exact Travelling Wave Solutions O f The Benjamin-Bona-Mahony-Burgers Type(Bbmb) Nonlinear Pseudoparabolıc Equations By Using The (G ’/G) Expansion Method

INTRODUCTION

Nonlinear partial differential equations arise in a large number of physics, mathematics and engineering problems. In the soliton theory, the study of exact solutions to these nonlinear equations plays a very germane role, as they provide much information about the physical models they describe. Various powerful methods have been employed to construct exact travelling wave solutions to nonlinear partial differential equations. These methods include the inverse scattering transform[1], the Backlund transform[2,3], the Darboux transform[4], the Hirota bilinear method[5], the tanh-function method[6,7], the sine-cosine method[8], the exp-function method[9], the generalized Riccati equation[10], the homogeneous balance method[11], the first integral method[12,13], the (G'/G) expansion method[14,15], and the modified simple equation method[16,18].The objective of this paper is to use a powerful method called the (G'/G) expansion method to obtain travelling wave solution for the a class of nonlinear pseudoparabolic equations. The method, first introduced by Wang and Zhang[19], has been widely used to obtain exact solutions of nonlinear equations[20,25].Equations with one-time derivative appearing in the highest order term are called pseudoparabolic and arise in many areas of mathematics and physics. They have been used, for instance, for fluid flow in fissured rock, consolidation of clay, shear in second-order fluids, thermodynamics and propagation of long waves of small amplitude. For more details, we refer reader to [26,30] and references therein.An important special case of pseudoparabolic-type equations is the Benjamin-Bona-Mahony-Burgers (BBMB) equation

U - u x - C u xx + ux + uux = 0 (1)

where u (x , t ) represents the fluid velocity in the horizantal direction x and a are positive constants.A generalized Benjamin-Bona-Mahony-Burgers (BBMB) equation

ut - u„ t - a u xx + P ux + (g ( u ) ) x = 0 (2)

has been considered and a set of new solitons, kinks, antikinks, compactons and Wadati solitons have been derived using by the classical Lie method, where a is a positive constant, ¡ e R , and g (u ) is a C2-smooth

nonlinear function. Equation (2) with the dissipative term c u xx arises in the phenomena for both the bore propagation and the water waves. Peregrine and Benjamin, Bona, Mahony have proposed Eq.(2) with the parameters g (u ) = uux ,C = 0, ¡ = 1 [31,32]. Moreover, Benjamin, Bona, Mahony have proposed Eq.(2)as an alternative regularized long-wave equation with the same parameters. Tari and Ganji implemented variational iteration and homotopy perturbation methods obtaining approximate explicit solutions for

U 2 U 2 11Eq.(2) with g (u ) = ^~ [33]. In addition, for g (u ) = uux, g (u ) = ~ ^ and g (u ) = ^~ Akgagil and

Gozukizil obtain some exact soltions by using tanh method[34].As stated before, pseudoparabolic-type equation arise in many areas of mathematics and physics to describe many physical phenomena. In recent years considerable attention has been paid to the study of

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pseudoparabolic -type equations. In this paper, (G'/G) expansion method is used to find the solutions for the pseudoparabolic-type equations stated above.The main ideas are that the travelling wave solutions of nonlinear equation can be expressed by a polynomial in (G'/G), where G = G (g ) satisfies the second order linear ordinary differential equation: G '1 + A G '+ juG = 0, where ^ = x — c t and A, /u,c are constants. The degree of this polynomial can be determined by considering the homogenous balance between the highest order derivative and nonlinear terms appearing in the given nonlinear equations. The coefficients of the polynomial A,/uand c can be obtained by solving a set of algebraic equations resulting from the process of using the proposed method. Moreover, the travelling wave solutions obtained via this method are expressed by the hyperbolic functions, the trigonometric functions and the rational functions.

DESCRIPTION OF THE (G'/G) EXPANSION METHOD

In this section, we describe the (G'/G) expansion method for finding travelling wave solutions of nonlinear partial differential equations. Suppose that a nonlinear partial differential equation(PDE), say in two independent variables x and t, is given by

P ( u , Ut , ux , uxt , uxx , Utt ,...) = 0 (3)

where u(x,t) is an unknown function, P is a polynomial in u = u (x, t ) and its various partial derivatives,in which highest order derivatives and nonlinear terms are involved.

The summary of the (G'/G) expansion method, can be presented in the following six steps:

Step 1: To find the travelling wave solutions of Eq.(3) we introduce the wave variable

u (x , t ) = U ( E ) , E = x - c t (4)

where the constant c is generally termed the wave velocity. Substituting Eq.(4) into Eq.(3), we obtain the following ordinary differential equations(ODE) in T (which illustrates a principal advantage of a travelling wave solution, i.e., a PDE is reduced to an ODE).

P (U , cU ' ,U ' , cU '', c2U '' ,U '' ,...) = 0 (5)

Step 2: If necessary we integrate Eq.(5) as many times as possible and set the constants of integration to be zero for simplicity.Step 3: We suppose the solution of nonlinear partial differential equation can be expressed by a polynomial in (G'/G) as

(6)

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AYDEMİR and GÖZÜKIZIL, Exact Travelling Wave Solutions O f The Benjamın-Bona-Mahony-Burgers Type(Bbmb) Nonlinear Pseudoparabolıc Equations By Using The (G ’/G) Expansion Method

where G=G(T) satisfies the second-order linear ordinary differential equationG ' ' + A G '+<uG = 0 (7)

dCj d 2(jrwhere G ' = ---, G " = --- —, and a ,^-and u are real constants with a ^ 0. Here the prime denotes

d f d fthe derivative with respect to î. Using the general solutions of Eq.(7), we have

G

_ a J a 2 - A ß

2 2

f

+ c2 coshV A 2 - 4ß

t

c cosh

A y l4 ß - A 2

2 2

A— +

2

-C sin

■ A2 - 4 ß

2

■ \]4ß-A

\

t

t

+ c sinh

+ c2 cos

■v/A2 - 4ßt

A 2 - 4 ß > 0

J J

c cos t + c2 sin ‘\J4 ß - At

, A 2 - 4 ß < 0 (8)

J J

C1 + C2tA 2 - 4 ß = 0

where c and c2 are arbitrary constants.Step 4:The positive integer m can be accomplished by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq.(5) as follows: if we define the degree ofu ( 0 as D [u (£ ) ] = m , then the degree of other expressions is defined by

D

D

d qu

f d qu ^

d t

= M + q,

= s ( M + q ) + M r ,

Therefore, we can get the value of m in Eq.(2.4).Step 5:Substituting Eq.(6) into Eq.(5) using general solutions of Eq.(7) and collecting all terms with the same order of (G'/G) together, then setting each coefficient of this polynomial to zero yield a set of algebraic equations for a , c, A and u .

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Step 6:Substitute a t,c, X and ß obtained in step 5 and the general solutions of Eq.(7) into Eq.(6). Next,

depending on the sign of discriminant (X2 — 4 ß ) , we can obtain the explicit solution of Eq.(3)

immediately.

BENJAMIN-BONA-MAHONY-BURGERS (BBM B) EQUATION

The Benjamin-Bona-Mahony-Burgers (BBMPB) equation is given by

u t — u xxt — a u xx + u x + u u x = 0 (9)

where a is a positive constant. Using the wave variable E = x - c t in Eq.(9), then integrating this equation and considering the integration constant to be zero, we obtain

U 2(1 - c )U + c U " - a U ' + — = 0 (10)

According to step 4, balancing U and U " gives N=2. Therefore, the solutions of Eq.(10) can be written in the form

G ' f G 'VU = a + a — a I — I (1 1 )

0 1 G 21 G )

where a0, a and a2 are constants which are unknowns to be determined later. By Eq.(7) we derive

U ' = —2a

U " = 6a,

— I — ( 2 a2X + a ) — I — ( 2 a2U + a X )G

V G

G— i—a u ,

V G

GG

( g < y+ (l0a2X + 2a ) — I +(&a2¡u + 4a2X 2 + 3ax X) (12)

+ ( 6 a2 X ß + a X + 2ax ß ) j — | + 2 a2 ¡U + a l X ß .

4 2

Substituting Eq.(11) and its derivatives Eq.s(12) into Eq.s(10) and equating each coefficient of (G'/G) to zero, we obtain a set of nonlinear algebraic equations for a ,a ,a ,^ a n d c. Solving this system using Maple, we obtain

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AYDEMİR and GÖZÜKIZIL, Exact Travelling Wave Solutions O f The Benjamin-Bona-Mahony-Burgers Type(Bbmb) Nonlinear Pseudoparabolic Equations By Using The (G ’/G) Expansion Method

Set 1.1 V25 + 24«

c = — -2 10

-, — I C - 1 „ -i 12aÀ = +i l4 / j + —— , a n = - \ 2 c , a 1= -1 2 cA ~ -6c 5

a — — 12c u + c — 1 — ■6 aA

Set 2.1 V25 + 24«

c — — i 2 10

2-, — I C - 1 „ -1 12«À = +i l4 / j + —— , a n = - \ 2 c , a 1= -1 2 cA ~ -

6c 5

a — — 12c u i c — 1 — ■6 aA

Set 3.1 V25 — 24«

c — — i 2 10

2J l — c 12«

4jU + —— , a 2 = - \ 2 c , cil = - l2 c A ~ -6c 5

a — — 12c u i c — 1 —6 aA

5

5

5

Set 4.1 V25 — 24«

c — 2 10

2J l — c 12«

4/J. + —— , a 2 = - \ 2 c , cil = - l2 c A ~ -6c 5

a — — 12c u i c — 1 —6 aA

Set 5. c = 0 , A = +. 14// + — , a 2 = 0, a x = - 2 a , a (l = -1 - a A ;«

Using these values in Eq.(11) when A 2 — 4j > 0, we obtain the hyperbolic solutions respectively:

5

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q sinh 1 c -1f

v 2 V 6c y+ q cosh

1 c -1f

v 2 V 6c y

q cosh1 c -1

fv 2 V 6c y

+ c2 sinh1 c -1

f

c -1q sinh

1 c -1f

v 2 V 6c y+ q cosh

1 c -1 A A(13)

fv 2 V 6c y

q cosh1 c -1

fv 2 V 6c y

+ q sinh1 c -1

fv 2 V 6c y y

2

2

fwhere f — x -

1 V25 + 24g2 10

2 1 V25 + 2 4 «2i and c — — 1-----------

2 10yr

3 ( c - 1) 6 a lc -1q sinh 1 c -1

fv 2 V 6c y

+ q coshc -16c

f

c, cosh1 c -1

fv 2 V 6c y

+ q sinh1 c -1

fv 2 V 6c yy

c -1q sinh 1 c -1

fv 2 V 6c y

+ q cosh12 V

c -16c

f

c, cosh1 c -1

fv 2 V 6c y

+ q sinh1 c -1

fv 2 V 6c yy

(14)

where f — x -1 V25 + 24a2 10

2 1 V25 + 2 4 a2t and c —--------------

2 10y

U ( x, t ) —c -1 6a 1 - c

2 5 V 6c

q sinh1 1 - c

fv 2 V 6c y

+ q cosh1 1 - c

fv 2 V 6c y

c, cosh1 1 - c

fv 2 V 6c y

+ q sinh1 1 - c

fv 2 V 6c yy

+c -1

f { q sinh

12 V

1 - c6c

f + q cosh12 V

1 - c W6c

f

q cosh1 1 - c

fv 2 V 6c y

+ q sinh1 1 - c

fv 2 V 6c yy

(15)

2

2

2

2

2

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where t = x -2 1 V25 - 2 4 a2

t and c = — I-----------2 10

1 V25 - 24a2 10V

c -1 6 a 1 - c

2 5 V 6c

1

2 V

1 - c

6ct

(c2 cosh

1

2 V

1 - c

6c

wt

( 1 1 - ct

V2 V 6c j

(- c2 sinh

1 1 - ct

V 2 V 6c j j

+ ■c -1

1

2 V1 - c6c

t c2 cosh1 1 - c

t2^ 6c ,V j

1 1 - ct

V 2 V 6c jc2 sinh

1 1 - ct

V 2 V 6c j j

(16)2

2

where t = x -1 V25 - 24a22 10

t and c = ---2

V 2 5 - 24a210

u5 ( x( x, t ) = -1 - 2a-c2cosh

2a

2a- c2 sinh

V 2 a j j

(17)

In particular, if we take q ^ 0, c < q2 , then Eq.s(13)-(17) lead the formal solitary wave solutions to Eq.(9) as

U (x, t ) = c -1 - lc— 1 tanh5 6c

1 c -1 A

2 V 6c t + tc - 1 ,2 +-----sec h

1 c -1 A

2 V 6ct + t (18)

2

where t = x -2 1 V25 + 2 4 a2

t and c = — !-----------2 10

1 V25 + 24a2 10v j

u2 ( x.( x, t ) = c - 1 ----6a c -15 6c

tanh1 c - 12 V 6c t + t0 + -

c -1sec h 2

1 c -1

2 \ 6ct + 4 (19)

2

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where % = x —f 1 V 2 5 + 24a2

2 1O1 V 2 5 + 2 4 a2

t and c = 2 1O

/ \ 1 6 a 1 — cI x, t ) = c — 1---- . ---- tanhV 1 5 V 6c

1 1 — c A

2 V 6c £ + &c — 1

2sec h

1 1 — c2 V 6c

(20)

1 V 2 5 — 24a22 1O

t and c = — 2

24aİO

u4 ( x.( x, t ) — c — 1 —6a İ — c

tanh1 1 — c2 V 6c Ï + &

c —İ2

sec h21 1 — c2 V 6c Ï + & (21)

1 V 2 5 — 24a2 1O

2 1 1 V 2 5 — 24a2t and c = --------------

2 1O

f xu5 ( x ,t ) = —1 — t a n h ----h£0 I,y 2a

(22)

where — tanhc_ 2

V c 1 y

THE GENERALIZED BENJAMIN-BONA-MAHONY-BURGERS (BBM B) EQUATIONS

Consider the Oskolkov-Benjamin-Bona-Mahony-Burgers(OBBMB) equation

u t — u xxt —a u xx + ß u x + ( g (u ) ) x = O (23)

where a is positive and J3 e R .

Case 1. g (u ) = uuxUsing the wave variable E = x — c t carries (23), then integrating this equation and considering the integration constant to be zero, we obtain

M A N A S Journal of Engineering, Volume 4 (Issue 1) © 2016 www.journals.manas.edu.kg29

B a M JE N



( ß - c ) U - a U '+ c U "+ U U ' = 0 (24)

According to step 4, balancing UU' and U" gives N=1. Therefore, the solutions of Eq.(24) can be written in the form

A G '\(25)U (£ ) = a o + a

G

where a and a are constants which are unknowns to be determined later. Substituting Eq.(25) and its derivatives into Eq.s(24) and equating each coefficient of (G'/G) to zero, we obtain a set of nonlinear algebraic equations for a ,a , and c.Solving this system using Maple, we obtain

c = P , a = 2c, a = P ^ + a ;

Using these values in Eq.(25) when X 2 — 4 a > 0, we obtain the hyperbolic solution:f f r r — r ~ ^ f r r — r ~ M

( x, t ) = a + ß y j Ä 2 - 4 a

tJ â 2 - 4 aZ + c2cosh

^ ä 2 - 4 aZ

fc cosh

■sJ ä 2 - 4 aZ + c2 sinh

<Jä 2 - 4 aZ

(26)

where % = x — c t .

When X 2 — 4 a < 0, we obtain the trigonometric solution:

u2 ( x, t ) = a + ß ^ J 4 a - Ä 2

y j4 a - Ä 2Z + c2 cos Z

C cos + a sinJ J

(27)

u

where Z = x - c t .

When X 2 — 4 a = 0, we obtain the rational solution:

u3 ( x.( x, t ) = a + 2 ßO j + c 2Z

(28)

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where Ç = x - c t .In particular, if ci * 0 and c2 = 0, then the solutions in Eq.s(26)-(27) become respectively:

U ! ( x, t ) — a + ß ^ Ä 2 - 4 ß tanhf

y ]À 2 - 4 ß\

f (29)

where f — x - c t and Ä - 4 ß > 0,

U ! ( x, t ) — a - ß y j 4 ß - Ä- À tan•y/4^-À2

f (30)

where Ç = x - c t and A 2 - 4^ < 0 .The same manner, if c = 0 and c2 * 0 , then the solutions in Eq.s(26)-(28) become respectively:

( n r-;—— N\U 2 ( x, t ) — a + ß< jÄ2 - 4^ coth • Ä - 4^

f (31)

where f — x - ct and Ä2 - 4^ > 0,

U 2 ( x ,t ) — a + ß^/4 ß - Äf

- À cotV 4 ^ - r

f (32)

where f — x - ct and À2 - 4 ß < 0,

w31 ( x, t ) —a + 2ß x - ß t

(33)

where À 2 - 4 ß — 0.

uCase 2. g ( u ) = y

Using the wave variable E = x - c t , then integrating this equation and considering the integration constant to be zero, we obtain

U( c - ß ) U + a U '- c U "- y — 0 (34)

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According to step 4, balancing U2 and U' ' gives N=2. Therefore, the solutions of Eq.(34) can be written in the form

U (£ ) = a 0 + a ^G j + a 2 ^G j , (35)

where a0, a and a2 are constants which are unknowns to be determined later. Substituting Eq.(35) and its derivatives Eq.s(12) into Eq.s(34) and equating each coefficient of (G'/G) to zero, we obtain a set of nonlinear algebraic equations for a0,a ,a2, A and c.Solving this system using Maple, we obtain

P J 25J32+24a2 , _ f c - p „ 12aSet 1. c = — h—----------- , A = + A 4 u a------, a, = -12c, a, = -\2cA----- ,

2 10 V 6c 2 1 5

a0 = c — 12c u — f t — A ;

p J2 5 p 2+24a2 - _ r c - P „ 12aSet2. c = — — —----------- , A = +A4u-\----— , an =-12c, a, =-12 cA -----,

2 10 V 6c ' 1 5

a0 = c — 12c u — f t — 6~ A ;

P J2 5 p 2- 2 4 a2 . _ r p - c „ 12aSet 3. c = — h—----------- , A = +A4u-\----- , a-, = -12c, a, = -12c/l----- ,

2 10 V 6c - 1 5

a = c — 12c u — f t — ~ ~ A ;

p J2 5 p 2- 2 4 a2 , _ f /?-c _ _ 12aSet4. c = ----------------, A = + A 4 u a----- , a, =-12c, a, = -12cl----- ,

2 10 V 6c ‘ 5

a = c —12c u —f t — A ;

- /?2Se t5. c = 0, A = +a|4>uh— 7 , a 2 =0, ^ = - 2 a ,a 0 = ~ P~ aA .V a" “

Using these values in Eq.(35) when A 2 — 4 j > 0, we obtain the hyperbolic solutions respectively:

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(

c —ß

2 5 V 6c

İ c — ßE

İ \c—Z

v 2 V 6 C Jİ c — ß

E + c2 sinh

+ c2 sinh İ İ c ßV 2 V 6c E j

(36)

where E = x —A

Z ■s]25ß2 + 24a2 İOV J

2 ß J 2 5 ß 2 + 24a t and c = ^- + -*— -------

2

( x, t ) =_ 3 ( c — ß ) 6a /c — ß

5 V 6c

q sinh c — ß6c

E

İO

+ c2 cosh İ c — ßE

q cosh İ c — ßE + q sinh İ c — ß

E

c —ßq sinh İ c — ß

E + q cosh İ c — ßE

V 2 V 6c J

q cosh İ c — ßE + q sinh İ c — ß

EV2 V 6c yy

(37)

2

2

2

2

2

where E = x — ß ^ 2 5 ß 2 + 24aİO

21 ß J 2 5 ß 2 + 24a t and c = --- —-----------

2

y

u3 ( x.c — ß 6 a ß — c

2 5 6c

q sinh İ ß — cE

İO

+ c cosh İ ß — cE

C cosh İ ß — cE

V 6c J+ c2 sinh İ ß — c

E

+c —ß

( ( C sinh İ

2 VZ —c

6c Ez'

+ c cosh İ

2 Vß —c

6c E

C cosh İ ß — cE + c2 sinh İ ß — c

E

(38)

2

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where Z = x -ß ^ 2 5 ß 2 - 24a2 10v J

2 ß J 2 5 ß 2 - 24at and c = -- + -----------

2 10

2

(

/ \ c - ß 6 a ß - cu4 ( x, t ) = --------- J -----

4V ' 2 5 V 6c

(1 ß - c

AZ

(+ c2 cosh 1 ß - c

Z

C cosh 1 ß - cZ

v 2 V 6c j

+ c2 sinh 1 ß - cZ

v 2 y 6c j j

+c - ß

( 12 V

ß - c6c

Z + c2cosh 1 ß - cZ

(C cosh 1 ß - c

Zv 2 V 6c J

+ c2 sinh 1 ß - cZ

v 2 V 6c J

(39)

2

where Z = x -ß ^/25ß2 - 24a

102 ß J2 5 ß 2 - 24a

t and c = ^ --3 — t--------2

J10

u5 ( x( x t ) = - ß - ßc sinh i x I + c2 cosh | x

v 2 a J v 2 a y

c cosh i x I + c2 sinh | x v v 2 a J v 2 a j j

(40)

In particular, if we take c ^ 0, c2 < c1 , then Eq.s(36)-(40) lead the formal solitary wave solutions to Eq.(23) as

u1 ( x t ) = c - ß -6 a c - ß f

5 V 6ctanh 1 c - ß

2 \ 6cZ + Z0 + c - ß 72---- sec h

2

f1 c - ß2 \ 6c

Z + Z0 (41)

where Z = x -ß ■s/25ß2 + 24a2 10v J

2 ß J 2 5 ß 2 + 24at and c = — + !■— - -------

2 10

2

i \ r, 6 a lc - ß ( x, , ) = c - ß - tanh 1 c - ß

2 \ 6c Z + Z0c - ß

sec h2 1 c - ß2 \ 6c

Z (42)

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where £ = x - ß _ 4 25ß 2 + 2 4 a ' 2 10

"Aß J 2 5 ß 2 + 2 4 a2

t and c = --- —------------2 10

u.t \ r, 6 a I ß - c ( x ,t ) = c - ß - T ^ - tanh 1 ß - c

2 \ 6c £ + £ 0c - ß sec h 2 1 ß - c

2 \ 6c£ (43)

where £ = x -ß ^ 2 5 ß 2 - 2 4 a22 10v

f

u,

ß J 2 5 ß 2 - 2 4 a2t and c = -- + -----------

2 10

/ \ r, 6 a I ß - c(x ,t ) = c - ß - ^ ^ / ^ 6 c - tanh 1 ß - c

2V 6c

A£ + £0

c - ßsec h 2 1 \ß—

2 \ 6c£ (44)

where £ = x -ß _ ^ 2 5 ß 2 - 24a 2 10

2 ß J2 5 ß 2 - 24at and c = --- —-----------

2 10

2

( x, t ) = - ß - ß ta n h — x + £ß2 a

(45)

where £0 = tanh 1V c 1 J

CONCLUSIONS

In this paper, we implemented the (G'/G) expansion method to solve some nonlinear pseudoparabolic Benjamin-Bona-Mahony-Burgers equations and obtained new solutions which could not be obtained in the past. Besides, we have seen that the (G'/G) expansion method is applied successfully and reliable to solve not only for the class of nonlinear evolution equations but also for the a class of nonlinear pseudoparabolic and Sobolev-type equations.

We have seen that three types of travelling wave solutions were successfully found, in terms of hyperbolic, trigonometric and rational functions. It will be more important to seek solutions of higherorder nonlinear equations which can be reduced to ODEs of the order greater than 2. We have noted that this method changes the given difficult problems into simple problems which can be solved easily. The method yields a general solution with free parameters which can be identified by the above conditions in section 2. Moreover, some numerical methods like the Adomian decomposition method and homotopy perturbation method depend on the initial conditions and obtain a solution in a series which converges to the exact solution of the problem. However,it is obtained by the (G'/G) expansion method a general solution without approximation and there is no need to apply the initial and boundary conditions at the outset. The (G'/G) expansion method is also a standard, direct and computerizable method, which allows us to solve complicated and tedious algebraic calculation. The solution procedure can be easily implemented in Mathematica or Maple.

u

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REFERENCES[1] M.J.Ablowitz, "H.Segur, Solitons and the inverse scattering transform", SIAM, Philadelphia, Pa, USA,

1981.

[2] M.R.Miura, "Backlund transformation", Springer, Berlin, Germany, 1978.

[3] C.Rogers, W.F. Shadwick, "Backlund transformations", Academic Press, New York, NY, USA, 1982.

[4] V.B.Matveev, M.A.Salle, "Darboux transform and Solitons", Springer, Berlin, Germany, 1991.

[5] R.Hirota, "The direct method in soliton theory", Cambridge University Press, Cambridge, 2004.

[6] §.Akgagil, O.F.Gozukizil, "The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions", Advances in Difference Equations 143, 2013.

[7] W.Malfiet, "The tanh method: a tool for solving certain classes of nonlinear PDEs, Mathematical Methods in the Applied Sciences" , Vol 28, no.17, pp. 2013-2935, 2005.

[8] C.T.Yan, "A simple transformation for nonlinear waves", Physics Letter A 224, 77-84,1996.

[9] J.H.He, X.H.Wu, "Exp-function method for nonlinear wave equations", Chaos,Solitons and Fractals, Vol.30, no.3, pp. 700-708, 2006.

[10] Z.Y.Yan, H.Q.Zhang, "New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water", Physics Letter A 285, 355-362,2001.

[11] M.L.Wang, "Exact solutions for a compound KdV-Burgers equation", Physics Letter A 213, 279287,1996.

[12] F.Tascan, A.Bekir, M.Koparan, "Travelling wave solutions of nonlinear evolution equations by using the first integral method", Commun. Nonlinear Sci. Numer. Simul. 14 1810-1815, 2009.

[13] Z.S.Feng, "The first integral method to the two dimensional Burgers-Korteweg-de-Vries equation", Physics Letter A 308, 173-178, 2003.

[14] M.L.Wang, X.Li, J.Zhang, "The (frac(G',G)-expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics", Physics Letter A 372, 417-423, 2008.

[15] E.M.E.Zayed, "Travelling wave solutions for higher dimensional nonlinear evolution equations using the (G'/G)-expansion method", Journal of Applied Mathematics and Informatics, Vol. 28, pp. 383-395, 2010.

[16] A.J.M.Jawad, M.D.Petkovic, A.Biswas, "Modified simple equation method for nonlinear evolution equations", Applied Mathematics and Computation, Vol.217, no.2, pp. 869-877, 2010.

[17] E.M.E.Zayed, "A note on the modified simple equation method applied to Sharma-Tasso-Olver equation", Applied Mathematics and Computation, Vol. 218, no.7, pp. 3962-3964, 2011.

[18] E.M.E.Zayed, S.A.H.Ibrahim, "Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method", Chinese Physics Letters, Vol.29, no.6, Article ID060201, 2012.

[19] M.Wang, X.Li, J.Zhang, "The (G'/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics", Physics Letter A 372, 417-423, 2008.

[20] A.Bekir, "Application of the (G'/G)- expansion method for nonlinear evolution equations", Physics Letter A 372, 3400-3406, 2008.

BaMJEN36




[21] J.Zhang, X.Wei, "A generalized (G'/G)-expansion method and its applications", Physics Letter A 372, 3653-3658, 2008.

[22] I.Aslan, T.Ozi§, "Analytical study on two nonlinear evolution equations by using (G'/G)-expansion method", Appl. Math. Comp. 209, 425-429, 2009.

[23] E.M.E.Zayed, "The (G'/G)-expansion method and its application to some nonlinear evolution equations", J. Appl. Math. Comp. 30, 89-103, 2009.

[24] E.M.E.Zayed, K.A.Gepreel, "Some applications of the (G'/G)-expansion method to nonlinear partial differential equations", Appl. Math. Comp. 212, 1-13, 2009.

[25] A.Borhanifar, A.M.Zamiri, "Application of the (G'/G)-expansion method for the Zhiber-Shabat equation and other related equations", Math. Comp. Model. 549 (9-10), 2109-2116, 2011.

[26] A.Quarteroni, "Fourier spectral methods for pseudo-parabolic equations", SIAM J.Numer.Anal.24(2),323-335, 1987.

[27] Korpusov, MO,Sveshnikov, "AG: Blow-up solutions of strongly nonlinear equations of pseudoparabolic type", J.Math.Sci. 148 (1), 1-142, 2008.

[28] SA.Dubey,"Numerical solution for nonlocal Sobolev-type differential equations", Electron J. Differ.Eq.Conf. 19, 75-83, 2010.

[29] El.Kalkina, Pl.Naumkin, lA.Shishmarev, "The Cauchy Problem for an equation of Sobolev-type with power non-linearity", Izv.Math. 69 (1), 59-111, 2005.

[30] G.Karch, "Asymtotic behaviour of solutions to some pseudoparabolic equations", Math.Methods Appl.Sci.20,271-289, (1997).

[31] Peregrine, "DH: Calculations of the development of an undular bore", J.Fluid Mech. 25, 321-330, 1996.

[32] T.B.Benjamin, J.L.Bona, J.J.Mahony, "Model equations for long waves in nonlinear dispersive systems", Philos.Trans.R.Soc.Lond.Ser.A 272, 47-78, 1972.

[33] H.Tari, D.D.Ganji, "Approximate explicit solutions of nonlinear BBMB equations by He's methods comparison with the exact solution", Physics Lett. A 367, 95-101, 2007.

[34] §.Akgagil, O.F.Gozukizil,"Exact solutions of Benjamin-Bona-Mahony-Burgers-type nonlinear pseudo-parabolic equations", Boundary Value Problems 144, 2012.

B aM JEN37




Oz:

Xn+ı =ı y n

x « - ı x «y n+ı = ■

ı x „

Iy n~i y n J Maksimumlu Fark Denklem Sisteminin Çözümleri

Dağıstan Ş İM ŞEKKırgızistan-Türkiye Manas Üniversitesi, Bişkek, Kırgızistan; Selçuk Üniversitesi, Konya, Türkiye

[email protected] Mustafa ERÖZ

Sakarya Üniversitesi, Sakarya, Türkiye [email protected]

Burak OĞUL Kırgızistan-Türkiye Manas Üniversitesi, Bişkek, Kırgızistan

burak [email protected]

Recei ved: 15.04.2016; Accepted: 26.05.2016

Aşağıdaki fark denklem sisteminin çözümlerinin periyodikliği ve davranışları incelenmiştir.

X n+ı =ı y „

x « - ı x «y « +ı =

ı x „

y « - ı y « (1)

Başlangıç şartları pozitif reel sayılardır.

Anahtar Kelimeler: Fark Denklemi, Maksimum Operatörü, Yarı Dönmeler, Periyodiklik

xn+ı =i— \ ; y«+ı = ı— ,-x«-ı x« J I y«-ı y« Solutions O f The System O f Maximum Difference Equations

Abstract: The behaviour and periodicity of the solutions o f the following system of difference equations is examined

X n+i = {— , —} ; y n+i = \ — , — } I Xn-1 X n J I yn -1 ^ n J (1)

where the initial conditions are positive real numbers.

Keywords: Difference Equation, Maximum Operations, Semicycle, Periodicity_____________________

E a M JE N M A N A S Journal of Engineering, Volume 4 (Issue 1) © 2016 www.journals.manas.edu.kg





ŞİMŞEK, vd., Maksimumlu Fark Denklem Sisteminin Çözümleri

GİRİŞ

Son zamanlarda, lineer olmayan fark denklemlerinin periyodikliği ile ilgili ilginç çalışmalar yapılmaktadır. Özellikle fark denklem sisteminin periyodikliği, pozitif ve negatif yarı dönmeleri gibi çözümlerin davranışları incelenmektedir. Birçok araştırmacı, son yıllarda özellikle maksimumlu fark denklemleri ve maksimumlu fark denklem sistemleri ile ilgili araştırma yapmışlardır. Örneğin [1-29].

Tanım 1 :

Xn+1 = f ( x n , x n-1,■■■, X n s ) n = 0,1,2, ... için (2)

fark denkleminde x = f (x,.„, x) oluyorsa x ye denge noktası denir.

Tanım 2 : x (2) denkleminin pozitif bir denge noktası olsun. (2) denkleminin bir {x n } çözümünün bir

pozitif yarı dönmesi {x t , x l x m } terimlerinin bir dizisinden oluşur ve bunların hepsi x denge

noktasına eşit veya büyük bütün terimlerdir. Öyle ki l > 0 ve m < ro olur ve burada ya l = 0 ya da l > 0

ve x l_l < x ; ve, ya m = ro ya da m <<x> ve xm+1 < x ve xm+1 < x dir.

Tanım 3: x (2) denkleminin negatif bir denge noktası olsun. (2) denkleminin bir {x n } çözümünün bir

negatif yarı dönmesi {x t , x l+1v.., x m } terimlerinin bir dizisinden oluşur ve bunların hepsi x denge noktasından daha küçük terimlerdir. Öyle ki l > 0 ve m <ro olur ve burada ya

l = 0 ya da l > 0 ve xM > x veya m = ro y a da m < ro ve x m+1 > x dir.

Tanım 4 : Eğer {x n } dizisi için x = x n ise, {x n } dizisi p periyotludur denir ve p bu şartı sağlayan en

küçük pozitif tam sayıdır.

ANA SONUÇLAR

V ı = | — ,— I; .y„+ı = | — , M (1)I x„-1 J l yn-1 yn JŞimdi (1) denkleminin pozitif denge noktasını bulalım.

x = maxl 1 , y I; y = maxJ X I olur. Buradanl x x J l y y J

- 1 - y - 1 - x (~\2 (~\2x = — veya x = — ; y = — veya y = — elde edilir. \x) = 1 ve \y ) = 1 bulunur. Buradan da

x x y y

x = 1 ve y = 1 elde edilir.

B a M JEN39




Lemma 1 :

1 < xA <x0 <y_ <y0,1 < x <x_ <y_1 <^,1 <x <y_1 <y < x_ı,1 < x <y_! <x_, <y0,1 < x_ < x0 <y_, <y0,

1 < x_1 <y_1 < X0 <y0,1 < x_1 < X0 <y <y_1,1 <y_1 < x < x_1 <y,,1 <y_ <x_1 < x <y0,1 <x <y_ < x_1 <y,,

1 < x0 < x_1 <y_ <y ^ İ< x_1 < x0 <y_1 <y ^ İ <x_1 <y_1 < x0 <y ^ İ <y_1 <x < x_1 <y ^ İ <y_1 < x0 <y < x_1,

1 <y_1 < x_1 < x0 <y^1 < x0 <x_1 <y_1 <y^1 <x0 <y_1 <y < x_1,1 < x0 <y_1 <x_1 <y ,Yukarıdaki başlangıç şartları için aşağıdakiler doğrudur : n > 2 için xn çözümleri ve n > 1 için y n

çözümlerindea) Her negatif yarı dönme bir terimden oluşur.b) Her pozitif yarı dönme üç terimden oluşur.c) Bir uzunluğundaki her negatif yarı dönmeyi üç uzunluğundaki pozitif yarı dönme takip eder.d) Üç uzunluğundaki her pozitif yarı dönmeyi bir uzunluğundaki negatif yarı dönme takip eder.

İspat : xw, y N çözümü N >0 ve 1 < x0 < y _x < y < x_j için aşağıdaki gibi elde edilir.

y = max •

y 2 = max •

y = max <

x4 = max •

y = max •

1 y ,1 = ^ x xx_1 , x0 J x0

i , y0 ,i = * < y

y 0

A } = max \1 x2 1

y02 J

xl

y0< x

V x1 J 1x0 ,

, x }y 1 }J

= max < y l , x02,

1 z i1 x02

x y^0 1y

1- }

= max <x 0 y0414 i> = Z îl x x

xı , x2 J 1y o , x0 J x04

A } = max \ y0. x l, A

L y 0 . x yy 2 J 1x0 : y 0 ,1 x0

~ }= max \

2020

xÇ_ , ^

2020II x x

x*2 ^x3 J 1 y0 ,

1 = max <\ 4

WÂ , 31 = y x y

y 2, y3 J x03J x0

x 4 X x , x 5 xx , x 6 < x ,

3

1 2 3 7 8x 9 > x , x lo < x

B a M JEN40




görüldüğü gibi x n çözümleri PNPPPNPPPNPPPNPPPN... şeklinde devam eder.

Yi < y, y2 > y, Ys > y, Y4 > y, Ys < y Ya > y, Yt > y, Ys > y, Y9 < y, Yig > yBuradan görüldüğü gibi Yn çözümleri NPPPNPPPNPPP... şeklinde devam eder.

Görüldüğü üzere n > 2 için xn çözümleri ve n > 1 için y n çözümlerinde; her negatif yarı dönme bir terimden oluşur. Her pozitif yarı dönme üç terimden oluşur. Bir uzunluğundaki her negatif yarı dönmeyi üç uzunluğundaki pozitif yarı dönme takip eder. Üç uzunluğundaki her pozitif yarı dönmeyi bir uzunluğundaki negatif yarı dönme takip eder.

Lemma 2:i < y_1 < yG < -_1 <*0,1 < y_1 < -_1 < yG < xG,1 < y_1 < yG < < x_1,1 < < y_1 < yG < xG,1 < < yG < y_1 < xG,1 < x_1 < yo < -0 < y_1,1 < yo < -o < y_1 < x_1,1 < yo < -o < x_1 < y_1,1 < yo < < -o < y_1,1 < yo < y_1 < x_1 < -g,1 < y0 < -_1 < y_1 < -0,1 < y0 < y_1 < -_1 < -0,1 < y0 < y_1 < -_1 < - 0,1 < y0 < y_1 < - 0 < -_1,1 < y_1 < y0 < -_1 < -0,

1 < y_1 < -_1 < y0 < - 0,1 < -_1 < y_1 < y0 < -0,1 < -_1 < y0 < y_1 < -0,1 < -_1 < y0 < - 0 < y_1,1 < y0 < y_1 < - 0 < -_1

1 < yo < -_1 < y_1 < -o,1 < yo < y_1 < -_1 < -o,1 < yo < y_1 < -o < -_1,1 < y_1 < yo < < -o,1 < yo < < -o < y_1,

1 < yo < *1 < y_1 < -o,Yukarıdaki başlangıç şartları için aşağıdakiler doğrudur : n > 1 için -n çözümleri ve n > 2 için y n

çözümlerinde

a) Her negatif yarı dönme bir terimden oluşur.b) Her pozitif yarı dönme üç terimden oluşur.c) Bir uzunluğundaki her negatif yarı dönmeyi üç uzunluğundaki pozitif yarı dönme takip eder.d) Üç uzunluğundaki her pozitif yarı dönmeyi bir uzunluğundaki negatif yarı dönme takip eder.

İspat : Lemma 2 nin ispatı Lemma 1 in ispatına benzer şekilde elde edilir.

Teorem 1:1 < -_1 < -o < y_1 < y o , 1 < -_1 < y_1 < -o < y , , 1 < -_1 < -o < yo < y_1,1 < y _ < -o < < yo,

1 < y_1 < -o < yo < -_1,1 < y_1 < < -o < yo ,1 < -o < -_1 < yo < y_1,1 < -o < yo < -_1 < y_1,

1 < -o < yo < y -1 < - _ 1,1 < -o < y _1 < yo < - _ 1,1 < -o < y _1 < < y 0, 1 < -o < < y A < yo

Yukarıdaki başlangıç şartları için

- -Iyo _1 y L - X • y - I ^ yO. y. v- n | , ••• f • y n | ’ 2’ ’ 0, *** [ -0 -0 -0 J l yo -0 -0

veya

-y o o y o y o

’ 2 y o

■; yn =■. y o

2 3y o y o y o

çözümleri elde edilir.

o 0 g 0 0 0

B a M JEN41




İspat :

x = max1 y i . = 2 0

x-1 !' x0 J x0

^ 1> = ^0.y-1 ’ y0 J y0

1 ¥ i.x ’ x2

1 x2y ’ y2

1 1x2 x31 x y 4 = max 1 —

^ 2 ys

y 3 = max

= max

= max

= max

= max

1 4 1_ 1

x0 ’ y0 J x0

y f lL = y Ly '’ x02 J1 x02

x0 y L l . = y0-y0 ’ x0 J x0

y 1>=y0x0 '’ y02 J x0

x0 4y0 J

x0

\x _U 02

' y 0

x x

İterasyon işlemine devam edilirse

x J * . l y [ x l ; y yL?± y Xx n | , x0, ■■■ X ; y n 1 ’ 2 ’ , y 0, ■■■ X[ x0 x0 x0 J l y0 x0 x0 J

çözümleri elde edilir. İkinci çözümde benzer şekilde elde edilir.

Teorem2:1 < x-1 < x0 < y -1 < y 0,1 < x-1 < y -1 < x0 < y 0,1 < y -1 < x0 < x-1 < y 0,1 < y -1 < x0 < y 0 < x-15

1 < y -1 < x-1 < x0 < y 0,1 < x0 < x-1 < y -1 < y 0,1 < x0 < y -1 < y 0 < x-151 < x0 < y -1 < x-1 < y 0 Yukarıdaki başlangıç şartları için

f y e 1 1 f 1 y ^ . y - 1 1 xn = 1 — , y ^ . y - 1 ’ x 0 ’ - r ; yn = 1 — >--------->y - ^ y 0 ’ - Xx 0 x 0 J l y -1 x 0


B a M JEN42




İspat: Teorem 2 nin ispatı Teorem 1 e benzer şekilde elde edilir.

Teorem 3:1 <y_1 <y0 < x_1 < x0,1 <y_1 < x_1 < y0 < x0,1 <y_1 <y0 < x0 < x_p1 < x_1 < y_1 <y0 < x0 1 < x_1 <y0 <y_1 < x0,1 < x_1 < y0 < x0 < y_p1 <y0 < x0 <y_1 < x_p1 <y0 < x0 < x_1 < yA.

1 < y0 < x_1 < x0 < y_1,1 < y0 < y_1 < x_1 < xq,1 < y0 < x_1 < y_1 < xq,1 < y0 < y_1 < x0 < x_1


y ,, 2 , , x0,

x0 y0 y0

x0 1 x0

y 0 y 0 y 0

veyaf 2 3 1 Ç 2 4 2 1

y I x c _ x <l L v = J y o . x q l x q l Ixn 1 5 2 5 5 3 7 . . . | ; yn 1 5 2 5 4 5 2 ’ — I1 x 0 y 2 q 2 q J 1 2 q x 0 2 q 2 q J


İspat: Teorem 3 ün ispatı Teorem 1 e benzer şekilde elde edilir.

Teorem 4:1 < y_1 < yo < x_1 < xq,1 < y_1 < x_1 < y < xq,1 < x_ < y_ < y < x(l,1 < x < y < y_ < xq,

1 < x_ < y < x, < y_1,1 < y < x_ < x, < y_1, 1 < y < x < y < x(l,1 < y < y^ < x_ < x,


f 1 x _ x 1 [ x , 1 1x n =1 --- x_15 ^ . l ; yn = 1 5 x_1 x05 2 o 5 ^ }1 x_1 2 q J 1 2q 2q J


İspat: Teorem 4 ün ispatı Teorem 1 e benzer şekilde elde edilir.

ÖRNEKLER

Örnek 1: Başlangıç şartları Teorem 1 dekine uygun bir şekilde seçilirse

x[-1] = 2; x[0] = 3; y[-1] = 4; y[0] = 7

x(n)= {2.33333,0.333333,16.3333,3, 2.33333,0.333333,16.3333,3, 2.33333,0.333333,16.3333,3,..}

y(n)= {0.428571,5.44444,2.33333,7, 0.428571,5.44444,2.33333,7, 0.428571,5.44444,2.33333,7,...} çözümleri elde edilir ve çözümlerin grafikleri aşağıda gösterilmiştir.

B a M JEN43




x[n[

yJnl

n

n

B a M JEN44





x[-1] = 2; x[0] = 3; y[-1] = 4; y[0] = 5 x(n) = {1.66667,0.36,7.71605,2.77778, 1.66667,0.36,7.71605,2.77778, 1.66667,0.36,7.71605,2.77778, ...}

y(n) = {0.6,2.77778,1.66667,4.62963, 0.6,2.77778,1.66667,4.62963, 0.6,2.77778,1.66667,4.62963, ...} çözümleri elde edilir ve çözümlerin grafikleri aşağıda gösterilmiştir.

xjnl

y.n.

n

n

B a M JEN45





x[-1] = 2; x[0] = 4; y[-1] = 3; y[0] = 13 x(n)= {3.25,0.25,39,4, 3.25,0.25,39,4, 3.25,0.25,39,4, 3.25,0.25,39,4, ...} y(n)= { 0.333333,9.75,3,13, 0.333333,9.75,3,13, 0.333333,9.75,3,13, ...}

çözümleri elde edilir ve çözümlerin grafikleri aşağıda gösterilmiştir.

x.n.

lııi

n

y

n

B a M JEN46





x[-1] = 4; x[0] = 3; y[-1] =5; y[0]=2 x(n)= {0.666667,2.25,1.5,3, 0.666667,2.25,1.5,3, 0.666667,2.25,1.5,3, ...}

y(n)= {1.5,0.5,4.5,2, 1.5,0.5,4.5,2, 1.5,0.5,4.5,2, 1.5,0.5,4.5,2, ...} çözümleri elde edilir ve çözümlerin grafikleri aşağıda gösterilmiştir.

x j n j

lııi

m M JEN47


n

y

n




x [- l] = 6; x[0] = 5; y [- 1] = 7; y [0] = 4 x(n)= {0.8,1.5625,1.25,1.95313, 0.8,1.5625,1.25,1.95313, 0.8,1.5625,1.25,1.95313, ...}

y(n)= {1.25,0.64,2.44141,1.5625, 1.25,0.64,2.44141,1.5625, 1.25,0.64,2.44141,1.5625, ...} çözümleri elde edilir ve çözümlerin grafikleri aşağıda gösterilmiştir.

x[ıı[

Şekil 9. x(n) çözümlerinin grafiği

yjnj

m M JEN48


n

n




x[-1] = 4; x[0] = 13; y[-1] = 2; y[0]= 3 x(n)= {0.25,17.3333,4,13, 0.25,17.3333,4,13, 0.25,17.3333,4,13, ...}

y(n)= {4.33333,0.333333,52,3, 4.33333,0.333333,52,3, 4.33333,0.333333,52,3, ...} çözümleri elde edilir ve çözümlerin grafikleri aşağıda gösterilmiştir.

x[ıı[

yi nJ

m M JEN49


n

n



TARTIŞMA VE SONUÇ

Bu çalışmada, x_1;x ,;y_ 1;y q başlangıç şartları birden büyük pozitif reel sayılar olmak üzere,

incelenmiştir. Bu fark denklem sisteminde katsayıları değiştirilerek yeni maksimumlu fark denklem sistemleri oluşturulabilir. Oluşturulacak yeni maksimumlu fark denklem sisteminin çözüm davranışları incelenebilir.

Bu makale, Akdeniz Üniversitesinde 7-9 Eylül 2015 tarihinde düzenlenen XXVIII. ULUSAL MATEMATİK SEMPOZYUM' unda sunulmuştur.

[1] A. M. Amleh, "Boundedness Periodicity and Stability of Some Difference Equations", PhD Thesis, University of Rhode Island, 1998.

[2] C. Çinar, S. Stevic and İ.Yalçınkaya, "On the positive solutions of reciprocal difference equation with minimum", Journal of Applied Mathematics and Computing, 17, (1-2), 307-314, 2005.

[3] S. Elaydi, "An Introduction to Difference Equations", Spinger-Verlag, New York, 1996.[4] E. M. Elsayed and S. Stevic, "On the max-type equation x_{n+1}=max{A/x_{n},x_{n-2}}", Nonlinear

Analysis, TMA 71, 910-922, 2009.[5] E. M. Elsayed, B. Iricanin and S. Stevic, "On the max-type equation x_{n+1} = max{A_{n}/x_{n},x_{n-

1}}", ARS Combin., 2010.[6] J. Feuer, "Periodic solutions of the Lyness max equation", Journal of Mathematical Analysis and

Applications, 288, 147-160, 2003.[7] A. Gelişken, C. Çinar and R. Karataş, "A note on the periodicity of the Lyness max equation",

Advances in Difference Equations, Vol. (2008), Article ID 651747, 5 pages, 2008.[8] A.Gelişken, , C. Çinar and İ. Yalçınkaya, "On the periodicity of a difference equation with maximum",

Discrete Dynamics in Nature and Society, Vol. 2008, Article ID 820629, 11 pages, 2008.[9] A. Gelişken , Çinar, C. and A.S. Kurbanlı, "On the asymptotic behavior and periodic nature of a

difference equation with maximum", Computers & Mathematics with Applications, 59, 898-902,2010.

[10] B. Iricanin and E. M. Elsayed, "On a max-type equation x_{n+1}=max{A/x_{n},x_{n-3}}", Discrete Dynamics in Nature and Society, Vol. 2010, Article ID 675413, 2010.

[11] M. R. S. Kulenevic and G. Ladas, "Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture",Boca Raton, London, 2002.

[12] D. P. Mishev, W. T. Patula, and H. D. Voulov, "A reciprocal difference equation with maximum", Computers & Mathematics with Applications, 43, 1021-1026, 2002.

[13] L. A. Moybe, "Difference Equations with Public Health Applications", New York, USA, 2000.

_____J 1 y n _ 4 |, f 1 x j[14] EypaK Ory^, flarbicraH WMMweK, " x n+1 = max , K y n+1 = max^----, n 4 [■ CwcreMa

I x n _4 x n _4 J [ y n_4 y n_4 JpemeHMe pa3Hocmoro ypaBHeHMfl", BecHMK Kbiprbi3CKoro rocyflapcraeHHoro TexHMMecKoro yHMBepcMTeTa, N 34, EMWKeK, Kbiprbi3craH, 2015.

maksimumlu fark denklem sisteminin çözümlerinin davranışları

KAYNAKLAR

B a M JEN50




[15] B. Ogul, D. Simsek, " xn+1 ~ max1 xn-4

y n-4 y n-4Maksimumlu Fark

Denklem Sisteminin Çözümleri", Manas Journal of Engineering, 3(1): 35-57, 2015.

[16] G. Papaschinopoulos and V. Hatzifilippidis, "On a max difference equation", Journal of Mathematical Analysis andApplications, 258, 258-268, 2001.

[17] G. Papaschinopoulos, J. Schinas and V. Hatzifilippidis, "Global behaviour of the solutions of a max- equation and of a system of two max-equation", Journal of Computational Analysis and Applications, 5, 2, 237-247, 2003.

[18] W. T. Patula and H. D. Voulov, "On a max type recursive relation with periodic coefficients", Journal of Difference Equations and Applications, 10, 3, 329-338, 2004.

[19] G. Stefanidou and G. Papaschinopoulos, " The periodic nature of the positive solutions of a nonlinear fuzzy max--difference equation", Information Sciences, 176, 3694-3710, 2006.

[20] S. Stevie, "On the recursive sequence x_{n+1}=max{c,x_{n}A{p}/x_{n-1}A{p}}", Applied Mathematics Letters, vol. 21, No: 8, 791--796, 2008.

[21] D. Simsek, C. Cinar and I. Yalçınkaya, "On the solutions of the difference equation x_{n+1}=max{1/x_{n-1},x_{n-1}}, International Journal of Contemporary Mathematical Sciences, Vol.1, No: 9--12, 481--487, 2006.

[22] D. Simsek, B. Demir and A. S. Kurbanlı, "x_{n+1}=max{(1/(x_{n})),((y_{n})/(x_{n}))}, y_{n+1}=max{(1/(y_{n})),((x_{n})/(y_{n}))} Denklem Sistemlerinin Çözümleri Üzerine", Ahmet Keleşoğlu Eğitim Fakültesi Dergisi, 28, 91-104, 2009.

[23] D. Simsek, B. Demir and C. Cinar, "On the Solutions of the System of Difference Equations x_{n+1}=max{(A/(x_{n})),((y_{n})/(x_{n}))}, y_{n+1}=max{(A/(y_{n})),((x_{n})/(y_{n}))}", Discrete Dynamics in Nature and Society, Volume 2009, Article ID 325296, 11 pages, 2009.

[24] Simsek D., Kurbanlı A. S., Erdoğan M. E. , "x(n+1) = max{1 \ x(n-1) ; y(n-1) \ x(n-1)} ; y(n+1) = max{1 \ y(n-1) ; x(n-1) \ y(n-1)} Fark Denklem Sisteminin Çözümleri", XXIII. Ulusal Matematik Sempozyumu, 153 pp, .04-07 Ağustos 2010, Erciyes Üniversitesi, 2010.

[25] Dağıstan Şimşek and Ahmet Dogan , "Solutions Of The System Of Maximum Difference Equations", Manas Journal of Engineering, 2(2): 9-22, 2014.

[26] Dağıstan Şimşek and Mustafa Eröz, " x(n+1) = max {1/x(n-1), y(n)/x(n-1) }, y(n+1) = max {1/y(n-1), x(n)/y(n-1) } Maksimumlu Fark Denklem Sisteminin Çözümleri",, Manas Journal of Engineering, 3(2)

[27] H. D. Voulov, "On the periodic character of some difference equations", Journal of Difference Equations and Applications, 8, 799-810, 2002.

[28] I. Yalçinkaya, B. D. Iricanin and C. Çinar, "On a max-type difference equation", Discrete Dynamics in Nature and Society, Vol. 2007, Article ID 47264, 10 pages, 2007.

[29] I. Yalçinkaya, C. Çinar and M. Atalay, "On the solutions of systems of difference equations", Advances in Difference Equations, Vol. 2008, Article ID 143943, 9 pages, 2008.

2015.

B a M JEN51




The Tanh-Coth Method for Two System of Sobolev Type Equations In Mathematical Physics

Şamil A K Ç A Ğ ILŞeyh Edebali University, Faculty of Economics and Administrative Sciences, Bilecik, Turkey.

[email protected] Ömer Faruk GÖZÜKIZIL

Sakarya University, Department of Mathematics, Sakarya, Turkey. [email protected]

Recei ved: 19.04.2016; Accepted: 06.05.2016

Abstract:

Keywords:

Sobolev equations have been used to describe many physical phenomena and they are characterized by having mixed time and space derivatives appearing in the highest-order terms o f an partial differential equation. In this work we consider two important system o f Sobolev type equations namely improved Boussinesq and higher-order improved Boussinesq. B y using tanh-coth method, we obtain abundant new travelling wave solutions o f these important physical structures.Sobolev equation; The tanh-coth method; The system o f improved Boussinesq equations; The system o f higher-order improved Boussinesq equations

Matematiksel Fizikteki Sobolev Tipi İki Denklem Sistemi için Tanh-CothYöntemi

Öz: Sobolov denklemleri en yüksek mertebeden türevinde zaman ve boyuta göre türevleri beraberbulunduran denklemler olarak tanımlanır. Bu çalışmada, Boussinesq ve yüksek mertebeden geliştirilm iş Boussinesq adlı iki önemli Sobolev denklem sistemini ele aldık. Tanh-coth yöntemi kullanarak bu iki önemli denklem sisteminin bir çok yeni hareketli dalga çözümünü elde ettik.

. , „ , Sobolev denklemi; Tanh-coth metot; Geliştirilm iş Boussinesq denklem sistemi; YüksekAnahtar Kelimeler: , , , ■ i n ■ + ■_______________________ mertebeden geliştirilm iş Boussınesq denklem sistemi.________________________________





AKQAGIL and GOZUKIZIL, The Tanh-Coth Method for Two System o f Sobolev Type Equations In MathematicalPhysics

INTRODUCTION

The Tanh-Coth Method for Two System of Sobolev Type Equations in Mathematical Physics

We have focused on travelling wave solutions of the system of two improved Boussinesq equations.

The system of improved Boussinesq equations

The system of improved Boussinesq equations

uxxtt — u tt + u xx + ( f l (u , w )) xx = 0 (1)

Wxxtt - Wtt + Wxx + ( f 2( U, w ) ) xx = 0

where and f 2 are given nonlinear functions, u (x , t ) and w (x , t ) are unknown functions and subscripts x and t indicate partial derivatives, has been derived to describe bidirectional wave propagation in several study. For instance, a transversal degree of freedom is introduced in the Toda lattice model and the initial value problem for the system (1) have been studied. Travelling wave solutions and numerical solutions of the system are obtained [1]. Khusnutdinova et al. have studied nonlinear longitudinal waves in a two- layered structure with a soft bonding layer using a relatively simple long wave model in the form of coupled Boussinesq-type equations [2]. Wattis has used (1) to find waves of general speed in a diatomic lattice by using the quasi-continuum method of approximation [3]. In [4], Godefroy has studied (1) as the Cauchy problem under certain conditions and showed that the solution for the Cauchy problem of this system blows up in finite time. Wang and Li have considered the Cauchy problem for (1) and proved the existence and uniqueness of the global solution and given sufficient conditions of blow-up of the solution in finite time by convex methods [5]. Rosenau has studied transversal degree of freedom for the propagation of non-linear wave in Toda lattice via the system (1) [6]. Sergei has analysed the system and stated that the solution of the system would blow up if there is a non-positive Hamiltonian [7]. Pego et al. have studied the stability of solitary waves of two coupled Boussinesq equations which model weakly nonlinear vibrations in a cubic lattice [8].

The system of higher-order improved Boussinesq equations

P u xxxxtt + a u xxtt u tt + u xx + (g 1(u ,w ) ) XX= 0 (2)

~fiw xxxxtt + a w xxtt — w tt + w xx + ( 9 2 ( U, w )) xx = 0

where g ± and g 2 are given nonlinear functions, have been considered and blow-up results were obtained in [9]. Schneider and Wayne have shown that in the longwave limit the water wave problem without surface tension can be described approximately by single component form of equations (2) [10]. Duruk et al. have proved in [11] that the Cauchy problem for the single component form of (2) is globally well-posed in Sobolev spaces H s for s > 1/2 under certain conditions on nonlinear term and initial data.In this paper, we focused on travelling wave solutions of the system of two improved Boussinesq equations

B a M JEN53




^xxtt ^ tt + ^xx + ( 'U'W) XX 0 (3)Wxxtt ^ tt + WXX + ( u w ) xx 0

and the system of higher-order improved Boussinesq equations

P^XXXXtt + xxtt ^ tt + ^XX + (M'W~) Xx 0 (4)

P^xxxxtt + MWXxtt ^ tt + WXx + ( UW^)Xx 0

where a and ft are positive constants, u ( x , t ) and w ( x , t ) are unknown functions. We described outline of the tanh-coth method in following section and derived various exact travelling wave solutions of these physical structures in section 3 and 4 by using tanh-coth method. Finally, we summarized our conclusions in section 5.

OUTLINE OF THE TANH-COTH METHOD

Wazwaz has summarized the tanh method in the following manner:

i. First consider a general form of nonlinear equation

P (U ,U^,U x ,U xx , •••) °.

ii. To find the traveling wave solution of Eq. (5), the wave variable ( — x + y + z + ••• — V t is introduced so that

u (x ,t ) — U (^ ^ ). (6)

Based on this one may use the following changes

d d — — —y — d t d%

d d

(7)92 2 d2

~dx2 = ^ d f 2 d 3 „ d 3

- V 3-d x 3 * d% 3

and so on for other derivatives. Using (7) changes the PDE (5) to an ODE

Q (U ,U ',U " , . . . ) — 0 (8)

iii. If all terms of the resulting ODE contain derivatives in then by integrating this equation, and by considering the constant of integration to be zero, one obtains a simplified ODE.

iv. A new independent variable

era M JEN54




Y = t a n h (^ ) (9)

is introduced that leads to the change of derivatives:

(10)

where other derivatives can be derived in a similar manner.

v. The ansatz of the form

(11)

is introduced where M is a positive integer, in most cases, that will be determined. If M is not an integer, then a transformation formula is used to overcome this difficulty. Substituting (10) and (11) into the ODE(8) yields an equation in powers of Y.

highest order nonlinear terms are balanced. With M determined, one collects the all coefficients of powers of Y in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the a k and bk, (k = 0 ,..., M ), V, and ¡â . Having determined these parameters, knowing that M is a positive integer in most cases, and using (11) one obtains an analytic solution in a closed form.

THE SOLUTIONS OF THE SYSTEM OF IMPROVED BOUSSINESQ EQUATIONS

The system of improved Boussinesq equations is given by

^XXtt ^ tt + ^XX + (uw^xx 0 (12)

Wxxtt Wtt + Wxx + ( u w )xx °.

Using the wave variable % = x — V t then by integrating this equation twice, and considering the constants of integration to be zero, the system (12) is carried to a system of ODEs

vi. To determine the parameter M , the linear terms of highest order in the resulting equation with the

V 2U " - V 2U + U + U W = 0 (13)

55M A N A S Journal of Engineering, Volume 4 (Issue 1) © 2016 www.journals.manas.edu.kg



V 2W " - V 2W + W + U W = o.

Balancing V 2U " with U W and V 2W " with U W in (13) gives

2M = M + N (14)

2N = M + N ,

so that

M = N.(15)

We take M = N = 2 and consider solutions in the form

2 2(16)

2 2

Substituting (16) into the two components of (13), and collecting the coefficients of Y gives two systems of algebraic equations for a 0, a t, a 2, b t, b2, c0, Ci, c2, d t, d2, V and

6 a2V 2 2 + C2a2 = 0,6c2V2^2 + C2a2 = 0,2aiV2^2 + Cl(X2 + C2at = 0,2c1V 2 2 + c t a2 + C2at = 0

a2 — 0.2V 2 — 8a2V2^2 + CoU2 + C ia i + C2a0 = 0 ,C 2 - C2V 2 — 8c2V2 2 + c0a2 + C ia i + C2a0 = 0

a i — a iV 2 — 2 a iV 2 2 + diÜ2 + biC2 + Coai + Ciao = 0 ,C i — C iV 2 — 2 c iV 2 2 + diÜ2 + biC2 + Coai + Ciao = 0

ao — V 2ao + d iü i + d2Ü2 + b iC i + b2C2 + Coao + 2 V 2b2^2 + 2V 2a2^2 = 0

Co — V 2Co + d ia i + ba2 + b iC i + b2C2 + Coao + 2V 2d2^2 + 2 V 2C2^2 = 0

b i — b iV 2 — 2biV2^2 + d iao + b a i + biCo + b2Ci = 0

d i — d iV 2 — 2 d iV 2 2 + d iao + d 2 a i + biCo + b2Ci = 0

b2 — b2V2 — 8b2V2 2 + d ib i + d2a.o + b2Co = 0

d2 — d2V2 — 8d2V2 2 + d ib i + d2&o + b2Co = 0

2 b iV 2 2 + d2bi + dib2 = 0 ,2 d iV 2 2 + d2bi + dib2 = 0

6b2V2^2 + d2b2 = 0,6d2V2^2 + d2b2 = 0 (17)

B a M JEN56




Solving these systems leads to the following sets:

a 0 — 6^ 2,b 2 — d 2 — —6^ 2,c 0 — 2^ 2,V — ± 1 (18)2y 2 6y.2 1

&Q — Cq — --~--- , b2 — d 2 — n----, V — ± 1 r\\0 0 4^ 2 + 1 ’ 2 2 4^ 2 + 1 ± V V + I (19)

aQ — 2y ? ,b 2 — d 2 — —6y?,C o — 6^ 2,V — ± 1 (20)6 ^ 2 6^ 2 1 1

ao — Co — — - 2-- T ,b 2 — d 2 — — ^-- - ,V — ± — (21)V — 1 4^ 2 — 1 ^ —4yi 2 + 1 2

ao — 6^ 2,a ,2 — c 2 — —6^ 2,Co — 2^ 2,V — ± 1 (22)2y.2 6 ^ 2 1

a o — Co — , a 2 — C2 — — . J , „ , V — ± ^ = (23)4 v 2 + 1 4^ 2 + 1 V V + T ( )

6y.2 6y.2 1 1a o — Co = —W = ~ ı ’ a2 = C2 — 4H 2— ¡■v = ± V = 4; j s + i M < 2 (24)

4\i2 6\i2 1ao — Cq — — , A ,a .2 — b2 — C2 — d 2 — — 2 ^ . , V — ± - (25)

16\i2 + 1 16\i2 + 1 V 1 6 p 2 + 1ao — —4^ 2,a2 — b2 — C2 — d 2 — —6^2,Cq — 12^ 2,V — ± 1 (26)

12\i2 12\i2 1 <%o — Co — — — —=— —, a 2 — b2 — C2 — d 2 — — — —5— —,V — ±- 1 1

16^2 - 1 1 6 ^ — 1 ’ - ^ - 1 6 ^ + 11

(27)

< 4

where ^ is left as a free parameter and a t = b t = c t = d^ = 0. Consequently, we obtain the following travelling wave solutions:

u ±(x , t ) = 6^2 — 6^2co th 2 {x + t ) t ) = 2^2 — 6^2coth2^{x + t )

2y.2 6^2 ( _ 1U2(x ’ 0 = A.,2 ■ 1 - A „2 ■ 1 COth2V I * +

(28)

4n + 1 4n + 1 \ ^ 4 ^ 2 + 1

2y 2 6y.2 ( _ 14 iJ T Ï - 4i22 + 1 COtk +W2(X, t ) = 1 COth2V \ X + . J l (29)

u 3(x , t ) = 2\i2 — 6^2co th 2^ (x + t ) w 3(x , t ) = 6^2 — 6y2coth2y.(x + t )

6y.2 6y.2 ( _ 1 \ 1u 4(x , t ) = —^ ^ -- 7 + ^ -- - m th 2^ l x + —— t \ t h i]< —

(30)

4^ 2 — 1 4^ 2 — 1 \ ^ —4^2 + 1 y 26u2 6u2 _ / _ 1 \ 1 (31)

w A{x ,t ) = —— 2-- ~ + — 2-- - c o th 2v [ x + = t l , ^ 1 < - ( )4y 2 — 1 4y 2 — 1 \ ^ —4pf2 + 1 ) 2

u 5(x , t ) = 6y.2 — 6^2ta n h 2^ (x + t ) w 5(x, t ) = 2^ 2 — 6^2ta n h 2^.(x + t ) (32)

B a M JEN57




2^2 6^2 ( _ 1 u 6(x , t ) = — =— - — — =— - ta n h 2ß I x T ■

4H2 + 1 4ß 2 + 1 \ ^ 4 [ l2 + 1 2^ 2 6y.2 ( _ 1

w 6(x ’ ( ) = + (33)

6y.2 6y.2 ( _ 1 \ 1i 7 ( x , t )= —— ^-- - + r- ^-- ~ ta n h 2/ u ix + = t ] , I^ İ < -

4^ 2 — 1 4^ 2 — 1 \ J —4pL2 + 1 ) 26y.2 ( 6y.2 \ / _ 1 \ 1

W 7 (x ,t) = — — ;— t + ^ t ~2— T ]t a n h 2n [x + = t ] , In I < - (34)4V 2 — 1 \4^ 2 — 1) V ^ —4 ß 2 + 1 J 2

4 ß 2 6y.2

16ß 2 + 1 16ß 2 + 1 T J1 6 ^ 2 + 1 " ) ' * \ C T J1 6 ^ 2 + 1u8(x, t) = — ^^ \ , t — A \ i -, {tanh2 [ x T t ^---- 1. ] + coth2 [ x T

4^2w 8(x, t ) = — „ _ mt2 | a

2ta n h 2^ I x + --- 1 ] + co th 2ß I x T

16 [i2 + 1e f < i 1 \ / 1 (35)

16P 2 + 1 t V J1 6 [ i2 + 1 / V -J16p2 + 1

u 9(x , t ) = —4^2 — 6^2{tanh2p,(x T t ) + coth¡2^(x T t ) }

w 9(x , t ) = 12^2 — 6/j2 [tanh2iJ.(x T t )+ co th 2 ß (x T t )

(36)

12y.2 ( ( _ 1 \ n i - 1 X ) 1 u 10 (x , t ) = — 7— - {1 + ta n h 2^ I x + t ] + co th 2^ I x + £ ] } , İMİ < 7

16^ 2 — 1 l \ ^ —16^ + 1 ) V J —16n2 + 1 ) ) ’ 4

12\i2 ( , / 1 w 10 (x , t ) = — 7— - {1 + ta n h 2ß I x T ■

16V 2 — H V J —16n2 + 11 1

+ co th 2u I x T t. ] } , |u| < -V 4 —16^2 + 1 n 4

(37)

The system of higher-order improved Boussinesq equations

The system of higher-order improved Boussinesq equations reads:

ß^XXXXtt + ^^XXtt ^ tt + ^XX + (M'W')xx O (38)

—ß w xxxxtt + a w xxtt — w tt + w xx + (u w ^xx = O,

where a and ft are positive constants, u (x , t ) and w (x , t ) are unknown functions. Using the wave variable % = x — V t then by integrating twice, considering the constants of integration to be zero we find

—ß V 2U (4) + a V 2U " — V 2U + U + U W = O (39)

—ß V 2W (4 + a V 2W " — V 2W + W + U W = O.

t

B a M JEN58




Balancing the linear term of highest order with the nonlinear terms, we find

M + 4 = M + N (40)

N + 4 = M + N

so that M = N = 4. This gives the solutions in the form:

4 4

(41)

4 4

Substituting (41) into the two components of (39), and collecting the coefficients of Y gives two systems of algebraic equations for a t ,b t ,(i = 0 ,... ,4), V and ^:

b4a4 — 840V2a4p^4 = 0, b4a4 — 840V2b4p^4 = 0, b3a4 — 360a3p V 2 4 + b4a3 = 0,

b3a4 — 360b3@V2 4 + b4a3 = 0

b2a4 + b3Ü3 + b4Ü2 + 20V2 a4a ^2 + 2080V2 a4P .4 — 120V2 a2p^4 = 0

b2a.4 + b3Ü3 + b4Ü2 — 120V2b2^^4 + 20V2b4<xn2 + 2080V2b4fiy4 = 0

b2Ü3 + b3Ü2 + b4Üt + btÜ4 + 12V2Ü3a^ 2 + 816V2a3^^4 — 24V2a 1fiy4 = 0

b2Ü3 + b3Ü2 + b4Üt + btÜ4 + 12V2b3ap2 + 816V2b3fiy4 — 24V2b1p^4 = 0

a4 — V2ü4 + b2Ü2 + b3at + b4a0 + b0Ü4 + b^ 3 — 32V2a4<xn2 — 1696V2a4^^4 + 6V2Ü2a^ 2 + 240V2a2^^4 = 0

b4 — V2b4 + b2Ü2 + b3at + b4a0 + b0Ü4 + b^ 3 + 6V2b2a^2 + 240V2ap^4 — 32V2b4ap2 — 1696V2b4^^4 = 0

Ü3 — V2ü3 + b2a t + b3a0 + b0Ü3 + bta2 — 18V2a 3 < x — 576V2a3P^4 + 2V2a 1a ^2 + 40V2a 1p^4 = 0

b3 — V2b3 + b2a t + b3a0 + b0a3 + b ^ — 18V2b3a^2 — 576V2b3P^4 + 2V2b1a^2 + 40V2b1fiy4 = 0

a2 — V2ü2 + b2a 0 + b0a 2 + btüt + 12V2a4ay2 + 480V2a4^^4 — 8V2Ü2a^2 — 136V2a2P^4 = 0

b2 — V2b2 + b2a0 + b0a2 + b ^ t — 8V2b2&n2 — V36V2b2P^4 + 12V2b4a^ 2 + 480V2b4fiy4 = 0

a i — V2a-i + b0at + b ta0 + 6V2a3a^2 + 120V2ü3P^4 — 2V2a 1a^2 — 16V2a1fiy4 = 0

bt — V2b1 + b0at + b ta0 + 6V2b3<xn2 + 120V2b3fiy4 — 2V2b1a ^2 — 16V2btP^4 = 0

a0 — V2a0 + b0a 0 — 24V2ü4P^4 + 2V2a 2<x 2 + 16V2a2^^4 = 0

b0 — V2bo + boa.o + 2V2b2a^ 2 + 16V2b2^^4 — 24V2b4fiy4 = 0

B a M JEN59




Using Maple or Mathematica we get the following four solutions:

1 0 5 a2 1 0 5 a2 3 3 a 2 1ao — a 4 — b4 — ---- 7T,a2 — b2 — --------,bo — --- - ,V — ± 1 ,u — ±---

338/3 169ft 338ft ~ 26

ao — bo —

V — ±13

3 3 a 22 (3 6 a 2 + 169ft)

,a2 — b2 — —1 0 5 a2

3 6 a 2 + 169ft,a 4 — b

N

1 0 5 a2

13a

ft

3 6 a 2 + 169ft’

ft3 6 a 2 + 1 6 9 ft’ ^ ± 2 6 _

13a

ft

3 3 a 2 1 0 5 a2 1 0 5 a2a 0 — n , a 2 — b2 — — „ rn n , a 4 — b4 — — ——, V338ft 169ft 338ft ± '- tl — ± 2b

13a

ft

(42)

(43)

(44)

(45)

Q-o — U4. — bo — b4 — —1 0 5 a2 1 0 5 a2

-,0.2 — b2 —2 (3 6 a 2 — 169 ft) 3 6 a 2 — 169ft’

± 1 3 V —(3 6 a 2 — 169 ft ) ft t 1V — 3 6 a 2 — 169ft , ^ — ± 2 6

13a

ft- ,3 6 a2 < 169ft

where a ± — a 3 — b± — b3 — 0. As a result we find following solutions:

1 0 5 a2( x , t ) —

w x ( x , t ) —

338ft

3 3 a 2 1 0 5 a2+

338ft 338ft

{1 — 2 ta n h 2y.(x + t ) + ta n h 4^ (x + t ) }

{ —2 ta n h 2 (x + t ) + ta n h 4^ (x + t ) }

(46)

(47)

U 2 (x ,t) —3 3 a2 1 0 5 a2

2 (3 6 a 2 + 169 ft) 3 6 a 2 + 169ft

1 0 5 a2

ta n h 2^ ( x + 13 ft3 6 a 2 + 169ft

+3 6 a 2 + 169ft

ta n k 4 ( x + 13 ft3 6 a 2 + 169ft

W 2 (x ,t) —3 3 a2

+ —1 0 5 a2

2 (3 6 a 2 + 169 ft) 3 6 a 2 + 169ft

1 0 5 a2

ta n h 2^ ( x + 13 ft3 6 a 2 + 169ft

+3 6 a 2 + 169ft

ta n k 4 ( x + 13 ft3 6 a 2 + 169ft

(48)

1

B a M JEN60




3 3 a 2 1 0 5 a2 r _ . _U3 (X’ l ) = 338/3 + 169/? { - tan h ^ (x + t ) + ta n h 4^ (x + t ) } (49)

3 3 a 2 1 0 5 a2 _W 3(X ’ l ) = ^ 0 /3 + 169$ { - ta n h 2^ (x + t ) + ta n h ^ (x + t ) }

1 0 5 a2 1 0 5 a2 _ J ( —3 6 a 2 +169/3)13 \Ua ( x , t ) = ------- =--------- 1-----=------- ta n h 2u ( x + 1 3 ------- =---------- 1 )

2 (3 6 a 2 — 169 f i ) 3 6 a 2 — 169fi ( 3 6 a 2 — 169fi ) (5 0 )

1 0 5 a2 ( J ( - 3 6 ü 2 + Ï6 9 p jp \ta n h 4^ ( x + 1 3 ---——~— ---1 ), 169 p

2 (3 6 a 2 — 169f i) \ 3 6 a 2 — 169fi

> 3 6 a 2

1 0 5 a2 1 0 5 a2 f _ J ( —3 6 a 2 + 169/3)/3

W4(X ’ ( ) = — 2 (3 6 a 2 — 1 6 9 fi) + 36a2 — 1 6 9 fit a n h ^ [ X + 13 36a 2 — 169/3 ‘

1 0 5 a2 . _ J ( —3 6 a 2 + 169/3)/3 \ _ ---------- ta n h 4^ ( x + 1 3--- ——=— ---1 ), 169 > 36a

2 (3 6 a 2 — 169 f i) V 3 6 a 2 — 169/3

Moreover, we obtained some complex solutions and will not be considered these solutions in this work.

CONCLUSION

In this article, with aid of the Maple and Mathematica, the tanh--coth method has been successfully implemented to find new traveling wave solutions for two Sobolev type system of equations, namely, the system of improved Boussinesq equations and the system of higher-order improved Boussinesq equations and exact solutions are obtained. The solutions show that the tanh-coth method is a powerful mathematical tool for solving Sobolev type partial differential equations.

REFERENCES[1] P. L. Christiansen, P. S. Lomdahl and V. Muto, "On a Toda lattice model with a transversal degree of

freedom", Nonlinearity, vol. 4, pp. 477-501, 1991.[2] K. R. Khusnutdinova, A. M. Samsonov and A. S. Zakharov, "Nonlinear layered lattice model and

generalized solitary waves in imperfectly bonded structures", Physical Review E, pp. 1-14, vol. 79, 2009

[3] J. A. D. Wattis, "Solitary waves in a diatomic lattice: analytic approximations for a wide range ofspeeds by quasi-continuum methods", Physics Letters A, vol. 284, pp. 16-22, 2001

[4] A. De Godefroy, "Blow up of solutions of a generalized Boussinesq equation", IMA Journal of AppliedMathematics, vol. 60, pp.123-138,1998

[5] S. Wang and M. Li, "The Cauchy problem for coupled IMBq equations", IMA Journal of AppliedMathematics, vol. 74 ,pp. 726-740, 2009

[6] D. Rosenau, "Dynamics of dense lattice", Phys. Rev. B, vol.36, pp. 5868-5876, 1987

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[7] K. T. Sergei, "On a Toda lattice model with a transversal degree of freedom, sufficient criterion ofblow-up in the continuum limit", Phys. Lett.A, vol.173, pp. 267-269, 1993

[8] R. L. Pego, P. Smereka and M.I.Weinstein, "Oscillatory instability of solitary waves in a continuummodel of lattice vibrations", Nonlinearity, vol. 8, pp. 921-941, 1995

[9] N. Duruk, H. A. Erbay, and A. Erkip, "Blow-up and global existence for a general class of nonlocalnonlinear coupled wave equations", J. Differential Equations, vol. 250, pp. 1448-1459, 2011

[10] G. Schneider and C. E. Wayne, "Kawahara dynamics in dispersive media", Physica D, vol. 152, pp. 384-394, 2001

[11] N. Duruk, A. Erkip and H. A. Erbay, "A higher-order Boussinesq equation in locally nonlinear theory of one-dimensional nonlocal elasticity", IMA Journal of Applied Mathematics, vol. 74, pp. 97-106, 2009

62M A N A S Journal of Engineering, Volume 4 (Issue 1) © 2016 www.journals.manas.edu.kgB a M JEN



Determination of Escherichia coli O157:H7 in Chicken Meats Sold in Sanliurfa Region

Ahmet Hulusi D IN£O GLU ,Mehmet Akif Ersoy University, Faculty of Health Sciences, Department of Nutrition and Dietetics,

Burdur,TURKEY [email protected]

Zafer G O N U LA LA NErciyes University, School of Veterinary Medicine, Department of Food Hygiene and Technology,

Kayseri,[email protected]


Abstract: Escherichia coli O157:H7 has been an important problem ofpublic health in most countries o f the world since 1982. This study was therefore aimed to investigate the presence o f E . coli O157:H7 in chicken meat samples collected from various markets in Sanliurfa region which was located in Southern Turkey. Fo r this purpose, 155 chicken meat samples were analyzed between September 2005 and February 2006. The samples were plated onto Cefxime Tellurite Supplement and Sorbitol M ac Conkey Agar after enrichment process. Suspected colonies were then analyzed fo r identification o f E. coli O157:H7 as given in materials and methods section. E. coli O157 and E. coli O157:H7 were found in 9 (5.81%) and 3 (1.94%) o f the total o f 155 samples, respectively. The results showed that control measures should be developed to prevent contamination with this pathogen in chicken meats in this region. To our knowledge, this is the first report o f isolation o f E. coli O157:H7 from chicken meat samples in Southern Turkey.

Keywords: Chicken meat, cultural method, E. coli O157:H7, prevalence, public health

Şanlıurfa Bölgesinde Satılan Tavuk Etlerinde Escherichia coli O157:H7’nin Tespiti

Öz: Escherichia coli O157:H71982’den beri dünyanın birçok ülkesinde önemli bir halk sağlığıproblemi olmuştur. Bu nedenle, bu çalışma Türkiye ’nin güneyinde yer alan Şanlıurfa bölgesindeki çeşitli marketlerden toplanan tavuk eti örneklerinde E. coli O157:H7’nin varlığını araştırmak amacıyla yapıldı. Bu amaçla, 2005 Eylü l ile 2006 Şubat döneminde toplanan 155 adet tavuk eti örneği analiz edildi. Zenginleştirme işleminden sonra örnekler Cefixime Tellurite Supplement içeren Sorbitol M ac Conkey Agara aktarıldı. Daha sonra şüpheli koloniler E. coli O157:H7’nin identifıkasyonu için materyal ve metot kısmında belirtildiği gibi analiz edildi. Toplam 155 örneğin 9 ’unda (%5.81) E. coli O157 ve 3 ’ünde (%1.94) E. coli O157:H7 tespit edildi. Bu sonuçlar, bu bölgedeki tavuk etlerinin bu patojenle kontaminasyonunu önlemek için kontrol önlemlerinin artırılması gerektiğini gösterdi. Bildiğimiz kadarıyla, elde ettiğimiz sonuçlar Türkiye’nin güney bölgesindeki tavuk eti örneklerinden E. coli O157:H7’nin izole edildiği ilk rapordur.

Anahtar Kelimeler:_____ Tavuk eti, kültürel metot, E. coli O157:H7, prevalans, halk sağlığı____________________

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DlNQOGLU and GONULALAN, Determination o f Escherichia coli O157:H7 in Chicken Meats Sold in SanliurfaRegion

INTRODUCTION

E. coli O157:H7 is one of the well-known and the most serious bacterial agent among the food borne pathogens [1]. This was first recognized as a cause of illness in 1982 when it caused two major outbreaks of hemorrhagic colitis traced to the consumption of hamburgers in the USA [2]. Undercooked hamburgers from the same fast food restaurant chain were identified as the vehicle, and E. coli O157:H7 was isolated from patients and a frozen ground beef patty [3]. Such outbreaks increased dramatically and became widespread in the following years and this bacteria has become one of the most important foodborne pathogens. E. coli O157:H7 infections have continued to occur in large outbreaks and sporadic cases, although outbreaks were decreased after 1999 [4]. It has been estimated that E. coli O157:H7 causes 73,000 illnesses and 250 deaths annually in the United States [5].This pathogenic microorganism causes hemolytic uremic syndrome (HUS), hemorrhagic colitis (HC) and trombotic trombositopenic purpura (TTC). The pathogen is likely to be responsible for 85-95 % of hemolytic uremic syndrome cases [6, 7]. The natural reservoirs of pathogen are many kind of animals especially cattle, sheep, goats, and wild animals. Consumption of undercooked or contaminated foods of animal origin is often implicated in foodborne outbreaks of E. coli O157:H7. However, fecal contaminations of other food products or direct contact with infected animals have also been linked as routes of transmission for human illness [5, 8].Ground beef is the main source of these infections. Microbial contamination of raw and ready-to-eat (RTE) meat products with human pathogens is a consequence of a wide array of pre-harvest, harvest, and postharvest processes. E. coli O157:H7 can colonize the intestinal tract of cattle and other animals[9]. This microorganism was also detected in chicken caeca and feces and it has also been noted that chicken meat is a vehicle for transmission of E. coli O157:H7 to humans [10].It is highly probable that foods produced, stored and/or marketed under unhygienic conditions be contaminated with E. coli O157:H7. Since the foods sold in the market in the Southeastern Anatolia region are often produced by ignoring the basic hygienic rules, these foods are likely available for E. coli O157:H7 contamination. This eventually threats the health of the people living in this region.This study was carried out to determine the E. coli O157:H7 contamination level of chicken meats sold in the local butchers and the supermarkets, if any.

MATERIALS AND METHODS

Collection of Samples

During the six month period, 155 chicken meat samples were collected from the markets in Sanliurfa region located in Southern part of Turkey. The samples were taken between September 2005 and February 2006. After purchase, the samples in their original packaging were placed directly into cool boxes and transported to the laboratory within 2 h on the day of collection. Samples were stored at 4 °C in the laboratory prior to processing on the day following receipt. The samples were analyzed to determine if any E. coli O157:H7 serotype exist, according to the procedure proposed by Food and Drug Administration [11].

Enrichment and Isolation

Modified novobiocine EC Broth (mEC+n, Merck 14582, Berlin, GERMANY) was used as enrichment medium. Enrichment cultures for each sample were carried out by combining 25 g of each sample with 225 ml of EC Broth supplemented with 20 mg/l novobiocin (Novobiocin/ N1628, Sigma, GERMANY)

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sterile bag, homogenized for at least 2 min into a stomacher and incubated at 37 °C for 24 h. CT-SMAC (Cefixime-Tellurite Supplement and Sorbitol MacConkey Agar, Oxoid CM 813 and SR172 E, Basingstoke, UK) were used for as selective solid medium. A swap of the enrichment broth was then spread onto selective CT-SMAC and incubated at 42 °C for 24-48 h. At the end of the incubation, colorless, sorbitol negative (-), suspected colonies were streaked onto Fluorocult Violet Red Bile (VRB) Agar (Merck 1.04030, GERMANY) and these plates were incubated at 42 °C for 24-48 h aerobically.

Identification and Serological Confirmation

Colonies grown on VRB were checked under UV light. Gram stain and IMVIC tests were performed on suspected colonies. The colonies were then subjected to the agglutination test to determine the serotype of the bacteria using specific antisera to E. coli O157 (Oxoid, 200075, UK) and Dryspot E. coli O157 latex agglutination test (Oxoid, UK) for E. coli O157 carried out in parallel. Cultures identified as E. coli O157 were tested with antisera H7 (Oxoid, 211057, UK) as described by the manufacturer.

RESULTS

The results showed that 9 (5.81%) of 155 chicken meat samples were contaminated with E. coli O157 serotype. E. coli O157:H7 was detected in 3 (1.94%) of total samples. Distribution of those serotypes detected in chicken meats is shown in Table 1. A six months of chicken meat survey showed that E.coli O157:H7 serotype was not detected between November and February. Between the same months, E. coli O157 was detected only in December. The highest prevalence of E. coli O157 and E. coli O157:H7 were detected in September and October. The climate is warm from June to November in Sanliurfa region. Pathogen incidence is the highest level, contributing high warm weather at this season.

Table 1. Distribution of E. coli O157:H7 and E. coli O157 serotypes isolated in chicken meats according tomonths.

SampledMonths

SampleNumbers

Positive Sample Numbers for E. co li O157:H7

Positive Sample Numbers for E. co li O157

September 29 2 (6.90%) 5 (17.24%)October 27 1 (3.70%) 3 (11.11%)

November 28 - -December 24 - 1 (4.17%)January 25 - -February 22 - -TOTAL 155 3 (1.93%) 9 (5.81%)

DISCUSSION

Infection with E. coli O157:H7 has become emerging foodborne disease in developed countries. E. coli O157:H7 has been isolated from dairy cattle, calves, chickens, swine and even sheep and from their meat. However, its incidence and prevalence shows intensely instability because of different reasons [12, 13]. In these, chicken has been considered as vehicles of transmission of E. coli O157:H7, since chicks can be colonized by small populations of this pathogen and continue to be long-term shedders [14].There are a lot of studies to show importance of E. coli O157:H7 for public health from different countries concerning the incidence of E. coli O157:H7 on a variety of foods. Some of them show similarity to the prevalence found in our investigation while some researchers reported dissimilar results.

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One of these was carried out by Samedpour and Liston [15], 4 (12%) of the 33 chicken samples obtained from local grocery stores in the Seattle area were positive for E. coli O157:H7. In the other study [16] this pathogen was isolated from 3 (2%) of the 150 chicken giblets purchased from markets in Costa Rica. Researchers in this article pointed out cross contamination possibility.Contrasting Reuben et al. [16], Miri [17] reported E. coli O157:H7 wasn't isolated using microbiological culture and PCR from the 70 chicken nugget samples collected in Isfahan, Iran. Similarly, this bacteria wasn't isolated in 80 chicken samples examined by Soriano et al. [18]. Griffin and Tauxe [19] did not isolate this bacterium from raw chicken. Jo et al. [20] detected no E.coli O157 in 2843 different meat samples including 52 chickens by cultural method. Tutenel et al. [21] reported same prevalence for E.coli O157 in 241 chicken samples.El-Safey [22] detected no E. coli O157:H7 in 100 Austrian food samples, including 20 chicken by cultural and immunomagnetic separation (IMS) methods. In a study [23] on some Egyptian foods, 5 (21.7%) of 23 chicken samples were positive for E.coli O157:H7. These results are meaningful to show in countries having different hygienic conditions for this pathogen.Diseases occurring with E. coli O157:H7 aren't problem only in underdeveloped countries. This pathogen was detected from various meats and meat products in different countries both developed and underdeveloped [24, 25, 26, 27, 28, 29, 30].In our country, some other studies also reported the presence of this pathogen on different meat and meat products. Aksu et al. [31] examined 500 food samples for this bacteria in a study carried out to investigate the presence of E. coli O157:H7 in various foods of animal origin. Among the group of meat products, they determined E. coli O157:H7 in 3 (6%) out of 50 ground beef samples, 1 out of 25 ground lamb samples and 1 (2%) out of 50 meatball samples. Abay et al. [32] reported none of 400 chicken meat samples collected from various markets in Kayseri between March 2010 and February 2011 was positive for E. coli O157 via cultural method and PCR. E. coli O157:H7 and E. coli O157 were investigated from 330 meat samples (120 beef, 105 chicken's meat and 105 turkey's meats) sold in butchers and markets using cultural methods by Unsal [33] who found in 1 (0.9%) and 3 (2.7%) of 105 chicken samples respectively. Baran and Gulmez [34] examined 100 samples of animal origin (50 ground beefs and 50 chicken hams) for E. coli O157: H7 in a study carried out in Kars Province. Consequently, they isolated bacteria from 3 ground beef samples (6%), but they didn't isolate any bacteria in chicken samples.Akkaya et al. [35] determined that 2 (1.05%) out of 190 chicken meat samples were contaminated with E. coli O157:H7. In Afyonkarahisar, Akkaya et al. [36] took samples using swaps from four different points of 250 beef carcasses each. Prevalence of E. coli O157 and E. coli O157:H7 in the taken samples was found at 3.2% and 0.8%, respectively.Using the cultural method, Mercanoglu and Aytac [37] detected E. coli O157 in 1 (1.8) of 57 chicken samples collected in Ankara during 8 month period. In their study, Alisarli and Akman [38] investigated 300 ground meat samples (150 ground beef and 150 ground lamb) sold in butcher shops and markets in Van. They determined E. coli O157 serotype in 7 ground beef samples (4.66%) and 3 ground lamb samples (2%), and concluded that ground meat had high risks of this bacteria.Cebiroglu and Nazli [39] examined 115 hamburger and meatball samples sold in Istanbul for E. coli O157:H7 and isolated bacteria from 4 samples (3.47%). In the conclusion of the study, they determined that these foods had high risks for public health, and hygiene rules should be followed in every stage of production and consumption.Obtained data indicated that the chicken meats marketed under uncontrolled conditions have threatening effects on public health with regard to E. coli O157:H7 serotype. One should bear in mind that meat is always considered intrinsically unsafe regarding E. coli O157:H7 contamination. Therefore, adequate standards of hygiene must be observed and controlled to avoid the presence of this pathogen in meat. To our knowledge, this is the first report of isolation of these bacteria from chicken meat samples in Southern part of Turkey.

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REFERENCES

[1]. Fedio WM, Jinneman KC, Yoshitomi KJ, Zapata R, Wendakoon CN, Browning P and Weagant SD, Detection of E. coli O157:H7 in raw ground beef by Pathatrix™ Immunomagnetic-separation, realtime PCR and cultural m methods. Int J Food Microbiol, 148, 87-92 (2011).

[2]. Anonymous, Epidemiologic Notes and Reports Isolation of Escherichia coli O157:H7 from sporadic cases of hemorrhagic colitis, United States. CDC MMWR Weekly, 46(30), 700-704 (1997) http://www.cdc.gov/mmwr/preview/mmwrhtml/00048579.htm. [Accessed 25/11/2015]

[3]. Riley LW, Remis RS, Helgerson SD, McGee HB and Wells JG, Hemorrhagic colitis associated with a rare Escherichia coli Serotype. N Engl J Med, 308, 681-685 (1983).

[4]. Lim JY, Yoon J and Hovde CJ, A brief overview of Escherichia coli O157:H7 and its plasmid O157. J Microbiol Biotechnol, 20(1), 5-14 (2010).

[5]. Mead PS, Slutsker L, Dietz V, McCaig LF, Bresee JS, Shapiro C, Griffin PM and Tauxe RV, Food- related illness and death in the United States. Emerging Infect Dis, 5, 607-625 (1999).

[6]. Gordillo R, Cordoba JJ, Andrade MJ, Luque MI and Rodriguez M, Development of PCR assays for detection of Escherichia coli O157:H7 in meat products. Meat Sci, 88, 767-773 (2011).

[7]. Griffin PM, Escherichia coli O157:H7 and other enterohemorrhagic Escherichia coli. In "Infections of the gastrointestinal tract", Blaser MJ, Smith PD, Ravdin JI, Greenberg HB and Guerrant RL, ed., Raven Press Ltd., New York, 1995, pp. 739-761.

[8]. Kiranmayi B, Krishnaiah N, Naga Mallika E, Escherichia coli O157:H7 - An emerging pathogen in foods of animal origin. Vet World, 3(8), 382-389 (2010).

[9]. Buchanan RL and Doyle MP, Foodborne disease significance of Escherichia coli O157:H7 and other enterohemorrhagic E. coli. Food Technology, 51(10), 69-76 (1997).

[10]. Beery JT, Doyle MP and Schoeni JL, Colonization of Chicken Cecae by Escherichia coli Associated with Hemorrhagic Colitis. Appl Environ Microbiol, 49(2), 310-315 (1985).

[11]. FDA (Food & Drug Administration) 2001. Escherichia coli and the Coliform Bacteria. Chapter 4. Bacteriological Analytical Manuel Online.

[12]. Fu An-Hung, Sebranek J and Murano E, Survival of Listeria m onocylogenes, Yersinia enterocolitica and Escherichia coli 0157: H7 and quality changes after irradiation of beef steaks and ground beef. J of Food Sci, 60, 972-977 (1995).

[13]. Doyle MP and Schoeni J, Isolation of Escherichia coli O157:H7 from retail fresh meats and poultry. Appl Environ Microbiol, 53(10), 2394-2396 (1987).

[14]. Schoeni JL and Doyle MP, Variable Colonization of chicken plerorally inoculated with Escherichia coli O157:H7 and subsequent contamination of eggs. Appl Environ Microbiol, 60, 2958-62 (1994).

[15]. Samedpour M and Liston J, Occurrence of Shiga-like toxin producing Escherichia coli in retail fresh sea food, beef, lamb, pork and poultry from grocery stores in Seattle, Washington. Appl Environ Microbiol, 60, 1038-1040 (1994).

[16]. Reuben A, Treminio H, Arias ML and Villalobos L, Isolation of Escherichia.coli O157:H7 from Costa Rican food. Rev Biomed, 13, 273-276 (2002).

[17]. Miri A, Rahimi E, Mirlohi M, Mahaki B, Jalali M and Safaei HG, Isolation of Shiga toxin producing Escherichia coli O157:H7/NM from hamburger and chicken nugget. Int J of Environ Health Eng, 3(1), 19-23 (2014).

[18]. Soriano JM, Rico H, Molto JC and Manes J, Incidence of microbial flora in lettuce, meat and Spanish potato omelette from restaurants. Food Microbiol, 18, 159-163 (2001).

[19]. Griffin P and Tauxe R, The epidemiology of infection caused by Escherichia coli O157:H7, other enterohaemorrhagic E. coli and the associated haemolytic uraemic syndrome. Epidemiologic Reviews, 13, 60-98 (1991).

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http://www.cdc.gov/mmwr/preview/mmwrhtml/00048579.htm



[20]. Jo M, Kim JH, Lim JH, Kang MY, Koh HB, Park YH and Yoon DY, Prevalence and characteristics of Escherichia coli O157 from major food animals in Korea. Int J of Food Microbiol, 95, 41-49 (2004).

[21]. Tutenel AV, Pierard D, Van Hoof J, Cornelis M and De Zuttera L, Isolation and molecular characterization of Escherichia coli O157 isolated from cattle, pigs and chickens at Slaughter. Int J of Food Microbiol, 84, 63-69 (2003).

[22]. El-Safey EM, Incidence of Salmonella and E. coli O157:H7 in some Austrian foods. International Conferance of Food Microorganism, Lillehammar, Norway; 17-18 August, 2002, pp:415 (2002).

[23]. El-Safey EM and Abdul Raouf UM, Detection of Escherichia coli O157:H7 in some Egyptian foods. Assiust J of Agricultural Science, 34(6), 373-378 (2003).

[24]. Ahmed MA and Shimamoto T, Isolation and molecular characterization of Salm onella enterica, Escherichia coli O157:H7 and Shigella spp. from meat and dairy products in Egypt. Int J of Food Microbiol, 168-169, 57-62 (2014).

[25]. Bosilevac M, Gassem MA, Al-Sheddy IA, Almaiman SA, Al-Mohizea IS, Alowainer A and Koohmaraie M, Prevalence of Escherichia coli O157:H7 and Salm onella in camels, cattle, goats, and sheep harvested for meat in Riyadh. J of Food Prot, 78(1), 89-96 (2015).

[26]. El-Shrek YM and Ali MR, Microbiological study of spiced chicken burgers in Tripoli City, Libya. East Mediterr Health J, 18(6), 653-662 (2012).

[27]. Gjurgji A and Sulaj K, Occurrence of E. coli O157:H7 of ground meat samples collected from butcher shops in Tirana market. Albanian J Agric Sci, 12(3), 349-352 (2013).

[28]. Mohammed HO, Stipetic K, Salem A, McDonough P, Chang YF and Sultan A, Risk of Escherichia coli O157:H7, Non-0157 Shiga toxin-producing Escherichia coli, and Cam pylobacter spp. in food animals and their products in Qatar. J of Food Prot, 78(10), 1812-1818 (2015).

[29]. Osaili MT, Al-Nabulsi AA, Shaker RR, Jaradat ZW, Taha M, Al-Kherasha M, Meherat M and Holley R, Prevalence of Salm onella Serovars, Listeria m onocytogenes, and Escherichia coli O157:H7 in Mediterranean ready-to-eat meat products in Jordan. J of Food Prot, 77(1), 106-111 (2014).

[30]. Phillips D, Tholath S, Jenson I and Sumner J, Microbiological quality of Australian sheep meat in2011. Food Control, 31(2), 291-294 (2013).

[31]. Aksu H, Ozgen-Arun O and Colak H, Yoğurdun inkübasyonu ve soğuk muhafazasının Escherichia coli O157: H7 canlılığı üzerine etkisi. Pendik Vet Mikrobiyol Derg, 30(2), 71-75 (1999).

[32]. Abay S, Aydin F, Ertas N, Hizlisoy H, Erdogdu S and Gonulalan Z, Kanatlılardan Escherichia coli O157 izolasyonu üzerine çalışmalar. Erciyes Üniv Vet Fak Derg, 11(1), 1-6 (2014).

[33]. Unsal C, Erzurum bölgesinde satışa sunulan etlerde E. coli O157:H7'nin varlığının araştırılması. Yüksek Lisans Tezi, Erzurum, Türkiye. 2007.

[34]. Baran F and Gulmez M, The occurence of Escherichia coli O157:H7 in the ground beef and chicken drumsticks. Internet J of Food Safety, 2, 13-15 (2002).

[35]. Akkaya L, Atabay HI, Kenar B and Alisarli M, Prevalence of verocytotoxigenic Escherichia coli O157:H7 on chicken carcasses sold in Turkey. Bull Vet Inst Pulawy, 50, 513-516 (2006a).

[36]. Akkaya L, Cetinkaya Z, Alisarlı M, Telli R and Gok V, Afyonkarahisar ilinde sığır karkaslarında Escherichia coli O157/O157:H7, Listeria m onocytogenes ve Salm onella spp'nin varlığı. 2 nci Ulusal Veteriner Gıda Hijyeni Kongresi, İstanbul, Turkey, 18-20 Eylül (2006b).

[37]. Mercanoglu B and Aytac SA, Ankara bölgesinde satışa sunulan tavuk etlerinde Yersinia Enterocolitica ve Escherichia coli O157 varlığının araştırılması. Türkiye 9. Gıda Kongresi, Bolu, Turkiye, 24-26 Mayıs 2006.

[38]. Alisarli M and Akman HN, Perakende Satılan Kıymaların Escherichia coli O157 yönünden incelenmesi. YYÜ Vet Fak Derg, 15(1-2), 65-69 (2004).

[39]. Cebiroglu H and Nazli B, Dondurulmuş hamburger köfte ve diğer köfte çeşitlerinde enterohemorajik Escherichia coli O157:H7 suşunun varlığı üzerine araştırmalar. İstanbul Üniv Vet Fak Derg, 25(1), 107-121 (1999).

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Aibek BODO SHO V Kyrgyz-Turkish Manas University, Food Engineering Department, Bishkek, Kyrgyzstan

[email protected]

Received: 27-04-2016; Accepted: 23.05.2016

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Local monophonic white varieties of beans: some physical properties and especially the amino acid composition

Abstract: The aim of the work is to investigate some physical properties and amino acid composition o f lopatka,kitayanka, saharniy beans varieties. Investigation o f physical properties o f beans is essential for design of equipment fo r harvesting, processing, transportation, cleaning, sorting, separation and packaging. In this research some physical properties o f beans grains were studied such as shape, length, width and thickness o f grains, mass o f 1000 units and angle o f repose. The hardness, bulk density were 0.43 mm, 0.08 mm, 0.05 mm and 685.11 kg/m3, 719.60 kg/m3, 666.84 kg/m3 respectively. In the second stage of work was studied total content o f essential amino acids by high performance liquid chromatography. The investigation included sample preparation, derivatization and injection o f the sample. The total content o f essential amino acids in 100 g o f saharniy, lopatka, kitayanka beans varieties 11.88 g, 9.03 g, 7.31 g respectively. Methionine was the amino acid limiting the quality o f protein in the investigated samples. In addition the variety o f the kitayanka beans the limiting amino acids were leucine, lysine, threonine and phenylalanine.

Keywords:_____ Beans, monophonic bean varieties, protein, amino acid_________________________________________




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B a M JEN71



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MbiHga, A , y.ry - M3M.geHyyMy 6e.oKTyH İ r «e İ00 r KapMa.raH i - a.MamTbpb.rbC aMMHKMC.oTaCbiHbiH KoHU,eHTpau,MflCbi; A^m - «ugea.gyy 6e.oKTory» İ r «e İ00 r KapMa.raH i - a.MamTbpb.rbC aMMHKMC.oTaCbiHbiH KoHU,eHTpaU,MflCbl.«Mgea.gyy 6e.oKTory» ap 6up a.MamTbpb.rbC aMMHKUC.oTa yMyH aMMHKMC.oTa.WK CKop İ00% gen 6u.MHeT. Mbmgafi 6a.aHCTarb TaMaKTaHyy To.yK KaHgyy 6o.yn CaHa.aT. ^êpge, KaMCbi. 6up aMMHKMC.oTaHbH CKopy İ00% gaH a3 6o.Co 6y. aMMHKUC.oTa .MMMTTeHyyMy 6o.yn Ta6b.aT. Mbmgafi yMypga «eTumnereH aMMHKMC.oTaHbH ^ce6MHeH opraHM3MguH eCyyCy «afi «ypeT. A . smm aMMHKUC.oTa.bK CKop İ00% gaH amKaH yMypga ga opraHM3MguH CMûpyyCy HaMap.aMT [18-20 ]. HonaTKa, KMTaflHKa «aHa CaxapHbfi 6up e^Mefi aK CopTTory Tee 6yypMaK y.ry.epgyH «agb6a. 2 ge KepCeTy.reH a.MamTbpb.rbC aMMHKMC.oTa.bK KypaM e3reMe.yKTepyHyH Heru3MHge aMMHKMC.oTa.bK CKop KepCeTKyMTepy aHbiKTa.bin «agbi6a. 4 Te 6epu.gu.

Maflbi6a. 4. Wepm.MKTYY 6up e^nefi aK CopTTory óyypnaKTapgbrn aMMHKUC.oTa.NK CKop KaCueirepu.

TeeöyypnaK

AMMHKMC.OTa.MK CKOp, %

Ba.MH M30.e^MH .fleÛH HU3UH MeTUOHUH + „TpeOHUHÛCTeUH TpunT0$aH OeHu.a.aHUH+ TUpO3UH

HonaTKa 117,77 134,78 144,10 199,53 31,74 170,13 141,50 124,38KuTaflHKa 100,94 103,84 70,94 97,09 23,72 70,82 109,60 62,44CaxapHbü 126,48 141,56 162,31 235,89 52,01 185,23 138,42 266,67

Ba.MH, M3o.eMU,MH, TpMnToâH aMMHKMC.oTa.apb 3 y.ryge Terç aMMHKUC.oTa.bK CKopy İ00% gaH «oropy. HeMU,MH, . m3mh, TpeoHMH, $eHM.a.aHMH+Tupo3MH KMTaflHKa CopTyHgarb 6yypMaKTapga aMMHKMC.oTa.bK CKopy İ00% gaH a3 MbKKaH. MeTMoHMH+u,MCTeMH 6apgbK CopTTopgo aMUHKUC.oTa.bK CKopy İ00% gaH a3 Ta6b.raH. By. aMMHKMC.oTa 6yypMaKTyy.apga Ta6MraTb 6oroHMa Ken KapMa.6afiT, «aHb6ap TeKTyy a3bKTapga Ken.AMMHKMC.oTa.bK CKopgyH aübpMa.yy.yK Ko^^MU,MeHTM (KPAC, %) - .MMMTTeHyyMy aMMH KMC.oTa.bK CKopro KapaTa 6amKa aMMH KMC.oTa.apgbH optomo aübpMaCbH KepCeTyyMy MorçgyK. A . TeMeHKy ôpMy.a (2) 6oroHMa Ta6b.aT:

n

' Z à P A C i

K P A C = — ------- (2)n

MbHga, n - a.MamTbpb.rbC aMMHKMC.oTa.apgbH CaHb; APACi - 6upuHMM .MMMTTeHyyMy aMMHKMC.oTa.biK CKopro KapaTa i - a.MamTbpb.rbC aMMHKMC.oTaCbHbH afibpMa.yy.yry:

A P A C i = A C i - ACm in Be.oKTyH 6uo.orufl.bK 6aa.yy.yry (BB, %) ôpMy.a (3) TeH ^ce^Te.MHeT:BB = İ00 - KPAC (3)

TaMaK-am a3bKTapga a.MamTbpb.rbC aMMHKMC.oTa.apbHbH KapMa.yyCy 0A0/B03 ^Ta.oHyHga 6epu.reHgeH Ken «e TeCKepuCMHMe a3 6o.ymy MyMKyH. ^Ta.oH 6e.oKKo KapaTa TaMaK-amTapgarb a.MamTbpb.rbC aMMHKMC.oTa.apbHbH 6a.aHCTyy.yryH KepCeTKeH yTM.gyy.yK (ôpMy.a 4) Ko^^MU,MeHTM Ki MeHeH Ta6b.aT [19, 21].

B a M JEN74



E Ö ffÖ ^Ü ß, MepsUMUKmYY aK copmmo3y mee ôyypnaKmap, kss ÔUp $U3UKrnbiK KacUemmepU wanaaMUHKUcmmanbiK KypaM e.33enenYKmepY

_ - - w mi"A C" (4)

MblHfla, ACi - İ - a^MaWTbipbmrblC aMMHKMcnoTacblHblHgarbl aMMHKMC.nOTa.nblK CKOp; ACmin - a3blKTarbl we 6enoKTory 6mpmhmm nuMMTTeHYY^Y aMMHKMcnoTacbiHbiH aMMHKMcnoTanbiK CKopy. AnMawTbipbinrbic aMMHKMcnoTanapbHbH yTMngYY-nYK KO^^MU,MeHTM aMMHKMcnoTanapbiHbiH yTMngYY-nYrYH (U) ^ce^Tee Y^YH KongoHynaT. Byn KO^^MU,MeHT ^TanoHô KapaTa anMawTbipbinrbic aMMHKMcnoTanapbiHbiH 6anaHCTyynyryH (ôpMyna 5) KepceTeT:

nI (A K )

U = ^ ------ (5)I A ^ ti=1

BanaHCTan6araH anMawTbipbinrbic aMMHKMcnoTanap opraHM3M Tapa6biHaH yTMngeH6eMT, 6yn Ke3emHge awbiKMa KapManraH anMawTbipbinrbic aMMHKMcnoTanapbiHbiH 6anaHCTyynyry gereH KO^^MU,MeHT MeHeH KapanaT:

6c = 6n / ACmin MbiHga, ön 6yn ôpMyna (6) 6oroHMa Ta6binaT:

a n = I (A i - A C mi" A i,3T ) (6)i =1

TaMaK-awTapgbiH, mmmkm a3biKTapgbiH 6Monomfl.nbiK 6aanyynyryH aHbiKTOogo KongoHynraH 6awKa 6up bKMa 6yn anMawTbipbinrbic aMMHKMcnoTanapgbiH MHgeKcM (AAKM) (opyc TunuHge "MHgeK He3aMeHMMbix aMMHKMcnoT-MHAK") 6onyn caHanaT. Byn bKMa MeHeH ^ce^Tee Oser (1951) xmmma^nk bKMacuHbH MogM$MKau,MflCbi 6onyn caHanaT. MbiHga ôpMyna (7) KongoHynaT:

Ä A K H = .R u 3 J p u y * Tp ey

^ U33 tp u 3 ' ' Tp es (7)

MbHga, n - anMawTbipbinrbic aMMHKMcnoTanapgbiH caHbi, y, - M3MngeHMn waTKaH Y-nrYgerY waHa ^TanoH 6enoKTory anMawTbipbinrbic aMMHKMcnoTanapgbiH KapManyycy [18-22].Wannbi aHanM3gMH wbifibiHTbiKTapbi 6oroHMa wepmnMKTYY 6up e^Mefi aK coprrory 6yypMaKTapgaH 6enoK

Ken KapMaraH copT KMTaaHKa, aHaH kmmmhkm caxapHbm waHa a3 nonaTKa copTy 6onyn aHbiKTanraH. AnMawTbipbinrbic aMMHKMcnoTanapgbiH KoHU,eHTpau,Macbi 6oroHMa caxapHbm 6mpmhmm, nonaTKa KMH M waHa KMTaflHKa ymyhmy opyHgapgbi ^^nemeT. AMMHKMcnoTanbiK cKopgyH afibipManyynyK KO^^MU,MeHTM 6oroHMa nonaTKa waHa KMTaaHKa coprropy 6upgeM MaaHM apanbirbiHga waTaT (wagbi6an 5), an smm caxapHbm copTy woropypaaK. Byn caxapHbm copTyHgarb nMMMTTeHYYMY aMMHKMcnoTa 6onroH MeTMoHMHgMH 6awKa aMMHKMcnoTanapgaH canbiwTbipManyy aa6aM KoHU,eHTpau,MA.nbiK afibipMaMbinbirbi MeHeH TywYHgYpceK 6onoT. Wagbi6an 4 Te 6epMnreHgefi nM3MH waHa êHMnanaHMH aMMHKMcnoTanapbiHbiH caxapHbM copTyHga aMMHKMcnoTanbiK cKopy 2,3 waHa 2,7 ^ce Ken 6onroHgyry Hem3m Taacup 6epYYMY âKTop. BMonomflnbiK 6aanyynyry 6oroHMa KMTaaHKa MeHeH nonaTKa copTTopy caxapHbm copTyHaH 6upa3 raHa afibipManaHaT. AnMawTbipbinrbic aMMHKMcnoTanapgbiH MHgeKcM 6oroHMa nonaTKa waHa caxapHbM copTTopy KMTaaHKa copTyHaH optomo 0,55 Ke Ke6YpeeK 6onyn Ta6bngb.

B a M JEN75



EQflQWÔB, MepBmuKmYY aK copmmo3y mee 6yypuaKmap, K^^ 6up fyusuKanbiK Kacuemmepu waHaaMUHKucmmanbiK KypaM eszenenyKmepY

Magbi6afl 5. Ak coprrory Tee 6yypMaK gaHgapgbH 6MonomflnbiK 6aanyynyK KacMeTTepM

A3bK BeÔK, %AflMaWTbipblflrblC

aMMHKMCnOTanapflbHcyMMacb

KPA C ,% BB,% U AAKM

.HonaTKa 18,37 9,026 33,42 66,58 0,32 1,20KMTaaHKa 25,17 7,312 32,89 67,11 0,24 0,75CaxapHbM 21,15 11,875 37,09 62,91 0,52 1,32

WblMbl HTbIKWeprnnMKTyy 6up e^Metf aK coprrory Tee 6yypMaK gaHgap copTyHa KapaTa 18-25% 6enoK KaMTbMT. BenoK KapMoocy 6oroHMa T a3bKTapbHa Te^ KenMn, KOHOMMKa.nblK waKTaH optomo 8-9 ^ce ap3aH. CM^MpMMgyynyry ga woropy.AMMHKMcnoTanbK KypaM 6oroHMa caxapHbM copTy *a^nb anMamTbipbinrbic aMMHKMcnoTanapgb mo KoHU,eHTpau,Maga KapMaraH copT 6onyn aHbKTangb. AMMHKMcnoTanbK cKop 6oroHMa MeTMoHMHgeH 6amKa 6apgbK aMMHKMcnoTanap aMMHKMcnoTanbK cKopy 100% gaH woropy. Hm3mh waHa êHMnanaHMH+TpMnroâH 6oroHMa aMMHKMcnoTanbK cKop Mgeangyy 6enoK mKanacbHaH 2,4 waHa 2,7 ^ce amaT. AnMamTbipbinrbic aMMHKMcnoTanapbHbH 6anaHcTyynyry 6oroHMa caxapHbM copTyHga KMTaaHKa, nonaTKa copTTopyHa KaparaHga woropy. BMpoK omo^ .ne yMypga aMMHKMcnoTanbK cKopgyH afibipManyynyK KO^^MU,MeHT KepceTKyMy 6amKa copTTopgoH woropypaaK. Byn Ty3geH-Ty3 MeTMoHMH, nM3MH waHa $eHMnanaHMH aMMHKMcnoTanapbHa 6aMnaHbmTyy. HonaTKa waHa KMTaaHKa coprropy caxapHbM copTyHa KaparaHga 6uo.norMfl.nbK 6aa.nyy.nyry woropy, anMamTbipbinrbic aMMHKucnoTanapgb weTMm^^p.nMK KaMTbiraH copTTop. KepeKTeenynep KyHyMgyK pau,MoHgopyHga nonaTKa, KMTaflHKa waHa e3reMe caxapHbM copTyHgarb 6yypMaKTapgb weTumTyy KeneMge KongoHyy MeHeH anMamTbpbnrbc aMMHKMcnoTanapgb TMemenyy KeneMge ana anbmaT.

KOÂOHy^rAH a a a b h a t t a p

[1] Alimkulov B.B. Vodniy rejim fasoli obyknovennoi. Bishkek, Kyrgyzstan: Kut-Ber, pp. 15-146, 2010.[2] Study of the production and export of beans in Talas. Japan International Cooperation Agency (JICA) in

collaboration with the Public Foundation "Center for Peace". B.: Altin Print, pp.69-71, 2010.[3] V.N. Grisuk, N.P. Iluha, I.I. Suholotyuk i dr. Tovarovedeniye: Agricultural products and medicinal and

industrial raw materials. M.: Ekonomika, p. 399, 1988.[4] Vavilov N.I. Miroviye resursy sortov hlebnih zlakov, zernovyh bobovyh, lna i ih ispolzovaniye v seleksii.-

M.; L.: Izd-vo AN SSSR, pp. 460-462, 1957.[5] http://www.stat.kg/ru/statistics/selskoe-hozyajstvo.[6] Bodoshov A.U., Kydyraliev N. A. Investigation physical properties of beans grown in Kyrgyzstan.

Materials of International scientific conference Food security issues of CIS Member States: National and International Perspectives. Bishkek, pp.197-203, 2011.

[7] Kashaninejad, M., A. Amortazavi, A. Safefekordi, and L.G. Tabil. "Some Physical Properties of Pistachio(P istacia vera l.) Nut and its Kernel". J. Food Engr. 72(1): pp.30-38, 2006.

[8] Davies, R.M. Some physical properties of Groundnut Grains. Department of Agricultural and Environmental Engineering, Niger Delta University, Wilberforce Island. Bayelsa State. Journal of Applied Sciences, Engineering and Technology 1(2): pp.10-13, 2009.

[9] GOST 9404-88. Muka i otrubi. Metod opredelenya vlagi.

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http://www.stat.kg/ru/statistics/selskoe-hozyajstvo


EQflQW^OB, Mep¿unuKmYY aK copmmo3y mee 6yypuaKmap, K^^ 6up fyusuKanbiK Kacuemmepu waHaaMUHKucmmanbiK KypaM eszenenyKmepY

[10] GOST 13586.5-93. Zerno. Metody opredelenya vlajnosti.[11] KMS 927:2004. Fasol prodovelstvennaya.-Vved.M.: IPK. Izd-vo standartov, pp 26-28, 2004.[12] Dekapreleyevich L.L. Fasol. M.: Kolos, 1965.[13] M.J. Gonzalez-Castro, J. Lopez-Hernández, J. Simal-Lozano, and M.J. Oruña-Concha. Determination of

Amino Acids in Green Beans by Derivatization with Phenylisothiocianate and High-Performance Liquid Chromatography with Ultraviolet Detection. Journal of Chromatographic Science, Vol. 35, pp. 181185, 1997.

[14] David L. Jonesa, Andrew G. Owena, John F. Farrar. Simple method to enable the high resolution determination of total free amino acids in soil solutions and soil extracts. Soil Biology & Biochemistry Vol. 34, pp. 1893-1902, 2002.

[15] Bodoshov A.U., Kydyraliev N. A. Investigation some physical properties grain of beans grown in Talas region, Kyrgyzstan. Journal « Molodoy Ucheniy». Tatarstan: 5 ed., p.74, 2014.

[16] Bodoshov A.U. "Himicheskiy sostav zeren fasoli mestnih sortov". Journal «Nauka i noviye Tehnologi», Bishkek, No. N4, pp. 38-41, 2014.

[17] Bodoshov A.U. "Aminokislotniy sostav zeren fasoli vyrashivaemyh v Kyrgyzstane". Journal « Molodoy Ucheniy». Tatarstan: 24 ed., pp. 94-96, 2015.

[18] T.N. Sokolova, V.M. Prohorov, V.R. Kartashov. Definition of indicators of biological value foodstuffs calculation method. Nijniy Novgorod, 2015.

[19] M.V. Stepuro, E.N. Khaprova. Comparative Evaluation of Biological Value of Vegetable Raw Material's Proteins. Proceedings of the universities. Food Technology, N° 4, pp. 34-35, 2010.

[20] Ph.D. Nadtochii L.A. Designing a protein component of food in the Microsoft Excel spreadsheet editor. Processes and devices of food manufacture, N 2, 2013.

[21] I. A. Shestepalova, N. A. Uvarova. Biological value of laying hen protein. National Research University of Information Technologies, Mechanics and Optics. Institute of Refrigeration and Biotechnologies.

[22] Kolodjaznaya V.S., Baranenko D.A., Broyko J.V. Biological value of milk-fed veal protein. Saint- Petersburg state university of refrigeration and food engineering, 2015.

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