AN EXPANSION METHOD FOR FINDING TRAVELING WAVE

Transkript

İstanbul Ticaret Üniversitesi Fen Bilimleri Dergisi Yıl: 14 Sayı:27 Bahar 2015 s. 65-81
Araştırma Makalesi
AN EXPANSION METHOD FOR FINDING TRAVELING
WAVE SOLUTIONS TO NONLINEAR PDEs
Asıf YOKUŞ 12
ABSTRACT
In this study, we construct an expansion method. We have implemented this method for finding traveling
wave solutions of nonlinear Burgers equation, KdV equation, RLW-Burgers equation and Hirota-Satsuma
equation system. We also point out that this expansion method could implement the other nonlinear
equations.
Keywords: Burgers equation; KdV equation; RLW-Burgers equation; Hirota-Satsuma equation
system; an expansion method.
LİNEER OLMAYAN KISMİ DİFERANSİYEL DENKLEMLERDE YÜRÜYEN
DALGA ÇÖZÜMLERİ İÇİN BİR AÇILIM METODU
ÖZ
Bu çalışmada bir genişleme metodu oluşturuldu. Bu metodu lineer olmayan Burgers denklemi, KdV
denklemi, RLW-Burgers denklemi ve Hirota-Satsuma denklem sisteminin yürüyen dalga çözümünü
bulmak için bu metodu uyguladık. Biz ayrıca bu metodun diğer lineer olmayan denklemlere uygulanabilir
olduğunu gösterdik.
Anahtar kelimeler: Burgers denklemi; KdV denklemi; RLW-Burgers denklemi; Hirota-Satsuma
denklem sistemi; açılım metodu.
Makale Gönderim Tarihi: 03.03.2016
1
Kabul Tarihi: 06.04.2016
Fırat Üniversitesi, Fen Fakültesi, Aktüerya Bilimleri Bölümü, e-mail/e-ileti: [email protected]
Asıf YOKUŞ
1. INTRODUCTION
In this study, we present an expansion method by inspiring of (G ′ G ) -expansion
method which is introduced in (Wang et al., 2008), and the other similar methods
(Guo and Zhou, 2010; Fan, 2000). When we implemented this method we found
several analytic solutions form of rational trigonometric solutions of the classical
nonlinear Burgers equation, KdV equation, RLW-Burgers equation and HirotaSatsuma equation. However, there are many methods to obtain traveling wave
solutions of the nonlinear partial differential equation in literature (Clarkson, 1989;
Wazwaz, 2005; Parkes and Duffy, 1996; Fan, 2000; Elwakil et al., 2002; Zheng,
2003; He and Wu, 2006; Kaya, 2003; Inan, 2010).
2. AN ANALYSIS OF THE METHOD AND APPLICATIONS
Before starting to give the (1 G′) -expansion method, it is necessary to explain a
simple description of the (1 G′) -expansion method. In order to do this, one can
consider in a two-variable general form of nonlinear partial differential equation
(PDE)
Q(u , u t , u x , u xx ,) = 0 ,
(1)
and transform Eq. (1) with u ( x, t ) = u (ξ ) , ξ = x − Vt , where V is constant. After
transformation, we get a nonlinear ordinary differential equation (ODE) for u (ξ )
Q' (u ′, u ′′, u ′′′, ) = 0.
(2)
The solution of the Eq. (2) we are looking for is expressed as
i
m
 1 
u (ξ ) = a0 + ∑ ai   ,
 G′ 
i =1
(3)
where G = G (ξ ) satisfies the second order linear ODE in the form
G ′′ + λG ′ + µ = 0,
(4)
where ai ,  , bi ,  , µ and λ are constants to be determined later, the positive
integer m can be determined by balancing the highest order derivative and with the
highest nonlinear terms into Eq. (2). Substituting solution (3) into Eq. (2) and using
Eq. (4) yields a set of algebraic equations for the same order of (1 G ′) ; then all
coefficients same order of (1 G ′) have to vanish. After we have separated this
algebraic equation, we can find ai ,  , bi ,  and V constants. We know the general
solutions of the Eq. (4) well, and then substituting ai ,  , bi ,  , and the general
solutions of Eq. (4) into (3), we can get more traveling wave solutions of Eq. (1)
(Yokus, 2011; Inan 2010).
66
İstanbul Ticaret Üniversitesi
Yıl: 14
Fen Bilimleri Dergisi
Sayı: 27
Bahar 2015
Example 1. Let’s consider Burgers equation (Gorguis, 2006).
ut + uu x + δ u xx = 0,
(5)
for doing this example, we can use transformation u ( x, t )= u (ξ ) , ξ= x − Vt then
Eq. (5) become
1
c − Vu + u 2 + δ u ′ = 0,
2
When balancing
(6)
u 2 with u ′ then gives m = 1 . Therefore, we may choose
 1 
u (ξ ) = a 0 + a1  ,
 G′ 
(7)
substituting Eq. (7) into (6) yields a set of algebraic equations for a 0 , a1 , λ , c, δ , µ
and V . These systems are
0
 1 
  :
 G′ 
1
 1 
  :
 G′ 
2
 1 
  :
 G′ 
a 02
+ c − a 0V = 0,
2
a1 (a 0 − V + δλ ) = 0,
a12
+ a1δµ = 0 .
2
(8)
From the solutions system (8), we obtain the following with the aid of Mathematica.
Case 1:
a 0 = −δλ − 2c + δ 2 λ2 , a1 = −2δµ , V = − 2c + δ 2 λ2 ,
(9)
substituting Eq. (9) into (7) we have three types of traveling wave solutions of Eq.
(5):
ξ = x+
u1 (ξ ) = −δλ − 2c + δ 2 λ2 −
( 2c + δ λ )t,
2
2δλµ
.
− µ + λA(Cosh(ξλ ) − Sinh(ξλ ))
(10)
(11)
67
Asıf YOKUŞ
Fig 1: Traveling wave solution of Eq. (5) for case 1 when
λ � = 2 , 𝛿 = 0.8, µ
= 3, 𝑐 = 2, 𝐴 = 1.
Giving values to constants in Eq. (10) and (11), we obtain figure 1. Therefore, we
have showed traveling wave solution of Eq. (5) in figure 1.
Case2:
a 0 = −δλ + 2c + δ 2 λ2 , a1 = −2δµ , V = 2c + δ 2 λ2 ,
(12)
Substituting Eq. (12) into (7) we have three types of traveling wave solutions of Eq.
(5):
ξ = x−
u 2 (ξ ) = −δλ + 2c + δ 2 λ2 −
68
( 2c + δ λ )t,
2
2δλµ
.
(13)
(14)
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λ � = 2 , 𝛿 = 0.8, µ
= 3, 𝑐 = 2, 𝐴 = 1.
Example 2. Let’s consider KdV equation (Kaya and Aassila, 2002)
u t + uu x + δu xxx = 0.
(15)
For doing this example, we can use transformation u ( x, t )= u (ξ ) , ξ= x − Vt then Eq.
(15) become
− Vu ′ + uu ′ + δu ′′′ = 0.
(16)
When balancing u ′u , u ′′′ then gives m = 2 . Therefore, we may choose
2
 1 
 1 
u (ξ ) = a 0 + a1   + a 2   ,
′
G
 G′ 
 
(17)
substituting Eq. (17) into (16) yields a set of algebraic equations for
a 0 , a1 , a 2 , δ , µ , c, λ and V . These systems are
0
 1 
  :
 G′ 
1
 1 
  :
 G′ 
a 02
+ c − a 0V = 0,
2
a1 a 0 − a1V + a1δλ2 = 0,
69
Asıf YOKUŞ
a12 + 2a 0 a 2 − 2a 2V + 8a 2δλ2 + 6a1δµλ
= 0,
2
2
 1 
  :
 G′ 
3
 1 
  :
 G′ 
4
 1 
  :
 G′ 
a1 a 2 + 2a1δµ 2 + 10a 2δ µλ = 0,
a 2 (a 2 + 12δµ 2 )
= 0.
2
(18)
From the solutions of the system (18), we obtain the following with the aid of
Mathematica.
Case 1:
a 0 = −δλ2 − 2c + δ 2 λ4 , a1 = −12δλµ , a 2 = −12δµ 2 , V = − 2c + δ 2 λ4 .
(19)
(15):
ξ=
x+
(
u1 (ξ ) = −δλ2 − 2c + δ 2 λ4 −
)
2c + δ 2λ 4 t ,
12δλ2 µ
−
12δµ 2 λ2
(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )))
2
.
λ � = 0.2 , 𝛿 = 0.8, µ
70
(20)
= 0.3, 𝑐 = 0.2, 𝐴 = 1.
(21)
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Case 2:
a 0 = −δλ2 + 2c + δ 2 λ4 , a1 = −12δλµ , a 2 = −12δµ 2 , V = 2c + δ 2 λ4 .
(22)
Substituting Eq. (22) into (17) we have three types of traveling wave solutions of
Eq. (15):
ξ = x−
u 2 (ξ ) = −δλ2 +
−
( 2c + δ λ )t,
2c + δ 2 λ 4 −
(23)
2 4
12δλ2 µ
12δµ 2 λ 2
(24)
(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )))2
.
λ � = 0.2 , 𝛿 = 0.8, µ
= 0.3, 𝑐 = 0.2, 𝐴 = 1.
Example 3. Consider the RLW-Burgers equation (Kaya, 2004)
u t + u x + 12uu x − α u xx − bu xxt = 0.
(25)
Where α and b are positive constants. For doing this example, we can use
transformation u ( x, t )= u (ξ ) , ξ= x − Vt then Eq. (25) become
71
Asıf YOKUŞ
− Vu ′ + u ′ + 12uu ′ − α u ′′ + Vbu ′′′ = 0.
When balancing
(26)
uu x with u xxt then gives M =2. Therefore, we may choose
2
 1 
 1 
u (ξ ) = a 0 + a1   + a 2   ,
 G′ 
 G′ 
(27)
substituting Eq. (26) into (27) yields a set of algebraic equations for
a 0 , a1 , a 2 , δ , µ , c, λ , α , b and V . These systems are
1
 1 
  :
 G′ 
a1λ + 12a 0 a1λ − a1Vλ − a1λ2α + a1bVλ3 = 0,
12a12 λ + 2a 2 λ + 24a 0 a 2 λ − 2a 2Vλ − 4a 2αλ2
 1 
  :
 G′ 
+ 8a 2 bVλ3 + a1 µ + 12a 0 a1 µ − a1Vµ − 3a1αλµ + 7 a 1 bVλ2 µ = 0,
2
36a1 a 2 λ + 12a12 µ + 2a 2 µ + 24a 0 a 2 µ − 2a 2Vµ
3
 1 
  :
 G′ 
− 10a 2αλµ + 38a 2 bVλ2 µ − 2a1αµ 2 + 12a1bVλµ 2 = 0,
4
 1 
  :
 G′ 
24a 22 λ + 36a1 a 2 µ − 6a 2αµ 2 + 54a 2 bVλµ 2 + 6a1bVµ 3 = 0,
5
 1 
  :
 G′ 
24a 22 µ + 24a 2 bVµ 3 = 0.
(28)
Mathematica.
Case 1:
a0 =
1 
2αµ
α
αµ 2
α

, a2 =
, V =−
.
+ 6αλ , a1 =
− 5 −
60 
5
5
5
bλ
λ
bλ

(29)
(25):
 α 
ξ = x+
t ,
 5bλ 
u1 (ξ ) =
2αµλ
1 
α

+ 6αλ  +
− 5 −
60 
bλ
 5(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )) )
+
72
(30)
αµ 2 λ
.
2
5(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )) )
(31)
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λ � = 2 , 𝛿 = 0.8, µ
= 0.3, 𝑐 = 0.2, 𝑏 = 6, 𝛼 = 2, 𝐴 = 1.
Case 2:
a0 =
1 
α
αµ 2
α

, V =
,
+ 6αλ , a1 = 0, a 2 = −
− 5 −
60 
5λ
5bλ
bλ

(32)
Eq. (25):
 α 
ξ = x−
t ,
 5bλ 
u 2 (ξ ) =
1 
α
αµ 2 λ

+ 6αλ  −
.
− 5 −
2
bλ
60 
 5(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )) )
(33)
(34)
73
Asıf YOKUŞ
λ � = 2 , 𝛿 = 0.8, µ
= 0.3, 𝑐 = 0.2, 𝑏 = 6, 𝛼 = 2, 𝐴 = 1.
Case 3:
a0 =
1
(− 1 + αλ ), a1 = αµ , a 2 = 0, V = 0.
12
6
(35)
Eq. (25):
u 3 (ξ ) =
74
αµλ
1
(− 1 + αλ ) +
.
12
6(− µ + λA(Cosh( xλ ) − Sinh( xλ )) )
(36)
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Bahar 2015
λ � = 2 , 𝛿 = 0.8, µ
= 0.3, 𝑐 = 0.2, 𝑏 = 6, 𝛼 = 2, 𝐴 = 1.
Example 4. Let’s consider Hirota-Satsuma system (Wazwaz, 2008)
ut =
1
u xxx + 3uu x − 6 gg x ,
2
g t = − g xxx − 3ug x .
(37)
For doing this example, we can use transformation u ( x, t )= u (ξ ) , ξ= x − Vt then Eq.
(37) become
1
3
c − Vu − u ′′ − u 2 + 3 g 2 = 0,
2
2
− Vg ′ + g ′′′ + 3ug ′ = 0.
(38)
When balancing u ′u , u ′′′ then gives m = 2 . Therefore, we may choose
2
 1 
 1 
 1
 1
u (ξ ) =
a0 + a1   + a2   and g (ξ ) = b0 + b1   + b2   .
 G′ 
 G′ 
 G′ 
 G′ 
2
(39)
Substituting Eq. (38) into (39) yields a set of algebraic equations for
a 0 , a1 , a 2 , b0 , b1 , b2δ , µ , c, λ and V . This system is
75
Asıf YOKUŞ
0
 1 
  :
 G′ 
−
3a 02
+ 3b02 + c − a 0V = 0,
2
1
 1 
  :
 G′ 
− 3a 0 a1 + 6b0 b1 − a1V −
2
 1 
  :
 G′ 
−
a1λ2
= 0,
2
3a 0 b1λ − b1Vλ + b1λ3 = 0,
3a12
3a λµ
− 3a 0 a 2 + 3b12 + 6b0 b2 − a 2V − 2a 2 λ2 − 1
= 0,
2
2
3a1b1λ + 6a 0 b2 λ − 2b2Vλ + 8b2 λ3 + 3a 0 b1 µ − b1Vµ + 7b1λ2 µ = 0,
3
 1 
  :
 G′ 
− 3a1 a 2 + 6b1b2 − 5a 2 λµ − a1 µ 2 = 0,
3a 2 b1λ + 6a1b2 λ + 3a1b1 µ + 6a 0 b2 µ − 2b2Vµ + 38b2 λ2 µ + 12b1λµ 2 = 0.
4
3a 2
 1 
2
2
  : − 2 + 3b2 − 3a 2 µ = 0,
2
 G′ 
6a 2 b2 λ + 3a 2 b1 µ + 6a1b2 µ + 54b2 λµ 2 + 6b1 µ 3 = 0,
5
 1 
3
  : 6a 2 b2 µ + 24b2 µ = 0,
 G′ 
(40)
Mathematica.
Family 1:
(
)
(
)
a0 =
1
− 10λ2 + 10 − 12c + λ4 , a1 = −4λµ , a 2 = −4 µ 2 ,
30
b0 =
1
− 5λ2 + 2 10 − 12c + λ4 , b1 = −2λµ , b2 = −2 µ 2 ,
30
V =
− 12c + λ4
10
(41)
.
Eq. (37) as following
 − 12c + λ4

10

ξ = x+
76

t.


(42)
u1 (ξ ) =
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Bahar 2015
)
(
1
4λ 2 µ
− 10λ + 10 − 12c + λ4 −
30
−
g1 (ξ ) =
Yıl: 14
4 µ 2 λ2
(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )) )2
,
)
(
1
2 µλ2
− 5λ2 + 2 10 − 12c + λ4 −
30
−
2µ λ
2
(43)
(44)
2
(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )) )2
.
Giving values to constants in Eq. (42)-(44), we obtain figure 8. Therefore, we have
showed traveling wave solution of Eq. (37) in figure 8.
Family 2:
a0 =
)
(
1
− 8λ2 µ 2 + 2 36cµ 2 + 5λ4 µ 4 ,
18µ 2
a1 = −2λµ ,
 14
4
2λ 36cµ 4 + 5λ4 µ 4
3 − λ3 µ −
9
9
µ
b0 = − 
4 7λ2 µ 2 − 2 2 36cµ 4 + 5λ4 µ 4
b1 = −


,
a 2 = −2 µ 2 ,
b2 = 0,
1
7λ2 µ 2 − 2 2 36cµ 4 + 5λ4 µ 4 ,
3
36cµ 4 + 5λ4 µ 4
1
V =  − λ2 +
3
2µ 2


.


(45)
77
Asıf YOKUŞ
1
36cµ 4 + 5λ4 µ 4
ξ = x +   − λ2 +
3
2µ 2
 
u 2 (ξ ) =

 t ,


(46)
)
(
1
2λ2 µ
,
− 8λ2 µ 2 + 2 36cµ 2 + 5λ4 µ 4 −
2
(
− µ + λA Cosh(ξλ ) − Sinh(ξλ ))
18µ
−
2µ λ
2
(47)
2
,
 14
4
3 − λ3 µ −
2λ 36cµ 4 + 5λ4 µ 4
9
9µ

g 2 (ξ ) = −
4 7λ2 µ 2 − 2 2 36cµ 4 + 5λ4 µ 4


2 2
4
4 4
 − λ 7λ µ − 2 2 36cµ + 5λ µ .
3(− µ + λA(Cosh(ξλ ) − Sinh(ξλ )) )
(48)
Family 3:
a0 =
)
(
1
− 8λ2 µ 2 − 2 36cµ 2 + 5λ4 µ 4 ,
18µ 2
b0 = −
14λ3 µ 4 2λ 36cµ 4 + 5λ4 µ 4
+
9
9µ
7λ2 µ 2 2 2
+
4
36cµ 4 + 5λ4 µ 4
9
9
b1 = −
a 2 = −2 µ 2 ,
, b2 = 0,
7λ 2 µ 2 2 2
+
36cµ 4 + 5λ4 µ 4 ,
9
9
2 36cµ 4 + 5λ4 µ 4
1
V =  − 2λ 2 −
6
µ2

78
a1 = −2λµ ,

.


(49)
Yıl: 14
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Bahar 2015
1
2 36cµ 4 + 5λ4 µ 4
ξ = x +   − 2λ 2 −
6
µ2


u 3 (ξ ) =

 t.


)
(
1
2λ 2 µ
− 8λ2 µ 2 − 2 36cµ 2 + 5λ4 µ 4 −
2
18µ
−
g 3 (ξ ) = −
2µ λ
2
4
4 4
14λ3 µ 4 2λ 36cµ + 5λ µ
+
9
9µ
4
7λ µ
2 2
+
36cµ 4 + 5λ4 µ 4
9
9
2
(51)
2
2
(50)
,
7λ 2 µ 2 2 2
+
36cµ 4 + 5λ4 µ 4
9
9
.
λ−
−
(52)
3. CONCLUSIONS
By using a relatively new expansion method, we derive the traveling wave
solutions of several nonlinear equations such as Burgers equation, KdV equation,
RLW-Burgers equation and Hirota-Satsuma equation. We provide that the method
can be applied to many other nonlinear equations or coupled ones.
The solutions obtained via the method used in the present paper are different
with the solutions of the method known as (G′ G ) - expansion method in literature. In
addition, this method is also computerizable, i.e., it allows us to perform
complicated algebraic calculations on a computer.
79
Asıf YOKUŞ
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AN EXPANSION METHOD FOR FINDING TRAVELING WAVE

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finding the traveling wave solutions of some nonlinear partial