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1
Exp-Function Method for Finding Exact Solutions of
Nonlinear Evolution Equations
A Thesis
Submitted by
Lakesh Kumar Ravi
Roll No: 409MA5004
In partial fulfillment of the requirements for the award of the degree
of
Master of Science in Mathematics
Under the supervision
of
Prof. Santanu Saha Ray
Department Of Mathematics
National Institute Of Technology
Rourkela 769008 Odisha
May 2014
2
DECLARATION
I hereby declare that the thesis entitled “Exp-Function Method for Finding Exact Solutions of
Nonlinear Evolution Equations” which is being submitted by me to National Institute of
Technology for the award of degree of Master of Science is original and authentic work conducted
by me in the Department of Mathematics, National Institute of Technology Rourkela, under the
supervision of Prof. Santanu Saha Ray, Department of Mathematics, National Institute of
Technology, Rourkela. No part or full form of this thesis work has been submitted elsewhere for
a similar or any other degree.
Place: Lakesh Kumar Ravi
Date: Roll no 409MA5004
3
CERTIFICATE
This is to certify that the project report entitled “Exp-Function Method for Finding Exact
Solutions of Nonlinear Evolution Equations” submitted by Lakesh Kumar Ravi for the partial
fulfillment of M.Sc. degree in Mathematics, National Institute of Technology Rourkela, Odisha,
is a bona fide record of review work carried out by him under my supervision and guidance. The
content of this report, in full or in parts, has not been submitted to any other institute or university
for the award of any degree or diploma.
(Prof. Santanu Saha Ray)
Associate professor
Department of Mathematics
NIT Rourkela- 769008
4
ACKNOWLEDGEMENT
I would like to warmly acknowledge and express my deep sense of gratitude and indebtedness to
my guide Prof. Santanu Saha Ray, Department of Mathematics, NIT Rourkela, Odisha, for his
keen guidance, constant encouragement and prudent suggestions during the course of my study
and preparation of the final manuscript of this project.
I would like to thank the faculty members of Department of Mathematics NIT Rourkela,
who have always inspired me to work hard and helped me to learn new concepts during our stay
at NIT Rourkela. Also I am grateful to my senior Ashrita Patra, Arun Kumar Gupta, Prakash
Kumar Sahu, Subhadarshan Sahoo, research scholar, for their timely help during my work.
I would like to thanks my family members for their unconditional love and support. They
have supported me in every situation. I am grateful for their support.
My heartfelt thanks to all my friends for their invaluable co-operation and constant
inspiration during my project work.
Rourkela, 769008
May 2014 Lakesh Kumar Ravi
5
ABSTRACT
In this report, we applied Exp-function method to some nonlinear evolution equations to obtain its
exact solution. The solution procedure of this method, by the help of symbolic computation of
mathematical software, is of utter simplicity. The prominent merit of this method is to facilitate
the process of solving systems of partial differential equations. These methods are straightforward
and concise by themselves; moreover their applications are promising to obtain exact solutions of
various partial differential equations. The obtained results show that Exp-function method is very
powerful and convenient mathematical tool for nonlinear evolution equations in science and
engineering.
6
CONTENTS
TITLE PAGE NUMBER
DECLARATION 2
CERTIFICATE 3
ACKNOWLEDGEMENT 4
ABSTRACT 5
INTRODUCTION 7
CHAPTER 1 Basic idea of Exp-function method 8
CHAPTER 2 Exp-function method for solving 14
nonlinear evolution equation
CHAPTER 3 Application of Exp-function to 20
find the exact solution of coupled Boussinesq
Burgers equations
BIBLIOGRAPHY
7
INTRODUCTION
Recently a variety of powerful methods has been proposed and analyzed to obtain the exact
solutions to nonlinear evolution equations such as the homogeneous balance principle, F-
expansion method, 𝑡𝑎𝑛ℎ method, auxiliary equation method, the coupled Riccati equations, the
Jacobi elliptic equations, etc. Among these methods, the Exp-function method introduced by He
and Wu in 2006, is successfully applied to many kind of nonlinear problems. Up to now, the Exp-
function method has been applied to find the solutions of a class of nonlinear evolution equations,
such as the nonlinear Schrodinger equation, KdV equation with variable coefficient, the discrete
(2 + 1) − dimensional Toda lattice equation and the Maccari’s system. So it is easy to see that the
Exp-function method is very powerful technique and can be used to study the exact solutions of
the high-dimensional system, the discrete system and the system with variable coefficients. All
applications verified that Exp-function method is straightforward, concise and effective in
obtaining generalized solitary solutions and periodic solutions of nonlinear evolution equations.
The main merits of this method over other method are that it gives more general solutions with
some free parameters. Also this method can be applied to a wide class of nonlinear evolution
equations including those in which the odd and even order derivative terms coexist.
8
CHAPTER 1
1.1 Basic idea of Exp-Function Method:
In order to illustrate the basic idea of the Exp-function method, consider a given nonlinear
dispersive equation of the form:
02 xxxxt uuuu (1.1.1)
This equation is called modified KdV equation, which arises in the process of understanding the
role of nonlinear dispersion and in the formation of structure like liquid drops, and it exhibits
compactors: solitons with compact support.
Introducing a complex variation defined as
tkx (1.1.2)
We have
032 ukukuu (1.1.3)
where prime denotes the differential with respect to 𝜂.
The Exp-function method is very simple and straightforward, it is based on the assumption that
the travelling wave solution can be expressed in the following form:
q
pm
mm
d
cn
nn
eb
ea
u
)( (1.1.4)
Where 𝑐, 𝑑, 𝑝 and 𝑞 are positive integers which are unknown to be further determined, na and mb
are unknown constants.
9
We suppose that the solution of eq. (1.1.3) can be expressed as
)exp()exp(
)exp()exp()(
qbpb
dacau
qp
dc
(1.1.5)
To determine values of c and p , we balance the linear term of highest order in eq. (1.1.3) with
the highest order nonlinear term. By simple calculation, we have
]8exp[
])7exp[(
2
1
pc
cpcu (1.1.6)
and
]8exp[
])35exp[(
]4exp[
])3exp[(
4
3
4
32
pc
cpc
pc
cpcuu (1.1.7)
where ic are known coefficients.
Balancing highest order of Exp-function in eqs. (1.1.6) and (1.1.7), we have
cpcp 357
which yields cp (1.1.8)
Similarly, to determine values of d and q , we balance the linear term of lowest order in eq. (1.1.3)
with the lowest order nonlinear term.
]8exp[
])7(exp[
2
1
qd
dqdu
(1.1.9)
and
10
]8exp[
])35(exp[
]4exp[
])3(exp[
4
3
4
32
qd
dqd
qd
dqduu
(1.1.10)
where id are known coefficients.
Balancing lowest order of Exp-function in eqs. (1.1.9) and (1.1.10), we have
)35()7( dqdq
which yield dq (1.1.11)
For simplicity, we set 1 cp and 1 dq , so eq. (1.1.5) reduces to
)exp()exp(
)exp()exp()(
10
101
bb
aaau (1.1.12)
Substituting eq. (1.1.12) into eq. (1.1.3), and by the help of Mathematical software, we have
)2exp()exp()exp()2exp()3exp((1
210123 CCCCCCA
0))3exp(3 C (1.1.13)
where,
401 ))exp()(exp( bbA
013
02100
30
31013 bakakaaakbkabaC
12100
311
2011
201
31
3111
32 24222428 akabakbaakaabakbkabakC
1
3
00
2
0100
2
1 8222 akbababaka ,
3
00
2
01101101
32
00
32
00101
3
011 6186 kabakaaakabbakbakbabbabaC
11
10
2
101
2
101
33
01
3
011010
3 55523 bakabakabakbakbababak ,
111
2
01
32
11
3
1
2
0111
32
11
2
110 43243244 babbakbakbakabakakabaC
1
2
0111
2
1
2
011
2
0
2
01
3 44444 bbabakabaakabak ,
2
101
2
10
2
10
2
010
33
01
3
101101
3
1 55618 bbaakababbakbakbbabbakC
1
3
0
2
101
3
01
2
0
2
10
3
0
2
11
3
01
2
010 523 bkabbakbakabakbakababba
11016 baaka ,
1
2
010
2
10
33
111
2
111
32
110
2
02 2422222 bbabbakbabakaakbabbaC
3
11
3
0
2
1011
2
0
2
11
3
1
2
01
3 82284 bakbakabakabakbbak ,
2
101
32
101
3
10
3
0
3
1
3
101
2
103 bbakbbabakbkababakaC ,
Equating the coefficients of )exp( n to be zero, we have
03210123 CCCCCCC (1.1.14)
Solving the system, eq. (1.14), simultaneously, we obtain
2
1
2
1
22
01
1
2
1
22
01
1
0
2
0108
)23(,
8
)23(,
3
a
akbb
a
akba
a
bkbaa
(1.1.15)
32
1 kka
where 1a and 0b are free parameters.
We, therefore, obtain the following solutions:
12
])(exp[8
)23(])(exp[
])(exp[8
)23(3])(exp[
),(32
121
21
220
032
1
321
1
21
220
1
02
0132
11
tkkakxa
akbbtkkakx
tkkakxa
akba
bkbatkkakxa
txu
])(exp[8
)23(])(exp[
3
3212
1
21
220
032
1
1
02
1
tkkakxa
akbbtkkakx
abk
a
(1.1.16)
Generally 1a , 0b and k are real numbers, and the obtained solution, eq. (1.1.16), is a generalized
solitonary solution.
In case k is an imaginary number, the obtained solitionary solution can be converted into periodic
solution or compact like solution. We write
iKk . (1.1.17)
Using the transformation
])(exp[])(exp[ 321
321 tKKaiiKxtkkakx
])(sin[])(cos[ 321
321 tKKaKxitKKaKx
and
])(exp[])(exp[ 321
321 tKKaiiKxtkkakx
])(sin[])(cos[ 321
321 tKKaKxitKKaKx
eq. (1.1.16) becomes
13
])(sin[)1(])(cos[)1(
3
),(32
1032
1
1
02
1tKKaKxpibtKKaKxp
abk
atxu
, (1.1.18)
where2
1
2
1
22
0
8
)23(
a
aKbp
,
If we search for a periodic solution or compact-like solution, the imaginary part in the denominator
of eq. (1.1.18) must be zero,
08
)23(11
2
1
2
1
22
0
a
aKbp (1.1.19)
Solving 0b from eq. (1.1.19) we obtain
)23(
821
20 aKb
(1.1.20)
Substituting eq. (1.1.20) into eq. (1.1.18) results a compact-like
121
232
1
21
22
1
238])(cos[2
)23(83
),(
aaK
tKKaKx
aKK
atxu
121
232
1
21
22
1
)23(2])(cos[
)23(23
aaK
tKKaKx
aKK
a
(1.1.21)
where 1a and K are free parameters and it requires that
221 32 Ka .
So the suggested Exp-function method can obtain easily the generalized solitionary
solution and compact-like solution for nonlinear equations.
14
CHAPTER 2
2.1 Exp-function method for solving nonlinear evolution equations
The two dimensional Bratu-type equation is given as:
0)exp( suuu yyxx (2.1.1)
and the generalized Fisher equation with higher order nonlinearity is given as:
)1( nxxt uuuu (2.1.2)
Two dimensional Bratu model stimulates a thermal reaction process in a rigid material where the
process depends on a balance between chemically generated heat addition and heat transfer by
conduction. The nonlinear reaction-diffusion equation was first introduced by Fisher as a model
for the propagation of a mutual gene. It has wide application in the fields of logistic population
growth, flame propagation, neurophysiology, autocatalytic chemical reactions and nuclear reactor
theory. It is well known that wave phenomena of plasma media and fluid dynamics are modelled
by kink-shaped and 𝑡𝑎𝑛ℎ solution or bell-shaped 𝑠𝑒𝑐ℎ solutions.
2.2 Exp-Function method and application to two dimensional Bratu type equation
Using a complex variation wykx and the transformation vs
u ln1
, we can convert eq.
(2.1.1) into ordinary differential equation
0)( 322222 svvwkvvwk (2.2.1)
According to Exp-function method, we assume that the solution of eq. (2.2.1) can be expressed in
the form
15
q
pm
m
d
cn
n
mb
na
v
)exp(
)exp(
)(
(2.2.2)
where c , p d and q are positive integers, na and mb are unknown constants.
For simplicity, we set 1 cp and 1 dq , then eq. (2.2.2) reduces to
)exp()exp(
)exp()exp()(
101
101
bbb
aaav (2.2.3)
Putting eq. (2.2.3) into eq. (2.2.1), we get
0)]4exp()exp()exp()4exp([1
41014 CCCCCA
(2.2.4)
Equating the coefficients of )exp( n in eq. (2.2.4) to be zero yields a set of algebraic equations:
0432101234 CCCCCCCCC
Solving the system of algebraic equations given above with the aid of symbolic computation
system of mathematical software, we obtain;
01 a , 00 aa , 1
20
14b
bb , 00 bb , ,
)(
0
02
0
bbksa
w
(2.2.5)
Substituting eq. (2.2.5) into eq. (2.2.3), we obtain the following exact solution
))(exp(4
)exp(
),(20
0
0
wykxb
bwykx
ayxv
(2.2.6)
So,
16
))(exp(4
)exp(
ln1
),(20
0
0
wykxb
bwykx
a
syxu (2.2.7)
Using the properties
)cosh(2))(exp()exp( wykxwykxwykx (2.2.8)
When 20 b , eq. (2.2.7) reduces to travelling wave solution as follows:
22
)2(cosh2
ln1
),(2
0
0
yksa
kx
a
syxu
(2.2.9)
2.3 The generalized Fisher equation:
The generalized Fisher equation above in eq. (2.1.2) is
)1( nxxt uuuu
Introducing the complex variation defined as ,wtkx we have
012 uuuwuk n (2.3.1)
where w and k are real parameter.
Making the transformation
nvu1
(2.3.2)
eq. (2.3.1) becomes
17
0)1( 2232222 vnvnvwnvvnkvnvk (2.3.3)
We assume that the solution of eq. (2.3.2) can be expressed in the form
)exp()exp(
)exp()exp()(
10
101
bb
aaav (2.3.4)
Substituting eq. (2.3.4) into (2.3.3) and by the help of mathematical software, equating the
coefficients of all powers of )exp( n to be zero, 4,3,...,3,4 n yields a set of algebraic equations
for 1a , 0a , 1a , 1b , 0b , 1b , k , w solving this system of algebraic equations by using mathematical
software, we obtain the following results,
Case I.
)2(2
)4(,
)2(2,,
4,1,0,0 00
20
1101
n
nnw
n
nkbb
bbaaa (2.3.5)
where 0b is a free parameter, substituting these results into eq. (2.3.4), from (2.3.2) we obtain the
following exact solution
n
wtkxb
bwtkx
wtkxtxu
1
20
0 ))(exp(4
)exp(
)exp(),(
(2.3.6)
Using the properties
)cosh(2))(exp()exp( wykxwykxwykx (2.3.7)
)sinh(2))(exp()exp( wykxwykxwykx (2.3.8)
when 20 b , eq. (2.3.5) becomes,
18
n
BtxA
BtxABtxAtxu
1
))(cosh(22
))(sinh())(cosh(),(
(2.3.9)
where )2(2
n
nA ,
)2(2
4
n
nB
Case II.
)2(2
)4(,
)2(2,,
4,0,0, 00
20
11011
n
nnw
n
nkbb
bbaaba (2.3.10)
Where 0b is a free parameter, substituting these results into eq. (2.3.4), and from (2.32) we obtain
the following exact solution
n
wtkxb
bwtkx
wtkxb
txu
1
20
0
20
))(exp(4
)exp(
))(exp(4
),(
(2.3.11)
Using the properties (2.3.7)-(2.3.8), when 20 b eq. (2.3.11) becomes
n
BtxA
BtxABtxAtxu
1
))(cosh(22
))(sinh())(cosh(),(
(2.3.12)
where )2(2
n
nA ,
)2(2
4
n
nB
19
Conclusion
In this chapter, we applied Exp-function method for obtaining the exact solutions of the two
dimensional Bratu-type equation and the generalized Fisher equation. The result show that this
method is a powerful and effective mathematical tool for solving nonlinear evolution equation in
science and engineering.
20
CHAPTER 3
Application of Exp-function to coupled Boussinesq-Burgers equations
A well-known model is the coupled Boussinesq-Burgers equations [2]
xxt vuuu2
12 , (3.1)
xxxxt uvuv )(22
1 , (3.2)
Using the transformation )(uu , )(vv , wtkx eqs. (3.1) and (3.2) become the ordinary
differential equations,
02
12 vkukuuw , (3.3)
0)(22
1 3 uvkukvw , (3.4)
On integrating eqs. (3.3) and (3.4) once with reference to 𝜂 and assuming that the constant of
integration is zero, we get
02
12 kvkuwu (3.5)
022
1 3 kuvukwv (3.6)
According to Exp-function method, we assume that the solution of eqs. (3.3) and (3.4) can be
expressed in the form,
21
)exp()exp(
)exp()exp(
)exp(
)exp(
)(
pbqb
haga
mb
na
upq
hg
p
qm
m
h
gn
n
(3.7)
)exp()exp(
)exp()exp(
)exp(
)exp(
)(
tcsc
jdid
mc
nd
vts
ji
t
sm
m
j
in
n
(3.8)
Where h , g , p , q , j , i , t and s are positive integers na , mb , nd and mc are unknown constants.
For simplicity, we set 1,1 qgph and 1,1 sitj , then eqs. (3.7) and (3.8) reduced to
)exp()exp(
)exp()exp()(
01
101
bb
aaau (3.9)
)exp()exp(
)exp()exp()(
01
101
cc
dddv (3.10)
Substituting eqs. (3.9) and (3.10) into eqs. (3.5) and (3.6), we have
)2exp()exp()exp()2exp()3exp({1
210123 A
0)}3exp(3 (3.11)
)exp()exp()2exp()3exp()4exp({1
101234 B
0)}4exp()3exp()2exp( 432 (3.12)
where
)exp()exp()exp()exp( 01
2
01 ccbbA
22
)exp()exp()exp()exp( 01
3
01 ccbbB
1211111
213
2
1 dkbcbwacka ,
0211010110
211011101012
2
12 dkbdbkbcbwackacbwacbwacaka ,
0011001111111201111
211 22 cakacbwacbwacakackacwabwaka
1
2
10011
2
0110010102
1
2
1dkbdbkbdkbdkbcbwacbwa ,
01101110100110010 2242224(2
1cwacbwacakacwabwabwaaka
02001100000110110
20 222242 dkbdkbdkbcbwacbwacakacka
)2 101 dbkb ,
0100012111001111
2011 22 cakacwackacwabwabwaakakawa
12011001001
2
1
2
1dkbdkbdkbkdcbwa
,
10002101011002
2
12 dkbkdckacwabwaakawa ,
12113
2
1kdkawa ,
1311
2114 2 dwbdbka ,
102110111
2101011
31
210
33 342
2
1
2
1 dbwbdbbkadbkacbbakcbak
23
0310
2112 dwbdbka
0
2
10
3
1
2
01
3
1010
3
1
2
11
3
111
3
22
1
2
1
2
122 cbakcbakcbbakcbakcbak
10101
2
111
2
11110011
3 42342
1dbbkadbkadwbdbkacbbak
002100110
2101
2011
201 34232 dbwbdbbkadbkadbwbdbka
1
3
11
2
112 dwbdbka ,
1011
3
101
3
110
3
011
32
10
3
12
3
2
33
2
1
2
1cbbakcbakcbakbbakbak
1100
2
01
3
0010
3
0
2
11
3
011
3 42
1
2
122 dbkacbakcbbakcbakcbak
0111
3
01
2
001011101101 42464 dbkadwbdbkadbbkadbwbdbka
12100
2010
20100100
2110
21 232423 dbkadbwbdbkadbbkadbkadwb
10
2
11011 34 dbwbdbbka ,
111
3
11
32
01
3
010
32
11
3
11
3
0 4444(2
1cbakcakbakbbakbakbak
1100113
0013
0103
1201
3100
3 4336 dkacbbakcbakcbakcbakcbak
0101
2
011
2
010011111 846886 dbkadbkadwbdbkadbkadwb
1110
3
00
2
000011001001 8248128 dbkadwbdbkadbbkadbwbdbka
)64846 12011
20110101
2111
21 dbwbdbkadbbkadbkadwb
24
013
1013
103
0113
013
103
1 22
1
2
1
2
3
2
33 cakcbakcakbbakbakbak
10110100201
3000
3011
3 4322
1
2
12 dbkadwbdkacbakcbakcbak
1100
2
010
3
00000110101 4234432 dbkadbkadwbdbkadbkadwbdka
1
3
01
2
001011101101 2464 dwbdbkadbbkadbwbdbka
10013
0032
013
003
113
13
22
1
2
1
2
1
2
122 wdcbakcakbakbakbakak
1111111001000011 4324322 dbkadwbdkadbkadwbdkadka
1
2
011
2
0100 234 dbkadwbdbka
101101001001
3
0
3
3 43222
1
2
1dbkadwbdkadkawdbakak
1114 2 dkawd
By setting,
03210123 , and
0432101234 ,
and solving these system of algebraic equations simultaneously with the aid of symbolic
computation system of mathematical software, we obtain the following results,
25
Case I.
,,0,,0,0,0,2
,0 2
011001110
01 bcdwbddbabw
aa
wkbc 2,2 00 ,
Substituting these values into eqs. (3.9) and (3.10), we get:
,)]sinh()cosh([2
),(
0
0
wtkxwtkxb
bwtxu
)sinh()sinh()cosh()cosh(2),(
200
0
wtkxwtkxwtkxbwtkxb
wbtxv
,
Fig. 3(a). Soliton solutions of eqs. (3.9) and (3.10) in case I, when 10 wb .
Case II.
,,0,,0,0,0,2
,0 2
011001110
01 bcdwbddbabw
aa
wkbc 2,2 00
26
Substituting these values into eqs. (3.9) and (3.10), we get:
)sinh()cosh(2
),(
0
0
wtkxwtkxb
bwtxu
,
)sinh()sinh()cosh()cosh(2
),(200
0
wtkxwtkxwtkxbwtkxb
wbtxv
Fig. 3(b). Soliton solutions of eqs. (3.9) and (3.10) in case II, when 10 wb .
Case III.
,,0,,0,0,2
,0,0 20110011101 bcdwbddb
waaa
,2,2 00 wkbc
Substituting these values into eqs. (3.9) and (3.10), we get:
,
)sinh()cosh(2
)sinh()cosh(),(
0 wtkxwtkxb
wtkxwtkxwtxu
)sinh()sinh()cosh()cosh(2),(
200
0
wtkxwtkxwtkxbwtkxb
wbtxv
,
27
Fig. 3(c). Soliton solutions of eqs. (3.9) and (3.10) in case III, when 10 wb .
Case IV.
,,0,,0,0,2
,0,0 20110011101 bcdwbddb
waaa
,2,2 00 wkbc
Substituting these values into eqs. (3.9) and (3.10), we get:
,
)sinh()cosh(2
)sinh()cosh(),(
0 wtkxwtkxb
wtkxwtkxwtxu
)sinh()sinh()cosh()cosh(2),(
200
0
wtkxwtkxwtkxbwtkxb
wbtxv
28
Fig. 3(d). Soliton solutions of eqs. (3.9) and (3.10) in case IV, when 10 wb .
Case V.
,0,0],)4(22[4
1,
211
20100
11
dawbwbbwa
bwa
,0,2
)4(1
201
23
02
0
dw
bbwwbwd
,
)4(
))4()(3[
201
20
25
201
201
230
25
012
5
1
bbwwbw
bbwbbwbwbbwc
wk
bbwwbw
bbwbwbwbwc 2,
)4(
])4(2[2
201
20
25
2010
220
25
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
)22sinh()22cosh()sinh()cosh(22
)cosh()sinh()4(2
),(
01
20101
wtkxwtkxwtkxwtkxbb
wtkxwtkxwbwbbwbw
txu
,
29
)}sinh()}{cosh(4224{),( 2
010201 wtkxwtkxwbwbbwwbwbwtxv
201
201
201
3001 244)(32
1
bwbwwbwbbbbwbbw
0
2010 {)}sinh(){cosh(42 bwwtkxwtkxwbwbb
)}22sinh()22}{cosh(4 2
01 wtkxwtkxwbwb
Fig. 3(e). Soliton solutions of eqs. (3.9) and (3.10) in case V, when 1,1,2 10 wbb .
Case VI.
,0,0],)4(22[4
1,
211
2
01001
1
dawbwbbwabw
a
,0,2
)4(1
2
012
3
0
2
0
dw
bbwwbwd
30
,
)4(
)4()(3
201
20
25
201
201
230
25
012
5
1
bbwwbw
bbwbbwbwbbw
c
wk
bbwwbw
bbwbwbwbw
c 2,
)4(
)4(22
201
20
25
2010
220
25
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
)22sinh()22cosh()sinh()cosh(22
)cosh()sinh()4(2
),(
01
20101
wtkxwtkxwtkxwtkxbb
wtkxwtkxwbwbbwbw
txu
,
)sinh()cosh(4224),( 2
010201 wtkxwtkxwbwbbwwbwbwtxv
201
201
201
3001 244)(32
1
bwbwwbwbbbbwbbw
0
2010 )}sinh(){cosh(42 bwwtkxwtkxwbwbb
)}22sinh()22{cosh(4 2
01 wtkxwtkxwbwb
31
Fig. 3(f). Soliton solutions of eqs. (3.9) and (3.10) in case VI, when 1,1,2 10 wbb .
Case VII.
,0,0],)4(22[4
1,
211
20100
11
dawbwbbwa
bwa
,0,2
)4(1
2
012
3
0
2
0
dw
bbwwbwd
,
)4(
)4()(3
201
20
25
201
201
230
25
012
5
1
bbwwbw
bbwbbwbwbbw
c
wk
bbwwbw
bbwbwbwbw
c 2,
)4(
)4(22
201
20
25
2010
220
25
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
)22sinh()22cosh()}sinh(){cosh(22
)cosh()sinh(42
),(
01
20101
wtkxwtkxwtkxwtkxbb
wtkxwtkxwbwbbwbw
txu
,
32
)}()(}{4224{),( 2
010201 wtkxSinhwtkxCoshwbwbbwwbwbwtxv
201
201
201
3001 24(4)(32
1
bwbwwbwbbbbwbbw
02010 ())sinh())(cosh(42 bwwtkxwtkxwbwbb
))22sinh()22)(cosh(4 2
01 wtkxwtkxwbwb
Fig. 3(g). Soliton solutions of eqs. (3.9) and (3.10) in case VII, when 1,1,0 10 wbb .
Case VIII.
,0,0],)4(22[4
1,
211
2
01001
1
dawbwbbwabw
a
,0,2
)4(1
2
012
3
0
2
0
dw
bbwwbwd
33
,
)4(
)4()(3
201
20
25
201
201
230
25
012
5
1
bbwwbw
bbwbbwbwbbw
c
wk
bbwwbw
bbwbwbwbw
c 2,
)4(
)4(22
201
20
25
2010
220
25
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
)22sinh()22cosh()sinh()cosh(22
)cosh()sinh(42
),(
01
20101
wtkxwtkxwtkxwtkxbb
wtkxwtkxwbwbbwbw
txu
)sinh()cosh(4224),( 2
010201 wtkxwtkxwbwbbwwbwbwtxv
201
201
201
3001 24(4)(32
1
bwbwwbwbbbbwbbw
02010 ())sinh())(cosh(42 bwwtkxwtkxwbwbb
))22sinh()22)(cosh(4 2
01 wtkxwtkxwbwb
34
Fig. 3(h). Soliton solutions of eqs. (3.9) and (3.10) in case VIII, when 1,1,2 10 wbb .
Case IX.
,0,2
),)4(22(4
1,0 11
2
01001 dw
awbwbbwaa
,0,2
)4(1
2
012
3
0
2
0
dw
bbwwbwd
,
)4(
))4()(3(
201
20
25
201
201
230
25
012
5
1
bbwwbw
bbwbbwbwbbwc
wkbbwwbw
bbwbwbwbwc 2,
))4((
))4(2(2
2
01
2
02
5
2
010
22
02
5
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
){cosh(2}4)}{sinh(){cosh(),( 2
010 wtkxwwbwbbwwtkxwtkxtxu
35
)}sinh(){cosh([22
1)}sinh(
01 wtkxwtkxbbwtkx
)]22sinh()22cosh( wtkxwtkx
)sinh()cosh(4224),( 2
010201 wtkxwtkxwbwbbwwbwbwtxv
201
201
201
3001 24{4)(32
1
bwbwwbwbbbbwbbw
}4{)}sinh()}{cosh(42 2010
2010 wbwbbwwtkxwtkxwbwbb
)}22sinh()22{cosh( wtkxwtkx
Fig. 3(i). Soliton solutions of eqs. (3.9) and (3.10) in case IX, when 1,1,2 10 wbb .
Case X.
,0,2
),)4(22(4
1,0 11
2
01001 dw
awbwbbwaa
36
,0,2
)4(1
2
012
3
0
2
0
dw
bbwwbwd
,)4(
))4()(3(
2
01
2
02
5
2
01
2
01
23
02
5
012
5
1
bbwwbw
bbwbbwbwbbwc
wkbbwwbw
bbwbwbwbwc 2,
))4((
))4(2(2
2
01
2
02
5
2
010
22
02
5
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
){cosh(24)}{sinh()[{cosh(),( 2010 wtkxwwbwbbwwtkxwtkxtxu
)}sinh(){cosh([22
1)}}]sinh(
01 wtkxwtkxbbwtkx
)]22sinh()22cosh( wtkxwtkx
)}sinh()}{cosh(4224{),( 2
010201 wtkxwtkxwbwbbwwbwbwtxv
201
201
201
3001 24{4)(32
1
bwbwwbwbbbbwbbw
}4{)}sinh()}{cosh(42 2010
2010 wbwbbwwtkxwtkxwbwbb
)}22sinh()22cosh({ wtkxwtkx ,
37
Fig. 3(j). Soliton solutions of eqs. (3.9) and (3.10) in case X, when 1,1,2 10 wbb .
Case XI.
,0,2
),)4(22(4
1,0 11
2
01001 dw
awbwbbwaa
,0,2
)4(1
2
012
3
0
2
0
dw
bbwwbwd
,
)4(
)4()(3
201
20
25
201
201
230
25
012
5
1
bbwwbw
bbwbbwbwbbw
c
wk
bbwwbw
bbwbwbwbw
c 2,
)4(
)4(22
201
20
25
2010
220
25
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
){cosh(2}4)}{sinh(){cosh(),( 2
010 wtkxwwbwbbwwtkxwtkxtxu
38
)}sinh(){cosh(22
1)}sinh(
01 wtkxwtkxbbwtkx
)22sinh()22cosh( wtkxwtkx
)}sinh()}{cosh(4224{),( 2
010201 wtkxwtkxwbwbbwwbwbwtxv
201
201
201
3001 24{4)(3{2
1
bwbwwbwbbbbwbbw
}4{)}sinh()}{cosh(42 2010
2010 wbwbbwwtkxwtkxwbwbb
)}22sinh()22{cosh( wtkxwtkx ,
Fig. 3(k). Soliton solutions of eqs. (3.9) and (3.10) in case XI, when 1,1,0 10 wbb
Case XII.
,0,2
),)4(22(4
1,0 11
2
01001 dw
awbwbbwaa
39
,0,2
)4(1
2
012
3
0
2
0
dw
bbwwbwd
,
)4(
)4()(3
201
20
25
201
201
230
25
012
5
1
bbwwbw
bbwbbwbwbbw
c
wk
bbwwbw
bbwbwbwbw
c 2,
))4((
)4(22
201
20
25
2010
220
25
12
5
0
,
Substituting these values into eqs. (3.9) and (3.10), we get:
){cosh(24)}{sinh()[{cosh(),( 2010 wtkxwwbwbbwwtkxwtkxtxu
)}sinh(){cosh([22
1)}}]sinh(
01 wtkxwtkxbbwtkx
)}]22sinh()22{cosh( wtkxwtkx
)}sinh()}{cosh(4224{),( 2
010201 wtkxwtkxwbwbbwwbwbwtxv
201
201
201
3001 24{4)(32
1
bwbwwbwbbbbwbbw
}4{)}sinh()}{cosh(42 2010
2010 wbwbbwwtkxwtkxwbwbb
)}22sinh()22{cosh( wtkxwtkx ,
40
Fig. 3(l). Soliton solutions of eqs. (3.9) and (3.10) in case XII, when 1,1,0 10 wbb .
CONCLUSION
In this chapter, we have successfully implemented Exp-function method developed by He and Wu
to derive the exact solutions of the coupled Boussinesq-Burgers equation. The results demonstrate
that Exp-function method is straightforward and concise mathematical tool to establish analytical
solutions (both solitary and periodic) on nonlinear evolution equations. Therefore, we hope that
this method can be more effectively used to investigate other nonlinear evolution equations which
are frequently take place in engineering, applied mathematics and physical sciences.
BIBLIOGRAPHY
[1] J. H. He, X. H. Wu, “Exp-function method for nonlinear wave equations”, Chaos, Solitons
and Fractals, 30 (2006) 700-708.
[2] P. Wang, B. Tian, W. J. Liu, X. L., Y. Jiang, “Lax Pair, Backlund transformation and multi-
soliton solutions for the Boussinesq-Burgers equations from shallow water waves”,
Applied Mathematics and Computation 218(2011) 1726-1734.
41
[3] A. E. H, Ebaid, “Generalization of He’s Exp-function method and new exact solutions for
Burgers equation”, 2008, Available online at
http://www.znaturforsch.com/aa/v64a/64a0604.pdf
[4] J. M. Kim and C. Chun, “New exact solutions to KdV-Burgers-Kuramoto equation with
the Exp-function method”, Hindawi Publishing Corporation, Abstract and Applied
Analysis, (2012), Article ID 892420.
[5] S. T. Mohyud.-Din, M. Aslam Noor and K. Inayat Noor, “Exp-function method for solving
higher-order boundary value problems”, Bulletin of the Institute of Mathematics, Academia
Sinica (New Series), 4 (2009), 219-234.
[6] E. Misirli and Y. Gurefe, “Exp-function method for solving nonlinear evolution equations”,
Mathematical and computation applications, 16 (2011), 258-266.
[7] S. T. Mohyud-Din, M Aslam Noor, K. Inayat Noor, “Exp-function method for traveling
wave solutions of modified Zakharov-Kuznetsov equation”, Journal of King Saud
University (Science), 22 (2010), 213-216.