A generalized expansion method for the loaded Burgers equation Introduction

Exact Solutions of loaded Burgers equation

In this section, we will show how to find the exact solution of the loaded modifiedKorteweg-de Vries equationusing the -expansion method.For doing this, we perform the stepsabove for equation (1). The travelling wave variable below

, , (7)
permits us convertingequation (1) into an ordinary differential equationfor
, (8)
integrating it with respect to once yields
, (9)
where is an integration constant that can be determined later.
We express the solution of equation (9) in the form of a polynomial in below
, (10)
where satisfies the second order ordinary differential equation in the form
. (11)
Using (10) and (11), and are easily derivedto
, (12)
. (13)
Considering the homogeneous balance between and in equation (9), based on (12) and (13) we required that . Taking into account considerations, the form of is as following
, (14)
then we know the exact view of
, (15)
using (14) and (11) iseasily derived to
. (16)
By substituting (14)-(16) into equation (9) and collecting allterms with the same power of , the left-hand sideof equation (9) is converted into another polynomial in .

. (17)
Equating each coefficient of expression (17) to zero,yields a set of simultaneous equationsfor , , and as following:
: ,
: ,
: .
By solvingthese the equations, we obtain the followings
,
,
, (18)
, , and are arbitrary constants.
Using (18), expression (14) can be rewritten as
, (19)
where .The function(19) is a solution of equation (9), provided that the integration constant in equation (9) is taken as that in (18).Substituting the general solutions of equation (11) into (19), we have three types of travelling wave solutions of the loaded Burgers equation (1) as follows:
When ,
, (20)
where , , and are arbitrary constants.It is obvious that the function can be easily found based on expression (20).
For example, let isgiven as below
,
where are constants, if , and ,then becomes
(21)
The function (21) is the solution of the following loaded Burgers equation.
.
When ,
, (22)
where , , , and are arbitrary constants.It is clear that it is not difficult for us to find based on expression (22). Let isgiven as below
,
where are constants, in particular, if and , then becomes
. (23)
The function (23) is the solution of the following loaded Burgers equation.
.
When ,
, (24)
where , , , and are arbitrary constants. The function is found based on expression (24).
If , , and isgiven as below
,
then becomes
. (25)
We know that the function (25) satisfies the following loaded Burgers equation.
.