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This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

Open Access Research

Exact solutions of two nonlinear partial differential equations by using the first integral method

Hossein Jafari12*, Rahmat Soltani1, Chaudry Masood Khalique2 and Dumitru Baleanu345

Author affiliations

1 Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran

2 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa

3 Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Çankaya University, Balgat, Ankara, 0630, Turkey

4 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia

5 Institute of Space Sciences, P.O. Box MG-23, Magurele, Bucharest, 76900, Romania

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Citation and License

Boundary Value Problems 2013, 2013:117  doi:10.1186/1687-2770-2013-117


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/117


Received:22 August 2012
Accepted:12 April 2013
Published:7 May 2013

© 2013 Jafari et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In recent years, many approaches have been utilized for finding the exact solutions of nonlinear partial differential equations. One such method is known as the first integral method and was proposed by Feng. In this paper, we utilize this method and obtain exact solutions of two nonlinear partial differential equations, namely double sine-Gordon and Burgers equations. It is found that the method by Feng is a very efficient method which can be used to obtain exact solutions of a large number of nonlinear partial differential equations.

Keywords:
first integral method; double sine-Gordon equation; Burgers equation; exact solutions

1 Introduction

With the availability of symbolic computation packages like Maple or Mathematica, the search for obtaining exact solutions of nonlinear partial differential equations (PDEs) has become more and more stimulating for mathematicians and scientists. Having exact solutions of nonlinear PDEs makes it possible to study nonlinear physical phenomena thoroughly and facilitates testing the numerical solvers as well as aiding the stability analysis of solutions. In recent years, many approaches to solve nonlinear PDEs such as the extended tanh function method [1-6], the modified extended tanh function method [7,8], the exp-function method [9-11], the Weierstrass elliptic function method [12], the Laplace decomposition method [13,14] and so on have been employed.

Among these, the first integral method, which is based on the ring theory of commutative algebra, due to Feng [15-19] has been applied by many authors to solve different types of nonlinear equations in science and engineering [20-23]. Therefore, in the present article, the first integral method is applied to analytic treatment of some important nonlinear of partial differential equations.

The rest of this article is arranged as follows. In Section 2, the basic ideas of the first integral method are expressed. In Section 3, the method is employed for obtaining the exact solutions of double sine-Gordon (SG) and Burgers equations, and finally conclusions are presented in Section 4.

2 The first integral method

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M1">View MathML</a>

(1)

be a nonlinear partial differential equation (PDE) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M2">View MathML</a> as its solution. We introduce the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M3">View MathML</a>

(2)

where c is constant. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M4">View MathML</a>

(3)

Thus PDE (1) is then transformed to the ordinary differential equation (ODE)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M5">View MathML</a>

(4)

We now introduce a new transformation, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M6">View MathML</a>

(5)

and this gives us the system of ODEs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M7">View MathML</a>

(6)

If we can find the integrals of (6) under the same conditions, the qualitative theory of differential equations [24] tells us that the general solutions of (6) can be obtained directly. But in general, it is very difficult even for a single first integral. Since for a plane autonomous system, there is no methodical theory which gives us first integrals, we will therefore apply the division theorem to find one first integral (6), which will reduce (4) to a first-order integral for an ordinary differential equation. By solving this equation, exact solutions of (1) will be obtained. We recall the division theorem.

Theorem 2.1 (Division theorem, see [25])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M8">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M9">View MathML</a>be polynomials of two variablesxandyin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M10">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M8">View MathML</a>be irreducible in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M10">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M9">View MathML</a>vanishes at all zero points of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M8">View MathML</a>, then there exists a polynomial<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M15">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M10">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M17">View MathML</a>.

The division theorem follows immediately from the Hilbert-Nullstellensatz theorem [26].

Theorem 2.2 (Hilbert-Nullstellensatz theorem)

LetKbe a field andLbe an algebraic closure ofK. Then:

(i) Every idealγof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18">View MathML</a>not containing 1 admits at least one zero in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19">View MathML</a>.

(ii) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M21">View MathML</a>be two elements of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19">View MathML</a>; for the set of polynomials of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18">View MathML</a>zero atxto be identical with the set of polynomials of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18">View MathML</a>zero aty, it is necessary and sufficient that there exists a K-automorphismSofLsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M25">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M26">View MathML</a>.

(iii) For an idealαof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18">View MathML</a>to be maximal, it is necessary and sufficient that there exists anxin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19">View MathML</a>such thatαis the set of polynomials of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18">View MathML</a>zero atx.

(iv) For a polynomialQof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18">View MathML</a>to be zero on the set of zeros in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M19">View MathML</a>of an idealγof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M18">View MathML</a>, it is necessary and sufficient that there exists an integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M33">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M34">View MathML</a>.

3 Applications

3.1 Exact solutions to the double sine-Gordon equation

Consider the double sine-Gordon (SG) equation [27,28]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M35">View MathML</a>

(7)

In order to apply the first integral method described in Section 2, we first introduce the transformations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M36">View MathML</a>

(8)

Using (2) and (3), Eq. (7) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M37">View MathML</a>

(9)

We next use the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M38">View MathML</a>

(10)

We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M39">View MathML</a>

(11)

Next, we introduce new independent variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M41">View MathML</a> which change (11) to the system of ODEs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M42">View MathML</a>

(12)

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M43">View MathML</a>, then (12) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M44">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M45">View MathML</a>.

According to the first integral method, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M47">View MathML</a> are nontrivial solutions of Eq. (13) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M48">View MathML</a>

is an irreducible polynomial in the complex domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M49">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M50">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M51">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M52">View MathML</a>) are polynomials in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M53">View MathML</a>. Equation (14) is called the first integral to Eq. (13). Applying the division theorem, one sees that there exists a polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M54">View MathML</a> in the complex domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M55">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M56">View MathML</a>

(15)

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M57">View MathML</a> in (14), and then, by comparing with the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M58">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M59">View MathML</a>) on both sides of (15), we have

(16)

(17)

(18)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63">View MathML</a> is a polynomial in X, from (16) we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63">View MathML</a> is a constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M65">View MathML</a>. For simplicity, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M66">View MathML</a>. Then Eq. (17) indicates that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M67">View MathML</a>. Thus, from Eq. (18) we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M68">View MathML</a>. Now suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M69">View MathML</a>

(19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M75">View MathML</a> are all constants to be determined. Substituting Eq. (19) into Eq. (17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M76">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M77">View MathML</a>

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80">View MathML</a> in (18) and setting all the coefficients of powers X to be zero, we obtain a system of nonlinear algebraic equations, and by solving it, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M81">View MathML</a>

(20)

Substituting (20) in (14), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M82">View MathML</a>

(21)

Combining Eq. (21) with (13), second-order differential Eq. (11) can be reduced to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M83">View MathML</a>

(22)

Solving Eq. (22) directly and changing to the original variables, we obtain the exact solutions to Eq. (7):

(23)

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M86">View MathML</a> is an arbitrary constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M45">View MathML</a>.

Therefore, the exact solutions to the double sine-Gordon (SG) equation can be written as

(25)

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M86">View MathML</a> is an arbitrary constant.

3.2 Exact solutions to the Burgers equation

The Burgers equation [29]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M91">View MathML</a>

(27)

is one of the most famous nonlinear diffusion equations. The positive parameter a refers to a dissipative effect.

Using (2) and (3), Eq. (27) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M92">View MathML</a>

(28)

We rewrite (28) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M93">View MathML</a>

(29)

By introducing new variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M95">View MathML</a>, Eq. (29) changes into a system of ODEs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M96">View MathML</a>

(30)

Now, the division theorem is employed to seek the first integral to (30). Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M98">View MathML</a> are nontrivial solutions to (30), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M99">View MathML</a> is an irreducible polynomial in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M100">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M101">View MathML</a>

(31)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M51">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M52">View MathML</a>) are polynomials in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M53">View MathML</a>. Equation (31) is called the first integral to Eq. (30). Due to the division theorem, there exists a polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M54">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M49">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M107">View MathML</a>

(32)

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M57">View MathML</a> in (31). By comparing the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M58">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M59">View MathML</a>) on both sides of (32), we have

(33)

(34)

(35)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63">View MathML</a> is a polynomial in X, from (33) we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63">View MathML</a> is a constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M116">View MathML</a>. For simplicity, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M66">View MathML</a>, and balancing the degrees of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M120">View MathML</a> or 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M121">View MathML</a>, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M122">View MathML</a>, then from (34), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M123">View MathML</a>

where B is an arbitrary integration constant.

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80">View MathML</a> in (35) and setting all the coefficients of powers X to be zero, we obtain a system of nonlinear algebraic equations, and by solving it, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M126">View MathML</a>

(36)

Using (36) in (31), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M127">View MathML</a>

(37)

Combining Eq. (37) with the first part of (30), we obtain the exact solutions of Eq. (29) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M128">View MathML</a>

(38)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129">View MathML</a> is an arbitrary constant.

Therefore, the exact solutions to the Burgers equation can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M130">View MathML</a>

(39)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129">View MathML</a> is an arbitrary constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M132">View MathML</a>.

Now suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M133">View MathML</a>. By an application of the division theorem, we can conclude that there exists a polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M134">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M49">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M136">View MathML</a>

(40)

Comparing the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M58">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M138">View MathML</a>) of both sides of (40) yields

(41)

(42)

(43)

(44)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M143">View MathML</a> is a polynomial of X, from (41) we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M143">View MathML</a> is a constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M145">View MathML</a>. For simplicity, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M146">View MathML</a>, and balancing the degrees of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M63">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M121">View MathML</a> or 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M121">View MathML</a>, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M152">View MathML</a>. Then from (42), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M153">View MathML</a>

where B is an arbitrary constant of integration. From (43) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M154">View MathML</a>

where D is an arbitrary constant of integration. Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M80">View MathML</a> in (44) and setting all the coefficients of powers of X to zero, we obtain a system of nonlinear algebraic equations. Solving these equations, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M157">View MathML</a>

(45)

Now using (45) in (31), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M158">View MathML</a>

(46)

Combining Eq. (46) with the first part of (30), we obtain the exact solutions to Eq. (29) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M159">View MathML</a>

(47)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129">View MathML</a> is an arbitrary constant of integration. Therefore, the exact solutions to the Burgers equation can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M161">View MathML</a>

(48)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M129">View MathML</a> is an arbitrary constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/117/mathml/M132">View MathML</a>.

4 Conclusions

The first integral method was employed successfully to solve some important nonlinear partial differential equations, including the double sine-Gordon and Burgers equations, analytically. Some exact solutions for these equations were formally obtained by applying the first integral method. Due to the good performance of the first integral method, we feel that it is a powerful technique in handling a wide variety of nonlinear partial differential equations. Also, this method is computerizable, which permits us to accomplish difficult and tiresome algebraic calculations on a computer with ease.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the manuscript was realized in collaboration with the same responsibility. All authors read and approved the final version of the manuscript.

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