Exact solutions of two nonlinear partial differential equations by using the first integral method
© Jafari et al.; licensee Springer. 2013
Received: 22 August 2012
Accepted: 12 April 2013
Published: 7 May 2013
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.
Keywordsfirst integral method double sine-Gordon equation Burgers equation exact solutions
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 , 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
If we can find the integrals of (6) under the same conditions, the qualitative theory of differential equations  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 )
Let and be polynomials of two variables x and y in , and let be irreducible in . If vanishes at all zero points of , then there exists a polynomial in such that .
The division theorem follows immediately from the Hilbert-Nullstellensatz theorem .
Theorem 2.2 (Hilbert-Nullstellensatz theorem)
Every ideal γ of not containing 1 admits at least one zero in .
Let , be two elements of ; for the set of polynomials of zero at x to be identical with the set of polynomials of zero at y, it is necessary and sufficient that there exists a K-automorphism S of L such that for .
For an ideal α of to be maximal, it is necessary and sufficient that there exists an x in such that α is the set of polynomials of zero at x.
For a polynomial Q of to be zero on the set of zeros in of an ideal γ of , it is necessary and sufficient that there exists an integer such that .
3.1 Exact solutions to the double sine-Gordon equation
where is an arbitrary constant and .
where is an arbitrary constant.
3.2 Exact solutions to the Burgers equation
is one of the most famous nonlinear diffusion equations. The positive parameter a refers to a dissipative effect.
where B is an arbitrary integration constant.
where is an arbitrary constant.
where is an arbitrary constant and .
where is an arbitrary constant and .
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.
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