1. Introduction

In the recent years, seeking exact solutions of nonlinear partial differential equations (NLPDEs) is of great significance, since the nonlinear complex physical phenomena related to the NLPDEs are involved in many fields from physics, mechanics, biology, chemistry and engineering. As mathematical models of the phenomena, the investigation of exact solutions of NLPDEs will help one to understand the mechanism that governs these physical models or to better provide knowledge of the physical problem and possible applications. To this aim, a vast variety of powerful and direct methods for finding the exact significant solutions of NLPDEs though it is rather difficult have been derived. Some of the most important methods are Bäcklund transformation ^{[1, 2, 3]}, Hirota’s method ^[4], tanh-function method ^{[5, 6, 7, 8]}, extended tanh-function method ^{[9, 10, 11, 12]}, homogeneous balance method ^{[13, 14]}, variational iteration methods ^{[15, 16]}, collocation method ^{[17, 18, 19]}, Adomian Padé approximation ^[20], Jacobi elliptic function expansion method ^[21], F-expansion method ^[22], auxiliary equation method ^{[23, 24, 25]}, Fan sub-equation method ^{[26, 27, 28, 29, 30]}, extended Fan sub-equation method ^{[31, 32]}, modified extended Fan sub-equation method ^{[33, 34, 35]} and so on. Recently, Sirendaoreji ^{[36, 37]} proposed a new auxiliary equation method by introducing a new first order nonlinear ordinary differential equation (NLODE) with six-degree nonlinear terms and its solutions to construct exact travelling wave solutions of NLPDEs in a unified way. Later, Zhang and Xia ^[38] improved this method and obtained new formal solutions of some NLPDEs.

The aim of this letter is to give a new auxiliary function method. And using the auxiliary function method to solve a class of reaction-diffusion equation.

(1)

and BBM-Burgers equation

(2)

2. Auxiliary Function Method

To illustrate the basic concepts of a generalized auxiliary equation method, we consider a given PDE in two independent variables and dependent variable :

Step 1. We use the wave transformation , and reduce the given NLPDE

(3)

To the following ODE

(4)

Step 2. We seek for the solution of Eq. (4) in the following generalized ansätze

(5)

with satisfying the following new auxiliary equation

(6)

where the positive integer can be determined by balancing the highest-order derivative term with the nonlinear terms ^[37], , and are constants to be determined and satisfies the variable separated ODE (6), where and are real constants. We have found that the auxiliary equation (6) possesses several types of following solutions

(7)

where and are undermined constants.

Step 3. Substitute ansatz (5) along with Eq. (6) into (4) and equate the coefficients of all powers of to zero yields a set of algebraic equations for unknowns and.

Step 4. Solve the set of algebraic equations by use of MAPLE can permit obtention of explicit expressions of and.

Step 5. Obtain exact solutions. By using the results obtained in the above steps, we can derive a series of travelling wave solutions of Eq. (3) depending on the solution of Eq. (6). Selecting appropriate and substituting it into the travelling wave solutions Eq. (5), we can obtain exact solutions of Eq. (3).

3. Application

As an application, we use the new auxiliary function method to solve a class of reaction-diffusion equation and BBM-Burgers equation.

We make transformation and change Eq. (1) into the form

(8)

Now balancing and , we obtain This suggests that

(9)

where and are constants to be determined. Substitution of (9) into (8), we have:

(10)

Substitution of (6) into (10), and equate the coefficients of all powers of to zero, we get a set of algebraic equation as following:

(11)

Solving the resulting algebraic system (11), we get the two group solutions

Case 1.

(12)

Substituting (12) and (7) into (9), we get the solutions of Eq. (1)

(13)

Figure 1. as

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Figure 2. as

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Figure 3. as , ,

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Case 2.

(14)

Substituting (14) and (7) into (9), we get the solutions of Eq. (1)

(15)

We make transformation and change Eq. (2) into the form

(16)

Now balancing and , we obtain This suggests that

(17)

where and are constants to be determined. Substitution of (17) into (16), we have:

(18)

Substitution of (6) into (18), and equate the coefficients of all powers of to zero, we get a set of algebraic equation, and solving the algebraic system, we get the four group solutions

Case 1.

(19)

Substituting (19) and (7) into (17), we get the solutions of Eq. (2)

(20)

Case 2.

(21)

Substituting (21) and (7) into (17), we get the solutions of Eq. (2)

(22)

Case 3.

(23)

Substituting (23) and (7) into (17), we get the solutions of Eq. (2)

(24)

Case 4.

(25)

Substituting (25) and (7) into (17), we get the solutions of Eq. (2)

(26)

4. Conclusion

In this letter, a new auxiliary function method is given, and some exact solutions of the auxiliary function are given too. Reaction-diffusion equation and BBM-Burgers equation are solved by using the new auxiliary function method. Obviously, the auxiliary function method can be applied to solve other type of nonlinear partial differential equations as well.

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