Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application

Yang, Xiao-Feng; Wei, Yi

doi:https://doi.org/10.1155/2020/4912159

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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Research Article | Open Access

Volume 2020 | Article ID 4912159 | https://doi.org/10.1155/2020/4912159

Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application

Xiao-Feng Yang¹and Yi Wei²

Academic Editor: Liguang Wang

Received04 Mar 2020

Accepted06 Apr 2020

Published28 Apr 2020

Abstract

The homogeneous balance of undetermined coefficient method is firstly proposed to derive a more general bilinear equation of the nonlinear partial differential equation (NLPDE). By applying perturbation method, subsidiary ordinary differential equation (sub-ODE) method, and compatible condition to bilinear equation, more exact solutions of NLPDE are obtained. The KdV equation, Burgers equation, Boussinesq equation, and Sawada-Kotera equation are chosen to illustrate the validity of our method. We find that the underlying relation among the -expansion method, Hirota’s method, and HB method is a bilinear equation. The proposed method is also a standard and computable method, which can be generalized to deal with other types of NLPDE.

1. Introduction

The nonlinear partial differential equation (NLPDE) is known to describe a wide variety of phenomena not only in physics but also in biology, chemistry, and several other fields [1–3]. The investigation of the exact solutions for NLPDE plays an important role on the study of nonlinear physical phenomena [4–9]. In recent years, many powerful methods are used to obtain the exact solutions of NLPDE, for example, the inverse scattering method [7], Bäcklund and Darboux transformation method [8], homotopy perturbation method [9], first integral method [10–12], variational iteration method [13, 14], sub-ODE method [15–17], Jacobi elliptic function method [18], tanh-sech method [19], -expansion method [20, 21], Hirota’s method [22–24], and homogeneous balance (HB) method [25–27].

As the direct methods, the -expansion method, Hirota’s method, and HB method are very effective for constructing the exact solutions of NLPDE. Exact traveling wave solutions, -soliton solutions, and solitary wave solutions of some NLPDE are obtained by using the above three methods [20–30]. Fan improved the HB method to investigate the BT, Lax pairs, symmetries, and exact solutions for some NLPDE [31, 32]. He also showed that there are many links among the HB method, Weiss-Tabor-Carnevale method, and Clarkson-Kruskal method.

However, there is no unified direct method which can be used to deal with all types of NLPDE. And also, no literature is available to illustrate the underlying relations among the three direct methods.

In the present paper, by improving some key steps in the HB method [26], we propose a new method, HB of undetermined coefficient method, which can be used to derive the bilinear equation of NLPDE. Based on the bilinear equation, by applying the perturbation method, sub-ODE method, and compatible condition, more exact solutions of NLPDE are obtained. We illustrate the real meaning of balance numbers. We show the underlying relations among the -expansion method, Hirota’s method, and HB method.

This paper is organized as follows: the HB of undetermined coefficient method is described in Section 2. In Sections 3 and 4, the KdV equation and Burgers equation are chosen as examples to illustrate the method, respectively. In Section 5, the bilinear equations of Boussinesq equation and Sawada-Kotera equation are derived, respectively. Some brief conclusions are given in Section 6.

2. Description of the HB of Undetermined Coefficient Method

Let us consider a general NLPDE, say, in two variables where is a polynomial function of its arguments and the subscripts denote the partial derivatives. The HB of undetermined coefficient method consists of three steps.

Step 1. Suppose that the solution of Equation (1) is of the form where , , , , (balance numbers), and (balance coefficients) are constants to be determined later. By balancing the highest nonlinear terms and the highest order partial derivative terms, balance numbers are obtained. Substituting Equation (2) into Equation (1) and balancing the terms with yield a set of algebraic equations for .

Step 2. Solving the set of algebraic equations and simplifying Equation (1), we can get the bilinear equation of Equation (1) directly or after integrating some times (Generally, integrating times equal to the orders of the lowest partial derivative of Equation (1).) with respect to .

Step 3. Generally, in order to obtain the exact solutions of Equation (1), there are three methods to deal with the bilinear equation of Equation (1): (i)Applying the perturbation method to the bilinear equation of Equation (1), -soliton solution of Equation (1) can be obtained.(ii)By using traveling wave transformations the bilinear equation of Equation (1) satisfies the following ODE: where the prime denotes the derivation with respect to and , , and are constants to be determined later. Substituting Equations (3) and (4) into Equation (1), it is converted into the following equation: where , , and are polynomial functions of , , and . Setting yields a set of algebraic equations for , , and . Solving the set of algebraic equations and using the solutions of Equation (4), can be determined. Substituting into Equation (2), the exact traveling wave solutions of Equation (1) are obtained.(iii)By applying the compatible condition to the bilinear equation of Equation (1), it is reduced to an ODE. Solving the ODE, more exact solutions of Equation (1) can be obtained.

Next, we choose the KdV equation and Burgers equation to illustrate our method.

3. Application to the KdV Equation

Let us consider the celebrated KdV equation in the form where is a constant. Suppose that the solution of Equation (6) is of the form where , , and are constants to be determined later.

Balancing and in Equation (6), it is required that and . Then, Equation (7) can be written as

From Equation (8), one can calculate the following derivatives:

Equating the coefficients of and on the left-hand side of Equation (6) to zero yields a set of algebraic equations for and as follows:

Solving the above algebraic equations, we get and . Substituting and back into Equation (8), we get where is an arbitrary constant. Substituting Equation (11) into Equation (6), we get where

Simplifying Equation (12) and integrating once with respect to , we get

Equation (14) is identical to where is an arbitrary function of and is an arbitrary constant.

In particular, taking as zero in Equation (15), we get the bilinear equation of Equation (6) as follows:

Equation (16) can be written concisely in terms of -operators as where

Remark 1. Applying Hirota’s method [22–24], the bilinear equation of Equation (6) can be written as Equation (19) is obtained by setting in Equation (17). Obviously, Equation (19) is a special case of Equation (17). (i)Now, we apply the perturbation method to Equation (17) to derive -soliton solution of Equation (6). Suppose that can be expanded as follows: where is a parameter and .Substituting Equation (20) into Equation (17) and arranging it at each order of , we get The order- equation can be rewritten as a linear differential equation for as follows: Solving Equation (22), we get where is an arbitrary constant.
The coefficient of can be rearranged as follows: Substituting Equation (23) into Equation (24), the right-hand side of Equation (24) equals zero. Therefore, we can choose Substituting Equations (23) and (25) into Equation (20), we get where , , and are arbitrary constants.
Substituting Equation (26) into Equation (11), 1-soliton solution of Equation (6) can be obtained. If we choose in Equation (22), similar to the process of obtaining 1-soliton solution, we can get 2-soliton solution of Equation (6) as follows: where , , , and are arbitrary constants.
Substituting Equation (27) into Equation (11), 2-soliton solution of Equation (6) can be obtained. Similarly, we can get -soliton solution of Equation (6).

Remark 2. Obviously, setting in Equations (23) and (27), 1-soliton and 2-soliton solutions of Equation (6) are identical to Hirota’s results [22–24].

Remark 3. By using the properties of -operators [22–24], a Bäcklund transformation of Equation (17) can be obtained as follows: where and satisfy Equation (17) and , , and are arbitrary constants. (ii)Now, we discuss Equation (16) by using the sub-ODE method.Using transformations , Equation (16) is reduced to where the prime denotes the derivation with respect to and is a constant to be determined later.
Noticing the bilinear property of Equation (16), suppose that satisfies the following ODE: where and are parameters.
Substituting Equation (30) into Equation (16), we get where Setting yields a set of algebraic equations for , , and . Solving this set of algebraic equations, we get where , , and are arbitrary constants.
Substituting Equation (30) into Equation (11), we get Substituting the general solutions of Equation (30) into Equation (34), we get three types of traveling wave solutions of Equation (6) as follows.
When , where and , , , , and are arbitrary constants.
Taking and , Equation (35) can be rewritten as where , , and are arbitrary constants and , , and are given by Equation (36).
In particular, if , then Equation (37) is reduced to where , , and are arbitrary constants and , , and are given by Equation (36), .
When , where , , and are arbitrary constants and , , and are given by Equation (36).
Obviously, Equation (39) can be written as where , , and are arbitrary constants and , , and are given by Equation (36), .
When , where , , , , and are arbitrary constants. (iii)Now, we discuss Equation (16) from the compatible condition. Equation (16) can be written as Notice ; otherwise, we can only get a trivial solution. Setting the second term of Equation (42) to zero and solving yield Substituting Equation (43) into Equation (42), we get Integrating Equation (44) once with respect to , we get where is an arbitrary function of .
Using transformations and , Equation (45) is reduced to Solving the above equation, we get namely, where and are arbitrary functions of .

Case 1. When , from Equation (48), we get where and are arbitrary functions of .
Substituting the above equation into Equation (48), we get Setting the coefficients of to zero in the above equation, we get Solving the above equations, we get where are arbitrary constants. Then, we get Substituting Equation (53) into Equation (11), we get an exact solution of Equation (6) as follows: where and are arbitrary constants.

Case 2. When and , similar to Case 1, we get and an exact solution of Equation (6) as follows: where and are arbitrary constants.

Case 3. When and similar to Case 1, we get and an exact solution of Equation (6) as follows: where and are arbitrary constants.

Case 4. When and , similar to Case 1, we get and an exact solution of Equation (6) as follows: where where , , and are arbitrary constants.

Case 5. When and , similar to Case 1, we get and an exact solution of Equation (6) as follows: where where and are arbitrary constants.

So far, based on the bilinear equation which is derived by using the HB of undetermined coefficient method, many exact solutions of the KdV are obtained by applying the perturbation method, sub-ODE method, and compatible condition. Our results can compare with the -expansion method, Hirota’s method, and HB method [20–30].

4. Application to the Burgers Equation

Let us consider the Burgers equation in the form where is a constant.

Suppose that the solution of Equation (65) is of the form where , , and are constants to be determined later.

Balancing and in Equation (65), it is required that and . Then, Equation (66) can be written as

From Equation (67), one can calculate the following derivatives:

Equating the coefficients of on the left-hand side of Equation (65) to zero yields an algebraic equation for as follows:

Solving the above algebraic equation, we get . Substituting back into Equation (67), we get where is an arbitrary constant.

Substituting Equation (70) into Equation (65), we get where

Simplifying Equation (71), we get

Equation (73) can be written concisely in terms of -operators as

Equation (74) is identical to where is an arbitrary function of and is an arbitrary constant.

In particular, taking as constant in Equation (75), we get

Remark 4. Applying Hirota’s method [22–24], the bilinear equation of Equation (74) can be written as Equation (77) is obtained by setting in Equation (74). Obviously, Equation (77) is a special case of Equation (74).

Remark 5. Equations (70) and (76) are general Cole-Hopf transformations. In fact, setting in Equations (70) and (76), we get the famous Cole-Hopf transformations (i)Now, we apply the perturbation method to Equation (74) to derive -soliton solution of Equation (65). Suppose that can be expanded as follows: where is a parameter and .Substituting Equation (79) into Equation (74) and arranging it at each order of , we get The order- equation can be rewritten as a linear differential equation for as follows: Solving Equation (81), we get where is an arbitrary constant.
The coefficient of can be rearranged as follows: Substituting Equation (82) into Equation (83), the right-hand side of Equation (83) equals zero. Therefore, we can choose Substituting Equations (82) and (84) into Equation (79), we get where , , and are arbitrary constants.
Substituting Equation (85) into Equation (70), 1-soliton solution of Equation (65) can be obtained.
If we choose in Equation (81), Equation (74) has no 2-soliton solution. But there exists a solution as follows: where , , and are arbitrary constants. (ii)Now, we discuss Equation (73) by using the sub-ODE method.Using transformations , , Equation (73) is reduced to where the prime denotes the derivation with respect to and is a constant to be determined later.
Noticing the bilinear property of Equation (87), suppose that satisfies the following ODE: where and are parameters.
Substituting Equation (88) into Equation (87), we get where Setting yields a set of algebraic equations for , , and . Solving the set of algebraic equations, we get where , , and are arbitrary constants.
Substituting the general solutions of Equation (88) into Equation (70), we get three types of traveling wave solutions of Equation (65) as follows.
When , where where , , , , and are arbitrary constants.
Taking and , Equation (93) can be written as where , , and are arbitrary constants and and are given by Equation (94).
In particular, if , then Equation (94) is reduced to where , , and are arbitrary constants and and are given by Equation (94), .
When , where , , and are arbitrary constants and and are given by Equation (94).
Obviously, Equation (96) can be written as where , , and are arbitrary constants and and are given by Equation (94), .
When , where , , , , , and are arbitrary constants. (iii)Now, we discuss Equation (73) from the compatible condition.Using the compatible condition, we can get nothing but Equation (76). Using transformations , Equation (76) is reduced to Substituting the general solutions of Equation (99) into Equation (70), we get three types of traveling wave solutions of Equation (65) as follows.
When , where where , , , , and are arbitrary constants.
Taking and , Equation (101) can be written as where , , , , and are arbitrary constants and and are given by Equation (101).
In particular, if , then Equation (102) is reduced to where , , , , , and are arbitrary constants and and are given by Equation (101).
When , where , , , , and are arbitrary constants and and are given by Equation (101).
Obviously, Equation (104) can be written as where , , , , , and are arbitrary constants and and are given by Equation (101).
When , where , , , , and are arbitrary constants.
Moreover, note that Equation (76) is linear, so we can get the solution of Equation (65) as follows: where where , ; , ; , ; ; ; and are arbitrary constants, and , , and are arbitrary but finite integers.

Remark 6. We can deal with Equation (76) by using some assumptions. For example, when we suppose that and , we get where , , , and are arbitrary constants.
When we suppose that and , we get where , , and are arbitrary constants.
Similarly, we can assume that ; then, a new solution of Equation (65) can be obtained. Being similar to above process, we omit it.

So far, applying the HB of undetermined coefficient method to the Burgers equation, we get the bilinear equation of Burgers equation. Moreover, we reduce the Burgers to a linear equation. Based on them, many exact solutions of the Burgers equation are obtained by applying the perturbation method, sub-ODE method, and compatible condition. Our results can compare with the -expansion method, Hirota’s method, and HB method [20–30].

5. Bilinear Equation of the Boussinesq Equation and Sawada-Kotera Equation

In this section, we derive the bilinear equations of the Boussinesq equation and Sawada-Kotera equation by using the HB of undetermined coefficient method. Being similar to Section 4, we omit the process of solving exact solutions.

Example 1. The generalized Boussinesq equation reads where , , , and are known constants.
In order to balance and in Equation (111), it is required that and . Then, we suppose that the solution of Equation (111) is of the form where are constants to be determined later.
Substituting Equation (112) into Equation (111) and equating the coefficients of and on the left-hand side of Equation (111) to zero yield a set of algebraic equations for and . Solving the algebraic equations, we get and . Substituting and back into Equation (112), we get where is an arbitrary constant.
Substituting Equation (113) into Equation (111), we get where Simplifying Equation (114) and integrating twice with respect to , we get Equation (116) is identical to where and are arbitrary functions of and is an arbitrary constant.
In particular, letting in Equation (117), we get the bilinear equation of Equation (111) as follows: Equation (118) can be written concisely in terms of -operators as where is an arbitrary constant.

Example 2. The Sawada-Kotera equation reads In order to balance and in Equation (120), it is required that and . Then, we suppose that the solution of Equation (120) is of the form where are constants to be determined later.
Substituting Equation (121) into Equation (120) and equating the coefficients of and on the left-hand side of Equation (120) to zero yield a set of algebraic equations for and . Solving the algebraic equations, we get and . Substituting and back into Equation (121), we get where is an arbitrary constant.
Substituting Equation (122) into Equation (120), we get where Simplifying Equation (123) and integrating once with respect to , we get Equation (125) is identical to where is an arbitrary function of and is an arbitrary constant.
In particular, letting in Equation (126), we get the bilinear equation of Equation (120) as follows: Equation (127) can be written concisely in terms of -operators as where is an arbitrary constant.

6. Conclusions

The HB of undetermined coefficient method is successfully used to establish the bilinear equation of NLPDE. By applying the perturbation method, sub-ODE method, and compatible condition to the bilinear equation, more exact solutions of NLPDE are obtained. We illustrate the real meaning of balance numbers. We show the underlying relations among the -expansion method, Hirota’s method, and HB method. Many well-known NLPDE can be handled by the HB of undetermined coefficient method. The performance of our method is found to be simple and efficient. The availability of computer systems like Maple facilitates the tedious algebraic calculations. Our method is also a standard and computable method, which allows us to solve complicated and tedious algebraic calculations.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no competing interests.

Authors’ Contributions

Xiao-Feng Yang analyzed and interpreted the data and wrote the manuscript; Yi Wei designed and optimized the algorithm and program. All authors read the manuscript.

Acknowledgments

The research is supported by the Fundamental Research Funds for the Central Universities (2452017373), the Doctoral Research Foundation of Northwest A&F University (2452017007), the Project of Shandong Province Higher Educational Science and Technology Program (J18KB100), the NSFC Cultivation Project of Jining Medical University (JYP2018KJ15), the Doctoral Research Foundation of Jining Medical University (2017JYQD22).

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Copyright © 2020 Xiao-Feng Yang and Yi Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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