Advances in Mathematical Physics
Volume 2018 (2018), Article ID 7628651, 15 pages
https://doi.org/10.1155/2018/7628651

Research Article

Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the -Expansion Method

Sekson Sirisubtawee1,2 and Sanoe Koonprasert1,2

¹Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Sekson Sirisubtawee; sekson.s@sci.kmutnb.ac.th

Received 29 January 2018; Revised 20 March 2018; Accepted 23 April 2018; Published 3 June 2018

Academic Editor: Guozhen Lu

Copyright © 2018 Sekson Sirisubtawee and Sanoe Koonprasert. 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.

Abstract.

We apply the -expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

1. Introduction

Various phenomena such as shallow water waves and multicellular biological dynamics arising in the nonlinear physical sciences [1, 2], engineering [3, 4], and biology [5] can be modeled by a class of integrable nonlinear evolution equations which can be expressed in terms of nonlinear partial differential equations (NPDEs) of integer orders. Consequently, study of traveling wave solutions of NPDEs plays a significant role in the investigation of behaviors of nonlinear phenomena. Due to the efficiency, reliability, and easy use of symbolic software packages such as Maple or Mathematica, many powerful methods have been constructed and developed to analytically solve NPDEs with their aid. Over the last few decades, exact solutions, analytical approximate solutions, and numerical solutions of many NPDEs have been successfully obtained. The methods for obtaining exact explicit solutions of NPDEs are, for example, the -expansion method [6–8], the -expansion method [9–11], the novel -expansion method [12], the tanh-function method [13], the exp-function method [14, 15], the F-expansion method [16], Hirota’s direct method [17, 18], Kudryashov method [19, 20], and the extended auxiliary equation method [21]. Examples of the methods for obtaining analytical approximate solutions to NPDEs are the variational iteration method [22, 23] (VIM), the Adomian decomposition method [24, 25] (ADM), the homotopy perturbation method [26, 27] (HPM), and the reduced differential transform method [28]. In addition, the examples of useful methods for solving NPDEs numerically are the generalized finite difference method [29], the finite volume method [30], the finite element method [31], the spectral collocation method [32], and the Galerkin finite element method [33]. However, we prefer, if possible, to obtain exact solutions of NPDEs.

In the recent decades, applications of the -expansion method for solving NPDEs have been proposed in various areas of applied sciences and engineering. For example, Chen [34] gave the application of the -expansion method for seeking exact solutions of the coupled nonlinear Klein-Gordon equation. Wen-An et al. [35] demonstrated the use of the -expansion method for finding traveling wave solutions of a nonlinear evolution equation. Zayed and Arnous [36] investigated the use of the modified -expansion method for finding traveling wave solutions of nonlinear evolution equations. The modified method can be thought as the generalization of the well-known -expansion method introduced in [37] with the special functions and including the case of and . In particular, Wen-An et al. [35] applied the modified method, i.e., the -expansion method, to find the traveling wave solutions of the Vakhnenko equation. Zhouzheng [38] applied the -expansion method to obtain the exact solutions of the modified Benjamin-Bona-Mahony (MBBM) and Ostrovsky-Benjamin-Bona-Mahony (OBBM) equations. They found that the explicit exact solutions of the equations obtained by the method are in terms of some trigonometric, hyperbolic, and rational functions. Gepreel [39] employed the extended rational -expansion method to obtain traveling wave solutions of the first equation of two integral members of nonlinear Kadomtsev-Petviashvili (KP) hierarchy equations in mathematical physics. Mohyud-Din and Bibi [40] used the -expansion method along with the fractional complex transform to analytically solve the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) and the space-time fractional coupled Burgers equations for innovative exact solutions. Their exact solutions include trigonometric, hyperbolic, and rational function solutions, while Zhang et al. [41] proposed the use of the -expansion method for solving the Schrödinger equation with third-order dispersion.

The rest of this article is organized as follows. In Section 2, the brief description of the -expansion method is given. In Section 3, we apply the method to some real world problems modeled by NPDEs in order to obtain their exact solutions. Finally, the conclusions are drawn in Section 4.

2. Algorithm of the -Expansion Method

In this section, we provide the description of the -expansion method which is discussed in [34, 40]. Consider a nonlinear evolution partial differential equation (NEPDE) in two independent variables and as follows:

where is an unknown function of independent variables and is a polynomial of and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.

The main steps of the method to obtain exact solutions of NPDEs can be given as follows [42, 43].

Step 1. Convert a nonlinear partial differential equation in (1) into an ordinary differential equation (ODE) using the traveling wave transformation in a variable as follows:

where and are nonzero arbitrary constants. It can be noted that another transform can be sometimes used for some certain problems. With the transformation in (2) and integrations with respect to as many as possible, (1) is reduced to an ODE in as follows:

where is a polynomial of and its various derivatives. The prime notation (′) denotes the derivative with respect to .

Step 2. Suppose that the formal solution of the ODE in (3) can be expressed in powers of as follows:

where satisfies the following nonlinear ODE:

in which and are integers. The unknown constants or may be zero, but both of them cannot be zero simultaneously. The coefficients are unknown constants to be determined at a later step.

Step 3. The value of the positive integer can be determined using the homogeneous balance principle, i.e., by balancing between the highest order derivatives and the nonlinear terms occurring in (3). More precisely, if the degree of is , then the degree of the other terms will be expressed as follows:

Step 4. Substituting (4) along with (5) into (3), we obtain a polynomial in . Collecting all coefficients of like-power of (, where is some positive integer) and setting all of the obtained coefficients to zero, we acquire a system of nonlinear algebraic equations for the unknown constants , and Assume that the resulting algebraic equations can be solved for the unknown constants using symbolic software packages such as Maple.

Step 5. The general solutions of (5) can be categorized into the following three cases when are arbitrary nonzero constants.

If , then we obtain the general solution

If , then we obtain the general solution

which is equivalent to

If and , then we obtain the general solution

The explicit exact solutions of (1) can be obtained by inserting the values of and the solutions in (7)-(10) into (4) with the transformation in (2).

3. Applications of the -Expansion Method

In this section, we will demonstrate the use of the -expansion method on three of the interesting problems in mathematical physics.

3.1. The Benney-Luke Equation

In this section, we will provide a use of the -expansion method for seeking exact solitary wave solutions of the Benney-Luke equation, which is used to approximate the full water wave equations and appropriately described two-way water wave propagation with surface tension. The equation can be written in the following form [44, 45]:

where are the positive integers such that their difference is in terms of the inverse bond number capturing the effects of gravity forces and surface tension.

Using the traveling wave transformation , (11) is converted into the following ODE in the variable :

Integrating (12) with respect to once and then choosing the constant of integration to be zero, we obtain the following ODE:

for which the homogeneous balance principle is applied. Following Step 3 of the mentioned method, the highest order derivative and the nonlinear term of the highest order are balanced via using formula (6) as follows:

which leads to . Hence, the form of exact solutions of the ODE in (13) using the -expansion method can be expressed as

where are unknown constants. Substituting (15) into (13) along with (5), then collecting all the coefficients with the same power of , and finally setting these resulting coefficients to be zero, we consequently obtain the following system of algebraic equations in :

Solving the obtained algebraic system (16) by use of Maple, we get the following three cases.

Case 1.

where are arbitrary constants.

Case 2.

where are arbitrary constants.

Case 3.

where are arbitrary constants.
When we substitute the above three cases of the obtained parameters along with the functions specified in (7)-(10) into the solution form (15), we can write three results of solutions of (11) as follows.

Result 1. From Case 1 in (17), we have and the following exact solutions.
When , the trigonometric function solution corresponding to the parameter values in Case 1 can be written as

When , the exponential function solution associated with the parameter values in Case 1 can be expressed as

When , the exact solution corresponding to the parameter values in Case 1 is , which is the constant solution.

Result 2. From Case 2 in (18), we have and the following exact solutions.
When , the trigonometric function solution corresponding to the parameter values in Case 2 can be written as

When , the exponential function solution associated with the parameter values in Case 2 can be expressed as

When , the rational function solution corresponding to the parameter values in Case 2 can be expressed as

for which the traveling wave transformation for this case is .

Result 3. From Case 3 in (19), we have and the following exact solutions.
When , the trigonometric function solution corresponding to the parameter values in Case 3 can be written as

When , the exponential function solution associated with the parameter values in Case 3 can be expressed as

When , the rational function solution corresponding to the parameter values in Case 3 can be expressed as

for which the traveling wave transformation for this case is .

Akter and Akbar [44] utilized the modified simple equation method to obtain exact solutions of (11) which were written as fractions of exponential functions. They can be transformed into the tanh and coth functions for which the arbitrary constants are selected appropriately as shown in their paper. Islam et al. [45] demonstrated the application of the improved -expansion method with Riccati equation to obtain exact traveling wave solutions of (11). Their results were expressed in terms of the following functions:

(i)tanh, coth, their reciprocals, and their summations(ii)tan, cot, their reciprocals, and their summations(iii)rational functions.

However, our obtained results in (20)-(27) are more generalized than the ones described above; i.e., selecting the appropriate constants and in our solutions can lead to the solutions obtained by other existing methods mentioned previously. Here we present the plots of the exact solution in (21) with the positive formula using the following parameter values: . The three-dimensional and two-dimensional plots of this solution are portrayed in Figure 1 demonstrating the solitary wave solution of kink type.

(a) 3D graph

(b) 2D graph when

Figure 1: The solitary wave solutions of kink type for in (21) when , , , , , , and .

3.2. Equation of Nanoionic Currents along Microtubules

In this section, we will show an application of the -expansion method in nanobiosciences. One of the important models in such fields is the nonlinear transmission line model for nanoionic currents along microtubules (MTs) segmented into identical elementary rings (ERs). The model is playing an important role in cellular signalling and the elaborated details of derivation of the equation can be found in [46, 47]. The equation of nanoionic currents along MTs is described as follows [47]:

where   Ω is the resistance of the ER with length m F is the total maximal capacitance of the ER, Si is conductance of pertaining nanopores (NPs), and   Ω is the characteristic impedance of the system. Parameters and represent conductance of NPs in ER and nonlinearity of ER capacitor, respectively.

Using the dimensionless wave variable , where is the dimensionless velocity of wave and s is the characteristic time of charging ER capacitor, we obtain the traveling wave transformation and then (28) is converted to the following new ODE:

where , , and .

Balancing between the highest order derivative and the nonlinear term of the highest order by using formula (6), we obtain

Thus, the form of exact solutions of (29) using the -expansion method can be expressed as

where are unknown constants. Substituting (31) into (29) along with (5), then collecting all the coefficients with the same power of , and finally setting these resulting coefficients to be zero, we consequently obtain the following system of algebraic equations in :

Using Maple to solve algebraic system (32), we obtain the following three cases.

Case 1.

with which are arbitrary constants.

Case 2.

with which are arbitrary constants.

Case 3.

with which are arbitrary constants.

Inserting the above three cases in (33)-(35) along with the functions described in (7)-(10) into the solution form (31), we attain three results of solutions of (28) as follows.

Result 1. Using parameter values of Case 1 in (33) and , we have the following exact solutions.
When , the trigonometric function solution of (28) can be written as

When , the exponential function solution of (28) can be expressed as

When , the exact solution of (28) is the constant solution .

Result 2. Using parameter values of Case 2 in (34) and , we have the following exact solutions.

When , the trigonometric function solution of (28) can be written as

When , the exponential function solution of (28) can be expressed as

When , the rational function solution of (28) can be expressed as

Result 3. Using parameter values of Case 3 in (35) and , we have the following exact solutions.

When , the trigonometric function solution of (28) can be written as

When , the exponential function solution of (28) can be expressed as

When , the rational function solution of (28) can be expressed as

Satarić et al. [46] firstly proposed model (28) and obtained its analytical solution by converting (28) with the change of variables and the use of the appropriate boundary conditions into the solvable ODE in a new variable. The analytical solution was expressed in terms of a square of the exponential secant function. Later, Sekulić et al. [47] applied the modified extended tanh-function method to solve (28) for exact traveling wave solutions. The solutions were written as the square of the following functions: tan, cot, tanh, and coth. It is not difficult to verify that the mathematical structures of their exact solutions are the particular cases of those of our solutions as shown in (36), (37), (38), (39), (41), and (42). Meanwhile our rational function solutions do not appear in their work. In addition, Zayed and Alurrfi [48] utilized the -expansion method to analytically obtain the exact traveling wave solutions of problem (28). Their solutions included the solitary wave solutions and the periodic wave solutions when the appropriate values of the parameters were particularly selected. Comparing the mathematical structures of their specific solutions with our obtained solutions, almost all of their specific solutions can be obtained from our solutions. For instance, solutions (20), (23), and (24) in their paper are structurally equivalent to the hyperbolic form of the solution in (39) via using and . The trigonometric solution (28) in their paper has the same mathematical structure as our solution in (38) with and . Solutions (29), (32), and (33) in their article can be equivalently transformed to our solution in (38) by choosing and . Finally, rational solutions (36) and (37) in their paper with of degree two in the denominators are structurally equivalent to our rational solution in (40). Furthermore, the exact solutions in (41) and (42) can generate other types of solutions different from the compared ones.

In [46, 47], they used the estimated dimensionless parameter and took for plotting their solutions. Using the parameter values mentioned above and choosing , we obtain the graphical representations of the exact traveling wave solution in (39) as demonstrated in Figure 2 describing the soliton solution of bell-type. The obtained three-dimensional graph shown in Figure 2(a) is similar to their results.

(a) 3D graph

(b) 2D graph when

Figure 2: The soliton solution of bell-type for in (39) when , , , , , , and .

3.3. The Generalized Hirota-Satsuma Coupled KdV System

In 1982 Satsuma and Hirota proposed the new system of equations which is called the generalized Hirota-Satsuma coupled KdV system as follows [49]:

The above system can be obtained from the four reductions of KP hierarchy. In particular, the well-known Hirota-Satsuma coupled KdV system [50], which was derived in 1981 by Hirota and Satsuma to describe interactions of two long waves with different dispersion relations, can be obtained by setting in (44). We want to obtain traveling wave solutions for system (44) which are in the following form:

where and are nonzero arbitrary constants to be determined later. Substituting (45) into (44), we yield a system of ODEs as follows:

Let [51]

where , and are constants to be also determined later.

Substituting (49) into (47) and (48) and then integrating once, we know that (47) and (48) give the same equation as follows:

where is a constant of integration. Multiplying (50) by and then integrating the resulting equation with respect to , we obtain

where is also a constant of integration.

From (49)-(51), we obtain

Integrating (46) once, we get

where is a constant of integration. Substituting (49) and (52) into (53), we yield the following system:

Let

We find from (54) that

From (50), we therefore obtain

Applying the homogeneous balance principle and (6) mentioned in Step 3 to the terms and , we then have that

which leads to . Hence, the form of exact solutions of the ordinary differential equation in (57) using the -expansion method is

where are unknown constants. Substituting (59) into (57) along with (5), then collecting all the coefficients with the same power of , and finally setting these resulting coefficients to be zero, we consequently attain the following system of algebraic equations in :

By solving the nonlinear system in (60) with Maple, we obtain the following cases.

Case 1.

where are arbitrary constants, and is expressed in (56).

Case 2.

where are arbitrary constants, and is expressed in (56).

Case 3.

where are arbitrary constants, and is expressed in (56).

Case 4.

where are arbitrary constants, and is expressed in (56).

Inserting the above four cases shown in (61)-(64) along with the functions described in (7)-(10) into the solution form (59), we obtain four results of solutions of system (44) as follows.

Result 1. Using parameter values specified in Case 1 as shown in (61) and , we have the following exact solutions.
When , the trigonometric function solution of system (44) can be written as

When , the exponential function solution of system (44) can be expressed as

When , the rational function solution of system (44) can be written as

Result 2. Using parameter values specified in Case 2 as shown in (62) and , we have the following exact solutions.
When , the trigonometric function solution of system (44) can be written as

When , the exponential function solution of system (44) can be expressed as

When , the rational function solution of system (44) can be written as

for which the traveling wave transformation for this case is .

Result 3. Using parameter values specified in Case 3 as shown in (63) and , we have the following exact solutions.
When , the trigonometric function solution of system (44) can be written as

When , the exponential function solution of system (44) can be expressed as

When , the rational function solution of system (44) can be written as

for which the traveling wave transformation for this case is .

Result 4. Using parameter values specified in Case 4 as shown in (64) and , we have the following exact solutions.
When , the trigonometric function solution of system (44) can be written as

When , the exponential function solution of system (44) can be expressed as

When , the rational function solution of system (44) can be written as

for which the traveling wave transformation for this case is .

Lu et al. [52] applied the new improved Riccati equation method to system (44) and then found that its exact solutions were expressed in terms of fractions of trigonometric functions and fractions of hyperbolic functions. El-Wakil and Abdou [53] found some exact solutions for system (44) by the modified extended tanh-function method. All of the obtained solutions had the forms of the following functions: tanh, coth, tan, and cot. It is not difficult to check that our exact solutions of system (44), which are shown in (65)-(76), are the functions whose mathematical forms are equivalent to those of most of the solutions found previously by choosing suitable parameters and some of them are obviously novel.

In Figures 3 and 4, we present the three-dimensional and two-dimensional plots of some exact solutions, which are and , expressed in (74) and (75), respectively. They are depicted using the sets of the appropriate parameter values as described below. Employing , which are solved implicitly using (56), (61) and the parameter values , and choosing , we obtain the plots of the selected exact traveling wave solution

as shown in Figure 3 which represents the periodic wave solutions.

(a) 3D graph

(b) 2D graph when

Figure 3: The periodic wave solutions of in (77) when , , , , , , , and .

(a) 3D graph

(b) 2D graph when

Figure 4: The singular wave solutions of in (78) when , , , , , , , , and .

Again implicitly solving (56) and (61) via using , we obtain . Then the graphical representations of the chosen exact solution

using the parameters as mentioned above and are simulated in Figure 4 describing the singular wave solutions.

4. Conclusions

In this paper, the -expansion method has been applied to find some new forms of the explicit exact solutions of the three problems, i.e., the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The explicit exact solutions of the problems, which are obtained by the method, can be considered as a part of the gigantic variety of possible solution forms; however, they have provided physical representations in each problem. As shown in Section 3, we have found that the obtained exact solutions of the problems are expressed in terms of trigonometric, exponential (or equivalently hyperbolic), and rational functions. The investigation demonstrates that the method is considerably efficient and practically appropriate for analytically solving such problems with the aid of Maple. All of the obtained exact solutions of each problem have ensured the correctness by substituting them back into the original equations. Moreover, the method could also be employed efficiently for a broad range of NPDEs of integer orders.

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 that they have no conflicts of interest.

Acknowledgments

This research was funded by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-KNOW-61-037).

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