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 solutionwhich is equivalent to

If and , then we obtain the general solutionThe 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 aswhere 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 asWhen , the exponential function solution associated with the parameter values in Case 1 can be expressed asWhen , 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 asWhen , the exponential function solution associated with the parameter values in Case 2 can be expressed asWhen , the rational function solution corresponding to the parameter values in Case 2 can be expressed asfor 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 asWhen , the exponential function solution associated with the parameter values in Case 3 can be expressed asWhen , the rational function solution corresponding to the parameter values in Case 3 can be expressed asfor 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

(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 aswhere 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.