#### Abstract

The two-variable ()-expansion method is employed to construct exact traveling wave solutions with parameters of nanobiosciences partial differential equation. When the parameters are replaced by special values, the solitary wave solutions and the periodic wave solutions of this equation have been obtained from the traveling waves. This method can be thought of as the generalization of well-known original -expansion method proposed by M. Wang et al. It is shown that the two-variable ()-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics. Comparison between our results and the well-known results is given.

#### 1. Introduction

In recent years, investigations of exact solutions to nonlinear partial differential equations (PDEs) play an important role in the study of nonlinear physical phenomena. Many powerful methods for finding these exact solutions have been presented, such as the inverse scattering method [1], the Hirota bilinear transform method [2], the truncated Painleve expansion method [36], the Backlund transform method [7, 8], the exp-function method [913], the tanh-function method [1417], the Jacobi elliptic function expansion method [1820], the -expansion method [2130], the modified -expansion method [31], the -expansion method [3235], the modified simple equation method [36], the multiple exp-function algorithm method [37], the transformed rational function method [38], the local fractional variation iteration method [39], and the local fractional series expansion method [40]. Further exact solutions to some real-life physical problems were already given in the recent articles [4144]. The key idea of the one-variable -expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in one variable in which satisfies the second-order linear ODE , where and are constants and . The key idea of the two variable -expansion method is that the exact traveling wave solutions of nonlinear PDEs can be expressed by a polynomial in two variables and in which satisfies the second-order linear ODE , where and are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in the given nonlinear PDEs. The coefficients of this polynomial can be obtained by solving a set of algebraic equations resulting from the process of using this method. Recently, Li et al. [32] have applied the -expansion method and determined the exact solutions of Zakharov equations, while Zayed et al. [3335] have used this method to find the exact solutions of the combined KdV-mKdV equation, the Kadomtsev-Petviashvili equation, and the potential YTSF equation, respectively.

The objective of this paper is to apply the two-variable -expansion method to find the exact traveling wave solutions of the following nonlinear partial differential equation of nanobiosciences [45, 46]: where is the resistance of the ER with length , is total maximal capacitance of the ER, is conductance of pertaining NPs, and is the characteristic impedance of our system. Parameters and describe nonlinearity of ER capacitor and conductance of NPs in RE, respectively.

The rest of this paper is organized as follows: In Section 2, we give the description of the two-variable -expansion method. In Section 3, we apply this method to solve (1). In Section 4, some conclusions are given.

#### 2. Description of the Two-Variable -Expansion Method

Before we describe the main steps of this method, we need the following remarks (see [3235]).

Remark 1. If we consider the second-order linear ODE and set , , then we get where and are constants.

Remark 2. If , then the general solution of (2) has the form where and are arbitrary constants. Consequently, we have where .

Remark 3. If , then the general solution of (2) has the form and hence where .

Remark 4. If , then the general solution of (2) has the form and hence Suppose we have the following nonlinear evolution equation: where is a polynomial in and its partial derivatives. In the following, we give the main steps of the -expansion method [3235].

Step 1. The traveling wave transformation where is a nonzero constant, reduces (10) to an ODE in the form where is a polynomial of and its total derivatives with respect to .

Step 2. Assume that the solution of (12) can be expressed by a polynomial in the two variables and as follows: where and are constants to be determined later.

Step 3. Determine the positive integer in (13) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in (12).

Step 4. Substitute (13) into (12) along with (3) and (5); the left-hand side of (12) can be converted into a polynomial in and , in which the degree of is not longer than one. Equating each coefficient of this polynomial to zero yields a system of algebraic equations which can be solved by using the Maple or Mathematica to get the values of , , , , , , and , where .

Step 5. Similar to Step 4, substituting (13) into (12) along with (3) and (7) for , (or (3) and (9) for ), we obtain the exact solutions of (12) expressed by trigonometric functions (or by rational functions), respectively.

#### 3. An Application

In this section, we will apply the method described in Section 2 to find the exact traveling wave solutions of (1) which describe models of microtubules (MTs) as nonlinear transmission lines. The physical details of derivation of (1) describing its ionic currents are elaborated in [45].

In order to solve (1) by the two-variable -expansion method, we see that the traveling wave transformation (11) with and the characteristic time of charging ER capacitor is . Thus (1) takes the form of ODE where , , .

By balancing with in (14), we get . Consequently, we get where , , , , and are constants to be determined later. There are three cases to be discussed as follows.

Case 1 (hyperbolic function solutions ). If , substituting (15) into (14) and using (3) and (5), the left-hand side of (14) becomes a polynomial in and . Setting the coefficients of this polynomial to be zero yields a system of algebraic equations in , , , , , , , and as follows:
On solving the above algebraic equations using the Maple or Mathematica, we get the following results.
Result 1. Consider
From (4), (15), and (17), we deduce the traveling wave solution of (1) as follows: where is a nonzero constant, .

In particular, by setting , , and in (18), we have the solitary wave solution while if , , and , then we have the solitary wave solution

Result 2. Consider

In this result, we deduce the traveling wave solution of (1) as follows: where is an arbitrary constant and .

In particular, by setting and in (22), we have the solitary wave solution (Figure 1) while if and , then we have the solitary wave solution

Note that our results (23) and (24) are in agreement with the results () obtained in page 1251 of [47] with interchanges and with the results (28) and (29) in page 1544 of [48] with the interchanges .

We close this case with the remark that our results (18)–(20) and (22) are new and are not reported elsewhere.

Case 2 (trigonometric function solution ). If , substituting (15) into (14) and using (3) and (7), the left-hand side of (14) becomes a polynomial in and . Setting the coefficients of this polynomial to be zero yields a system of algebraic equations in , , , , , , , and as follows:
On solving the above algebraic equations using the Maple or Mathematica, we get the following results.

Result 1. Consider

From (6), (15), and (26), we deduce the traveling wave solution of (1) as follows: where is a nonzero constant, .

In particular, by setting , , and in (27), we have the periodic solution while if , , and , then we have the periodic solution

Result 2. Consider

In this result, we deduce the traveling wave solution of (1) as follows: where is an arbitrary constant and .

In particular, by setting and in (31), we have the periodic solution (Figure 2) while if and , then we have the periodic solution

Note that our results (32) and (33) are in agreement with the results and obtained in page 1251 of [47] with interchanges and with the results (33) and (34) in page 1544 of [48] with the interchanges .

We close this case with the remark that our results (27)–(29) and (31) are new and are not reported elsewhere.

Case 3 (rational function solutions ). If , substituting (15) into (14) and using (3) and (9), the left-hand side of (14) becomes a polynomial in and . Setting the coefficients of this polynomial to be zero yields a system of algebraic equations in , , , , , and as follows:
On solving the above algebraic equations using the Maple or Mathematica, we get the following results.
Result 1. Consider
From (8), (15), and (35), we deduce the traveling wave solution of (1) as follows:
Result 2. Consider
In this result, we deduce the traveling wave solution of (1) as follows: where in (36) and (38) is given by . Our result (38) is new and is not reported elsewhere.

Remark 5. In all the above results, we have noted the term which is equivalent to the condition (21) of [45]. The condition (39) enables that the ohmic loss could be balanced by fresh ions injected from the nanopores in microtubules since these act ionic pumps in suitable voltage regime. Also, the authors [46] have obtained the condition (39) when they have discussed (1) using the extended tanh-function method. The same remark has been happened in [47, 48] using the improved Riccati equation mapping method and the improved -expansion method, respectively.

Remark 6. Since , then (14) becomes Integrating (40) once with respect to with zero constant of integration, we get
Multiplying (41) by and integrating once with respect to with zero constant of integration, we have and hence we have Consequently, we have where is a constant of integration.
From (44) we have the following solution (Figure 3): which is in agreement with the solution (17) obtained in [48] when , using the improved -expansion method. After some simple calculation, the solution (45) is equivalent to the solution (34) obtained in [45].

Remark 7. Integrating (40) twice with respect to , we get where and are nonzero constants of integration.Setting we obtain the Weierstrass equation [49, 50] where , , while is the Weierstrass elliptic -function.
If , (48) reduces to the equation , which admits the elementary solution where is a constant of integration. From (47) and (49), we get which is equivalent to the solution (36) for , .
From the well-known solutions of (48), see, for example, [50], we rewrite the solution (47) in terms of the Jacobi elliptic functions as follows: where , is the modulus of the Jacobi elliptic functions and , , are three roots of the cubic equation . When , that is, , we get and . Finally, we note that the solutions (51) are new and are not reported elsewhere.

#### 4. Conclusions

The two-variable -expansion method is used in this paper to obtain some new solutions of a selected nonlinear partial differential equation of nanoionic currents along MTs (1). As the two constants and take special values, we obtain the solitary wave solutions. When and in (2) and (13), the two-variable -expansion method reduces to the -expansion method. So the two-variable -expansion method is an extension of the -expansion method. The proposed method in this paper is more effective and more general than the -expansion method because it gives exact solutions in more general forms. In summary, the advantage of the two-variable -expansion method over the -expansion method is that the solutions using the first method recover the solutions using the second one. On comparing our results obtained in this paper with the well-known results obtained in [45, 47, 48] we deduce that some of these results are the same while the others are new which are not reported elsewhere. Finally, all solutions obtained in this paper have been checked with the Maple by putting them back into the original equation (14).

#### Glossary

 ER: elementary ring MT: microtubule NP: nanopore ODE: ordinary differential equation PDE: partial differential equation.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.