Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 521712, 10 pages

http://dx.doi.org/10.1155/2014/521712

Research Article

The ()-Expansion Method and Its Applications to Find the Exact Solutions of Nonlinear PDEs for Nanobiosciences

E. M. E. Zayed and K. A. E. Alurrfi

Mathematics Department, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

Received 14 November 2013; Accepted 10 February 2014; Published 25 March 2014

Academic Editor: Elmetwally Elabbasy

Copyright © 2014 E. M. E. Zayed and K. A. E. Alurrfi. 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.

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 [3–6], the Backlund transform method [7, 8], the exp-function method [9–13], the tanh-function method [14–17], the Jacobi elliptic function expansion method [18–20], the -expansion method [21–30], the modified -expansion method [31], the -expansion method [32–35], 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 [41–44]. 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. [33–35] 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 [32–35]).

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 [32–35].

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

Figure 1: The plot of solution (23) when , , , , .

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

Figure 2: The plot of solution (32) when , , , , .

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].

Figure 3: The plot of solution (45) when , , , , .

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.

References

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, New York, NY, USA, 1991.

R. Hirota, “Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.

J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983.

N. A. Kudryashov, “Exact soliton solutions of a generalized evolution equation of wave dynamics,” Journal of Applied Mathematics and Mechanics, vol. 52, no. 3, pp. 361–365, 1988.

N. A. Kudryashov, “Exact solutions of the generalized Kuramoto-Sivashinsky equation,” Physics Letters A, vol. 147, no. 5-6, pp. 287–291, 1990.

N. A. Kudryashov, “On types of nonlinear nonintegrable equations with exact solutions,” Physics Letters A, vol. 155, no. 4-5, pp. 269–275, 1991.

M. R. Miura, Backlund Transformation, Springer, Berlin, Germany, 1978.

C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, New York, NY, USA, 1982.

J. H. He and X. H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700–708, 2006.

E. Yusufoǧlu, “New solitonary solutions for the MBBM equations using Exp-function method,” Physics Letters A, vol. 372, no. 4, pp. 442–446, 2008.

S. Zhang, “Application of Exp-function method to high-dimensional nonlinear evolution equation,” Chaos, Solitons & Fractals, vol. 38, no. 1, pp. 270–276, 2008.

A. Bekir, “The Exp-function method for Ostrovsky equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 6, pp. 735–739, 2009.

A. Bekir, “Application of the Exp-function method for nonlinear differential-difference equations,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 4049–4053, 2010.

M. A. Abdou, “The extended tanh method and its applications for solving nonlinear physical models,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 988–996, 2007.

E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000.

S. Zhang and T. C. Xia, “A further improved tanh-function method exactly solving the -dimensional dispersive long wave equations,” Applied Mathematics E-Notes, vol. 8, pp. 58–66, 2008.

E. Yusufoğlu and A. Bekir, “Exact solutions of coupled nonlinear Klein-Gordon equations,” Mathematical and Computer Modelling, vol. 48, no. 11-12, pp. 1694–1700, 2008.

Y. Chen and Q. Wang, “Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to -dimensional dispersive long wave equation,” Chaos, Solitons & Fractals, vol. 24, no. 3, pp. 745–757, 2005.

S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001.

D. Lü, “Jacobi elliptic function solutions for two variant Boussinesq equations,” Chaos, Solitons & Fractals, vol. 24, no. 5, pp. 1373–1385, 2005.

M. L. Wang, X. Li, and J. Zhang, “The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.

S. Zhang, J. L. Tong, and W. Wang, “A generalized -expansion method for the mKdV equation with variable coefficients,” Physics Letters A, vol. 372, no. 13, pp. 2254–2257, 2008.

E. M. E. Zayed and K. A. Gepreel, “The -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics,” Journal of Mathematical Physics, vol. 50, no. 1, Article ID 013502, 2009.

E. M. E. Zayed, “The -expansion method and its applications to some nonlinear evolution equations in the mathematical physics,” Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 89–103, 2009.

A. Bekir, “Application of the -expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008.

B. Ayhan and A. Bekir, “The -expansion method for the nonlinear lattice equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 9, pp. 3490–3498, 2012.

N. A. Kudryashov, “A note on the -expansion method,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1755–1758, 2010.

I. Aslan, “A note on the -expansion method again,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 937–938, 2010.

E. M. E. Zayed, “Traveling wave solutions for higher dimensional nonlinear evolution equations using the -expansion method,” Journal of Applied Mathematics & Informatics, vol. 28, pp. 383–395, 2010.

H. Naher and F. A. Abdullah, “New approach of -expansion method and new approach of generalized -expansion method for nonlinear evolution equation,” AIP Advances, vol. 3, no. 3, Article ID 032116, 15 pages, 2013.

S. Zhang, Y. N. Sun, J. M. Ba, and L. Dong, “The modified - expansion method for nonlinear evolution equations,” Zeitschrift Für Naturforschung, vol. 66, pp. 33–39, 2011.

L. X. Li, E. Q. Li, and L. M. Wang, “The ()-expansion method and its application to travelling wave solutions of the Zakharov equations,” Applied Mathematics, vol. 25, no. 4, pp. 454–462, 2010.

E. M. E. Zayed and M. A. M. Abdelaziz, “The two-variable ()-expansion method for solving the nonlinear KdV-mKdV equation,” Mathematical Problems in Engineering, vol. 2012, Article ID 725061, 14 pages, 2012.

E. M. E. Zayed, S. A. H. Ibrahim, and M. A. M. Abdelaziz, “Traveling wave solutions of the nonlinear dimensional Kadomtsev-Petviashvili equation using the two variables ()-expansion method,” Journal of Applied Mathematics, vol. 2012, Article ID 560531, 8 pages, 2012.

E. M. E. Zayed and S. A. H. Ibrahim, “The two variable ()-expansion method for finding exact traveling wave solutions of the -dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation,” in Proceedings of the International Conference on Advanced Computer Science and Electronics Information, pp. 388–392, Atlantis Press, 2013.

E. M. E. Zayed and A. H. Arnous, “The modified simple equation method and its applications to -dimensional systems of nonlinear evolution equations,” Scientific Research and Essays, vol. 8, pp. 1973–1982, 2013.

W. X. Ma and Z. Zhu, “Solving the -dimensional generalized KP and BKP equations by the multiple exp-function algorithm,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11871–11879, 2012.

W. X. Ma and J. H. Lee, “A transformed rational function method and exact solutions to the dimensional Jimbo-Miwa equation,” Chaos, Solitons & Fractals, vol. 42, no. 3, pp. 1356–1363, 2009.

Y. J. Yang, D. Baleanu, and X. J. Yang, “A local fractional variational iteration method for Laplace equation within local fractional operators,” Abstract and Applied Analysis, vol. 2013, Article ID 202650, 6 pages, 2013.

A. M. Yang, X. J. Yang, and Z. B. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,” Abstract and Applied Analysis, vol. 2013, Article ID 351057, 5 pages, 2013.

M. Turkyilmazoglu and I. Pop, “Exact analytical solutions for the flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid,” International Journal of Heat and Mass Transfer, vol. 57, no. 3, pp. 82–88, 2013.

M. Turkyilmazoglu, “Exact solutions to heat transfer in straight fins of varying exponential shape having temperature dependent properties,” International Journal of Thermal Sciences, vol. 55, pp. 69–75, 2012.

M. Turkyilmazoglu, “Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids,” Chemical Engineering Science, vol. 84, pp. 182–187, 2012.

M. Turkyilmazoglu, “Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a porous rotating disk flow with Hall current,” International Journal of Mechanical Sciences, vol. 56, pp. 86–95, 2012.

M. V. Satarić, D. Sekulić, and M. Živanov, “Solitonic ionic currents along microtubules,” Journal of Computational and Theoretical Nanoscience, vol. 7, no. 11, pp. 2281–2290, 2010.

L. S. Dalibor, V. S. Miljko, and B. Z. Milos, “Symbolic computation of some new nonlinear partial differential equations of nanobiosciences using modified extended tanh-function method,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3499–3506, 2011.

E. M. E. Zayed, Y. A. Amer, and R. M. A. Shohib, “The improved Riccati equation mapping method for constructing many families of exact solutions for a nonlinear partial differential equation of nanobiosciences,” International Journal of Physical Sciences, vol. 8, no. 22, pp. 1246–1255, 2013.

E. M. E. Zayed, Y. A. Amer, and R. M. A. Shohib, “Improved -expansion method for constructing exact traveling wave solutions for a nonlinear PDE of nanobiosciences,” Scientific Research and Essays, vol. 8, pp. 1540–1546, 2013.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1965.

Y. Chen and Z. Yan, “The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 29, no. 4, pp. 948–964, 2006.