Abstract
An improvement of the expansion methods, namely, the improved -expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new -expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.
1. Introduction
Nonlinear evolution equations (NLEEs) are very important model equations in mathematical physics, engineering, and applied mathematics for describing diverse types of physical mechanisms of natural phenomena in the field of applied sciences, biochemistry, and engineering. Nonlinear wave equations play a major role in various fields such as plasma physics, fluid mechanics, optical fibers, solid state physics, chemical kinetics, geochemistry, and nonlinear optics [1, 2]. Much work has been done over the years on the subject of obtaining the analytical solutions to the nonlinear partial differential equations (NPDEs). One of the most exciting advances of nonlinear science and theoretical physics has been the development of methods to look for exact solutions for NPDEs. The advent of symbolic computation also enables performing some complicated and tedious algebraic processes coupled with differential calculations. With the rapid development of nonlinear sciences based on computer algebraic system, many effective methods have been presented, such as the homotopy analysis method [3, 4], the variational iteration method [5–7], the homotopy perturbation method [8, 9], the sine-cosine method [10], the tanh-coth method [11–13], the modified extended tanh-function method [14, 15], the Exp-function method [16, 17], the exp()-expansion method [18, 19], the -expansion method [20, 21], the modified simple equation method [22], the novel -expansion method [23], the new approach of the generalized -expansion method [21], the Jacobi elliptic function method [24], and the homogeneous balance method [25]. Recently, Naher and Abdullah [21] presented an effective and straightforward method, called the new approach of generalized -expansion method, to obtain exact travelling wave solutions of nonlinear evolution equations. Here, we use an effective method, namely, the improved -expansion method, for constructing a range of exact solutions for the following ordinary partial differential equations that in this paper we developed solutions as well. In this paper, we put forth the new approach of improved -expansion method to construct exact travelling wave solutions including solitons, kink, and periodic and rational solutions to the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. In [22, 26], the standard form of the Fitzhugh-Nagumo equation is given:where and is of the unknown function depending on the temporal variable and the spatial variable . When , (1) reduces to the real Newell-Whitehead equation which describes the dynamical behavior near the bifurcation point for the Rayleigh-Bénard convection of binary fluid mixtures [27]. The FN equation has various applications in the fields of flame propagation, logistic population growth, neurophysiology, branching Brownian motion process, autocatalytic chemical reaction, and nuclear reactor theory [28]. This equation combines diffusion and nonlinearity which is controlled by the term . Many physicists and mathematicians have paid much attention to the Fitzhugh-Nagumo equation in recent years due to its importance in mathematical physics. Remarkably, this nonlinear evolution equation is an important nonlinear reaction-diffusion equation and is usually used to model the transmission of nerve impulses [29–31]. Shih et al. utilized the approximate conditional symmetry method to determine approximate solutions admitted by a perturbation of (1). Equation (1) has been presented by Hariharan and Kannan via Haar wavelet method [32]. Kawahara and Tanaka [33] have found new exact solutions of (1), by applying the nonclassical symmetry reductions approach by using Hirota method. In [34], Nucci and Clarkson have obtained some new exact solutions with Jacobi elliptic function. Li and Guo [35] have obtained the new exact solutions of the Fitzhugh-Nagumo equation by using first integral method. In this paper, we consider that the generalized Fitzhugh-Nagumo equation with time-dependent coefficients is given aswhere , , and are arbitrary functions of . The nonlinear models with variable coefficients are needed to describe the propagation of pulses. Triki and Wazwaz [36] obtained a new variety of soliton solutions by means of specific solitary wave Ansatz and the tanh method for (2). Also, Jiwari et al. [37] applied polynomial differential quadrature method to find the numerical solution of the generalized Fitzhugh-Nagumo equation with time-dependent coefficients in one-dimensional space. In [38], the effect of diffusion on pattern formation in FitzHugh-Nagumo model has been surveyed. Van Gorder applied the method of homotopy analysis to study the Fitzhugh-Nagumo equation [39]. In [40], a FitzHugh-Nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. Abbasian et al. have searched symmetric bursting behaviors in the generalized FitzHugh-Nagumo model [41]. Ray and Sahooz used the fractional subequation method for solving the space-time fractional Zakharov-Kuznetsov and modified Zakharov-Kuznetsov equations [42]. Also, the authors of [43] applied the fractional subequation method to the time fractional KdV-Zakharov-Kuznets and space-time fractional modified KdV-Zakharov-Kuznetsov equations. The purpose of this paper is to obtain exact solutions of the generalized Fitzhugh-Nagumo equation and to determine the accuracy of the improved -expansion method in solving these kinds of problems. The paper is organized as follows. In Section 2, we describe the improved -expansion method. In Section 3, we examine the generalized Fitzhugh-Nagumo equation with method introduced in Section 2. Also, conclusion and advantages are given in Section 4. Finally, some references are given at the end of this paper.
2. Description of Improved -Expansion Technique
Step 1. We suppose the given nonlinear fractional partial differential equation for to be in the formwhich can be converted to an ODE:the transformation, , is wave variable. Also, is constant to be determined later.
Step 2. Suppose the traveling wave solution of (4) can be expressed as follows:where and are constants to be determined, such that , and satisfies the following ordinary differential equation:We will consider the following special solutions of (6): Family 1: When and , then Family 2: When and , then Family 3: When and , then Family 4: When and , then Family 5: When and , then Family 6: When and , then Family 7: When and , then Family 8: When , then Family 9: When , then Family 10: When and , then Family 11: When , then Family 12: When , then Family 13: When , then Family 14: When , then Family 15: When and , then Family 16: When and , then Family 17: When and , then , where , and are constants to be determined later. But the positive integer can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (6). If is not an integer, then a transformation formula should be used to overcome this difficulty.
Step 3. Substituting (5) into (4) with the value of obtained in Step 2, collecting the coefficients of , and then setting each coefficient to zero, we can get a set of overdetermined partial differential equations for , and with the aid of symbolic computation Maple.
Step 4. Solve the algebraic equations in Step 3, and then substitute in (5).
3. The Generalized Fitzhugh-Nagumo Equation
We consider the generalized Fitzhugh-Nagumo equation with time-dependent coefficients as follows:where , and are arbitrary functions of . By using the wave variable , reduce it to an ODE as follows:where and . Balancing the the linear term of the highest order with the highest order nonlinear term by using homogenous principal, we haveThen, the trail solutions areSubstituting (10) and (6) into (8) and collecting all terms with the same order of together, the left-hand side of (10) is converted into polynomial in . Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for , and as follows:Solving the set of algebraic equations using Maple, we get the following results.
Set 1. We have the followingwhere , and are arbitrary constants. By using (13), Families 7, 9, and 13 can be written, respectively, aswhere .
Set 2. We have the following: where , and are arbitrary constants. By using (17), Families 7, 9, and 13 can be written, respectively, aswhere .
Set 3. We have the following: where , and are arbitrary constants. By using (20), Families 7, 9, and 13 can be written, respectively, aswhere .
Set 4. We have the following: where , and are arbitrary constants. By using (23), Families 7, 9, and 13 can be written, respectively, aswhere .
Set 5. We have the following: where , and are arbitrary constants. By using (26), Families 7, 9, and 13 can be written, respectively, aswhere .
Set 6. We have the following: where , and are arbitrary constants. By using (29), Families 7, 9, and 13 can be written, respectively, aswhere .
Set 7. We have the following: where , and are arbitrary constants. By using (32), Family 2 can be written aswhere .
Set 8. We have the following: where , and are arbitrary constants. By using (35), Family 2 can be written aswhere .
Set 9. We have the following: where , and are arbitrary constants. By using (38), Family 2 can be written aswhere + + .
Set 10. We have the following: where , and are arbitrary constants. By using (41), Family 2 can be written aswhere .
Set 11. We have the following: where , and are arbitrary constants. By using (44), Family 2 can be written aswhere .
Set 12. We have the following: where , and are arbitrary constants. By using (47), Family 2 can be written aswhere .
Set 13. We have the following: By using (50), Families 1, 2, 7, and 14 can be written, respectively, aswhere .
Set 14. We have the following: By using (50), Families 1, 2, 7, and 14 can be written, respectively, aswhere .
Set 15. We have the following: By using (50), Families 1, 2, 7, and 14 can be written, respectively, aswhere .
Set 16. We have the following: By using (50), Families 1, 2, 7, and 14 can be written, respectively, aswhere .
Set 17. We have the following: By using (50), Families 1, 2, 7, and 14 can be written, respectively, aswhere .
Set 18. We have the following: By using (50), Families 1, 2, 7, and 14 can be written, respectively, aswhere .
Set 19. We have the following: By using (50), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 20. We have the following: By using (50), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 21. We have the following: By using (50), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 22. We have the following: By using (50), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 23. We have the following: By using (50), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 24. We have the following: By using (50), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 25. We have the following: By using (50), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 26. We have the following: where , and are arbitrary constants. By using (77), Families 7 and 9 can be written, respectively, aswhere .
Set 27. We have the following: where , and are arbitrary constants. By using (80), Families 7 and 9 can be written, respectively, aswhere .
Set 28. We have the following: where , and are arbitrary constants. By using (83), Families 7 and 9, respectively, can be written aswhere .
Set 29. We have the following: where , and are arbitrary constants. By using (86), Families 7 and 9 can be written, respectively, as where .
Set 30. We have the following: where , and are arbitrary constants. By using (89), Families 7 and 9 can be written, respectively, aswhere .
Set 31. We have the following: where , and are arbitrary constants. By using (92), Families 7 and 9 can be written, respectively, aswhere .
Set 32. We have the following: where is an arbitrary constant. By using (95), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 33. We have the following: where is an arbitrary constant. By using (99), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 34. We have the following: where is an arbitrary constant. By using (102), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 35. We have the following: where is an arbitrary constant. By using (105), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 36. We have the following: where is an arbitrary constant. By using (108), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 37. We have the following: where is an arbitrary constant. By using (111), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 38. We have the following: where is an arbitrary constant. By using (114), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 39. We have the following: where is an arbitrary constant. By using (117), Families 1, 2, and 14 can be written, respectively, aswhere .
Set 40. We have the following: where is an arbitrary constant. By using (120), Families 1, 2, and 14 can be written, respectively, aswhere .
Remark 1. In Figure 1, we plot two- and three-dimensional graphics of imaginary and real values of (14), which denote the dynamics of solutions with appropriate parametric selections. Also, in Figure 2, we draw two- and three-dimensional graphics of imaginary and real values of (15), which demonstrate the dynamics of solutions with convenient parametric choices. In Figure 3, we plot two- and three-dimensional graphics of (96), which represent the dynamics of solutions with proper parametric values. Moreover, in Figure 4, we draw two- and three-dimensional graphics of imaginary and real values of (97), which display the dynamics of solutions with suitable parametric selections. We plot two- and three-dimensional graphics of Figures 1–4, when and for Figures 5–8 when . To the best of our knowledge, these complex exponential function solutions have not been submitted to the literature in advance. The analytical solutions and figures obtained in this paper give us a different physical interpretation for the nonlinear generalized Fitzhugh-Nagumo equation.
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4. Conclusion
In this paper, the new approach of expansion method, namely, the improved -expansion method, has successfully been implemented to investigate the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions are attained. It is worth mentioning that some of the newly obtained solutions are identical to already published results. It has been shown that the applied method is effective and more wide ranging than the generalized and improved -expansion method because it gives many new solutions. Therefore, this method can be applied to study many other nonlinear partial differential equations which frequently arise in engineering, mathematical physics, and other scientific real time application fields.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.