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Solitons and Periodic Solutions of the Fisher Equation with Nonlinear Ordinary Differential Equation as Auxiliary Equation

Anika Tashin Khan, Hasibun Naher
American Journal of Applied Mathematics and Statistics. 2018, 6(6), 244-252. DOI: 10.12691/ajams-6-6-5
Received September 17, 2018; Revised November 04, 2018; Accepted November 14, 2018

Abstract

In this article the new extension of the generalized and improved (G’/G)-expansion method has been used to generate many new and abundant solitons and periodic solutions, where the nonlinear ordinary differential equation has been used as an auxiliary equation, involving many new and real parameters. We choose the Fisher Equation in order to explain the advantages and effectives of this method. The illustrated results belongs to hyperbolic functions, trigonometric functions and rational functional forms which show that the implemented method is highly effective for investigating nonlinear evolution equations in mathematical physics and engineering science.

1. Introduction

In physical sciences all essential equations are nonlinear and these are often complicated to interpret. So the exact solutions of nonlinear evolution equations (NLEEs) have turned out to be a chief concern for researchers. NLEE is one of the most powerful and important modelled equations among all equations in nonlinear sciences and it plays a vital role in the field of scientific work of engineering sciences such as chemical kinematics, fluid mechanics, chemistry, biology, nonlinear optics, optical fibers, plasma physics, solid state physics, biophysics, geochemistry, quantum mechanics, chemical physics, condensed matter physics, high-energy physics and so on. As they reveal a lot of physical information which help to understand the operation of the physical model better, that is why the explicit solutions of NLEEs play important role in the study of physical phenomena and remains a crucial field for researchers in the ongoing investigation.

For the past few decades, a vast research has been going on to construct explicit solutions of NLEEs, which are used as models in order to describe many important and problematics physical phenomena in various fields of science. So to figure out the exact solutions of NLEEs substantial work are being made by mathematicians and scientists and have developed effective and convincing methods such as the Hirota’s bilinear transformation method 1, the tanh-function method 2, 3, the exp-function method 4, 5, the F-expansion method 6, the Jacobi elliptic function method 7, the homogeneous balance method 8, the homotopy perturbation method 9, the tanh-coth method 10, the direct algebraic method 11, the Backlund transformation method 12, and others 13, 14, 15, 16.

Later in 2008, Wang et al 17 introduced a new method called the -expansion method for finding the solutions of traveling waves of NLEEs. This - expansion method shows that it is one of the most powerful and effective method to solve NLEEs since it gives a clear and short to the point results in terms of hyperbolic functions, trigonometric functions and rational functions which is why scientists have carried out a lot of researches to construct traveling wave solutions via this method 18, 19, 20, 21.

Further research of - expansion method has been carried out by many researchers to show the possible productivity of the application. For example- Zhang et al 22 expanded the original - expansion method and named as the improved - expansion method. Using this method many researches have been carried out in order to find travelling wave solutions for NLPDEs 23, 24, 25, 26, 27, 28, 29, 30. Then Akber et al 31 introduced the generalized and improved - expansion method, where the second order LODE were used as auxiliary equation to construct travelling wave solution, this method were also used in the study of higher dimensional NLPDEs 32. In the meanwhile, Naher ad Abdullah 33 demonstrated a new method that is the new approach of the - expansion method and new approach of the generalized - expansion method where nonlinear ODE were used as auxiliary equation and the resulted travelling wave solutions of this method were quite different. Many researchers still carrying out experiments using the new extension of - expansion method to generate more new travelling wave solutions of NLEEs.

2 Methodology of New Extension of the Generalized and Improved (G’/G) - Expansion Method

Recently a new application have been introduced called the new extension of the generalized and improved - expansion method for NLEEs. So to demonstrate this method, first a NLPDE is taken with real independent variables and i.e.

(2.1)

where is the polynomial and here is an unknown function. In the polynomial contains different partial derivatives of the function itself wherein involves the highest order derivatives and the highest nonlinear terms. Now the prime process of this method is being discussed in steps below.

Step 1: Suppose that,

(2.2)

where the constant term is known as the speed of wave, is substituted in Eq. (2.1), which allows a PDE to convert an ODE with respect to .

(2.3)

Step 2: Eq. (2.3) is being integrated term by term and if needed it can be integrated more than once and the integral constants may be set to zero to make easy to solve. Now the integrated travelling wave solution of Eq. (2.3) can be represented as.

(2.4)

where or can be zero but all cannot be zero at the same time, is the arbitrary constant to be determined later and is

(2.5)

where satisfies the nonlinear ordinary differential equation (ODE) i.e.

(2.6)

where and are the real parameters

Step 3: The positive integer appearing in the integrated solution of Eq. (2.3) is then determined by considering the homogeneous balance between the highest order derivative and the highest nonlinear term. The value of is substituted in Eq. (2.4) which gives a complete ODE. Then the completed ODE of Eq. (2.4), Eq. (2.5) along with Eq. (2.6) is substituted in the integrated solution of Eq. (2.3) and collecting all the powers of the term in descending order to the left hand side, thus transforms into another polynomial of here and .

Step 4: The coefficient of the polynomial is then equated to zero, hence generates a set of algebraic equation. By solving the algebraic equation gives the value for , , and obtained from Eq. (2.5). Now by solving Eq. (2.6) we obtain a general solution, which is then substituted with the values of constants into Eq. (2.4) we can achieve more general type and more new travelling wave solutions of NLPDE of Eq. (2.1).

Step 5: Using the general solution of Eq. (2.6), the following solutions for Eq. (2.5) are obtained:

Family 1: When and

(2.7)

Family 2: When and

(2.8)

Family 3: When and

(2.9)

Family 4: When and

(2.10)

Family 5: When and

(2.11)

3. Application of the Method

Let us consider the Fisher equation to investigate and construct new wave solutions by executing new extension of the generalized and improved expansion method 34.

The Fisher equation:

(3.1)

By using the wave transformation of Eq. (2.2), into Eq. (3.1), the above equation transforms into the following NLODE:

(3.2)

Now by taking the homogeneous balance between the nonlinear term and the highest order derivative term in Eq. (3.2), we obtain the value for i.e. . Therefore the solution of Eq. (3.2) can be written in the form:

(3.3)

where and are constants to be determined.

Substituting Eq. (3.3) along with Eq. (2.5) and (2.6) into Eq. (3.2) and by simplifying it transforms into polynomials in and By collecting the resulted polynomials, yields a set of simultaneous algebraic equations for and . After solving the systems of algebraic equations with the aid of Maple, we have obtained the following sets result for travelling waves.

3.1. Results of Travelling Waves

Set 1

(3.1.1)

where,

Set 2

(3.1.2)

where,

Set 3

(3.1.3)

Set 4

(3.1.4)

Set 5

(3.1.5)

Set 6

(3.1.6)

Set 7

(3.1.7)

where,

Set 8

(3.1.8)

where,

Set 9

(3.1.9)

where,

Set 10

(3.1.10)

where,

Set 11

(3.1.11)

Set 12

(3.1.12)

Set 13

(3.1.13)

Set 14

(3.1.14)

Set 15

(3.1.15)

Set 16

(3.1.16)
3.2. Solutions

Substituting Eq (3.1.1) in Eq (3.3), along with Eq (2.7) and simplifying, yields the following travelling wave solution, ( if and )

Substituting Eq. (3.1.1) in Eq. (3.3), along with Eq. (2.8) and simplifying, our obtained solution becomes, ( if and )

Substituting Eq (3.1.1) in Eq (3.3), along with Eq (2.9) and simplifying, our exact solution becomes

Similarly, substituting Eq. (3.1.1) in Eq. (3.3), along with Eq. (2.10) and simplifying, our obtained exact solution becomes, (if and )

Substituting Eq. (3.1.1) in Eq. (3.3), along with Eq. (2.11) and simplifying, we obtain the following solution, (if and )

where

Similarly, substituting Eq. (3.1.3) in Eq. (3.3), along with Eqs. (2.7) - (2.11) and simplifying, our travelling wave solutions become:

where

Similarly, substituting Eq (3.1.7) in Eq (3.3), along with Eq (2.7) - (2.11) and simplifying, our obtained travelling wave solutions become:

where

Similarly, substituting Eq. (3.1.9) in Eq. (3.3), along with Eqs. (2.7), (2.9) and (2.11) and simplifying, our obtained solutions become:

where

and

Similarly, substituting Eq. (3.1.11) in Eq. (3.3), along with Eqs. (2.7) - (2.11) and simplifying, the following travelling wave solutions become:

where

Similarly, substituting Eq. (3.1.14) in Eq. (3.3), along with Eqs. (2.7), (2.9) and (2.11) and simplifying, our solutions become:

where

and

4. Discussions

Various methods have been used to investigate for the solutions of Fisher Equation, such as Kudryashov 35 investigated by using simplest equation method, Wazwaz et al. 36 studied by using the Adomain decomposition method, in Ref. 37 Öziş et al. implemented by the Exp-function method, the homotopy analysis method executed by Tan et al. 38, and Ablowitz et al. 39 investigated solutions for a special wave speed. To the best our awareness the Fisher Equation has not been investigated by the new generalized and improved expansion method. It is important to point out that our some obtained solutions are new, simple, straightforward, and precise compared to the solutions obtained in the open literature.

5. Graphical Representations

With the help of the computational software, Maple, we have illustrated some of the obtained solutions for travelling waves solutions in below.

6. Conclusions

In this article, the new extension of the generalized and improved expansion method has been applied successfully in the Fisher Equation. The auxiliary equation used in the method that involves many arbitrary parameters and those can take any real values then the NLODE produces many new solutions. The obtained solutions show that the method is effective and gives precise and direct solutions. Therefore, we conclude that this method could be implemented for constructing various types of wave solutions of NLEEs those arise in the application of mathematical field.

References

[1]  Hirota, R. (1973). Exact envelope-soliton solutions of a nonlinear wave equation. Journal of Mathematical Physics, 14(7), 805-809.
In article      View Article
 
[2]  Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654.
In article      View Article
 
[3]  Abdou, M. A. (2007). The extended tanh method and its applications for solving nonlinear physical models. Applied Mathematics and Computation, 190(1), 988-996.
In article      View Article
 
[4]  Naher, H., Abdullah, F. A., & Akbar, M. A. (2011). The exp-function method for new exact solutions of the nonlinear partial differential equations. International Journal of Physical Sciences, 6(29), 6706-6716.
In article      
 
[5]  He, Y., Li, S., & Long, Y. (2012). Exact solutions of the Klein-Gordon equation by modified Exp-function method. In Int. Math. Forum (Vol. 7, No. 4, pp. 175-182).
In article      
 
[6]  Zhou, Y., Wang, M., & Wang, Y. (2003). Periodic wave solutions to a coupled KdV equations with variable coefficients. Physics Letters A, 308(1), 31-36.
In article      View Article
 
[7]  Ali, A. T. (2011). New generalized Jacobi elliptic function rational expansion method. Journal of computational and applied mathematics, 235(14), 4117-4127.
In article      View Article
 
[8]  Wang, M. (1995). Solitary wave solutions for variant Boussinesq equations. Physics letters A, 199(3-4), 169-172.
In article      View Article
 
[9]  Mohyud-Din, S. T., & Noor, M. A. (2006). Homotopy perturbation method for solving fourth-order boundary value problems. Mathematical Problems in Engineering, 2007.
In article      
 
[10]  Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654.
In article      View Article
 
[11]  Soliman, A. A., & Abdo, H. A. (2012). New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method. arXiv preprint arXiv:1207.5127.
In article      
 
[12]  Rogers, C., & Shadwick, W. F. (1982). Bäcklund transformations and their applications. Academic press.
In article      
 
[13]  Daga, A., & Pradhan, V. (2014). Variational homotopy perturbation method for the nonlinear generalized regularized long wave equation. American. Journal of Applied Mathematics and Statistics, 2, 231-234.
In article      View Article
 
[14]  Prasad, H. S., & Reddy, Y. N. (2015). A Fifth Order Compact Difference Method for Singularly Perturbed Singular Boundary Value Problems. American Journal of Applied Mathematics and Statistics, 3(2), 49-53.
In article      
 
[15]  Aboiyar, T., Luga, T., & Iyorter, B. V. (2015). Derivation of continuous linear multistep methods using Hermite polynomials as basis functions. American Journal of Applied Mathematics and Statistics, 3(6), 220-225.
In article      
 
[16]  Naher, H., & Begum, F. A. (2015). Application of Linear ODE as Auxiliary Equation to the Nonlinear Evolution Equation. American Journal of Applied Mathematics and Statistics, 3(1), 23-28.
In article      
 
[17]  Wang, M., Li, X., & Zhang, J. (2008). The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423.
In article      View Article
 
[18]  Zayed, E. M. E., & Gepreel, K. A. (2009). The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. Journal of Mathematical Physics, 50(1), 013502.
In article      View Article
 
[19]  Naher, H., Abdullah, F. A., & Akbar, M. A. (2011). The (𝐺′/𝐺)-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation. Mathematical Problems in Engineering.
In article      View Article
 
[20]  Feng, J., Li, W., & Wan, Q. (2011). Using (G′/G-)expansion method to seek the traveling wave solution of Kolmogorov-Petrovskii–Piskunov equation. Applied Mathematics and Computation, 217(12), 5860-5865.
In article      View Article
 
[21]  Naher, H., & Abdullah, F. A. (2012). The Basic (G'/G)-expansion method for the fourth order Boussinesq equation. Applied Mathematics, 3(10), 1144.
In article      View Article
 
[22]  Zhang, J., Jiang, F., & Zhao, X. (2010). An improved (G′/G)-expansion method for solving nonlinear evolution equations. International Journal of Computer Mathematics, 87(8), 1716-1725.
In article      View Article
 
[23]  Hamed, Y. S., Sayed, M., Elagan, S. K., & El-Zahar, E. R. (2011). The improved (G'/G)-expansion method for solving (3+1)-dimensional potential-YTSF equation. Journal of Modern Methods in Numerical Mathematics, 2(1-2), 32-39.
In article      View Article
 
[24]  Naher, H., Abdullah, F. A., & Bekir, A. (2012). Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method. AIP Advances, 2(4), 042163.
In article      View Article
 
[25]  Naher, H., & Abdullah, F. A. (2012). Some New Traveling Wave Solutions of the Nonlinear Reaction Diffusion Equation by Using the Improved (𝐺′/𝐺)-Expansion Method. Mathematical Problems in Engineering, 2012.
In article      View Article
 
[26]  Naher, H., & Abdullah, F. A. (2012). Some new solutions of the combined KdV-MKdV equation by using the improved G/G-expansion method. World Applied Sciences Journal, 16(11), 1559-1570.
In article      
 
[27]  Naher, H., & Abdullah, F. A. (2013). The Improved (G'/G)-Expansion Method to the (3 Dimensional Kadomstev-Petviashvili Equation. American Journal of Applied Mathematics and Statistics, 1(4), 64-70.
In article      View Article
 
[28]  Naher, H., Abdullah, F. A., & Rashid, A. (2014). The generalized riccati equation together with the (G'/G)-Expansion method for the (3+ 1)-dimensional modified KDV-zakharov-kuznetsov equation.
In article      
 
[29]  Naher, H., & Abdullah, F. A.. (2014). The improved (G'/G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis & Applications, 16(2), 220-235.
In article      
 
[30]  Naher, H., Abdullah, F. A., & Rashid, A. (2014). Some New Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation via the Improved (G'/G)-Expansion Method. Journal of Computational Analysis & Applications, 17(2) 287-296.
In article      
 
[31]  Akbar, M. A., Ali, N. H., & Zayed, E. M. E. (2012). A generalized and improved-expansion method for nonlinear evolution equations, Math. Prob. Eng., Article ID 459879, 22.
In article      
 
[32]  Naher, H., Abdullah, F. A., & Akbar, M. A. (2013). Generalized and improved (G′/G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. PloS one, 8(5), e64618.
In article      View Article  PubMed
 
[33]  Naher, H., & Abdullah, F. A. (2013). New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation. AIP Advances, 3(3), 032116.
In article      View Article
 
[34]  Naher, H., & Abdullah, F. A. (2016). Further extension of the generalized and improved (G′/G)-expansion method for nonlinear evolution equation. Journal of the Association of Arab Universities for Basic and Applied Sciences, 19(1), 52-58.
In article      View Article
 
[35]  Kudryashov, N. A. (2005). Exact solitary waves of the Fisher equation. Physics Letters A, 342(1-2), 99-106.
In article      View Article
 
[36]  Wazwaz, A. M., & Gorguis, A. (2004). An analytic study of Fisher's equation by using Adomian decomposition method. Applied Mathematics and Computation, 154(3), 609-620.
In article      View Article
 
[37]  Öziş, T., & Köroğlu, C. (2008). A novel approach for solving the Fisher equation using Exp-function method. Physics Letters A, 372(21), 3836-3840.
In article      View Article
 
[38]  Tan, Y., Xu, H., & Liao, S. J. (2007). Explicit series solution of travelling waves with a front of Fisher equation. Chaos, Solitons & Fractals, 31(2), 462-472.
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[39]  Ablowitz, M. J., & Zeppetella, A. (1979). Explicit solutions of Fisher's equation for a special wave speed. Bulletin of Mathematical Biology, 41(6), 835-840.
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Published with license by Science and Education Publishing, Copyright © 2018 Anika Tashin Khan and Hasibun Naher

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Normal Style
Anika Tashin Khan, Hasibun Naher. Solitons and Periodic Solutions of the Fisher Equation with Nonlinear Ordinary Differential Equation as Auxiliary Equation. American Journal of Applied Mathematics and Statistics. Vol. 6, No. 6, 2018, pp 244-252. http://pubs.sciepub.com/ajams/6/6/5
MLA Style
Khan, Anika Tashin, and Hasibun Naher. "Solitons and Periodic Solutions of the Fisher Equation with Nonlinear Ordinary Differential Equation as Auxiliary Equation." American Journal of Applied Mathematics and Statistics 6.6 (2018): 244-252.
APA Style
Khan, A. T. , & Naher, H. (2018). Solitons and Periodic Solutions of the Fisher Equation with Nonlinear Ordinary Differential Equation as Auxiliary Equation. American Journal of Applied Mathematics and Statistics, 6(6), 244-252.
Chicago Style
Khan, Anika Tashin, and Hasibun Naher. "Solitons and Periodic Solutions of the Fisher Equation with Nonlinear Ordinary Differential Equation as Auxiliary Equation." American Journal of Applied Mathematics and Statistics 6, no. 6 (2018): 244-252.
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[1]  Hirota, R. (1973). Exact envelope-soliton solutions of a nonlinear wave equation. Journal of Mathematical Physics, 14(7), 805-809.
In article      View Article
 
[2]  Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654.
In article      View Article
 
[3]  Abdou, M. A. (2007). The extended tanh method and its applications for solving nonlinear physical models. Applied Mathematics and Computation, 190(1), 988-996.
In article      View Article
 
[4]  Naher, H., Abdullah, F. A., & Akbar, M. A. (2011). The exp-function method for new exact solutions of the nonlinear partial differential equations. International Journal of Physical Sciences, 6(29), 6706-6716.
In article      
 
[5]  He, Y., Li, S., & Long, Y. (2012). Exact solutions of the Klein-Gordon equation by modified Exp-function method. In Int. Math. Forum (Vol. 7, No. 4, pp. 175-182).
In article      
 
[6]  Zhou, Y., Wang, M., & Wang, Y. (2003). Periodic wave solutions to a coupled KdV equations with variable coefficients. Physics Letters A, 308(1), 31-36.
In article      View Article
 
[7]  Ali, A. T. (2011). New generalized Jacobi elliptic function rational expansion method. Journal of computational and applied mathematics, 235(14), 4117-4127.
In article      View Article
 
[8]  Wang, M. (1995). Solitary wave solutions for variant Boussinesq equations. Physics letters A, 199(3-4), 169-172.
In article      View Article
 
[9]  Mohyud-Din, S. T., & Noor, M. A. (2006). Homotopy perturbation method for solving fourth-order boundary value problems. Mathematical Problems in Engineering, 2007.
In article      
 
[10]  Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654.
In article      View Article
 
[11]  Soliman, A. A., & Abdo, H. A. (2012). New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method. arXiv preprint arXiv:1207.5127.
In article      
 
[12]  Rogers, C., & Shadwick, W. F. (1982). Bäcklund transformations and their applications. Academic press.
In article      
 
[13]  Daga, A., & Pradhan, V. (2014). Variational homotopy perturbation method for the nonlinear generalized regularized long wave equation. American. Journal of Applied Mathematics and Statistics, 2, 231-234.
In article      View Article
 
[14]  Prasad, H. S., & Reddy, Y. N. (2015). A Fifth Order Compact Difference Method for Singularly Perturbed Singular Boundary Value Problems. American Journal of Applied Mathematics and Statistics, 3(2), 49-53.
In article      
 
[15]  Aboiyar, T., Luga, T., & Iyorter, B. V. (2015). Derivation of continuous linear multistep methods using Hermite polynomials as basis functions. American Journal of Applied Mathematics and Statistics, 3(6), 220-225.
In article      
 
[16]  Naher, H., & Begum, F. A. (2015). Application of Linear ODE as Auxiliary Equation to the Nonlinear Evolution Equation. American Journal of Applied Mathematics and Statistics, 3(1), 23-28.
In article      
 
[17]  Wang, M., Li, X., & Zhang, J. (2008). The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423.
In article      View Article
 
[18]  Zayed, E. M. E., & Gepreel, K. A. (2009). The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. Journal of Mathematical Physics, 50(1), 013502.
In article      View Article
 
[19]  Naher, H., Abdullah, F. A., & Akbar, M. A. (2011). The (𝐺′/𝐺)-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation. Mathematical Problems in Engineering.
In article      View Article
 
[20]  Feng, J., Li, W., & Wan, Q. (2011). Using (G′/G-)expansion method to seek the traveling wave solution of Kolmogorov-Petrovskii–Piskunov equation. Applied Mathematics and Computation, 217(12), 5860-5865.
In article      View Article
 
[21]  Naher, H., & Abdullah, F. A. (2012). The Basic (G'/G)-expansion method for the fourth order Boussinesq equation. Applied Mathematics, 3(10), 1144.
In article      View Article
 
[22]  Zhang, J., Jiang, F., & Zhao, X. (2010). An improved (G′/G)-expansion method for solving nonlinear evolution equations. International Journal of Computer Mathematics, 87(8), 1716-1725.
In article      View Article
 
[23]  Hamed, Y. S., Sayed, M., Elagan, S. K., & El-Zahar, E. R. (2011). The improved (G'/G)-expansion method for solving (3+1)-dimensional potential-YTSF equation. Journal of Modern Methods in Numerical Mathematics, 2(1-2), 32-39.
In article      View Article
 
[24]  Naher, H., Abdullah, F. A., & Bekir, A. (2012). Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method. AIP Advances, 2(4), 042163.
In article      View Article
 
[25]  Naher, H., & Abdullah, F. A. (2012). Some New Traveling Wave Solutions of the Nonlinear Reaction Diffusion Equation by Using the Improved (𝐺′/𝐺)-Expansion Method. Mathematical Problems in Engineering, 2012.
In article      View Article
 
[26]  Naher, H., & Abdullah, F. A. (2012). Some new solutions of the combined KdV-MKdV equation by using the improved G/G-expansion method. World Applied Sciences Journal, 16(11), 1559-1570.
In article      
 
[27]  Naher, H., & Abdullah, F. A. (2013). The Improved (G'/G)-Expansion Method to the (3 Dimensional Kadomstev-Petviashvili Equation. American Journal of Applied Mathematics and Statistics, 1(4), 64-70.
In article      View Article
 
[28]  Naher, H., Abdullah, F. A., & Rashid, A. (2014). The generalized riccati equation together with the (G'/G)-Expansion method for the (3+ 1)-dimensional modified KDV-zakharov-kuznetsov equation.
In article      
 
[29]  Naher, H., & Abdullah, F. A.. (2014). The improved (G'/G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis & Applications, 16(2), 220-235.
In article      
 
[30]  Naher, H., Abdullah, F. A., & Rashid, A. (2014). Some New Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation via the Improved (G'/G)-Expansion Method. Journal of Computational Analysis & Applications, 17(2) 287-296.
In article      
 
[31]  Akbar, M. A., Ali, N. H., & Zayed, E. M. E. (2012). A generalized and improved-expansion method for nonlinear evolution equations, Math. Prob. Eng., Article ID 459879, 22.
In article      
 
[32]  Naher, H., Abdullah, F. A., & Akbar, M. A. (2013). Generalized and improved (G′/G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. PloS one, 8(5), e64618.
In article      View Article  PubMed
 
[33]  Naher, H., & Abdullah, F. A. (2013). New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation. AIP Advances, 3(3), 032116.
In article      View Article
 
[34]  Naher, H., & Abdullah, F. A. (2016). Further extension of the generalized and improved (G′/G)-expansion method for nonlinear evolution equation. Journal of the Association of Arab Universities for Basic and Applied Sciences, 19(1), 52-58.
In article      View Article
 
[35]  Kudryashov, N. A. (2005). Exact solitary waves of the Fisher equation. Physics Letters A, 342(1-2), 99-106.
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