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Permanent URI for this collectionhttps://hdl.handle.net/10361/6846

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The exp-function method for new exact solutions of the nonlinear partial differential equations
(© 2011 International Journal of Physical Sciences, 2011) Naher, Hasibun; Abdullah, Farah Aini; Akbar, M. Ali; Department of Mathematics and Natural Sciences
In this article, the exp-function method is used to construct some new exact solitary wave solutions of the sixth-order Boussinesq equation and the regularized long wave equations. These equations play very important role in mathematical physics, engineering sciences and applied mathematics. The exp-function method is a powerful and straightforward mathematical tool for solving nonlinear evolution equations.
The (G'/G)-expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation
(© 2011 Mathematical Problems in Engineering, 2011) Naher, Hasibun; Abdullah, Farah Aini; Akbar, M. Ali; Department of Mathematics and Natural Sciences
We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation by the (G'/G) -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (G ′ / G) -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.
New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method
(© 2012 Journal of Applied Mathematics, 2012) Naher, Hasibun; Abdullah, Farah Aini; Akbar, M. Ali; Department of Mathematics and Natural Sciences
We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.
New traveling wave solutions of the higher dimensional nonlinear evolution equation by the improved (G′/G) expansion method
(© 2012 World Applied Sciences Journal, 2012) Naher, Hasibun; Abdullah, Farah Aini; Akbar, M. Ali; Department of Mathematics and Natural Sciences
In this article, we investigate the nonlinear evolution equation, namely, the (3+l)-dimensional modified KdV-Zakharov-Kuznetsev equation by applying the improved (G′/G)-expansion method to construct some new traveling wave solutions. The obtained solutions are expressed in terms of the hyperbolic, the trigonometric and the rational functions including solitons and periodic solutions. The attained solutions become some special functions when the arbitrary constants taken particular values. It is important to mention that some of our solutions are in good harmony with the existing results which certifies our other solutions.
The improved (G'/G) -expansion method for the (2+1)-dimensional modified Zakharov-Kuznetsov equation
(© 2012 Journal of Applied Mathematics, 2012) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
we apply the improved (G'/G) -expansion method for constructing abundant new exact traveling wave solutions of the (2+1)-dimensional Modified Zakharov-Kuznetsov equation. In addition, G'' + λ G' + μG = 0 together with b (α) = ∑ q=-w wp q (G'/G) q is employed in this method, where p q (q = 0, ± 1, ± 2,⋯, ± w), λ and μ are constants. Moreover, the obtained solutions including solitons and periodic solutions are described by three different families. Also, it is noteworthy to mention out that, some of our solutions are coincided with already published results, if parameters taken particular values. Furthermore, the graphical presentations are demonstrated for some of newly obtained solutions.
The modified benjamin-bona-mahony equation via the extended generalized riccati equation mapping method
(© 2012 Applied Mathematical Science, 2012) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
The generalized Riccati equation mapping is extended together with the (G'/G) -expansion method and is a powerful mathematical tool for solving nonlinear partial differential equations. In this article, we construct twenty seven new exact traveling wave solutions including solitons and periodic solutions of the modified Benjamin-Bona-Mahony equation by applying the extended generalized Riccati equation mapping method. In this method, G'(μ) = p + rG(μ) + sG 2 (μ) is implemented as the auxiliary equation, where r, s and p are arbitrary constants and called the generalized Riccati equation. The obtained solutions are described in four different families including the hyperbolic functions, the trigonometric functions and the rational functions. In addition, it is worth mentioning that one of newly obtained solutions is identical for a special case with already published result which validates our other solutions.
Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G′/G)-expansion method
(© 2012 Mathematical Problems in Engineering, 2012) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
We construct new exact traveling wave solutions involving free parameters of the nonlinear reaction diffusion equation by using the improved (G ′ /G)-expansion method. The second-order linear ordinary differential equation with constant coefficients is used in this method. The obtained solutions are presented by the hyperbolic and the trigonometric functions. The solutions become in special functional form when the parameters take particular values. It is important to reveal that our solutions are in good agreement with the existing results.
Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method
(© 2012 AIP Advances, 2012) Naher, Hasibun; Abdullah, Farah Aini; Bekir, Ahmet; Department of Mathematics and Natural Sciences
In this article, we investigate the compound KdV-Burgers equation involving parameters by applying the improved (G′/G)-expansion method for constructing some new exact traveling wave solutions including solitons and periodic solutions. The second order linear ordinary differential equation with constant coefficients is used, in this method. The obtained solutions are presented through the hyperbolic, the trigonometric and the rational functions. Further, it is significant to point out that some of our solutions are in good agreement for special cases with the existing results which validates our other solutions. Moreover, some of the obtained solutions are described in the figures.
New traveling wave solutions by the extended generalized Riccati equation mapping method of the (2 + 1) -dimensional evolution equation
(© 2012 Journal of Applied Mathematics, 2012) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
The generalized Riccati equation mapping is extended with the basic (G ′ / G) -expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equation G ′ (η) = w + u G (η) + v G 2 (η) is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.
New approach of (G′G)-expansion method and new approach of generalized (G′G)-expansion method for nonlinear evolution equation
(© 2013 AIP Advances, 2013) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
In this article, new (G′G)-expansion method and new generalized (G′G)-expansion method is proposed to generate more general and abundant new exact traveling wave solutions of nonlinear evolution equations. The novelty and advantages of these methods is exemplified by its implementation to the KdV equation. The results emphasize the power of proposed methods in providing distinct solutions of different physical structures in nonlinear science. Moreover, these methods could be more effectively used to deal with higher dimensional and higher order nonlinear evolution equations which frequently arise in many scientific real time application fields.
Generalized and Improved (G′/G)-Expansion Method for (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation
(© 2013 PLoS ONE, 2013) Naher, Hasibun; Abdullah, Farah Aini; Akbar, M. Ali; Department of Mathematics and Natural Sciences
The generalized and improved G′/G-expansion method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. In this article, we investigate the higher dimensional nonlinear evolution equation, namely, the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation via this powerful method. The solutions are found in hyperbolic, trigonometric and rational function form involving more parameters and some of our constructed solutions are identical with results obtained by other authors if certain parameters take special values and some are new. The numerical results described in the figures were obtained with the aid of commercial software Maple.
Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation
(© 2013 AIP Advances, 2013) Naher, Hasibun; Abdullah, Farah Aini; Mohyud-Din, Syed Tauseef; Department of Mathematics and Natural Sciences
In this article, the generalized Riccati equation mapping together with the basic (G′/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G′(φ) = h + f G(φ) + g G 2(φ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.
The extended generalized Riccati equation mapping method for the (1+1)-dimensional modified KdV equation
(© 2013 IDOSI Publications, 2013) Naher, Hasibun; Abdullah, Farah Aini; Akbar, M. Ali; Yildirim, Ahmet; Department of Mathematics and Natural Sciences
In this article, the generalized Riccati equation mapping is extended by the (G-/G)-expansion method. 2 In this method, the auxiliary equation G'- = r + pG + qG2 is used and called the generalized Riccati equation, where p,q and r are arbitrary constants. We construct twenty five exact traveling wave solutions of the (1+1)-dimensional modified KdV equation involving parameter by applying this method. The solutions are presented in terms of the hyperbolic, the trigonometric and the rational functional form including solitons and periodic solutions. Moreover, it is worth mentioning that one of our obtained solutions is in good agreement with the existing results which in turn validates our other solutions. In addition, some of newly obtained solutions are described in the figures.
The generalized riccati equation together with the (G' / G) -Expansion method for the (3+1)-dimensional modified KDV-zakharov-kuznetsov equation
(© 2014 Politechnica University of Bucharest, 2014) Naher, Hasibun; Abdullah, Farah Aini; Rashid, Abdur; Department of Mathematics and Natural Sciences
We construct new exact travelling wave solutions including solitons and periodic solutions of the (3+1)-dimensional modified KdV Zakharov-Kuznetsov equation involving parameter by applying the generalized Riccati equation together with the (G' /G) -expansion method. In addition, in this method, G' = n + lG + mG2 is used, as the auxiliary equation, called the generalized Riccati equation, where l, m and n are arbitrary constants. Further, the solutions are expressed in terms of the hyperbolic function, the trigonometric function and the rational functional form. Moreover, some of obtained traveling wave solutions are presented in the figures with the aid of commercial software Maple.
Some new solutions of the (3+1)-Dimensional Jimbo-Miwa equation via the improved (G'/G)-Expansion method
(© 2014 Eudoxus Press, LLC, 2014) Naher, Hasibun; Aini Abdullah, Farah; Rashid, Abdur; Department of Mathematics and Natural Sciences
The improved (G'/G)-expansion method is straightforward and effective mathematical tool for establishing exact traveling wave solutions of different nonlinear partial differential equations which arise in engineering sciences, applied mathematics and real time application fields. In this article, we have constructed some new traveling wave solutions of the nonlinear evolution equation, namely, the (3+1)-dimensional Jimbo-Miwa equation via the improved (G'/G)-expansion method. In this method, the general solution of the second order linear ordinary differential equation involving constants coefficients together with (Formula Presented) is employed, where (Formula Presented) are constants. Further, it is worth stating that the obtained solutions become in special functional forms for the particular values of the arbitrary constants. In addition, it is noteworthy declaring that, some of our solutions are in good agreement with already published results. Moreover, some of the solutions are described in the figures with the aid of commercial software Maple
Some new solutions of the (1+1)-dimensional PDE via the improved (G ′/ G)-expansion method
(© 2014 American Institute of Physics Inc., 2014) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
In this article, an improved (G′/G)-expansion method is implemented for the simplified Modified Camassa-Holm (MCH) equation involving parameters, with an aim to construct many new traveling wave solutions. In this method, second order linear ordinary differential equation with constant coefficients has been implemented as an auxiliary equation. The generated solutions including solitons and periodic solutions are demonstrated by the hyperbolic function, the trigonometric function and the rational forms. If the parameters take particular values, the solutions become in special functional form. Moreover, it is worth mentioning that, some of our solutions are in good agreement with already published results in the open literature by setting appropriate values of constants, which proves our other solutions. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple 13.
The improved (G'/G)-expansion method to the (2+1)-dimensional breaking soliton equation
(© 2014 Eudoxus Press, LLC, 2014-01) Naher, Hasibun; Aini Abdullah, Farah; Department of Mathematics and Natural Sciences
In this article, we generate abundant traveling wave solutions of partial differential equation, namely, the (2+1)-dimensional breaking soliton equation involving parameter by applying the improved (G'/G) -expansion method. In this method, G″+ψG′+ФG=0 together with (formula presented) is implemented, where Sf(f = 0,±1,±2,...,±U, ψ and Ф are constants. In addition, the obtained analytical solutions are illustrated in three different families including solitons and periodic solurions. Further, it is vital mentioning that, for a special case, some of our solutions are in good contract with those gained by other authors
New approach of (G'/G)-expansion method for RLW equation
(© 2014 Maxwell Science Publications, 2014) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
In this study, new extension of the (G'/G)-expansion method with nonlinear ODE has been introduced to investigate the generalized regularized long wave equation. Therefore, many new travelling wave solutions with many parameters are generated. The obtained solutions are more general and solutions include the solutions, the hyperbolic function, the trigonometric function and the rational forms. In addition, some of our solutions are identical with already published results, which validate newly generated solutions and others solutions have not been reported in the previous literature.
Further extension of the generalized and improved (G′/G)-expansion method for nonlinear evolution equation
(© 2016 Elsevier B.V, 2016) Naher, Hasibun; Abdullah, Farah Aini; Department of Mathematics and Natural Sciences
n this article, the generalized and improved (G'/G)-expansion method has been proposed for further extension to generate many new travelling wave solutions. In addition, nonlinear ordinary differential equation is implemented as auxiliary equation including many parameters instead of linear ordinary differential equation. Moreover, the presentation of the travelling wave solutions is quiet new. The effectiveness and reliability of the method are shown by its application to the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation. Some of our generated solutions turned into some known solutions, when parameters consider specific values and others are new.

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