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</svg>)-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

Shakeel, Muhammad; Ul-Hassan, Qazi Mahmood; Ahmad, Jamshad

doi:https://doi.org/10.1155/2014/601961

Abstract and Applied Analysis

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Volume 2014 | Article ID 601961 | https://doi.org/10.1155/2014/601961

Applications of the Novel ()-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

Muhammad Shakeel,¹Qazi Mahmood Ul-Hassan,¹and Jamshad Ahmad¹

Academic Editor: Dumitru Baleanu

Received17 Feb 2014

Revised24 Apr 2014

Accepted12 May 2014

Published18 Jun 2014

Abstract

We use the fractional derivatives in modified Riemann-Liouville derivative sense to construct exact solutions of time fractional simplified modified Camassa-Holm (MCH) equation. A generalized fractional complex transform is properly used to convert this equation to ordinary differential equation and, as a result, many exact analytical solutions are obtained with more free parameters. When these free parameters are taken as particular values, the traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. Moreover, the numerical presentations of some of the solutions have been demonstrated with the aid of commercial software Maple. The recital of the method is trustworthy and useful and gives more new general exact solutions.

1. Introduction

The class of fractional calculus is one of the most convenient classes of fractional differential equations which were viewed as generalized differential equations [1]. In the sense that much of the theory and, hence, applications of differential equations can be extended smoothly to fractional differential equations with the same flavor and spirit of the realm of differential equation, the seeds of fractional calculus were planted over three hundred years ago from a gracious idea of L’Hopital, who wrote a letter to Leibniz on 1695, asking about a rigorous description of the derivative of order . Fractional calculus is the theory of differentiation and integration of noninteger order and embodies the generality of the conventional differential and integral calculus. Therefore, some of the properties of the fractional integral and derivatives differ from the conventional ones in order to allow its implementation in a broader assortment of cases, which cannot be appropriately illustrated by the conventional integer-order calculus. Fractional calculus is painstaking to be a very authoritative tool to help scientists to unearth the concealed properties of the dynamics of multifaceted systems in all fields of sciences and engineering. In recent years, fractional calculus played an imperative role of a proficient, expedient, and elementary theoretical structure for more adequate modeling of multifaceted dynamic processes. Therefore, mounting applications of fractional calculus can be seen in modeling, signal processing, electromagnetism, mechanics, physics, biology, medicine, chemistry, bioengineering, biological systems, and in many other areas [2, 3]. Recently, it has turned out that those differential equations are involving derivatives of noninteger [4]. For example, the nonlinear oscillation of earthquakes can be modeled with fractional derivatives [5]. More recently, applications have included classes of nonlinear equation with multiorder fractional derivatives. We apply a generalized fractional complex transform [6–9] to convert fractional order differential equation to ordinary differential equation. Many important phenomena in electromagnetic, viscoelasticity, electrochemistry, and material science are well described by differential equations of fractional order [10–14]. A physical interpretation of the fractional calculus was given in [15–19]. With the development of symbolic computation software, like Maple, many numerical and analytical methods to search for exact solutions of NLEEs have attracted more attention. As a result, the researchers developed and established many methods, for example, the Cole-Hopf transformation [20], the Tanh-function method [21–24], the inverse scattering transform method [25], the variational iteration method [26, 27], Exp-function method [28–31], and -expansion method [32, 33] that are used for searching the exact solutions.

Recently, a straightforward and concise method, called -expansion method, was introduced by Wang et al. [34] and demonstrated that it is a powerful method for seeking analytic solutions of NLEEs. -expansion is a reliable technique, which gives various types of the solitary wave solutions including the hyperbolic functions, the trigonometric functions, and the rational functions. It is also evident from the literature that such solutions always satisfy the given nonlinear differential equations. For additional references, see the articles [35–40]. In order to establish the efficiency and assiduousness of -expansion method and to extend the range of applicability, further research has been carried out by several researchers. For instance, Zhang et al. [41] made a generalization of -expansion method for the evolution equations with variable coefficients. Zhang et al. [42] also presented an improved -expansion method to seek more general traveling wave solutions. Zayed [43] presented a new approach of -expansion method where satisfies the Jacobi elliptic equation, , where , , are arbitrary constants and obtained new exact solutions. Zayed [44] again presented an alternative approach of this method in which satisfies the Riccati equation , where and are arbitrary constants.

In this paper, we will apply novel -expansion method introduced by Alam et al. [45] to solve the time fractional simplified modified Camassa-Holm (MCH) equation in the sense of modified Riemann-Liouville derivative by Jumarie [46] and abundant new families of exact solutions are found. The Jumarie modified Riemann-Liouville derivative of order is defined by the following expression:

Some important properties of Jumarie’s derivative are

2. Description of the Method

Suppose that a fractional partial differential equation in the independent variables, say , is given by where is Jumarie’s modified Riemann-Liouville derivatives of is an unknown function, is a polynomial in , and its various partial derivatives including fractional derivatives in which the highest order derivatives and nonlinear terms are involved.

The main steps of the method are as follows.

Step 1. Li and He [7] proposed a fractional complex transformation to convert fractional partial differential equations into ordinary differential equations (ODE), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The traveling wave variable where , are arbitrary constants with , permits us to convert (5) into an ordinary differential equation of integer order in the form where the superscripts stand for the ordinary derivatives with respect to .

Step 2. Integrating (7) term by term one or more times if possible yields constant(s) of integration which can be calculated later on.

Step 3. Assume that the solution of (7) can be represented as where where both and cannot be zero simultaneously. and are constants to be determined later and satisfies the second order nonlinear ordinary differential equation as an auxiliary equation where , , and are real constants.
Equation (10) can be reduced to the following Riccati equation by making use of the Cole-Hopf transformation as
Equation (11) has twenty five solutions [47].

Step 4. The positive integer can be determined by balancing the highest order linear term with the nonlinear term of the highest order come out in (7).

Step 5. Substituting (8) together with (9) and (10) into (7), we obtain polynomials in and . Collecting each coefficient of the resulted polynomials to zero yields an overdetermined set of algebraic equations for , , , and .

Step 6. The values of the arbitrary constants can be obtained by solving the algebraic equations obtained in Step 4. The obtained values of the arbitrary constants and the solutions of (10) yield abundant exact traveling wave solutions of the nonlinear evolution equation (5).

3. Application of the Method to the Time Fractional Simplified (MCH) Equation

Now, consider the following time fractional simplified modified Camassa-Holm (MCH) equation: which is the variation of the equation

Many researchers investigated the simplified MCH equation by using different methods to establish exact solutions. For example, Liu et al. [48] were concerned about the -expansion method to solve the simplified MCH equation, whereas the second order linear ordinary differential equation (LODE) is considered as an auxiliary equation. Wazwaz [49] studied this equation by using the sine-cosine algorithm. Zaman and Sultana [50] used the -expansion method together with the generalized Riccati equation to MCH equation to find the exact solutions. Alam and Akbar [51] applied the generalized -expansion method to look for the exact solutions via the simplified MCH equation. Further details of MCH equation can be found in references [52, 53].

By the use of (4), (12) is converted into an ordinary differential equation of integer order and after integrating once, we obtain where is an integral constant which is to be determined later.

Considering the homogeneous balance between and in (14), we obtain ; that is, . Therefore, the trial solution formula (8) becomes

Using (15) into (14), left hand side is converted into polynomials in and . Equating the coefficients of same power of the resulted polynomials to zero, we obtain a system of algebraic equations for , , , , , , and (which are omitted for the sake of simplicity). Solving the overdetermined set of algebraic equations by using the symbolic computation software, such as Maple 13, we obtain the following four solution sets.

Set 1. Consider where , , , , and are arbitrary constants.

Set 2. Consider where , , , , and are arbitrary constants.

Set 3. Consider where , , , and are arbitrary constants.

Set 4. Consider where , , , and are arbitrary constants.

Substituting (16)–(19) into (15), we obtain where where where where

Substituting the solutions of (10) into (20) and simplifying, we obtain the following solutions.

When and (or ) (Figure 1), where and are real constants (Figure 2). Consider When and (or ), where and are real constants such that .

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

Consider

When and , where is an arbitrary constant.

When and , the solution of (12) is where is an arbitrary constant.

Substituting the solutions of (10) in (21) and simplifying, we obtain the following solutions.

When and (or ),

The other families of exact solutions of (12) are omitted for convenience.

When and (or ) (Figure 3),

(a)

(b)

(c)

(d)

When and , the solution of (12) is where is an arbitrary constant.

We can write down the other families of exact solutions of (12) which are omitted for practicality.

Similarly, by substituting the solutions of (10) into (22) and simplifying, we obtain the following solutions.

When and (or ),

Others families of exact solutions are omitted for the sake of simplicity.

When and (or ) (Figure 4),

(a)

(b)

(c)

(d)

When and , the solution of (12) is where is an arbitrary constant.

Other exact solutions of (12) are omitted here for convenience.

Finally, by substituting the solutions of (10) into (23) and simplifying, we obtain the following solutions.

When and (or ) (Figure 5),

(a)

(b)

(c)

(d)

Others families of exact solutions are omitted for the sake of ease.

When and (or ),

When and , the solution of (12) is where is an arbitrary constant.

Other exact solutions of (12) are omitted here for expediency.

4. Conclusions

A novel -expansion method is applied to fractional partial differential equation successfully. As applications, abundant new exact solutions for the time fractional simplified modified Camassa-Holm (MCH) equation have been successfully obtained. The nonlinear fractional complex transformation for is very important, which ensures that a certain fractional partial differential equation can be converted into another ordinary differential equation of integer order. The obtained solutions are more general with more parameters. Also comparison has been made in the form of table (Table 1), which shows that some of our solutions are in full agreement with the results obtained previously. Thus, novel -expansion method would be a powerful mathematical tool for solving nonlinear evolution equations.

Conflict of Interests

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

References

A. G. Nikitin and T. A. Barannyk, “Solitary wave and other solutions for nonlinear heat equations,” Central European Journal of Mathematics, vol. 2, no. 5, pp. 840–858, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Baleanu, J. A. Tenreiro Machado, and W. Chen, “Fractional differentiation and its applications I,” Computers & Mathematics with Applications, vol. 66, no. 5, p. 575, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
D. Baleanu, R. Garra, and I. Petras, “A fractional variational approach to the fractional Basset-type equation,” Reports on Mathematical Physics, vol. 72, no. 1, pp. 57–64, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
View at: MathSciNet
J. H. He, “Some applications of nonlinear fractional differential equations and their applications,” Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86–90, 1999.
View at: Google Scholar
Z. B. Li and J. H. He, “Application of the fractional complex transform to fractional differential equations,” Nonlinear Science Letter A, vol. 2, no. 3, pp. 121–126, 2011.
View at: Google Scholar
Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970–973, 2010.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J.-H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters A, vol. 376, no. 4, pp. 257–259, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. W. Ibrahim, “Fractional complex transforms for fractional differential equations,” Advances in Difference Equations, vol. 2012, article 192, pp. 1–12, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume I. Background and Theory, Nonlinear Physical Science, Higher Education Press, Beijing, China; Springer, Heidelberg, Germany, 2013.
View at: Publisher Site | MathSciNet
V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume II. Applications, Nonlinear Physical Science, Higher Education Press, Beijing, China; Springer, Heidelberg, Germany, 2013.
View at: Publisher Site | MathSciNet
D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Hackensack, NJ, USA, 2012.
View at: Publisher Site | MathSciNet
S. Das, R. Kumar, P. K. Gupta, and H. Jafari, “Approximate analytical solutions for fractional space- and time-partial differential equations using homotopy analysis method,” Applications and Applied Mathematics, vol. 5, no. 10, pp. 1641–1659, 2010.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. K. Golmankhaneh, T. Khatuni, N. A. Porghoveh, and D. Baleanu, “Comparison of iterative methods by solving nonlinear Sturm-Liouville, Burgers and Navier-Stokes equations,” Central European Journal of Physics, vol. 10, no. 4, pp. 966–976, 2012.
View at: Publisher Site | Google Scholar
S. Das, “A note on fractional diffusion equations,” Chaos, Solitons & Fractals, vol. 42, no. 4, pp. 2074–2079, 2009.
View at: Publisher Site | Google Scholar
X. J. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011.
View at: Google Scholar
X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
J. Ahmad, S. T. Mohyud-Din, and X. J. Yang, “Local fractional decomposition method on wave equation in fractal strings,” Mitteilungen Klosterneuburg, vol. 62, no. 2, pp. 98–105, 2014.
View at: Google Scholar
X. J. Yang and Y. D. Zhang, “A new Adomain decomposition procedure scheme for solving local fractional Volterra integral equation,” Advances in Information Technology and Management, vol. 1, no. 4, pp. 158–161, 2012.
View at: Google Scholar
A. H. Salas and C. A. Gómez, “Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 194329, 14 pages, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Malfliet, “The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations,” in Proceedings of the 10th International Congress on Computational and Applied Mathematics (ICCAM '02), vol. 164-165, pp. 529–541, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations,” Chaos, Solitons & Fractals, vol. 38, no. 5, pp. 1505–1516, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters. A, vol. 277, no. 4-5, pp. 212–218, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, Mass, USA, 1991.
View at: Publisher Site | MathSciNet
H. Jafari and H. Tajadodi, “He's variational iteration method for solving fractional Riccati differential equation,” International Journal of Differential Equations, vol. 2010, Article ID 764738, 8 pages, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, “Fractional variational iteration method via modified Riemann-Liouville derivative,” Journal of King Saud University—Science, vol. 23, no. 4, pp. 413–417, 2011.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Naher, F. A. Abdullah, and M. Ali Akbar, “New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method,” Journal of Applied Mathematics, vol. 2012, Article ID 575387, 14 pages, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. T. Mohyud-Din, “Solutions of nonlinear differential equations by Exp-function method,” World Applied Sciences Journal, vol. 7, pp. 116–147, 2009.
View at: Google Scholar
S. T. Mohyud-Din, M. A. Noor, and A. Waheed, “Exp-function method for generalized travelling solutions of calogero-degasperis-fokas equation,” Zeitschrift für Naturforschung Section A, vol. 65, no. 1, pp. 78–84, 2010.
View at: Google Scholar
M. Wang and X. Li, “Applications of $F$ -expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons & Fractals, vol. 24, no. 5, pp. 1257–1268, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. Abdou, “The extended $F$ -expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons & Fractals, vol. 31, no. 1, pp. 95–104, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
M. Wang, X. Li, and J. Zhang, “The $G^{'} / G$ -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. Akbar, N. H. M. Ali, and S. T. Mohyud-Din, “The alternative $G^{'} / G$ -expansion method with generalized Riccati equation: application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equation,” International Journal of Physical Sciences, vol. 7, no. 5, pp. 743–752, 2012.
View at: Google Scholar
M. A. Akbar, N. H. M. Ali, and E. M. E. Zayed, “Abundant exact traveling wave solutions of generalized Bretherton equation via improved $G^{'} / G$ -expansion method,” Communications in Theoretical Physics, vol. 57, no. 2, pp. 173–178, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
E. J. Parkes, “Observations on the basic $G^{'} / G$ -expansion method for finding solutions to nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1759–1763, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
M. Ali Akbar and N. H. M. Ali, “The modified alternative ( $G^{'} / G$ )-expansion method for finding the exact solutions of nonlinear PDEs in mathematical physics,” International Journal of Physical Sciences, vol. 6, no. 35, pp. 7910–7920, 2011.
View at: Publisher Site | Google Scholar
M. A. Akbar, N. H. M. Ali, and E. M. E. Zayed, “A generalized and improved ( $G^{'} / G$ )-expansion method for nonlinear evolution equations,” Mathematical Problems in Engineering, vol. 2012, Article ID 459879, 22 pages, 2012.
View at: Publisher Site | Google Scholar
E. Salehpour, H. Jafari, and N. Kadkhoda, “Application of ( $G^{'} / G$ )-expansion method to nonlinear Lienard equation,” Indian Journal of Science and Technology, vol. 5, no. 4, pp. 2554–2556, 2012.
View at: Google Scholar
J. Zhang, X. Wei, and Y. Lu, “A generalized $G^{'} / G$ -expansion method and its applications,” Physics Letters A, vol. 372, no. 20, pp. 3653–3658, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
J. Zhang, F. Jiang, and X. Zhao, “An improved $G^{'} / G$ -expansion method for solving nonlinear evolution equations,” International Journal of Computer Mathematics, vol. 87, no. 8, pp. 1716–1725, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
E. M. E. Zayed, “New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized $G^{'} / G$ -expansion method,” Journal of Physics A, vol. 42, no. 19, Article ID 195202, 13 pages, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
E. M. E. Zayed, “The $G^{'} / G$ -expansion method combined with the Riccati equation for finding exact solutions of nonlinear PDEs,” Journal of Applied Mathematics & Informatics, vol. 29, no. 1-2, pp. 351–367, 2011.
View at: Google Scholar | MathSciNet
M. N. Alam, M. A. Akbar, and S. T. Mohyud-Din, “A novel $G^{'} / G$ -expansion method and its application to the Boussinesq equation,” Chinese Physics B, vol. 23, no. 2, Article ID 020203, 2014.
View at: Publisher Site | Google Scholar
G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S.-D. Zhu, “The generalizing Riccati equation mapping method in non-linear evolution equation: application to $G^{'} / G$ -dimensional Boiti-Leon-Pempinelle equation,” Chaos, Solitons & Fractals, vol. 37, no. 5, pp. 1335–1342, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
X. Liu, L. Tian, and Y. Wu, “Application of $G^{'} / G$ -expansion method to two nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1376–1384, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
A.-M. Wazwaz, “New compact and noncompact solutions for two variants of a modified Camassa-Holm equation,” Applied Mathematics and Computation, vol. 163, no. 3, pp. 1165–1179, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. M. Zaman and S. Sultana, “Traveling wave solutions for the nonlinear evolution equation via the generalized riccati equation and the ( $G^{'} / G$ )-expansion method,” World Applied Sciences Journal, vol. 22, no. 3, pp. 396–407, 2013.
View at: Publisher Site | Google Scholar
M. N. Alam and M. A. Akbar, “Some new exact traveling wave solutions to the simplified MCH equation and the (1 + 1)-dimensional combined KdV-mKdV equations,” Journal of the Association of Arab Universities for Basic and Applied Sciences, 2014.
View at: Publisher Site | Google Scholar
R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Tian and X. Song, “New peaked solitary wave solutions of the generalized Camassa-Holm equation,” Chaos, Solitons & Fractals, vol. 19, no. 3, pp. 621–637, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

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Copyright © 2014 Muhammad Shakeel et al. 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.

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Abstract and Applied Analysis

Advances on Integrodifferential Equations and Transforms

Applications of the Novel ( )-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

Abstract

1. Introduction

2. Description of the Method

3. Application of the Method to the Time Fractional Simplified (MCH) Equation

4. Conclusions

Conflict of Interests

References

Copyright

Applications of the Novel ()-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation