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

and Applied Analysis 3 Step 4. The positive integer m 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 (k + (G/G))i and (k + (G/G))−i(i = 0, 1, 2, . . . , m). Collecting each coefficient of the resulted polynomials to zero yields an overdetermined set of algebraic equations for α i (i = 0, ±1, ±2, . . . , ±m), k, L, and V. 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: D α t u + 2δu x − u xxt + γu 2 u x = 0, where δ ∈ R, γ > 0, 0 < α ≤ 1, (12) which is the variation of the equation u t + 2δu x − u xxt + γu 2 u x = 0, where δ ∈ R, γ > 0. (13) 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 (G/G)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 (G/G)-expansion method together with the generalized Riccati equation to MCH equation to find the exact solutions. Alam and Akbar [51] applied the generalized (G/G)-expansionmethod to look for the exact solutions via the simplifiedMCH 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 (V + 2δL) u − VL 2 u 󸀠󸀠 + γL u 3 3 + C 1 = 0, (14) where C 1 is an integral constant which is to be determined later. Considering the homogeneous balance between u󸀠󸀠 and u 3 in (14), we obtain 3m = m + 2; that is, m = 2. Therefore, the trial solution formula (8) becomes u (ξ) = α −1 (k + Φ (ξ)) −1 + α 0 + α 1 (k + Φ (ξ)) . (15) Using (15) into (14), left hand side is converted into polynomials in (k + (G/G))i and (k + (G/G))−i (i = 0, 1, 2, . . . , m). Equating the coefficients of same power of the resulted polynomials to zero, we obtain a system of algebraic equations for α 0 , α 1 , α −1 , k, C 1 , L, and V (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 α 0 = ±i √6δL (A + 2k − 2Ck) √γ (L 2 (A 2 − 4BC + 4B) + 2) , α 1 = ±i 2√6δL (C − 1) √γ (L 2 (A 2 − 4BC + 4B) + 2) , V = − 4δL L 2 (A 2 − 4BC + 4B) + 2 , L = L, k = k, α −1 = 0, C 1 = 0, (16) where k, L, A, B, and C are arbitrary constants. Set 2. Consider α 0 = ∓i √6δL (A + 2k − 2Ck) √γ (L 2 (A 2 − 4BC + 4B) + 2) , α −1 = ±i 2√6δL (kA + k 2 − Ck 2 − B) √γ (L 2 (A 2 − 4BC + 4B) + 2) , V = − 4δL L 2 (A 2 − 4BC + 4B) + 2 , L = L, k = k, α 1 = 0, C 1 = 0, (17) where k, L, A, B, and C are arbitrary constants. Set 3. Consider α 1 = ±2i √3δL (C − 1) √γ (2L 2 (A 2 − 4BC + 4B) + 1) , α −1 = ±i √3δL (A 2 − 4BC + 4B) 2√γ (2L 2 (A 2 − 4BC + 4B) + 1) (C − 1) , V = − 2δL 2L 2 (A 2 − 4BC + 4B) + 1 , k = A 2 (C − 1) , L = L, α 0 = 0, C 1 = 0, (18) where L, A, B, and C are arbitrary constants. 4 Abstract and Applied Analysis Set 4. Consider α −1 = ±i √6δL (A 2 − 4BC + 4B) 2√γ (L 2 (A 2 − 4BC + 4B) + 2) (C − 1) , V = − 4δL L 2 (A 2 − 4BC + 4B) + 2 , k = A 2 (C − 1) , L = L, α 0 = 0, α 1 = 0, C 1 = 0, (19) where L, A, B, and C are arbitrary constants. Substituting (16)–(19) into (15), we obtain u 1 (ξ) = ± i √6δL (A + 2k − 2Ck) √γ (L 2 (A 2 − 4BC + 4B) + 2) ± i 2√6δL (C − 1) √γ (L 2 (A 2 − 4BC + 4B) + 2)


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  = 0.5.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][7][8][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][11][12][13][14].A physical interpretation of the fractional calculus was given in [15][16][17][18][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 Some important properties of Jumarie's derivative are

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 ,  ̸ = 0, 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.  ( = 0, ±1, ±2, . . ., ±) 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 Φ() = ln (())  =   ()/() as Equation ( 11) has twenty five solutions [47].
Abstract and Applied Analysis 3 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 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).

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 where  ∈ R,  > 0.

(13)
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  1 is an integral constant which is to be determined later.
Considering the homogeneous balance between   and  3 in ( 14), we obtain 3 =  + 2; that is,  = 2. Therefore, the trial solution formula (8) becomes Using ( 15) into ( 14), left hand side is converted into polynomials in ( + (  /)) and ( + (  /)) − ( = 0, 1, 2, . . ., ).Equating the coefficients of same power of the resulted polynomials to zero, we obtain a system of algebraic equations for  0 ,  1 ,  −1 , ,  1 , , 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.
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.
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.

Figure 1 :
Figure 1: (a)-(d) show the kink solution for  1 1 for different values of parameters.

Figure 2 :
Figure 2: (a)-(d) show the singular solution for  2 1 for different values of parameters.

Figure 3 :
Figure 3: (a)-(d) show the periodic solution for  12 2 for different values of parameters.