Theory and Modern Applications

Explicit Jacobi elliptic exact solutions for nonlinear partial fractional differential equations

Abstract

In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. An algebraic method is improved to construct uniformly a series of exact solutions for some nonlinear time-space fractional partial differential equations. We construct successfully a series of some exact solutions including the elliptic doubly periodic solutions with the aid of computerized symbolic computation software package such as Maple or Mathematica. This method is efficient and powerful in solving a wide classes of nonlinear partial fractional differential equations. The Jacobi elliptic doubly periodic solutions are generated by the trigonometric exact solutions and the hyperbolic exact solutions when the modulus $m\to 0$ and $m\to 1$, respectively.

1 Introduction

Nonlinear partial fractional equations are very effective for the description of many physical phenomena such as rheology, the damping law, diffusion processes, and the nonlinear oscillation of an earthquake can be modeled with fractional derivatives [1, 2]. Also many applications of nonlinear partial fractional differential equations can be found in turbulence and fluid dynamics and nonlinear biological systems . There are many methods for finding the approximate solutions for nonlinear partial fractional differential equations such as the Adomian decomposition method [5, 6], the variational iteration method , the homotopy perturbation method [8, 9], and the homotopy analysis method [3, 10, 11] and so on. No analytical method had been available before 1998 for nonlinear fractional differential equations. Li and He  have proposed the fractional complex transformation to convert the nonlinear partial fractional differential equations into ordinary differential equations so that all analytical methods devoted to advanced calculus can be applied to fractional calculus. Recently Zhang and Zhang  have introduced a direct method called the sub-equation method to look for the exact solutions for nonlinear partial fractional differential equations. He  has extended the exp-function method to fractional partial differential equations in the sense of a modified Riemann-Liouville derivative based on the fractional complex transform. There are many methods for solving the nonlinear partial fractional differential equations such as those in [15, 16]. Fan , Zayed et al.  and Hong and Lu [19, 20] have proposed an algebraic method for nonlinear partial differential equations to obtain a series of exact wave solutions including the soliton, and rational, triangular periodic, Jacobian, and Weierstrass doubly periodic solutions. In this paper, we will improve the extended proposed algebraic method to solve the nonlinear partial fractional differential equations. Also we use the improved extended proposed algebraic method to construct the Jacobi elliptic exact solutions for the following nonlinear time-space partial fractional differential equations:

1. (i)

First, we have the space-time fractional derivative nonlinear Korteweg-de Vries (KdV) equation 

$\frac{{\partial }^{\alpha }u}{\partial {t}^{\alpha }}+au\frac{{\partial }^{\beta }u}{\partial {x}^{\beta }}+\frac{{\partial }^{3\beta }u}{\partial {x}^{3\beta }}=0,\phantom{\rule{1em}{0ex}}t>0,0<\alpha ,\beta \le 1,$
(1)

where a is a constant. When $\alpha ,\beta \to 1$, the KdV equation has been used to describe a wide range of physics phenomena of the evolution and interaction of nonlinear waves. It was derived from the propagation of dispersive shallow water waves and is widely used in fluid dynamics, aerodynamics, and continuum mechanics, and as a model for shock wave formation, solitons, turbulence, boundary layer behavior, mass transport, and the solution representing the water’s free surface over a flat bed .

1. (ii)

Then we have the space-time fractional derivative nonlinear fractional Zakharov-Kunzetsov-Benjamin-Bona-Mahony (ZKBBM) equation ,

$\frac{{\partial }^{\alpha }u}{\partial {t}^{\alpha }}+\frac{{\partial }^{\beta }u}{\partial {x}^{\beta }}-2au\frac{{\partial }^{\beta }u}{\partial {x}^{\beta }}-b\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\left(\frac{{\partial }^{2\beta }u}{\partial {x}^{2\beta }}\right)=0,\phantom{\rule{1em}{0ex}}t>0,0<\alpha ,\beta \le 1,$
(2)

where a, b are arbitrary constants. When $\alpha ,\beta \to 1$, this system has been investigated by Benjamin et al.  for the first time, as an alternative model to the KdV equation for long waves and it plays an important role in the modeling of nonlinear dispersive systems. The Benjamin-Bona-Mahony equation is applicable to the study of drift waves in a plasma or Rossby waves in rotating fluids.

2 Preliminaries

There are many types of the fractional derivatives such as the Kolwankar-Gangal local fractional derivative , Chen’s fractal derivative , Cresson’s derivative , and Jumarie’s modified Riemann-Liouville derivative [27, 28].

In this section, we give some basic definitions of fractional calculus theory which will be used in this work.

Jumarie’s modified Riemann-Liouville derivative of order α is defined as [28, 29]

${D}_{x}^{\alpha }f\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}\frac{d}{dx}{\int }_{0}^{x}{\left(x-\xi \right)}^{-\alpha }\left(f\left(\xi \right)-f\left(0\right)\right)\phantom{\rule{0.2em}{0ex}}d\xi ,\phantom{\rule{1em}{0ex}}0<\alpha <1,$
(3)

where $f:R\to R$, $x↦f\left(x\right)$ denotes a continuous (but not necessarily first order differentiable) function.

Also the inverse of Jumarie’s modified Riemann-Liouville derivative to $f\left(x\right)$ of order α in the interval $\left[a,b\right]$ is defined by

${I}_{x}^{\alpha }f\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{x}{\left(x-\xi \right)}^{\alpha -1}f\left(\xi \right)\phantom{\rule{0.2em}{0ex}}d\xi =\frac{1}{\mathrm{\Gamma }\left(\alpha +1\right)}{\int }_{0}^{x}f\left(\xi \right){\left(dx\right)}^{\alpha },\phantom{\rule{1em}{0ex}}0<\alpha \le 1.$
(4)

Some properties for the proposed Jumarie’s modified Riemann-Liouville derivative are listed from  as follows:

${f}^{\left(\alpha \right)}\left[x\left(t\right)\right]=\frac{df}{dx}{x}^{\left(\alpha \right)}\left(t\right),$
(5)
$\begin{array}{c}{D}_{x}^{\alpha }{x}^{\beta }=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}\frac{d}{dx}{\int }_{0}^{x}{\xi }^{\beta }{\left(x-\xi \right)}^{-\alpha }\phantom{\rule{0.2em}{0ex}}d\xi =\frac{B\left(1+\beta ,1-\alpha \right)}{\mathrm{\Gamma }\left(1-\alpha \right)}\frac{d}{dx}\left({x}^{\beta -\alpha +1}\right)\hfill \\ \phantom{{D}_{x}^{\alpha }{x}^{\beta }}=\frac{\mathrm{\Gamma }\left(1+\beta \right)}{\mathrm{\Gamma }\left(\beta -\alpha +2\right)}\frac{d}{dx}\left({x}^{\beta -\alpha +1}\right)=\frac{\mathrm{\Gamma }\left(1+\beta \right)}{\mathrm{\Gamma }\left(\beta -\alpha +1\right)}{x}^{\beta -\alpha },\phantom{\rule{1em}{0ex}}\beta >\alpha -1,\hfill \end{array}$
(6)
${D}_{x}^{\alpha }\left(f\left(x\right)\cdot g\left(x\right)\right)=g\left(x\right){D}_{x}^{\alpha }\left(f\left(x\right)\right)+f\left(x\right){D}_{x}^{\alpha }\left(g\left(x\right)\right),$
(7)

where B is the beta function. The function $f\left(x\right)$ should be differentiable with respect to $x\left(t\right)$ and $x\left(t\right)$ is fractional differentiable in (6). The above results are employed in the following sections. The Leibniz rule is given (7) for modified Riemann-Liouville derivative which is modified by Wu in .

The modified Riemann-Liouville derivative has been successfully applied in probability calculus , fractional Laplace problems , the fractional variational approach with several variables , the fractional variational iteration method , the fractional variational approach with natural boundary conditions , and the fractional Lie group method .

3 The improved extended proposed algebraic method for nonlinear partial fractional differential equations

Consider the following nonlinear partial fractional differential equation:

$\begin{array}{c}F\left(u,{D}_{t}^{\alpha }u,{D}_{x}^{\beta }u,{D}_{y}^{\gamma }u,{D}_{z}^{\delta }u,{D}_{t}^{\alpha }{D}_{t}^{\alpha }u,{D}_{t}^{\alpha }{D}_{x}^{\beta }u,{D}_{x}^{\beta }{D}_{x}^{\beta }u,{D}_{x}^{\beta }{D}_{y}^{\gamma },{D}_{y}^{\gamma }{D}_{y}^{\gamma }u,\dots \right)=0,\hfill \\ \phantom{\rule{1em}{0ex}}0<\alpha ,\beta ,\delta ,\gamma \le 1,\hfill \end{array}$
(8)

where u is an unknown function, F is a polynomial in u and its partial fractional derivatives in which the highest order fractional derivatives and the nonlinear terms are involved.

We give the main steps of the modified extended proposed algebraic method for nonlinear partial fractional differential equations.

Step 1. The fractional complex transformation

$u\left(x,y,z,t\right)=u\left(\xi \right),\phantom{\rule{1em}{0ex}}\xi =\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{N{y}^{\gamma }}{\mathrm{\Gamma }\left(\gamma +1\right)}+\frac{M{z}^{\delta }}{\mathrm{\Gamma }\left(\delta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)},$
(9)

where K, L, M, and N are nonzero arbitrary constants, permits us to reduce (8) to a nonlinear ODE for $u=u\left(\xi \right)$ in the following form:

$P\left(u,{u}^{\prime },{u}^{″},{u}^{‴},\dots \right)=0.$
(10)

If possible, we should integrate (10) term by term one or more times.

Step 2. Suppose the solutions of (10) can be expressed by a polynomial in ϕ in the form [17, 18]:

$u\left(\xi \right)=\sum _{i=-m}^{m}{\alpha }_{i}{\varphi }^{i}\left(\xi \right),\phantom{\rule{1em}{0ex}}{\alpha }_{m}\ne 0,$
(11)

where ${\alpha }_{i}$ ($i=0,±1,\dots ,±m$) are arbitrary constants to be determined later, while $\varphi \left(\xi \right)$ satisfies the following nonlinear first order Jacobi elliptic differential equation:

${\left[{\varphi }^{\prime }\left(\xi \right)\right]}^{2}={e}_{0}+{e}_{1}{\varphi }^{2}\left(\xi \right)+{e}_{2}{\varphi }^{4}\left(\xi \right),$
(12)

where ${e}_{0}$, ${e}_{1}$, and ${e}_{2}$ are arbitrary constants.

Step 3. The positive integer ‘m’ can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Eq. (10).

Step 4. We must substitute (11) into (10) and using (12), collect all terms with the same order of $\varphi \left(\xi \right)$ together, then equating each coefficient of the resulting polynomial to be zero. This yields a set of algebraic equations for ${\alpha }_{i}$ ($i=0,±1,\dots ,±m$), ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, M, and N. We then solve this system of algebraic equation with the help of Maple software package to determine ${\alpha }_{i}$ ($i=0,±1,\dots ,±m$), ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, M, and N.

Step 5. The general solutions of (12) have been discussed in [37, 38]. We put some of the general solutions of (12) from  in Table 1.

There are other cases which are omitted here for convenience; see .

Step 6. Since the general solutions of (12) are discussed in Table 1, then substituting ${\alpha }_{i}$ ($i=0,±1,\dots ,±m$), ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, M, and N and the general solutions of (12) into (11), we obtain more new Jacobi elliptic exact solutions for the nonlinear partial fractional derivative equation (8).

4 Applications

In this section, we construct some new Jacobi elliptic exact solutions of some nonlinear partial fractional differential equations via the time-space fractional nonlinear KdV equation and the time-space fractional nonlinear Zakharov-Kunzetsov-Benjamin-Bona-Mahomy equation using the modified extended proposed algebraic method which has been paid attention to by many authors.

4.1 Example 1: Jacobi elliptic solutions for nonlinear fractional KdV equation

In this section, to demonstrate the effectiveness of this method, we use the complex transformation (9) to converting the nonlinear KdV equation with time-space fractional derivatives (1) to an ordinary differential equation; and we integrate twice, to find

$\frac{1}{2}L{U}^{2}+\frac{a}{6}K{U}^{3}+\frac{{K}^{3}}{2}{\left[{U}^{\prime }\right]}^{2}+{C}_{1}U+{C}_{2}=0,$
(13)

where ${C}_{1}$ and ${C}_{2}$ are the integration constants. Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in (13), we get

$U\left(\xi \right)={\alpha }_{0}+{\alpha }_{1}\varphi \left(\xi \right)+{\alpha }_{2}{\varphi }^{2}\left(\xi \right)+\frac{{\alpha }_{3}}{\varphi \left(\xi \right)}+\frac{{\alpha }_{4}}{{\varphi }^{2}\left(\xi \right)},$
(14)

where ${\alpha }_{0}$, ${\alpha }_{1}$, ${\alpha }_{2}$, ${a}_{3}$, ${a}_{4}$, L, and K are arbitrary constants to be determined later. Substituting (14) and (12) into (13), collecting all terms of $\varphi \left(\xi \right)$, and then setting each coefficient $\varphi \left(\xi \right)$ to be zero, we get a system of algebraic equations. With the aid of Maple or Mathematica we can solve this system of algebraic equations to obtain the following cases of solutions:

Case 1.

$\begin{array}{r}{a}_{0}=\frac{2{C}_{1}\left(L+4{K}^{3}{e}_{1}\right)}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}},\phantom{\rule{2em}{0ex}}{a}_{2}=\frac{24{C}_{1}{K}^{3}{e}_{2}}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}},\\ a=-\frac{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}}{4K{C}_{1}},\phantom{\rule{2em}{0ex}}{a}_{4}=\frac{24{C}_{1}{K}^{3}{e}_{0}}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}},\\ {C}_{2}=\frac{2{C}_{1}^{2}\left(-576L{e}_{2}{K}^{4}{e}_{0}+4\text{,}608{K}^{9}{e}_{0}{e}_{1}{e}_{2}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}\right)}{3{\left(192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}\right)}^{2}},\\ {a}_{1}={a}_{3}=0,\end{array}$
(15)

where ${C}_{1}$, L, K, ${e}_{0}$, ${e}_{1}$, and ${e}_{2}$ are arbitrary constants.

Case 2.

$\begin{array}{c}{C}_{1}=\frac{1}{2}\left(192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}\right)\hfill \\ \phantom{{C}_{1}=}×\sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ {a}_{0}=\left(L+4{K}^{3}{e}_{1}\right)\sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ {a}_{2}=12{K}^{3}{e}_{2}\sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ a=-\frac{1}{12K\sqrt{{C}_{2}}}\sqrt{6\left[-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}\right]},\hfill \\ {a}_{4}=12{K}^{3}{e}_{0}\sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ {a}_{1}={a}_{3}=0,\hfill \end{array}$
(16)

where ${e}_{0}$, ${e}_{1}$, ${e}_{2}$, K, L, and ${C}_{2}$ are arbitrary constants.

Note that there are other cases which are omitted here for convenience. Since the solutions obtained here are so many, we just list some of the Jacobi exact solutions corresponding to Case 1 to illustrate the effectiveness of the proposed method. Substituting (15) into (14) we have

$\begin{array}{rcl}u& =& \frac{2{C}_{1}\left(L+4{K}^{3}{e}_{1}\right)}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}}+\frac{24{C}_{1}{K}^{3}{e}_{2}}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}}{\varphi }^{2}\left(\xi \right)\\ +\frac{24{C}_{1}{K}^{3}{e}_{0}}{\left[192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}\right]{\varphi }^{2}\left(\xi \right)},\end{array}$
(17)

where

$\xi =\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}.$

According to the general solutions of (12) which are discussed in Table 1, we have the following families of exact solutions:

Family 1. If ${e}_{0}=1$, ${e}_{1}=-\left(1+{m}^{2}\right)$, ${e}_{2}={m}^{2}$ the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{1}& =& \frac{2{C}_{1}\left[L-4{K}^{3}\left(1+{m}^{2}\right)\right]}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{\left(1+{m}^{2}\right)}^{2}}\\ +\frac{24{C}_{1}{K}^{3}{m}^{2}}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{\left(1+{m}^{2}\right)}^{2}}s{n}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]\\ +\frac{24{C}_{1}{K}^{3}}{\left[192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{\left(1+{m}^{2}\right)}^{2}\right]}n{s}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right].\end{array}$
(18)

To illustrate the behavior of the Jacobi elliptic solution ${u}_{1}$ (18), see Figure 1.

Furthermore,

$\begin{array}{rcl}{u}_{2}& =& \frac{2{C}_{1}\left[L-4{K}^{3}\left(1+{m}^{2}\right)\right]}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{\left(1+{m}^{2}\right)}^{2}}\\ +\frac{24{C}_{1}{K}^{3}{m}^{2}}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{\left(1+{m}^{2}\right)}^{2}}c{d}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]\\ +\frac{24{C}_{1}{K}^{3}}{\left[192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{\left(1+{m}^{2}\right)}^{2}\right]}d{c}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right],\end{array}$
(19)

where ${C}_{2}=\frac{2{C}_{1}^{2}\left[-576L{m}^{2}{K}^{4}-4\text{,}608{K}^{9}{m}^{2}\left(1+{m}^{2}\right)+{L}^{3}-48{K}^{6}{\left(1+{m}^{2}\right)}^{2}L+128{K}^{9}{\left(1+{m}^{2}\right)}^{3}\right]}{3{\left[192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{\left(1+{m}^{2}\right)}^{2}\right]}^{2}}$.

Family 2. If ${e}_{0}=1-{m}^{2}$, ${e}_{1}=2{m}^{2}-1$, ${e}_{2}=-{m}^{2}$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{3}& =& \frac{2{C}_{1}\left(L+4{K}^{3}\left(2{m}^{2}-1\right)\right)}{\left\{-192{K}^{6}\left(1-{m}^{2}\right){m}^{2}-{L}^{2}+16{K}^{6}{\left(2{m}^{2}-1\right)}^{2}\right\}}\\ -\frac{24{C}_{1}{K}^{3}{m}^{2}c{n}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]}{\left\{-192{K}^{6}\left(1-{m}^{2}\right){m}^{2}-{L}^{2}+16{K}^{6}{\left(2{m}^{2}-1\right)}^{2}\right\}}\\ +\frac{24{C}_{1}{K}^{3}\left(1-{m}^{2}\right)n{c}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]}{\left\{-192{K}^{6}\left(1-{m}^{2}\right){m}^{2}-{L}^{2}+16{K}^{6}{\left(2{m}^{2}-1\right)}^{2}\right\}},\end{array}$
(20)

where ${C}_{2}$ = $\frac{1}{3{\left[-192{K}^{6}{m}^{2}\left(1-{m}^{2}\right)-{L}^{2}+16{K}^{6}{\left(2{m}^{2}-1\right)}^{2}\right]}^{2}}${$2{C}_{1}^{2}\left(576L{m}^{2}{K}^{4}\left(2{m}^{2}-1\right)$$4\text{,}608{K}^{9}{m}^{2}\left(2{m}^{2}-1\right)\left(1-{m}^{2}\right)$ + ${L}^{3}$$48{K}^{6}{\left(2{m}^{2}-1\right)}^{2}L$$128{K}^{9}{\left(2{m}^{2}-1\right)}^{3}\right)$}.

To illustrate the behavior of the Jacobi elliptic solution ${u}_{3}$ (20), see Figure 2.

Family 3. If ${e}_{0}={m}^{2}-1$, ${e}_{1}=2-{m}^{2}$, ${e}_{2}=-1$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{4}& =& \frac{2{C}_{1}\left[L+4{K}^{3}\left(2-{m}^{2}\right)\right]}{\left[192{K}^{6}\left(1-{m}^{2}\right)-{L}^{2}+16{K}^{6}{\left(2-{m}^{2}\right)}^{2}\right]}\\ -\frac{24{C}_{1}{K}^{3}d{n}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]}{\left[192{K}^{6}\left(1-{m}^{2}\right)-{L}^{2}+16{K}^{6}{\left(2-{m}^{2}\right)}^{2}\right]}\\ +\frac{24{C}_{1}{K}^{3}\left({m}^{2}-1\right)n{d}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]}{\left[192{K}^{6}\left(1-{m}^{2}\right)-{L}^{2}+16{K}^{6}{\left(2-{m}^{2}\right)}^{2}\right]},\end{array}$
(21)

where ${C}_{2}=\frac{2{C}_{1}^{2}\left(576L{K}^{4}\left({m}^{2}-1\right)-4\text{,}608{K}^{9}\left({m}^{2}-1\right)\left(2-{m}^{2}\right)+{L}^{3}-48{K}^{6}{\left(2-{m}^{2}\right)}^{2}L-128{K}^{9}{\left(2-{m}^{2}\right)}^{3}\right)}{3{\left[192{K}^{6}\left(1-{m}^{2}\right)-{L}^{2}+16{K}^{6}{\left(2-{m}^{2}\right)}^{2}\right]}^{2}}$.

Family 4. If ${e}_{0}=\frac{1}{4}\left(1-{m}^{2}\right)$, ${e}_{1}=\frac{1}{2}\left(1+{m}^{2}\right)$, ${e}_{2}=\frac{1}{4}\left(1-{m}^{2}\right)$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{5}& =& \frac{2{C}_{1}\left(L+2{K}^{3}\left(1+{m}^{2}\right)\right)}{\left[12{K}^{6}{\left(1-{m}^{2}\right)}^{2}-{L}^{2}+4{K}^{6}{\left(1+{m}^{2}\right)}^{2}\right]}\\ +\frac{6{C}_{1}{K}^{3}\left(1-{m}^{2}\right){\left[nc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±sc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}}{\left[12{K}^{6}{\left(1-{m}^{2}\right)}^{2}-{L}^{2}+4{K}^{6}{\left(1+{m}^{2}\right)}^{2}\right]}\\ +\frac{6{C}_{1}{K}^{3}\left(1-{m}^{2}\right)}{\left[12{K}^{6}{\left(1-{m}^{2}\right)}^{2}-{L}^{2}+4{K}^{6}{\left(1+{m}^{2}\right)}^{2}\right]}\\ ×\frac{1}{{\left[nc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±sc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}},\end{array}$
(22)

where ${C}_{2}=\frac{2{C}_{1}^{2}\left(-36L{K}^{4}{\left(1-{m}^{2}\right)}^{2}+144{K}^{9}{\left(1-{m}^{2}\right)}^{2}\left(1+{m}^{2}\right)+{L}^{3}-12{K}^{6}{\left(1+{m}^{2}\right)}^{2}L-16{K}^{9}{\left(1+{m}^{2}\right)}^{3}\right)}{3{\left[12{K}^{6}{\left(1-{m}^{2}\right)}^{2}-{L}^{2}+4{K}^{6}{\left(1+{m}^{2}\right)}^{2}\right]}^{2}}$.

To illustrate the behavior of the Jacobi elliptic solution ${u}_{5}$ (22), see Figure 3.

Family 5. If ${e}_{0}=\frac{{m}^{2}}{4}$, ${e}_{1}=\frac{1}{2}\left({m}^{2}-2\right)$, ${e}_{2}=\frac{{m}^{2}}{4}$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{6}& =& \frac{2{C}_{1}\left(L+2{K}^{3}\left({m}^{2}-2\right)\right)}{12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{\left({m}^{2}-2\right)}^{2}}\\ +\frac{6{C}_{1}{K}^{3}{m}^{2}{\left[sn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±icn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}}{\left[12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{\left({m}^{2}-2\right)}^{2}\right]}\\ +\frac{24{C}_{1}{K}^{3}{m}^{2}}{\left[12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{\left({m}^{2}-2\right)}^{2}\right]}\\ ×\frac{1}{{\left[sn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±icn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}},\end{array}$
(23)

where ${C}_{2}=\frac{2{C}_{1}^{2}\left(-36L{K}^{4}{m}^{4}+144{K}^{9}{m}^{4}\left({m}^{2}-2\right)+{L}^{3}-12{K}^{6}{\left({m}^{2}-2\right)}^{2}L-16{K}^{9}{\left({m}^{2}-2\right)}^{3}\right)}{3{\left[12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{\left({m}^{2}-2\right)}^{2}\right]}^{2}}$.

To illustrate the behavior of the Jacobi elliptic solution ${u}_{6}$ (23), see Figures 4 and 5.

Similarly, we can write down the other families of exact solutions of (1) which are omitted for convenience.

4.2 Example 2: Jacobi elliptic solutions for nonlinear fractional ZKBBM equation

In this section we use the proposed method to find the Jacobi elliptic solutions for the nonlinear fractional ZKBBM equation with time and space fractional derivatives (2). The complex fractional transformations (9) convert the nonlinear fractional ZKBBM equation (2) to the following nonlinear ordinary differential equation:

$\left(L+K\right)U-aK{U}^{2}-b{K}^{2}{U}^{″}+{C}_{1}=0,$
(24)

where ${C}_{1}$ is the integration constant. Considering the homogeneous balance between the highest order derivative and the nonlinear term in (24), we have

$U\left(\xi \right)={\alpha }_{0}+{\alpha }_{1}\varphi \left(\xi \right)+{\alpha }_{2}{\varphi }^{2}\left(\xi \right)+\frac{{\alpha }_{3}}{\varphi \left(\xi \right)}+\frac{{\alpha }_{4}}{{\varphi }^{2}\left(\xi \right)},$
(25)

where ${\alpha }_{0}$, ${\alpha }_{1}$, ${\alpha }_{2}$, ${a}_{3}$, ${a}_{4}$, L, and K are arbitrary constants to be determined later. Substituting (25) and (12) into (24), collecting all the terms of powers of $\varphi \left(\xi \right)$ and setting each coefficient $\varphi \left(\xi \right)$ to zero, we get a system of algebraic equations. With the aid of Maple or Mathematica we can solve this system of algebraic equations to obtain the following sets of solutions:

Case 1.

$\begin{array}{c}{a}_{0}=-\frac{4b{K}^{2}L{e}_{1}-K-L}{2aK},\phantom{\rule{2em}{0ex}}{a}_{2}=-\frac{6KbL{e}_{2}}{a},\phantom{\rule{2em}{0ex}}{a}_{4}=-\frac{6KbL{e}_{0}}{a},\hfill \\ {C}_{1}=\frac{1}{4aK}\left\{-{K}^{2}-2LK+192{b}^{2}{K}^{4}{L}^{2}{e}_{0}{e}_{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{e}_{1}^{2}\right\},\hfill \\ {a}_{1}={a}_{3}=0,\hfill \end{array}$
(26)

where b, L, K, ${e}_{0}$, ${e}_{1}$, and ${e}_{2}$ are arbitrary nonzero constants. There are many other cases which are omitted for convenience.

Substituting (26) into (25) we have

$u=-\frac{4b{K}^{2}L{e}_{1}-K-L}{2aK}-\frac{6KbL{e}_{2}}{a}{\varphi }^{2}\left(\xi \right)-\frac{6KbL{e}_{0}}{a{\varphi }^{2}\left(\xi \right)},$
(27)

where

$\xi =\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}$
(28)

and

${C}_{1}=\frac{1}{4aK}\left\{-{K}^{2}-2LK+192{b}^{2}{K}^{4}{L}^{2}{e}_{0}{e}_{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{e}_{1}^{2}\right\}.$

According to the general solutions of (12) which are discussed in Table 1, we have the following families of Jacobi elliptic exact solutions to the nonlinear ZKBBM equation:

Family 1. If ${e}_{0}=1$, ${e}_{1}=-\left(1+{m}^{2}\right)$, ${e}_{2}={m}^{2}$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{1}& =& \frac{4b{K}^{2}L\left(1+{m}^{2}\right)+K+L}{2aK}-\frac{6KbL{m}^{2}}{a}s{n}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]\\ -\frac{6KbL}{a}n{s}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right].\end{array}$
(29)

To illustrate the behavior of the Jacobi elliptic solution ${u}_{1}$ (29), see Figure 6.

Furthermore

$\begin{array}{rcl}{u}_{2}& =& \frac{4b{K}^{2}L\left(1+{m}^{2}\right)+K+L}{2aK}-\frac{6KbL{m}^{2}}{a}c{d}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]\\ -\frac{6KbL}{a}d{c}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right],\end{array}$
(30)

where ${C}_{1}=\frac{1}{4aK}\left\{-{K}^{2}-2LK+192{b}^{2}{K}^{4}{L}^{2}{m}^{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{\left(1+{m}^{2}\right)}^{2}\right\}$.

Family 2. If ${e}_{0}=1-{m}^{2}$, ${e}_{1}=2{m}^{2}-1$, ${e}_{2}=-{m}^{2}$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{3}& =& -\frac{4b{K}^{2}L\left(2{m}^{2}-1\right)-K-L}{2aK}+\frac{6KbL{m}^{2}}{a}c{n}^{2}\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\\ -\frac{6KbL\left(1-{m}^{2}\right)}{a}n{c}^{2}\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right),\end{array}$
(31)

where ${C}_{1}=\frac{1}{4aK}\left\{-{K}^{2}-2LK-192{b}^{2}{K}^{4}{L}^{2}\left(1-{m}^{2}\right){m}^{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{\left(2{m}^{2}-1\right)}^{2}\right\}$.

To illustrate the behavior of the Jacobi elliptic solution ${u}_{3}$ (31), see Figure 7.

Family 3. If ${e}_{0}={m}^{2}-1$, ${e}_{1}=2-{m}^{2}$, ${e}_{2}=-1$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{4}& =& -\frac{4b{K}^{2}L\left(2-{m}^{2}\right)-K-L}{2aK}+\frac{6KbL}{a}d{n}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right]\\ -\frac{6KbL\left({m}^{2}-1\right)}{a}n{d}^{2}\left[\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right],\end{array}$
(32)

where ${C}_{1}=\frac{1}{4aK}\left\{-{K}^{2}-2LK-192{b}^{2}{K}^{4}{L}^{2}\left({m}^{2}-1\right)-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{\left(2-{m}^{2}\right)}^{2}\right\}$.

Family 4. If ${e}_{0}=\frac{1}{4}\left(1-{m}^{2}\right)$, ${e}_{1}=\frac{1}{2}\left(1+{m}^{2}\right)$, ${e}_{2}=\frac{1}{4}\left(1-{m}^{2}\right)$, the exact traveling wave solution takes the form

$\begin{array}{rcl}{u}_{5}& =& -\frac{2b{K}^{2}L\left(1+{m}^{2}\right)-K-L}{2aK}\\ -\frac{3KbL\left(1-{m}^{2}\right)}{2a}{\left[nc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±sc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}\\ -\frac{3KbL\left(1-{m}^{2}\right)}{2a{\left[nc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±sc\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}},\end{array}$
(33)

where ${C}_{1}=\frac{1}{4aK}\left\{-{K}^{2}-2LK+12{b}^{2}{K}^{4}{L}^{2}{\left(1-{m}^{2}\right)}^{2}-{L}^{2}+4{b}^{2}{K}^{4}{L}^{2}{\left(1+{m}^{2}\right)}^{2}\right\}$.

To illustrate the behavior of the Jacobi elliptic solution ${u}_{5}$ (33), see Figure 8.

Family 5. If ${e}_{0}=\frac{{m}^{2}}{4}$, ${e}_{1}=\frac{1}{2}\left({m}^{2}-2\right)$, ${e}_{2}=\frac{{m}^{2}}{4}$, the exact traveling wave solution takes the form

$\begin{array}{rl}{u}_{6}=& -\frac{2b{K}^{2}L\left({m}^{2}-2\right)-K-L}{2aK}\\ -\frac{3KbL{m}^{2}}{2a}{\left[sn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±icn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}\\ -\frac{3KbL{m}^{2}}{2a{\left[sn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)±icn\left(\frac{K{x}^{\beta }}{\mathrm{\Gamma }\left(\beta +1\right)}+\frac{L{t}^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\right)\right]}^{2}},\end{array}$
(34)

where ${C}_{1}=\frac{1}{4aK}\left\{-{K}^{2}-2LK+12{b}^{2}{K}^{4}{L}^{2}{m}^{4}-{L}^{2}+4{b}^{2}{K}^{4}{L}^{2}{\left({m}^{2}-2\right)}^{2}\right\}$.

To illustrate the behavior of the Jacobi elliptic solution ${u}_{6}$ (34), see Figures 9 and 10.

Similarly, we can write down the other families of exact solutions of (2) which are omitted for convenience.

5 Conclusion

In this article we constructed the Jacobi elliptic exact solutions for the nonlinear partial fractional differential equations with the help of the complex fractional transformation and the improved extended proposed algebraic method. This method is effective and powerful for finding the Jacobi elliptic solutions for nonlinear fractional differential equations. Jacobi elliptic solutions are generalized to the hyperbolic exact solutions and trigonometric exact solutions when the modulus $m\to 1$ and $m\to 0$, respectively.

Author’s contributions

The author read and approved the final manuscript.

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The author would like to thank the editors and the anonymous referees for their careful reading of the paper.

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