Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics

1.   Introduction

Noticeable natural complex phenomena of the real world are described and formulated in the course of the differential equations. Fractional calculus has drawn the great interest of many researchers for their importance to depict the inner mechanisms of various complex physical phenomena of real world in broad sense. The fractional order nonlinear partial differential equations are more effective to explain widely the mechanisms of the nature of world than the classical differential equations of integer order. That is why; many researchers have recently paid deep attention to seek for the exact solutions to the nonlinear evolution equations (NLEEs) of fractional order. The study for revealing fractional order NLEEs is mainly due to their important appearance in different fields such as biology, physics, engineering, signal processing, systems identification, control theory, the finance, fractal dynamics and many other areas of science ^[1,2,3,4,5]. Several productive methods have been put forward to construct closed form analytic solutions to NLEEs of fractional order; namely the $(G'/G)$-expansion method and it's various modifications ^[6,7,8,9,10], the sub-equation method ^[11,12], the exp-function method ^[13,14], the first integral method ^[15,16], the functional variable method ^[17], the modified trial equation method ^[18,19], the simplest equation method ^[20], the Lie group analysis method ^[21], the characteristics method ^[22], the auxiliary equation method ^[23,24], the finite difference method ^[25], the finite element method ^[26], the differential transform method ^[27], the homotopy perturbation method ^[28], the Adomian decomposition method ^[29,30], the variational iteration method ^[31], the Tzou and Stehfest's algorithm ^[45], the spectral Gelarkin method ^[46], modified logistic model ^[47] and others ^{[48,49,50,51]}.

The fractional order NLEEs can depict the physical phenomena more accurately than that of the integer order NLEEs ^[1,2,3,5]. Consequently, the aim of this study is to construct new and further general closed form analytic wave solutions to the fractional order nonlinear evolution equations mentioned above in the sense of fractional derivative. Sousa and Oliveira have recently introduced new fractional derivatives ^[52,53]. There are also some definitions of fractional derivative in fractional calculus. Some of them are given below:

(ⅰ) The derivative of non-integer order defined by Caputo ^[42] is

(ⅱ) Riemann-Liouville fractional derivative is given as ^[42]

This definition is modified by Jumarie as ^[43]

(ⅲ) Ji-Huan He introduced the fractional derivative ^[44]

where $f_{0} (x)$ is a known function.

(ⅳ) The conformable fractional derivative of order $\alpha$ is defined as follows ^[32]:

If the above limit exists, then $f$ is called $\alpha$-differentiable. Let $\alpha \in (0, \, 1]$ and $f, \, \, g$ be $\alpha$-differentiable at a point $t > 0$, then $T_{\alpha }$ satisfies the following properties:

(ⅰ) $T_{\alpha } (a\, f+b\, g) = aT_{\alpha } (f)+bT_{\alpha } (g)$, for all $a, \, b\in R$

(ⅱ) $T_{\alpha } (t^{p}) = pt^{p-\alpha }$, for all $p\in R$

(ⅲ) $T_{\alpha } (\lambda) = 0$, for all constant functions $f(t) = \lambda$

(ⅳ) $T_{\alpha } (f\, g) = f\, T_{\alpha } (g)+g\, T_{\alpha } (f)$

(ⅴ) $T_{\alpha } (f/g) = \{ g\, T_{\alpha } (f)-f\, T_{\alpha } (g)\} /g^{2}$

(ⅵ) If, in addition, $f$ is differentiable, then $T_{\alpha } (f)(t) = t^{1-\alpha } \frac{df}{dt} (t)$.

In this article, the rational $(G'/G)$-expansion method ^[33] is used for searching the exact analytic traveling wave solutions to the suggested equations in the sense of conformable fractional derivative ^[32].

2.   Description of the rational $(G'/G)$-expansion method

Consider the following nonlinear evolution equation of fractional order in the independent variables $t, \, x_{1}, \, x_{2}, \, ..., \, x_{n}$:

where $u_{i} = u_{i} (t, \, x_{1}, \, x_{2}, \, ..., \, x_{n}), \, \, i = 1, \, ..., \, k$ are unknown functions, $F$ is a polynomial in $u_{i}$ and it's various partial derivatives including the derivatives of fractional order.

Now, the main steps of the rational $(G'/G)$-expansion method are presented as follows:

Step 1: Making use of the traveling wave variable ^[34]

where $t$ is the temporal variable and $x_i's$ are the spatial variables and $\xi$ is called wave variable, Eq. (2.1) is turned into the following ordinary differential equation of integer order with respect to the variable $\xi$:

Step 2: If possible take anti-derivative of Eq. (2.3) one or more times and integral constant can be set to zero as soliton solutions are hunted.

Step 3: Suppose the solution of Eq. (2.3) can be expressed as,

where $a_{i}, \, \, b_{i} \, \, (i = 0, \, 1, \, \, 2, \, ..., n\, )$ are constants with at least one of $a_{n}$ and $b_{n}$ is non-zero, while $G(\xi)$ satisfies the following second order linear ordinary differential equation:

where $\lambda$ and $\mu$ are real parameters.

Eq. (2.5) provides the solutions,

where $C_{1}$ and $C_{2}$ are arbitrary constants.

Step 4: The positive integer $n$ in Eq. (2.4) is fixed by taking homogeneous balance between the highest order derivative terms and the nonlinear terms in Eq. (2.3).

Step 5: Substituting Eq. (2.4) together with Eq. (2.5) into Eq. (2.3), we obtain a polynomial in $(G'/G)$. By equating each coefficient of this polynomial to zero yields a set of algebraic equations for $a_{i}, \, \, b_{i} \, \, (i = 0, \, 1, \, \, 2, \, ..., n\, )$, $\lambda$ and $\mu$. Solve this set of equations by the symbolic computation software, such as Maple for the parameters $a_{i}, \, \, b_{i} \, \, (i = 0, \, 1, \, \, 2, \, ..., n\, )$, $\lambda$ and $\mu$.

Step 6: Using the values of $a_{i}, \, \, b_{i} \, \, (i = 0, \, 1, \, \, 2, \, ..., n\, )$, $\lambda$ and $\mu$ obtained in step 5 together with Eq. (2.6) into Eq. (2.4) provide the closed form traveling wave solutions of the nonlinear fractional partial differential Eq. (2.1).

3.   Formulation of the solutions

In this section, the rational $(G'/G)$-expansion method is employed to derive the exact analytic solitary wave solutions to the (3+1)-dimensional space-time fractional mKdV-ZK equation, the time fractional biological population model and the space-time fractional modified regularized long-wave equation.

3.1.   The (3+1)-dimensional space-time fractional mKdV-ZK equation

Consider the (3+1)-dimensional space-time fractional mKdV-ZK equation

which is in the sense of conformable fractional derivative and $\delta$ is nonzero real constant. This equation is derived for plasma comprised of cool and hot electrons and a species of fluid ions ^[35].

The fractional complex transformation

where $l, \, \, m, \, \, n$ and $\omega$ are non-zero parameters, reduces Eq. (3.1) to the following ordinary differential equation with respect to the variable $\xi$:

The anti-derivative of Eq. (3.3) with integral constant zero gives

Due to the homogeneous balance between $u^{3}$ and $u''$ the solution Eq. (2.4) takes the form

with at least one of $a_{1}$ and $b_{1}$ is non-zero.

The substitution of Eq. (3.5) with the help of Eq. (2.5) into Eq. (3.4) yields a polynomial in $(G'/G)$. Setting like terms of this polynomial to zero makes available a set of algebraic equations for $a_{0}, \, \, a_{1}, \, \, b_{0}, \, \, b_{1}, \, \, \omega, \, \, \lambda$ and $\mu$. Solving these equations by Maple gives the following set of solutions:

Set-1: $a_{0} = \pm \frac{1}{2\delta } (2b_{1} \mu -b_{0} \lambda)\sqrt{-6\delta (l^{2} +m^{2} +n^{2})}$, $a_{1} = \pm \frac{1}{2\delta } (b_{1} \lambda -2b_{0})\sqrt{-6\delta (l^{2} +m^{2} +n^{2})}$,

where $b_{0}, \, \, b_{1}, \, \, \lambda$ and $\mu$ are arbitrary constants.

Set-2: $a_{0} = \pm \frac{b_{1} }{4\delta } (\lambda ^{2} -4\mu +\lambda \sqrt{4\mu -\lambda ^{2} })\sqrt{-6\delta (l^{2} +m^{2} +n^{2})}$, $b_{0} = \frac{b_{1} }{2} (\lambda +\sqrt{4\mu -\lambda ^{2} })$,

where $b_{1}, \, \, \lambda$ and $\mu$ are arbitrary constants.

Set-3: $a_{0} = \pm \frac{b_{1} (4\mu -\lambda ^{2})\sqrt{-6\delta (l^{2} +m^{2} +n^{2})} }{4\delta }$, $\omega = \frac{l}{2} (4\mu -\lambda ^{2})(l^{2} +m^{2} +n^{2})$, $a_{1} = 0$,

where $b_{1}, \, \, \lambda$ and $\mu$ are arbitrary constants.

Set-4: $a_{0} = \pm \frac{b_{0} \lambda }{2\delta } \sqrt{-6\delta (l^{2} +m^{2} +n^{2})}$, $a_{1} = \pm \frac{b_{0} }{\delta } \sqrt{-6\delta (l^{2} +m^{2} +n^{2})}$, $b_{1} = 0$,

where $b_{0}, \, \, \lambda$ and $\mu$ are arbitrary constants.

Using Eqs. (3.6)-(3.9) in Eq. (3.5) possess the following respective results:

where $\xi = \frac{1}{2\alpha } \{ 2(l\, x^{\alpha } +m\, y^{\alpha } +n\, z^{\alpha })-l(4\mu -\lambda ^{2})(l^{2} +m^{2} +n^{2})\, t^{\alpha } \}$.

Eq. (3.10) with the aid of Eq. (2.6) grants three types of traveling wave solutions of Eq. (2.1) as follows:

When $\lambda ^{2} -4\mu > 0$, the expression for the hyperbolic function solution is

In particular case, for $C_{1} \ne 0, \, \, C_{2} = 0$ Eq. (3.14) becomes

where $\xi = \frac{1}{2\alpha } \{ 2(l\, x^{\alpha } +m\, y^{\alpha } +n\, z^{\alpha })-l(4\mu -\lambda ^{2})(l^{2} +m^{2} +n^{2})\, t^{\alpha } \}$.

When $\lambda ^{2} -4\mu < 0$, the trigonometric function solution is gained as

For particular case, if we choose $C_{1} \ne 0, \, \, C_{2} = 0$ Eq. (3.16) turns into

where $\xi = \frac{1}{2\alpha } \{ 2(l\, x^{\alpha } +m\, y^{\alpha } +n\, z^{\alpha })-l(4\mu -\lambda ^{2})(l^{2} +m^{2} +n^{2})\, t^{\alpha } \}$.

The above acquired closed form solutions to the nonlinear space-time fractional mKdV-ZK equation are new and more general. If we choose $b_{1} = 0$, the solutions (3.15) and (3.17) coincide with those constructed by the $(G'/G)$-expansion method ^[36]. Furthermore, Eqs. (3.11)- (3.13) under the same procedure as above also provide much more new and general solutions which are not recorded here to keep the readers away from the inconvenience.

3.2.   The time fractional biological population model

The nonlinear time fractional biological population model is

where $\rho, \, \eta$ are constants, $u$ represents the population density and $\rho (u^{2} -\eta)$ represents the population supply due to births and deaths. A biological population model is a mathematical model which helps us to understand the dynamical procedure of population changes and provides valuable predictions. The universe that range from simple to dynamic is full of interactions. Most of the earth's processes affect human life. Procedures in population modeling have significantly enhanced our understanding of biology and the natural world. A population model that is applied to the study of population dynamics is a type of mathematical model which provides us a good understanding of how complicated interactions and procedures work.

Making use of the fractional compound transformation

Eq. (3.18) is converted into the integer order ODE,

Balancing the highest order derivative and the nonlinear term appearing in Eq. (3.20), the solution Eq. (2.4) reduces to the form

where at least one of $a_{1}$ and $b_{1}$ is nonzero.

Substituting Eq. (3.21) along with Eq. (2.5) into Eq. (3.20) yields a polynomial in $(G'/G)$. Setting each coefficient of this polynomial to zero, offered a system of algebraic equations for $a_{0}, \, \, a_{1}, \, \, b_{0}, \, \, b_{1}, \, \, c, \, \, \lambda$ and $\mu$. Solving this set of equations with the aid of computer algebra, like Maple, provides the following results:

where $b_{1}, \, c$ and $\lambda$ are arbitrary constants.

Inserting the values appearing in Eq. (3.22) into Eq. (3.21) possesses

where $\, \xi = k\, x+i\, k\, y-\frac{c\, t^{\alpha } }{\alpha }$, $i^{2} = -1$.

Eq. (3.23) along with Eq. (2.6) makes available the following three types of closed form traveling wave solutions:

When $\lambda ^{2} -4\mu > 0$, the hyperbolic function solution is gained as

We might choose the arbitrary constants as $C_{1} = r_{1} \cosh \theta, \, \, C_{2} = r_{1} \sinh \theta$ in Eq. (3.24) and simplify the solution to

where $r_{1} = \sqrt{C_{1} ^{2} -C_{2} ^{2} }$, $\theta = \tanh ^{-1} (C_{2} /C_{1})$ and $\, \xi = k\, x+i\, k\, y-\frac{c\, t^{\alpha } }{\alpha }$, $i^{2} = -1$.

For $\lambda ^{2} -4\mu < 0$, the trigonometric function solution is found as follows:

If the arbitrary constants are assigned as $C_{1} = r_{2} \cos \phi, \, \, C_{2} = r_{2} \sin \phi$ in Eq. (3.26), then it becomes to the simplest form

where $r_{2} = \sqrt{C_{1} ^{2} -C_{2} ^{2} }$, $\phi = \tanh ^{-1} (C_{2} /C_{1})$ and $\, \xi = k\, x+i\, k\, y-\frac{c\, t^{\alpha } }{\alpha }$, $i^{2} = -1$.

If $\lambda ^{2} -4\mu = 0$, the rational function solution is

In particular, for $C_{1} = 0$ Eq. (3.26) becomes

where $\, \xi = k\, x+i\, k\, y-\frac{c\, t^{\alpha } }{\alpha }$, $i^{2} = -1$.

The exact solutions obtained above to the biological population model are new and general. Abdel Salam and Gumma ^[37] constructed two traveling wave solutions in terms of hyperbolic function only. But we gained six closed form traveling wave solutions in terms of hyperbolic function, trigonometric function and rational function. So far we hunt; no one achieved these results ever.

3.3.   The space-time fractional modified regularized long-wave equation

The following nonlinear space-time fractional modified regularized long-wave equation is considered to be examined for further exact traveling wave solutions:

where $\delta, \, \, \tau$ and $\eta$ are all constants. This equation proposed by Benjamin et al. to describe approximately the unidirectional propagation of long waves in certain dispersive systems is supposed to be alternative to the modified KdV equation. Eq. (3.30) is formulated to demonstrate some physical phenomena like transverse waves in shallow water and magneto hydrodynamic waves in plasma and photon packets in nonlinear crystals ^[38,39,40].

The wave variable transformation

reduces Eq. (3.30) to the ODE

Integrating Eq. (3.32) and setting integral constant to zero gives

Considering homogeneous balance for Eq. (3.33) the solution Eq. (2.4) is appeared as

in which at least one of $a_{1}$ and $b_{1}$ is nonzero.

Put Eq. (3.34) with the help of Eq. (2.5) in Eq. (3.33); collect the coefficients of like powers of $(G'/G)$ and equate them to zero we obtain a set of equations for $a_{0}, \, \, a_{1}, \, \, b_{0}, \, \, b_{1}, \, \, c, \, \, \lambda$ and $\mu$. Calculating these equations by Maple gives the results

Set-1:

where $b_{0}, \, \, b_{1}, \, \, \lambda$ and $\mu$ are arbitrary constants.

Set-2:

where $b_{1}, \, \, \lambda$ and $\mu$ are arbitrary constants.

Set-3:

where $b_{0}, \lambda$ and $\mu$ are arbitrary constants.

Set-4:

$a_{0} = \pm \frac{b_{1} \delta \{ (4\mu -\lambda ^{2})(\lambda \eta \pm \sqrt{6\eta -3\eta ^{2} (4\mu -\lambda ^{2})})-2\lambda \} }{\sqrt{-\delta \tau } (4\mu \eta -\lambda ^{2} \eta -2)}$, $a_{1} = \pm \frac{b_{1} \delta }{\sqrt{-\delta \tau } }$,

where $b_{1}, \, \, \lambda$ and $\mu$ are arbitrary constants.

Utilizing the values appeared in Eqs. (3.35)-(3.38), the Eq. (3.34) provide the following expressions for desired solutions:

where $\xi = \frac{\, x^{\alpha } }{\alpha } +\frac{2\delta \, t^{\alpha } }{\alpha (4\mu \eta -\lambda ^{2} \eta -2)}$.

Each of Eqs. (3.39)-(3.42) together with Eq. (2.6) makes available exact traveling wave solutions to the space-time fractional modified regularized long-wave equation of three types, such as hyperbolic function solution, trigonometric function solution and rational function solution. For convenience of the readers, we record here the solutions only for Eq. (3.39) as follows:

When $\lambda ^{2} -4\mu > 0$, the hyperbolic function solution is formed as follows:

Assigning the arbitrary constants as $C_{1} \ne 0, \, \, \, C_{2} = 0$ to Eq. (3.43) and simplifying one may obtain

where $\xi = \frac{\, x^{\alpha } }{\alpha } +\frac{2\delta \, t^{\alpha } }{\alpha (4\mu \eta -\lambda ^{2} \eta -2)\alpha }$.

When $\lambda ^{2} -4\mu < 0$, the trigonometric function solution is

Since $C_{1}$ and $C_{2}$ are arbitrary constants, if we choose $C_{1} \ne 0, \, \, \, C_{2} = 0$, Eq. (3.45) after simplification becomes

where $\xi = \frac{\, x^{\alpha } }{\alpha } +\frac{2\delta \, t^{\alpha } }{\alpha (4\mu \eta -\lambda ^{2} \eta -2)\alpha }$.

For $\lambda ^{2} -4\mu = 0$, the rational function solution is

In particular, if $C_{1} \ne 0, \, \, \, C_{2} = 0$, Eq. (3.47) is simplified to the form

where $\xi = \frac{\, x^{\alpha } }{\alpha } +\frac{2\delta \, t^{\alpha } }{\alpha (4\mu \eta -\lambda ^{2} \eta -2)}$.

The closed form traveling wave solutions to the nonlinear space-time fractional modified regularized long-wave equation were successfully constructed in this effort. The solutions obtained by Kaplan et al. ^[41] and also by Abdel Salam and Gumma ^[37] are only in terms of hyperbolic, where as we achieved those in terms of hyperbolic, trigonometric and rational. On comparison, our solutions are general and much more in number than those of ^[37,41].

4.   Conclusion

In this article, our core aim was to explore further new and general closed form traveling wave solutions to the (3+1)-dimensional space-time fractional mKdV-ZK equation, the time fractional biological population model and the space-time fractional modified regularized long-wave equation. The desired solutions have been successfully achieved by the rational $(G'/G)$-expansion method in terms of hyperbolic, trigonometric and rational. To the best of our knowledge, the results obtained throughout this article are not recorded in the literature. The suggested method has shown high performance to construct traveling wave solutions in closed form which will be helpful to analyze important phenomena in the nature of real world.

Acknowledgments

This work is supported by the USM Research University Grant 1001/PMATHS/8011016 and the authors acknowledge this support.

Conflict of interest

The authors declare that they have no competing interests.