Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics Open Access

Fractional calculus originating from some speculations of Leibniz and L'Hospital in 1695 has gradually gained profound attention of many researchers for its extensive appearance in various fields of real world. Exact traveling wave solutions to fractional order nonlinear evolution equations (FNLEEs) are of fundamental and important in applied science because of their wide use to depict the nonlinear fractional phenomena and dynamical processes of nature world. The FNLEEs and their solutions in closed form play fundamental role in describing, modeling and predicting the underlying mechanisms related to the biology, bio-genetics, physics, solid state physics, condensed matter physics, plasma physics, optical fibers, meteorology, oceanic phenomena, chemistry, chemical kinematics, electromagnetic, electrical circuits, quantum mechanics, polymeric materials, neutron point kinetic model, control and vibration, image and signal processing, system identifications, the finance, acoustics and fluid dynamics [1–3]. The closed form wave solutions of these equations [4–6] are greatly helpful to realize the mechanisms of the complicated nonlinear physical phenomena as well as their further applications in practical life. Some attractive powerful approaches take into account in the recent research area related to fractional derivative associated problems [7–9]. Therefore, it has become the core aim in the research area of fractional related problems that how to develop a stable approach for investigating the solutions to FNLEEs in analytical or numerical form. Many researchers have offered different approaches to construct analytic and numerical solutions to FNLEEs as well as integer order and put them forward for searching traveling wave solutions, such as the He-Laplace method [10], the exponential decay law [11], the reproducing kernel method [12], the Jacobi elliptic function method [13], the $(G′/G)$-expansion method and its various modifications [14–18], the exp-function method [19], the sub-equation method [20, 21], the first integral method [22], the functional variable method [23], the modified trial equation method [24], the simplest equation method [25], the Lie group analysis method [26], the fractional characteristic method [27], the auxiliary equation method [28, 29], the finite element method [30], the differential transform method [31], the Adomian decomposition method [32, 33], the variational iteration method [34], the finite difference method [35], the homotopy perturbation method [36] and the He's variational principle [37], the new extended direct algebraic method [38, 39], the Jacobi elliptic function expansion method [40], the conformable double Laplace transform [41] etc. But each method does not bear high acceptance for the lacking of productivity to construct the closed form solutions to all kind of FNLEEs. That is why; it is very much indispensable to establish new techniques.

In this study, we offer a newly established technique, called the rational fractional $(DξαG/G)$-expansion method [42], to investigate closed form analytic wave solutions to some FNLEEs in the sense of conformable fractional derivative [43]. This effectual and reliable productive method shows its high performance through providing abundant fresh and general solutions to the suggested equations. The obtained solutions might bring up their importance through the contribution to analyze the inner mechanisms of physical complex phenomena of real world and make an acceptable record in the literature.

A new and simple definition of derivative for fractional order introduced by Khalil et al. [43] is called conformable fractional derivative. This definition is analogous to the ordinary derivative

where $ψ(x):[0,∞]→R$ and $x>0$⁠. According to this classical definition, $d(xn)dx=nxn−1$⁠. According to this perception, Khalil has introduced $α$ order fractional derivative of $ψ$ as

If the function $ψ$ is $α$ differentiable in $(0,r)$ for $r>0$ and $limx→0+Tαψ(x)$ exists, then the conformable derivative at $x=0$ is defined as $Tαψ(0)=limx→0+Tαψ(x)$⁠. The conformable integral of $ψ$ is

This integral represents usual Riemann improper integral.

The conformable fractional derivative satisfies the following useful properties [43]:

If the functions $u(x)$ and $v(x)$ are $α$ -differentiable at any point $x>0$⁠, for $α∈(0,1]$⁠, then

1. $Tα(au+bv)=aTα(u)+bTα(v)∀a,b∈R.$

2. $Tα(xn)=nxn−α∀n∈R.$

3. $Tα(c)=0$⁠, where $c$ is any constant.

4. $Tα(uv)=uTα(v)+vTα(u).$

5. $Tα(u/v)=vTα(u)−uTα(v)v2.$

6. if $u$ is differentiable, then $Tα(u)(x)=x1−αdudx(x)$⁠.

Many researchers used this new derivative of fractional order in physical applications due to its convenience, simplicity and usefulness [44–46].

In this subsection, we discuss the main steps of the rational fractional $(DξαG/G)$-expansion

method to examine exact traveling wave solutions to FNLEEs. A fractional partial differential equation in the independent variables $t,x1,x2,…,xn$ is supposed to be as follows:

where $0<α,β≤1$⁠; $ui=ui(t,x1,x2,…,xn)$⁠, $i=1,2,3,…,k$ are unknown functions, $F$ is a polynomial in $ui$ and it's various partial derivatives of fractional order. Maintain the following steps to unravel Eqn (2.2.1) by the rational fractional $(DξαG/G)$-expansion technique.

Let us consider the nonlinear fractional composite transformation

which reduces Eqn (2.2.1) to the following ordinary differential equation of fractional order with respect to the variable $ξ$⁠:

We might take anti-derivative of Eqn (2.2.3) term by term as many times as possible and integral constant can be set to zero as soliton solutions are sought.

• Step 1: Suppose the traveling wave solution of Eqn (2.2.1) can be expressed as follows:

where $ai′s$ and $bi,s$ are unknown constants to be determined later and $G=G(ξ)$ satisfies the following auxiliary nonlinear ordinary differential equation of fractional order:

where $λ$⁠, $μ$ are arbitrary constants and $DξαG(ξ)$ denotes the conformable fractional derivative of order $α$ for $G(ξ)$ with respect to $ξ$⁠.

The nonlinear fractional complex transformation $G(ξ)=H(η)$⁠, $η=ξα/Γ(1+α)$ reduces Eqn (2.2.5) into the following second order ordinary differential equation:

whose solutions are well-known. Since $DξαG(ξ)=DξαH(η)=H′(η)Dξαη=H′(η)$⁠, with the aid of the solutions of Eqn (2.2.6), we can obtain the solutions of Eqn (2.2.5) as follows:

where $C1$ and $C2$ are arbitrary constants.

• Step 2: The positive constant $n$ can be determined by taking homogenous balance between the highest order linear and nonlinear terms appearing in Eqn (2.2.3).

• Step 3: Substitute (2.2.4) and (2.2.5) into Eqn (2.2.3) with the value of $n$ obtained in step 2, we obtain a polynomial in $(DξαG/G)$⁠. Setting each coefficient of the resulted polynomial to zero gives a set of algebraic equations for $ai′s$ and $bi,s$ by means of the symbolic computation software, such as Maple, provides the values of constants.

• Step 4: Inserting the values of $ai′s$ and $bi,s$ into (2.2.4) along with (2.2.7)–(2.2.9), the closed form traveling wave solutions to the nonlinear evolution Eqn (2.2.1) are obtained.

In this section, the exact analytic traveling wave solutions to the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation are constructed.

This well-known equation is given as

With the aid of the fractional compound transformation

Eqn (3.1.1) is turned into the following ordinary differential equations of fractional order due to the variable $ξ$⁠:

Taking anti-derivative of (3.1.3) yields

Considering the homogenous balance to Eqn (3.1.4), the solution (2.2.4) becomes

Eqn (3.1.4) together with (3.1.5) and (2.2.5) becomes a polynomial in $(DξαG/G)$ equating whose coefficients to zero and solving provides the following outcomes:

where $a1,b0,b1,λandμ$ are free parameters.

where $a0,b0,λandμ$ are free parameters.

Insert the values appeared in (3.1.6) and (3.1.7) in the solution (3.1.5) provide the following expressions for exact analytic solutions:

where $ξ=x+y+{(4μ−λ2−3)/4}1/αt$⁠.

The expressions (3.1.8) and (3.1.9) along with (2.2.7)–(2.2.9) make available the following closed form traveling wave solutions in terms of hyperbolic function, trigonometric function and rational function:

When $λ2−4μ>0$⁠,

Choose $c1≠0,c2=0$⁠, then (3.1.10) becomes

where $ξ=x+y+{(4μ−λ2−3)/4}1/αt$⁠.

When $λ2−4μ<0$⁠,

The choice of $c1≠0,c2=0$ gives way

where $ξ=x+y+{(4μ−λ2−3)/4}1/αt$⁠.

When $λ2−4μ=0,$

Choosing $c1=0,c2≠0$ yields

where $ξ=x+y+{(−3)/4}1/αt.$

When $λ2−4μ>0,$

Assigning $c1≠0, c2=0$ provides

where $ξ=x+y+{(4μ−λ2−3)/4}1/αt$⁠.

When $λ2−4μ<0,$

Conveying $c1≠0, c2=0$ offers

where $ξ=x+y+{(4μ−λ2−3)/4}1/αt.$

When $λ2−4μ=0$⁠,

The transmission $c1=0,c2≠0$ puts forward

where $ξ=x+y+{(−3)/4}1/αt$⁠.

Consider the nonlinear space-time fractional STO equation

Using the complex fractional transformation

Eqn (3.2.1) reduces to the following fractional order ordinary differential equation with respect to the variable $ξ$⁠:

Taking anti-derivative of Eqn (3.2.3) yields

Applying the homogeneous balance method to Eqn (3.2.4) the solution (2.2.4) takes the form (3.1.5).

Eqn (3.2.4) under the use of solution (3.1.5) and Eqn (2.2.5) creates a polynomial in $(DξαG/G)$ whose coefficients assigning to zero and solving yields the outcomes:

where $b0,b1,k,c,βandλ$ are all arbitrary constants.

where $b0, k, c, β and λ$ are all unknown parameters.

Utilizing the values available in (3.2.5) and (3.2.6) in (3.1.5) provide the following expressions for analytic solutions:

where $ξ=k1/αx+c1/αt$⁠.

The expressions (3.2.7) and (3.2.8) along with (2.2.7)–(2.2.9) make available the following closed form traveling wave solutions in terms of hyperbolic function, trigonometric function and rational function:

When $λ2−4μ>0,$

Fixing $c1≠0, c2=0$ serves

where $ξ=k1/αx+c1/αt$⁠.

When $λ2−4μ<0,$

Setting up $c1≠0, c2=0$ provides

where $ξ=k1/αx+c1/αt$⁠.

When $λ2−4μ=0$⁠,

Putting $c1=0, c2≠0$ gives out

where $ξ=k1/αx+c1/αt$⁠.

When $λ2−4μ>0,$