I. INTRODUCTION

The partial differential equations of fractional order (FPDEs) have been widely employed in recent years to explain a wide range of physical effects and complicated nonlinear phenomena. This is because they accurately describe nonlinear phenomena in the fields of fluid mechanics, viscoelasticity, electrical chemistry, quantum biology, physics, and engineering mechanics [1]-[6] as well as other scientific domains. As a result, the research of PDEs has received a lot of interest as many physical events may be explained using the idea of fractional derivatives and integrals [7]-[9]. Add to that, when the exact solutions to the majority of FPDEs are difficult to find, analytical and numerical methods [10]-[29] which are proposed and developed by many authors must be used.

Lie symmetry analysis is extremely important in many fields of science, particularly in integrable systems with an infinite number of symmetries. Thus, Lie symmetry analysis is regarded as one of the most effective methods for obtaining analytical solutions to nonlinear partial differential equations (NLPDEs). Also, many FPDEs have been studied using this analysis [30]-[36]. Add to that, this analysis is used to build conservation laws, which are crucial in the study of nonlinear physical phenomena. Conservation laws are mathematical formulations state that the total amount of a certain physical quantity remains constant as a physical system evolves. Furthermore, conservation laws are used in the development of numerical methods to establish the existence and uniqueness of a solution. There are many studies that discuss conservation laws for time FPDEs, which are mentioned in the references [37]-[43].

In this article, we focus on the following time-fractional MZKB equation of the form:

where ${\partial }_t^{\alpha }$ is the fractional derivative of order $\alpha$ (with ), ${\kappa }_i(i=0,1,\dots ,4)$ are respectively the dispersion, non linearity, mixed derivative, and dissipation. The $q(x,y,z,t)$ is the potential function of space $x$, $y$, $z$ and time $t$. If $\alpha =1$, Eq. (1) is reduced to the classical MZKB equation [44, 45], which describes the nonlinear plasma dust ion acoustic waves DIAWs in a magnetized dusty plasma and it is derived using the standard reductive perturbation technique in small amplitude.

The article is organized as follows: The introduction is presented in Section 1. In Section 2, some definitions and description of Lie symmetry analysis for fractional partial differential equations (FPDEs) are briefly presented. Lie symmetry analysis and similarity reduction of the Eq. (1) are obtained in Section 3. We construct two sets of analytical solutions for Eq. (1) by using fractional sub-equation method and extended Kudryashov method in Section 4 and 5 respectively. In Section 6, the conservation laws of the Eq. (1) are obtained. Finally, the discussions and conclusions of this article are presented in Section 7.

II. PRELIMINARIES

Here in this section, we focus on some of the concepts that revolve around the subject of our article Definition 1: Let . The operator $I^{\alpha }$ defined by

is called the Riemann-Liouville (R-L) fractional integral operator of order $\alpha$, and $\mathrm{\Gamma }(.)$ denotes the gamma function.

Definition 2: Let . The operator $D_t^{\alpha }$ is defined by

is called the R-L fractional partial derivative [7, 8].

a) Description of Lie symmetry analysis

Let's consider the symmetry analysis for a FPDE of the form

Now, let Eq. (4) is invariant under the following one-parameter Lie group of point transformation acting on both the dependent and independent variables, given as

Notes where $\epsilon \ll 1$ is the Lie group parameter and $\xi ,\zeta ,\nu ,\tau ,\eta$ are the infinitesimals of the transformations for dependent and independent variables respectively. The explicit expressions of ${\eta }^x$, ${\eta }^{xx}$, ${\eta }^{xxx}$, ${\eta }^{xyy}$, ${\eta }^{xzz}$ are given by

where $D_x$, $D_y$, $D_z$, and $D_t$ are the total derivatives with respect to $x$, $y$, $z$, and $t$ respectively that are defined for $x^1=x$, $x^2=y$, $x^3=z$ as

where $q_j=\frac{\partial q}{\partial x^j}$ $q_{jk}=\frac{{\partial }^{2}q}{\partial x^k}$ and so on.

The corresponding Lie algebra of symmetries consists of a set of vector fields of the form

The invariance condition of Eq. (4) under the infinitesimal transformations is given as

where

Also, the invariance condition gives

The $\alpha$ th extended infinitesimal related to RL fractional time derivative with Eq. (7) can be represented as follows

where $D_t^{\alpha }$ is the total fractional derivative operator and by using the generalized Leibnitz rule

By applying the Leibnitz rule, Eq. (8) becomes

Now by using the chain rule for the compound function which is defined as follows

By applying this rule and the generalized Leibnitz rule with $f(t)=1$, we have

where

Therefore, Eq. (9) yields

Definition 3: The function $q=\theta (x,y,z,t)$ is an invariant solution of Eq. (4) associated with the vector field $W$, such that

1. $q=\theta (x,y,z,t)$ is an invariant surface of Eq. (4), this means

2. $q=\theta (x,y,z,t)$ satisfies Eq.4

III. LIE SYMMETRY ANALYSIS AND SIMILARITY REDUCTION OF Eq.(1)

In this section, we implemented Lie group method for Eq. (1) and then, used these symmetries to reduce Eq. (1) to be a FODE as shown in the next two subsections

a) Lie symmetry analysis for Eq. (1)

Let us consider Eq. (1) is an invariant under Eq. (5), we get

such that $q=q(x,y,z,t)$ satisfies Eq. (1), then using the point transformations Eq. (5) in Eq. (11), we get the invariant equation

By substituting Eq. (6) and Eq. (10) into Eq. (12), grouping the coefficients of all derivatives and various powers of $u$ and equating them to zero we get an algebraic system of equations. Solving this system, we obtain a set of infinitesimal symmetries as below:

Case 1: When ${\kappa }_i\ne 0$, $i=1,2,3$, ${\kappa }_4=0$

where $c_i$, $i=1,2,3,4,5$ are arbitrary constants. Thus, the infinitesimal generator of Eq. (1) can be expressed as follows

which can be spanned by the five vector fields listed below.

Case 2: When ${\kappa }_i\ne 0$, $i=1,2,3,4$

hence, there are four vector fields as below

b) The

In this part of the article, we used the symmetries defined by Eq. (14) and Eq. (15) to construct the similarity reduction for Eq. (1) as presented in the next cases

Case 1: For $V_1=\frac{\partial }{\partial t}$, ${\dot{V}}_2=\frac{\partial }{\partial x}$, $V_3=\frac{\partial }{\partial y}$, $V_4=\frac{\partial }{\partial z}$, $V_5=\frac{3t}{2\alpha }\frac{\partial }{\partial t}+\frac{x}{2}\frac{\partial }{\partial x}+\frac{y}{2}\frac{\partial }{\partial y}+\frac{z}{2}\frac{\partial }{\partial z}-2q\frac{\partial }{\partial q}$ with ${\kappa }_4=0$ we have a set of characteristic equations that arranged in the following subcases

Case 1.1: $V_1=\frac{\partial }{\partial t}$ we have a characteristic equation of the form

by integrating this equation and appoint the solutions $q$ as function of the dependent variables $x,y,z$, that is

this implies $\frac{{\partial }^{\alpha }q}{\partial t^{\alpha }}=0$ and Eq. (1) becomes

Case 1.2: For $V_2=\frac{\partial }{\partial x}$ we have a characteristic equation of the form

by solving this equation we have $q(x,y,z,t)={\mathrm{\Phi }}_2(y,z,t)$ which makes all the derivatives of $u(x,y,z,t)$ with respect to $x$ equal to zero and where $B_0$ is a constant

Case 1.3: For $V_3=\frac{\partial }{\partial y}$ the characteristic equation is of the form

thus $q(x,y,z,t)={\mathrm{\Phi }}_3(x,z,t)$ and all the derivatives of $q(x,y,z,t)$ with respect to $y$ equal to zero, therefore

Case 1.4: For $V_4=\frac{\partial }{\partial z}$ the characteristic equation is of the form

thus $q(x,y,z,t)={\mathrm{\Phi }}_4(x,y,t)$ and all the derivatives of $q(x,y,z,t)$ with respect to $z$ equal to zero, therefore

Case 1.5: For $V_5=\frac{3t}{2\alpha }\frac{\partial }{\partial t}+\frac{x}{2}\frac{\partial }{\partial x}+\frac{y}{2}\frac{\partial }{\partial y}+\frac{z}{2}\frac{\partial }{\partial z}-2q\frac{\partial }{\partial q}$, the characteristic equation becomes

Where solving this equation we can get the next similarity variables and similarity solution for Eq. (1) as below

By using the above transformation, Eq. (1) can be turned into a nonlinear FODE with a set of independent variable ${\gamma }^{\prime }s$. Consequently, one can conclude the next theorem.

Theorem 1: The transformation Eq. (17) reduces the time-fractional generalized Z-K Eq. (1) to the following equation

with the E-K fractional differential operator $\left(P_{\beta }^{\tau ,\alpha }\varphi \right)({\gamma }_1,{\gamma }_2,{\gamma }_3)$ which is defined as

where (19)

is the E-K fractional integral operator.

The proof of theorem 1: Depending on the definition of the R-L fractional derivatives provided with , $n=1,2,3,\dots$, then we have

Let $\mathrm{\Lambda }=\frac{t}{g}$, one can get $dg=-\frac{t}{{\mathrm{\Lambda }}^2}$, thus Eq. (21) can be written as

following the definition of E-K fractional differential operator, then Eq. (22) becomes

it is time to deal with the right-hand side of Eq. (23). Where

From that, we have

\frac{\partial^{n}}{\partial t^{n}} \left[ t^{n - \frac{4}{3} \alpha} \left(K_{\frac{3}{\alpha}}^{1 - \frac{\alpha}{3}, n - \alpha} \phi\right) (\gamma_{1}, \gamma_{2}, \gamma_{3}) \right] \",

according to the above result provided with the same steps for $(n-1)$ times, we get

\frac{\partial^{n}}{\partial t^{n}} \left[ t^{n - \frac{4}{3} \alpha} \left(K_{\frac{3}{\alpha}}^{1 - \frac{\alpha}{3}, n - \alpha} \phi\right) (\gamma_{1}, \gamma_{2}, \gamma_{3}) \right] \",

this implies

thus

At last, Eq. (1) can be reduced into the below equation and the proof is completed

Therefore, the proof is completed.

Case 2: For $V_1=\frac{\partial }{\partial t}$, $V_2=\frac{\partial }{\partial x}$, $V_3=\frac{\partial }{\partial y}$, $V_4=\frac{\partial }{\partial z}$ with ${\kappa }_4\ne 0$, we have the following sub-cases

Case 2.1: $V_6=\frac{\partial }{\partial t}$ we have a characteristic equation of the form

by integrating this equation and appoint the solutions $q$ as function of the dependent variables $x,y,z$, that is

this implies $\frac{{\partial }^{\alpha }q}{\partial t^{\alpha }}=0$ and Eq. (1) becomes

Case 2.2: For $V_7=\frac{\partial }{\partial x}$ we have a characteristic equation of the form

by solving this equation we have $q(x,y,z,t)={\mathrm{\Psi }}_2(y,z,t)$ which makes all the derivatives of $q(x,y,z,t)$ with respect to $x$ equal to zero and

$\frac{{\partial }^{\alpha }q}{\partial t^{\alpha }}=0,$ this equation has the following solution where $C_0$ is a constant.

Case 2.3: For $V_3=\frac{\partial }{\partial y}$ the characteristic equation is of the form

thus $q(x,y,z,t)={\mathrm{\Psi }}_3(x,z,t)$ and all the derivatives of $q(x,y,z,t)$ with respect to $y$ equal to zero, therefore

Case 2.4: For $V_4=\frac{\partial }{\partial z}$ the characteristic equation is of the form

thus $q(x,y,z,t)={\mathrm{\Psi }}_4(x,y,t)$ and all the derivatives of $q(x,y,z,t)$ with respect to $z$ equal to zero, therefore

IV. FRACTIONAL SUB-EQUATION METHOD FOR FPDES

In this section, we introduced a summary explanation of the fractional sub-equation method [15, 16] as shown in the following steps

Step 1: Let $q(x,y,z,t)=q(\xi ),\xi =x+y+z-\lambda t$ is the traveling wave transformation which can be used to reduce the below equation

to be a non-linear FODE of the form

Step 2: Assume that, the above equation has a solution of the form

where $A_i(i=0,1,\dots ,n)$ are constants to be detected and the positive integer $n$ can be obtained by balancing the nonlinear terms and the highest order derivatives in Eq. (28). Also, the function $\phi (\xi )$ satisfy the following fractional Riccati equation

where $\phi (\xi )$ has a set of solutions as shown below

where all the previous trigonometric and hyperbolic functions are expressed by the following Mittag-Leffler function

Step 3: Now, substituting Eq. (29) along with (30) into (28) and equate all the coefficients of all powers of $(\phi (\xi ){)}^i$ by zero. Then, we get a system of algebraic equations. Solving this system via the Mathematica program to determine the value of $A_i(i=0,1,\dots ,n)$. Consequently, we use these values with the solutions of Eq. (30) to construct the analytical solutions for Eq. (28) which is considered the main aim for this section.

a) Fractional sub-equation method for Eq. (1)

In this section, we apply the fractional sub-equation method for Eq. (1) therefore, we rewrite this equation by using $q(x,y,z,t)=v^2(x,y,z,t)$ as follows:

Let us introduce an important transformation

thus, Eq. (31) has the following form

According to the previous analysis of the considered method. We have the following solution for the reduced Eq. (33)

substituting Eq. (34) along with (30) into (33) and equate all the coefficients $(\phi (\xi ){)}^i$ by zero to get a system of algebraic equations. Solving this system with the aid of the Mathematica program we have

then, Eq. (1) has the below solutions

thus, we have a set of analytical solutions for Eq. (1) which is presented as follows

where $v_0$ is a constant and $\xi =x+y+z-\lambda t$

The following figures show the 3D and 2D plots for the solution Eq. (31):

(a) $\alpha =0.6$

(b) $\alpha =0.7$

(c) $\alpha =0.9$ Figure 1: The 3D double-layer solution (31). (a) The solution at $\alpha =0.6$ with the parameters $\lambda =0.4$, $\delta =-0.5$, ${\kappa }_1=0.6$, ${\kappa }_2=0.3$ (b) The solution at $\alpha =0.7$ with the same parameters. (c) The solution at $\alpha =0.9$ with the same parameters.

(a)

(b) Figure 2: The effect of $\alpha$ on the analytical solution (36) (a) The 3D plot at a fixed time $t=0.6$ and $0.65\le \alpha \le 0.9$ with the parameters $\lambda =0.4$, $\delta =-0.5$, ${\kappa }_1=0.6$, ${\kappa }_2=0.3$. (b) The 2D plot at $\alpha =0.8$, 0.85, 0.9 and the same values of the others parameters.

V. THE METHODOLOGY OF THE EXTENDED KUDRYASHOV METHOD

We briefly display the main steps of the extended Kudryashov method [46, 47] to construct analytical solutions for Eq. (1) as below.

Step 1: Consider a non-linear FODE Eq.(28) with the same traveling wave transformation as section 4 and assume that the solution of Eq.(28) can be expressed as follows:

where $a_i$, $i=0,1,2,\dots ,n$ are constants to be determined, and $\phi =\phi (\xi )$ satisfies the following equation:

Step 2: Determining the value of the positive integer $M$ by balancing the highest order derivatives with the nonlinear terms which appear in Eq.(28) by using the relation $M=\frac{2(s-rp)}{r-l-1}$ since, $(q^{(p)}(\xi ,\phi ){)}^r$ and $q^l(\xi ,\phi )q^{(s)}(\xi )$ are the balanced terms.

Step 3: Substituting Eq.(37) into Eq.(28) and using Eq.(38), collecting all terms with the same order of $\phi (\xi )$ together to zero yields a set of algebraic equations. Solving the equations system and using Eq.(38) to construct a variety of analytical solutions for Eq.(28).

a) Extended Kudryashov method for Eq. (1)

In this section, we apply the extended Kudryashov method method for Eq. (33) which is considered a reduced form of Eq. (1) and according to the previous analysis of the considered method. We have the following solution

substituting Eq. (39) along with (38) into (33) and equate all the coefficients $(\phi (\eta ){)}^i$ by zero to get a system of algebraic equations. Solving this system with the aid of the Mathematica program we have

then, Eq. (1) has the below solutions

thus, we have the next solutions for Eq. (1) which is presented as follows

where $\xi =x+y+z-\lambda t$

The 3D and 2D plots for the solution Eq. (41) are plotted in the following Figures:

(a) $\alpha =0.6$

(b) $\alpha =0.7$

(c) $\alpha =0.9$ Figure 3: The 3D double-layer solution (41). (a) The solution at $\alpha =0.6$ with the parameters $\lambda =0.7$${\kappa }_1=0.5$${\kappa }_2=0.3$ (b) The solution at $\alpha =0.7$ with the same parameters. (c) The solution at $\alpha =0.9$ with the same parameters.

(a)

(b) Figure 4: The effect of $\alpha$ on the analytical solution (41) (a) The 3D plot at a fixed time $t=0.3$ and $0.65\le \alpha \le 0.9$ with the parameters $\lambda =0.7$, ${\kappa }_1=0.5$, ${\kappa }_2=0.3$. (b) The 2D plot at $\alpha =0.7$, 0.8, 0.9 and the same values of the others parameters.

VI. CONSERVATION LAWS FOR EQ.(1)

In this section, the conservation laws of the time fractional MZKB equation (1) were derived, based on the formal lagrangian and Lie point symmetries as described in the following explanation: Consider a vector $C=(C^t,C^x,C^y,C^z)$ admits the following conservation equation

where $C^t=C^t(x,y,z,t,u,\dots )$, $C^x=C^x(x,y,z,t,u,\dots )$, $C^y=C^y(x,y,z,t,u,\dots )$, and $C^z=C^z(x,y,z,t,u,\dots )$ are called the conserved vectors for Eq. (1). According to the new conservation theorem for Ibragimov [37], the formal Lagrangian for Eq. (1) can be given by

here $\omega ((x,y,z,t)$ is a new dependent variable. Depending on the definition of the Lagrangian, we get an action integral as follows

The Euler-Lagrange operator is defined as

where $(D_t^{\alpha }{)}^{\ast }$ denotes to the adjoint operator of $D_t^{\alpha }$, and the adjoint equation to the nonlinear by means of the Euler-Lagrange equation is given by

Adjoint operator $(D_t^{\alpha }{)}^{\ast }$ for R-L is defined by

where $I_T^{n-\alpha }$ is the right-sided operator of fractional integration of order $n-\alpha$ that is defined by

Considering the case of one dependent variable $u(x,y,z,t)$ with four independent variables $x,y,z,t$, we get

where $\overline{X}$ is defined by

and the Lie characteristic function $W$ for case 1 and 2 in the subsection 3.1 is defined as

where $W$ can be expanded to

For the R-L time-fractional derivative, the density component $C^t$ of conservation law is defined as:

where the operator $J(\cdot )$ defined by

and the other (flux) components are defined as

where ${\xi }^1=\xi$, ${\xi }^2=\zeta$, ${\xi }^3=\nu$ and $m=1,2,\dots ,5$.

Now by using Eq. (44) with the help of Eqs. (45) and (46), we obtain the components of conservation laws for the time-fractional MZKB equation as the follows

Case 1: When ${\kappa }_4=0$ we have the following subcases according to the vector fields Eq. (14)

Case 1.1: $W_1=-q_x$ where ${\xi }^x=1$, ${\xi }^t=0$, ${\xi }^y=0$, ${\xi }^z=0$ and $\eta =0$ we get

Case 1.2: $W_2=-q_t$ where ${\xi }^x=0$, ${\xi }^t=1$, ${\xi }^y=0$, ${\xi }^z=0$ and $\eta =0$ we obtain

Case 1.3: $W_3=-q_y$ where ${\xi }^x=0$, ${\xi }^t=0$, ${\xi }^y=1$, ${\xi }^z=0$ and $\eta =0$ we have

Case 1.4: $W_4=-q_z$ where ${\xi }^x=0$, ${\xi }^t=0$, ${\xi }^y=0$, ${\xi }^z=1$ and $\eta =0$ we obtain

Case 1.5: $W_5=-2q-\frac{3t}{2\alpha }\frac{\partial }{\partial t}-\frac{x}{2}\frac{\partial }{\partial x}-\frac{y}{2}\frac{\partial }{\partial y}-\frac{z}{2}\frac{\partial }{\partial z}$ then

Case 2: When ${\kappa }_4\ne 0$ we have the following subcases according to the vector fields Eq. (14)

Case 2.1: $W_1=-q_x$ where ${\xi }^x=1$, ${\xi }^t=0$, ${\xi }^y=0$, ${\xi }^z=0$ and $\eta =0$ we get

Case 2.2: $W_2=-q_t$ where ${\xi }^x=0$, ${\xi }^t=1$, ${\xi }^y=0$, ${\xi }^z=0$ and $\eta =0$ we obtain

Case 2.3: $W_3=-u_y$ where ${\xi }^x=0$, ${\xi }^t=0$, ${\xi }^y=1$, ${\xi }^z=0$ and $\eta =0$ we have

Case 2.4: $W_4=-q_z$ where ${\xi }^x=0$, ${\xi }^t=0$, ${\xi }^y=0$, ${\xi }^z=0$ and $\eta =0$ we obtain

Remark: We have verified that all the cases satisfy the equation of conservation laws Eq. (42).

VII. CONCLUSION

In this article, we considered the time-fractional modified Zakharov-Kuznetsov-Burgers (MZKB) equation of $(3+1)$ dimensions. With the help of the Riemann-Liouville derivatives, the Lie symmetry analysis was successfully applied to study this equation. This analysis generated the symmetries and vector fields, which aided us in constructing the similarity reductions of the considered equation. Consequently, we constructed two sets of new analytical solutions via two powerful methods which are the fractional subequation method and extended Kudryashov method. Furthermore, to gain a better understanding of the dynamics of these solutions, we graphed the 3D and 2D plots of obtained solutions using appropriate parameters. Figure 1 described the double-layer solution (31) at $\alpha =0.6$, 0.7, 0.9 with the parameters $\lambda =0.4$, $\delta =-0.5$, ${\kappa }_1=0.6$, ${\kappa }_2=0.3$ at $-5\le x\le 5$. Figure 2 showed the effect of $\alpha$ on the solution (31) at a fixed time $t=0.6$ with $\alpha =0.8$, 0.85, 0.9 and the parameters $\lambda =0.4$, $\delta =-0.5$, ${\kappa }_1=0.6$, ${\kappa }_2=0.3$ for $-2.5\le x\le 4$. Figure 3 represented the double-layer solution (41) at $\alpha =0.6$, 0.7, 0.9 with the parameters $\lambda =0.7$, ${\kappa }_1=0.5$, ${\kappa }_2=0.3$ where this Figure was traced at $-5\le x\le 5$. Figure 4 was graphed by taking suitable parameters as $\lambda =0.7$, ${\kappa }_1=0.5$, ${\kappa }_2=0.3$ and described the stable behavior of the solution (41) at $\alpha =0.7$, 0.8, 0.9 for $0\le x\le 5$ at a fixed time $t$ and decreases as the fractional-order $\alpha$ increases outside this interval. As a consequence, we presume that the obtained results will be more useful in explaining the physical meaning of the time fractional MZKB equation. Furthermore, we obtained four kinds of conservation laws with independent variables laying the groundwork of Lie symmetries. Finally, because of accuracy, ease of application, and relevance of the used methods in this paper, they could be generalized to many FPDEs.

Data availability

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Funding

This research work is not supported by any funding agencies.

Competing interests

The author has declared that no competing interests exist.

Contributions

The authors declare that the work was realized in collaboration with a specified responsibility for each author. All authors read and approved the final paper.