Similarity Solution for a System of Fractional-Order Coupled Nonlinear Hirota Equations with Conservation Laws

Abstract

The analysis of differential equations using Lie symmetry has been proved a very robust tool. It is also a powerful technique for reducing the order and nonlinearity of differential equations. Lie symmetry of a differential equation allows a dynamic framework for the establishment of invariant solutions of initial value and boundary value problems, and for the deduction of laws of conservations. This article is aimed at applying Lie symmetry to the fractional-order coupled nonlinear complex Hirota system of partial differential equations. This system is reduced to nonlinear fractional ordinary differential equations (FODEs) by using symmetries and explicit solutions. The reduced equations are exhibited in the form of an Erdelyi–Kober fractional (E-K) operator. The series solution of the fractional-order system and its convergence is investigated. Noether’s theorem is used to devise conservation laws.

1. Introduction

Fractional calculus is related to the area of applied mathematics. It involves integrals and differentiation of arbitrary order. Fractional calculus is used to adequately describe the memory and inherent property of numerous substances and mechanisms. Lately, it has proved to be a robust tool for solving many complex processes concerning natural sciences and the theory of applied mathematics. Fractional calculus allures the assiduity of many physicists and mathematicians such as Lagrange, Euler, Abel, Gronwald, Letnikov, Riemann, and Caputo. They have made a huge contribution to the evolution of theory of fractional-order derivatives [1,2,3,4]. Fractional calculus has found a wide range of applications in natural sciences, engineering, technology, bioscience, fluid flow, electromagnetic, viscoelasticity, pandemic analysis, medicine, infectious modeling, drug therapy, image processing, and diffusion wave equations [5,6,7,8,9].

Fractional-order differential equations play an extensive role in the modeling of complex processes. Frenandez et al. [10] examined an integral transform established by Prabhakar, assuming Mittag–Leffler functions to investigate many various models involving fractional-order derivatives. They attained a new form of series representation for the transform, considering the Riemann–Liouville (R-L) fractional-order derivative. Fractional derivatives of the Riemann zeta function were specifically calculated by Guariglia et al. [11] a series representation of the fractional derivative was established. Srivastva et al. [12] employed homotopy techniques related to the Sumudu transform to examine a fractional-order mathematical model of a biological population with the property of carrying capacity. Similarity analyses for strong shocks in a nonideal gas and a numerical simulation of its coupled system have been analyzed by Arora et al. [13,14]. The Laplace transform method, sine–cosine technique, homotopy analysis, variation iteration method, homotopy perturbation method, reduced differential transform, and power series technique were suggested by authors [15,16,17,18,19,20,21] as semianalytic and computational methodologies for obtaining the series and exact solutions of fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs).

In our most recent work, we specifically applied Lie symmetry analysis [22] to determine an explicit and precise solution of fractional- and integral-order mathematical models. The invariance analysis and E-K operators were imposed on the system of FPDEs involving the R-L fractional derivative in [23,24,25,26,27,28,29].

The survey of construction of conservation laws for fractional-order differential equations is favorable for the apprehension as well as for the analysis of the physical problems. The conservation laws play a very crucial role in determining the consistency of a given system. Noether’s theorem for symmetries was proposed by Cicogna et al. [30], while Ibragimov [31] studied the applicability of the conservation theory in linear and nonlinear classical models. Invariance structure, explicit precise solutions with a power series solution, and a conservation analysis of the Boussinesq system, Burger’s coupled KdV and m–KdV equations, and Drinfeld–Sokolov–Satsuma–Hirota equations are explored in [32,33,34,35,36,37]. Biswas et al. [38,39,40] worked on dual dispersion, power laws, conservation laws and optimal quasi-solitons using Lie symmetry analysis. Chauhan and Arora [41] obtained the complete analysis of the time fractional Kupershmidt equation. Recently, Gandhi et al. [42,43,44] applied symmetry reduction on multiordered time-fractional KdV equations and Hirota–Satsoma-coupled Korteveg–de Vries equations to obtain the explicit solutions with convergence and conservation laws. Zhang et al. [45] exploited the conservation laws of the Fokkar–Plank equation with power diffusion. Bruzon et al. [46] focused on the study of similarity solutions and application of a new conservation theorem to the Cooper–Shepard–Sodano equation. A new soliton solution of the time-fractional Drinfeld–Sokolov–Satsuma–Hirota system in dispersive water waves was illustrated by Ray et al. [47]. A generalized two-component Hunter–Saxton system was studied by Yang et al. [48]. Dubey et al. [49] suggested the coupling of the fractional homotopy perturbation and fractional homotopy analysis methods with Sumudu transforms on the local fractional Laplace equation. The Haydon analysis was used by Chatibi et al. [50] to obtain the solution of ordinary and partial fractional-ordered systems and applied discrete symmetries to construct distinct physical solutions of FDEs. The generalized invariant structure of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation was proposed by Chauhan et al. [51]. They applied Lie point symmetry and described the numerical simulation of propagation of nonlinear dispersive waves in an inhomogeneous medium. Wang et al. [52] successfully derived the three-component coupled Hirota hierarchy, and first obtained the explicit soliton solutions of the equations via the dressing method. Tian et al. [53] successfully proposed an effective and direct approach to study the symmetry-preserving discretization for a class of generalized higher-order equations and proposed an open problem about symmetries and the multipliers of conservation law. Li and Tian [54] systematically solved the Cauchy problem of the general n-component nonlinear Schrödinger equations based on the Riemann–Hilbert method and given the N-soliton solutions. Moreover, they proposed a conjecture about the law of nonlinear wave propagation. Yang et al. [55] successfully solved the soliton solutions of the focusing nonlinear Schrödinger equation with multiple higher-order poles under nonzero boundary conditions for the first time. Wu and Tian [56] successfully solved the long-time asymptotic problem of the solution to the nonlocal short pulse equation with the Schwartz-type initial data: without solitons. With respect to soliton resolution conjecture, Li et al. [57,58,59] carried out some interesting work in deriving the solutions of the Wadati–Konno–Ichikawa equation and complex short pulse equation with the help of the Dbar-steepest descent method. They solved the long-time asymptotic behavior of the solutions of these equations and proved the soliton resolution conjecture and the asymptotic stability of solutions of these equations. Biswas et al. [60] interpreted the current advancement in the mathematical perspective of dispersion-managed optical solitons, quasi-linear pulses, Gabitov–Turitsyn equations, etc., and reduced the difference between the ideas of engineering and mathematics. Ito calculus was applied by Zahed et al. to acquire optical solitons of the Sasa–Satsuma model involving multiplicative noise [61].

In the present article, a nonlinear time fractional complex Hirota system of partial differential equations in the sense of Riemann–Liouville (RL) is examined by employing the Lie symmetry approach. The main goal of this work is to elaborate the utilization of symmetry analysis on a nonlinear fractional-order system and to design laws with their conservation for the given system of differential equations. Hirota equations have many applications in the propagation of sound and optical pulses in water crystal waveguides along with the study of single-mode fibers. The complex Hirota system [25] is explained below:

$$\left\{\begin{array}{l}\frac{{\partial }^{\theta }{z}_1}{\partial t^{\theta }}+\frac{{\partial }^3{z}_1}{\partial x^3}+6\left(|z_1{|}^2+|z_2{|}^2\right)\frac{\partial z_1}{\partial x}=0,\\ \frac{{\partial }^{\theta }{z}_2}{\partial t^{\theta }}+\frac{{\partial }^3{z}_2}{\partial x^3}+6\left(|z_1{|}^2+|z_2{|}^2\right)\frac{\partial z_2}{\partial x}=0,\end{array}\right.$$

(1)

where $z_1(x,t)=u(x,t)+iv\hspace{0.17em}(x,t)\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{z}_2(x,t)=w(x,t)+iz(x,t),$ and $\left(|z_1{|}^2+|z_2{|}^2\right)=u^2+v^2+w^2+z^2={\displaystyle \sum u^2}$.

In terms of the real variable, the system is:

$$\left\{\begin{array}{l}{\partial }_t^{\theta }u+u_{xxx}+6u_x\left({\displaystyle \sum u^2}\right)=0,\\ {\partial }_t^{\theta }v+v_{xxx}+6v_x\left({\displaystyle \sum u^2}\right)=0,\\ {\partial }_t^{\theta }w+w_{xxx}+6w_x\left({\displaystyle \sum u^2}\right)=0,\\ {\partial }_t^{\theta }z+z_{xxx}+6z_x\left({\displaystyle \sum u^2}\right)=0.\end{array}\right.$$

(2)

Here, ${\partial }_t^{\theta }u$, ${\partial }_t^{\theta }v$, ${\partial }_t^{\theta }w$ and ${\partial }_t^{\theta }z$ are partial derivatives, and the fractional parameter θ lies in 0 < θ < 1.

This article begins with the introduction in Section 1. Some fundamental definitions are presented in Section 2. The Lie symmetry approach for solving fractional-order coupled mathematical models is explained in Section 3. Section 4 is devoted to the application of Lie symmetry reduction in a nonlinear system of the Hirota system. Erdelyi–Kober operators are devised for the conversion of a system of FPDEs into FODEs in Section 5; thereafter, a power series solution is processed with convergence analysis in Section 6. Section 7 is dedicated to an adjoint system of the Hirota system and conservation laws by using Noether’s theorem, and eight cases for conserved quantities related to physical systems are successfully discussed. The conclusion and remarks related to the present work are given in Section 8.

2. Preliminaries

The fractional derivative of a function need not be unique. There are some acknowledged and established definitions of fractional derivatives in the literature. In the following section, some of the significant definitions of fractional-order derivatives are mentioned.

Definition 1.

The Caputo fractional-order derivative of function F(t) is defined as:

$$D_t^{\theta }(F(t))=\frac{1}{\mathrm{\Gamma }(\lambda -\theta )}{\displaystyle \underset{0}{\overset{t}{\mathrm{\int }}}{(t-\rho )}^{\lambda -\theta -1}{F}^{\lambda }(}\rho )d\rho \hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\lambda -1<\theta \le \lambda ;\hspace{0.17em}\hspace{0.17em}\lambda \in N\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}t>0.$$

(3)

Definition 2.

The Riemann–Liouville fractional-order derivative of F(t) is given by:

$$D_t^{\theta }(F(t))=\frac{1}{\mathrm{\Gamma }(\lambda -\theta )}\frac{d^{\lambda }}{d{t}^{\lambda }}{\displaystyle \underset{0}{\overset{t}{\int }}{(t-\rho )}^{\lambda -\theta -1}F(}\rho )d\rho \hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\lambda -1<\theta \le \lambda \hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\lambda \in N;\hspace{0.17em}\hspace{0.17em}t>0.$$

(4)

Definition 3.

Let the function be $u(x,t)$ with t > 0, then the Riemann–Liouville fractional partial order derivative is given below:

$${\partial }_t^{\theta }(u(x,t))=\left\{\begin{array}{l}\frac{1}{\mathrm{\Gamma }(\lambda -\mu )}\frac{{\partial }^{\lambda }}{\partial t^{\lambda }}{\displaystyle \underset{0}{\overset{t}{\int }}{(t-\rho )}^{\lambda -\theta -1}u(}\rho ,x)d\rho \hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\lambda -1<\theta <\lambda ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\lambda \in N,\\ \frac{{\partial }^{\lambda }u}{\partial t^{\lambda }}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\theta =\lambda .\end{array}\right.$$

(5)

Definition 4.

The product rule for RL fractional-order derivatives is provided below:

$$D_t^{\theta }(U.V)={\displaystyle \sum _{\lambda =0}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}{D}_t^{\theta -\lambda }(U).D_t^{\lambda }(V)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\theta >0\hspace{0.17em}\hspace{0.17em}with\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)=\frac{{(-1)}^{\lambda }\theta \hspace{0.17em}\mathrm{\Gamma }(n-\theta )}{\mathrm{\Gamma }(1-\theta )\mathrm{\Gamma }(\theta +1)}\hspace{0.17em}.$$

(6)

Definition 5.

The E–Kober generalized fractional differential operator $\left(E_{\partial }{}^{\tau ,\mu }\omega \right)(\zeta )$ is given by:

$$\begin{array}{l}\left(E_{\partial }^{\tau ,\mu }\omega \right)(\zeta )={\displaystyle \prod _{\lambda =0}^{m-1}\left(\tau +\lambda -\frac{1}{\partial }z\frac{d}{dz}\right)(K_{\partial }^{\tau +\mu ,m-\mu }\omega })(z)\hspace{0.17em}\hspace{0.17em}with\hspace{0.17em}\hspace{0.17em}\zeta >0,\hspace{0.17em}\hspace{0.17em}\partial >0\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\mu >0\hspace{0.17em};\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m=\left\{\begin{array}{c}[\mu ]+1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mu \notin N\hfill \\ \mu \hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mu \in N\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \end{array}\right.\end{array}$$

(7)

Definition 6.

The E–Kober generalized fractional-order integral operator $(K_{\partial }{}^{\tau ,\mu }\omega )(\zeta )$ is given as:

$$(K_{\partial }^{\tau ,\mu }\omega )(\zeta )=\left\{\begin{array}{l}\frac{1}{\mathrm{\Gamma }(\mu )}{\displaystyle \underset{1}{\overset{\infty }{\int }}{(\upsilon -1)}^{\mu -1}{\upsilon }^{-(\tau +\mu )}}\omega (\zeta {\upsilon }^{1/\partial })d\upsilon ,\hspace{0.17em}\hspace{0.17em}\mu >0,\\ \omega (\zeta )\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mu =0.\end{array}\right.\hspace{0.17em}\hspace{0.17em}$$

(8)

3. Methodology

In this section, a detailed description of the Lie symmetry reduction method for a coupled time fractional-order system is given.

Let us assume the system of FPDEs is with fractional-order θ.

$$\left\{\begin{array}{l}{\partial }_t^{\theta }u=F_1(t,x,u,v,\hspace{0.17em}w,\hspace{0.17em}{u}_x,v_x,w_x,z_x,u_{xx},v_{xx},w_{xx},z_{xx}\dots );\hspace{0.17em}\\ {\partial }_t^{\theta }v=F_2(t,x,u,v,\hspace{0.17em}w,\hspace{0.17em}{u}_x,v_x,w_x,z_x,u_{xx},v_{xx},w_{xx},z_{xx}\dots );\\ {\partial }_t^{\theta }w=F_3(t,x,u,v,\hspace{0.17em}w,\hspace{0.17em}{u}_x,v_x,w_x,z_x,u_{xx},v_{xx},w_{xx},z_{xx}\dots );\\ {\partial }_t^{\theta }z=F_4(t,x,u,v,\hspace{0.17em}w,\hspace{0.17em}{u}_x,v_x,w_x,z_x,u_{xx},v_{xx},w_{xx},z_{xx}\dots );\hspace{0.17em}\hspace{0.17em}0<\theta <1.\end{array}\right.$$

(9)

The infinitesimal transformations with single-parametric notation in the fractional Lie symmetry analysis is expressed as:

$$\left\{\begin{array}{l}\overline{t}=\overline{\hspace{0.17em}t\hspace{0.17em}}(x,t,u,v,w,z;\epsilon )=t+\epsilon \tau (x,t,u,v,w,z)+o({\epsilon }^2);\\ \overline{x}=\overline{x}\hspace{0.17em}(x,t,u,v,w,z;\epsilon )=x+\epsilon \xi (x,t,u,v,w,z)+o({\epsilon }^2);\\ \overline{u}=\overline{u}\hspace{0.17em}(x,t,u,v,w,z;\epsilon )=u+\epsilon \eta (x,t,u,v,w,z)+o({\epsilon }^2);\\ \overline{v}=\overline{v}\hspace{0.17em}(x,t,u,v,w,z;\epsilon )=v+\epsilon \phi (x,t,u,v,w,z)+o({\epsilon }^2);\\ \overline{w}=\overline{w}\hspace{0.17em}(x,t,u,v,w,z;\epsilon )=w+\epsilon \mu (x,t,u,v,w,z)+o({\epsilon }^2);\\ \overline{z}=\overline{z}(x,t,u,v,w,z;\epsilon )=z+\epsilon \mu (x,t,u,v,w,z)+o({\epsilon }^2).\end{array}\right.$$

(10)

The vector field which generates the infinitesimal transformation is taken as:

$$X=\tau \hspace{0.17em}{\partial }_t+\xi \hspace{0.17em}{\partial }_x+{\eta }_1\hspace{0.17em}{\partial }_u+{\eta }_2\hspace{0.17em}{\partial }_v\hspace{0.17em}+{\eta }_3\hspace{0.17em}{\partial }_w\hspace{0.17em}+{\eta }_4\hspace{0.17em}{\partial }_z$$

(11)

with $\tau ={\left.\frac{d\overline{t}}{d\epsilon }\right|}_{\epsilon =0},\hspace{0.17em}\hspace{0.17em}\xi ={\left.\frac{d\overline{x}}{d\epsilon }\right|}_{\epsilon =0}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{\eta }_1={\left.\frac{d\overline{u}}{d\epsilon }\right|}_{\epsilon =0},{\eta }_2={\left.\frac{d\overline{v}}{d\epsilon }\right|}_{\epsilon =0},{\eta }_3={\left.\frac{dw}{d\epsilon }\right|}_{\epsilon =0},{\eta }_4={\left.\frac{dz}{d\epsilon }\right|}_{\epsilon =0}.$ Here, ξ, τ, ${\eta }_1,{\eta }_2,{\eta }_3\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{\eta }_4$ are the obtained infinitesimal operators from (11); ${\eta }_1^{\theta ,t},{\eta }_2^{\theta ,t},{\eta }_3^{\theta ,t}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{\eta }_4^{\theta ,t}$ are the fractional extended infinitesimals of order θ; and ${\eta }_1^x,{\eta }_1^{xx},{\eta }_1^{xxx},{\eta }_2^x,{\eta }_2^{xx},{\eta }_2^{xxx},\hspace{0.17em}{\eta }_3^x,{\eta }_3^{xx},{\eta }_3^{xxx}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{\eta }_4^x,{\eta }_4^{xx},{\eta }_4^{xxx}$ are the extended infinitesimals of the described integer order.

$$\begin{array}{l}{\eta }_1^x=D_x({\eta }_1)-u_x{D}_x(\xi )-u_t{D}_x(\tau )\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\\ {\eta }_1^{xx}=D_x({\eta }_1^x)-u_{xx}{D}_x(\xi )-u_{xt}{D}_x(\tau )\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\\ {\eta }_1^{xxx}=D_x({\eta }_1^{xx})-u_{xxx}{D}_x(\xi )-u_{xxt}{D}_x(\tau )\\ {\eta }_2^x=D_x({\eta }_2)-v_x{D}_x(\xi )-v_t{D}_x(\tau )\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\\ {\eta }_2^{xx}=D_x({\eta }_2^x)-v_{xx}{D}_x(\xi )-v_{xt}{D}_x(\tau )\hspace{0.17em};\hspace{0.17em}\\ \hspace{0.17em}{\eta }_2^{xxx}=D_x({\eta }_2^{xx})-v_{xxx}{D}_x(\xi )-v_{xxt}{D}_x(\tau )\\ {\eta }_3^x=D_x({\eta }_3)-w_x{D}_x(\xi )-w_t{D}_x(\tau )\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\\ {\eta }_3^{xx}=D_x({\eta }_3^x)-w_{xx}{D}_x(\xi )-w_{xt}{D}_x(\tau )\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\hspace{0.17em}\\ {\eta }_3^{xxx}=D_x({\eta }_3^{xx})-w_{xxx}{D}_x(\xi )-w_{xxt}{D}_x(\tau )\\ {\eta }_4^x=D_x({\eta }_4)-z_x{D}_x(\xi )-z_t{D}_x(\tau )\hspace{0.17em}\hspace{0.17em};\hspace{0.17em}\\ {\eta }_4^{xx}=D_x({\eta }_4^x)-z_{xx}{D}_x(\xi )-z_{xt}{D}_x(\tau )\hspace{0.17em};\hspace{0.17em}\\ \hspace{0.17em}{\eta }_4^{xxx}=D_x({\eta }_4^{xx})-z_{xxx}{D}_x(\xi )-z_{xxt}{D}_x(\tau ),\end{array}$$

(12)

where D_x is the total derivative operator, defined as:

$$D_x=\frac{\partial }{\partial x}+u_x\frac{\partial }{\partial u}+u_{xx}\frac{\partial }{\partial u_{xx}}+\dots +v_x\frac{\partial }{\partial v_x}+v_{xx}\frac{\partial }{\partial v_{xx}}+.....+w_x\frac{\partial }{\partial w_x}+w_{xx}\frac{\partial }{\partial w_{xx}}+....$$

(13)

The extended infinitesimal function of θ-th order ${\eta }_1{}^{\theta ,\hspace{0.17em}t}$ involved in the R-L fractional derivative is described by:

$${\eta }_1{}^{\theta ,\hspace{0.17em}t}=D_t^{\theta }({\eta }_1)+\xi .D_t^{\theta }(u_x)-D_t^{\theta }(\xi \hspace{0.17em}{u}_x)+D_t^{\theta }(D_t(\tau )u)-D_t^{\theta +1}(\tau \hspace{0.17em}u)+\tau .D_t^{\theta +1}(u).$$

(14)

Additionally, $D_t^{\theta +1}(f(t))=D_t^{\theta }(D_t(f(t))$; then, the above expression is simplified to

$${\eta }_1{}^{\theta ,t}=D_t^{\theta }({\eta }_1)+\xi .D_t^{\theta }(u_x)-D_t^{\theta }(\xi \hspace{0.17em}{u}_x)+\tau .D_t^{\theta }(u)-D_t^{\theta }(\tau \hspace{0.17em}{u}_t).$$

(15)

Applying the generalized Leibnitz rule on (15), we obtain

$${\eta }_1{}^{\theta ,t}=D_t^{\theta }({\eta }_1)-\theta .D_t^{\theta }(\tau )\frac{{\partial }^{\theta }u}{\partial t^{\theta }}-{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ n\end{array}\right)}{D}_t^{\lambda }(\xi )D_t^{\theta -\lambda }(u_x)-{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda +1\end{array}\right)}{D}_t^{\lambda +1}(\tau )D_t^{\theta -\lambda }(u).$$

(16)

Using the generalized Leibnitz rule and chain rule (5), the term $D_t^{\theta }(\eta )$ in (16) can be defined as

$$\begin{array}{l}{D}_t^{\theta }({\eta }_1)=\frac{{\partial }^{\theta }{\eta }_1}{\partial t^{\theta }}+\left({\eta }_{1u}\frac{{\partial }^{\theta }u}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{1u})}{\partial t^{\theta }}\right)+\left({\eta }_{1v}\frac{{\partial }^{\theta }v}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{1v})}{\partial t^{\theta }}\right)+\left({\eta }_{1w}\frac{{\partial }^{\theta }w}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{1w})}{\partial t^{\theta }}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left({\eta }_{1z}\frac{{\partial }^{\theta }z}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{1z})}{\partial t^{\theta }}\right)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^n({\eta }_{1u})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(u)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{1v})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(v)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{1w})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(w)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{1z})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(z)+{\sigma }_1+{\sigma }_2+{\sigma }_3+{\sigma }_4,\end{array}$$

(17)

where

$$\left\{\begin{array}{l}{\sigma }_1={\displaystyle \sum _{\lambda =2}^{\infty }{\displaystyle \sum _{m=2}^{\lambda }{\displaystyle \sum _{k=2}^m{\displaystyle \sum _{r=0}^{k-1}\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\left(\begin{array}{c}\lambda \\ m\end{array}\right)\left(\begin{array}{c}k\\ r\end{array}\right)}\frac{t^{\lambda -\theta }}{k!\mathrm{\Gamma }(\lambda +1-\theta )}}}}{(-u)}^r\frac{{\partial }^m}{\partial t^m}(u^{k-r})\frac{{\partial }^{\lambda -m+k}{\eta }_1}{\partial t^{\lambda -m}\partial u^k}\hspace{0.17em}\hspace{0.17em}\\ {\sigma }_2={\displaystyle \sum _{\lambda =2}^{\infty }{\displaystyle \sum _{m=2}^{\lambda }{\displaystyle \sum _{k=2}^m{\displaystyle \sum _{r=0}^{k-1}\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\left(\begin{array}{c}\lambda \\ m\end{array}\right)\left(\begin{array}{c}k\\ r\end{array}\right)}\frac{t^{\lambda -\theta }}{k!\mathrm{\Gamma }(\lambda +1-\theta )}}}}{(-v)}^r\frac{\partial t^m}{{\partial t}^m}(v^{k-r})\frac{{\partial }^{\lambda -m+k}{\eta }_1}{\partial t^{\lambda -m}\partial v^k}\\ {\sigma }_3={\displaystyle \sum _{\lambda =2}^{\infty }{\displaystyle \sum _{m=2}^{\lambda }{\displaystyle \sum _{k=2}^m{\displaystyle \sum _{r=0}^{k-1}\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\left(\begin{array}{c}\lambda \\ m\end{array}\right)\left(\begin{array}{c}k\\ r\end{array}\right)}\frac{t^{\lambda -\theta }}{k!\mathrm{\Gamma }(\lambda +1-\theta )}}}}{(-w)}^r\frac{{\partial }^m}{\partial t^m}(w^{k-r})\frac{{\partial }^{\lambda -m+k}{\eta }_1}{\partial t^{\lambda -m}\partial w^k}\\ {\sigma }_4={\displaystyle \sum _{\lambda =2}^{\infty }{\displaystyle \sum _{m=2}^{\lambda }{\displaystyle \sum _{k=2}^m{\displaystyle \sum _{r=0}^{k-1}\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\left(\begin{array}{c}\lambda \\ m\end{array}\right)\left(\begin{array}{c}k\\ r\end{array}\right)}\frac{t^{\lambda -\theta }}{k!\mathrm{\Gamma }(\lambda +1-\theta )}}}}{(-z)}^r\frac{{\partial }^m}{\partial t^m}(z^{k-r})\frac{{\partial }^{\lambda -m+k}{\eta }_1}{\partial t^{\lambda -m}\partial z^k}\end{array}\right..$$

(18)

Finally, the expression for the θ-th order extended infinitesimal ${\eta }_1{}^{\theta ,t}$ of the form

$$\begin{array}{l}{\eta }_1{}^{\theta ,t}=\frac{{\partial }^{\theta }{\eta }_1}{\partial t^{\theta }}+\left({\eta }_{1u}-\theta \hspace{0.17em}{D}_t(\tau )\right)\frac{{\partial }^{\theta }u}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{1u})}{\partial t^{\theta }}+\left({\eta }_{1v}\frac{{\partial }^{\theta }v}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{1v})}{\partial t^{\theta }}\right)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left({\eta }_{1w}\frac{{\partial }^{\theta }w}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{1w})}{\partial t^{\theta }}\right)+\hspace{0.17em}\left({\eta }_{1z}\frac{{\partial }^{\theta }z}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{1z})}{\partial t^{\theta }}\right)\hspace{0.17em}+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\hspace{0.17em}\hspace{0.17em}\left[\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\right.}\frac{{\partial }^n{\eta }_{1u}}{\partial t^n}-\left(\begin{array}{c}\theta \\ \lambda +1\end{array}\right)D\left.{}_t^{\lambda +1}(\tau )\right]D_t^{\theta -\lambda }(u)+\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{1v})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(v)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{1w})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(w)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{1z})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(z)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}{D}_t^{\lambda }(\xi )D_t^{\theta -\lambda }(u_x)+{\sigma }_1+{\sigma }_2+{\sigma }_3+{\sigma }_4.\end{array}$$

(19)

Similarly, expressions for ${\eta }_2{}^{\theta ,t}$ ${\eta }_3{}^{\theta ,t}$ and ${\eta }_4{}^{\theta ,t}$ are also obtained.

$$\begin{array}{l}{\eta }_2{}^{\theta ,t}=\frac{{\partial }^{\theta }{\eta }_2}{\partial t^{\theta }}+\left({\eta }_{2v}-\theta \hspace{0.17em}{D}_t(\tau )\right)\frac{{\partial }^{\theta }v}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{2v})}{\partial t^{\theta }}+\left({\eta }_{2u}\frac{{\partial }^{\theta }u}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{2u})}{\partial t^{\theta }}\right)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left({\eta }_{2w}\frac{{\partial }^{\theta }w}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{2w})}{\partial t^{\theta }}\right)+\hspace{0.17em}\left({\eta }_{2z}\frac{{\partial }^{\theta }z}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{2z})}{\partial t^{\theta }}\right)\hspace{0.17em}+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\hspace{0.17em}\hspace{0.17em}\left[\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\right.}\frac{{\partial }^n{\eta }_{2v}}{\partial t^n}-\left(\begin{array}{c}\theta \\ \lambda +1\end{array}\right)D\left.{}_t^{\lambda +1}(\tau )\right]D_t^{\theta -\lambda }(v)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{2u})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(v)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{2w})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(w)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{2z})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(z)-\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}{D}_t^{\lambda }(\xi )D_t^{\theta -\lambda }(v_x)+{\sigma }_5+{\sigma }_6+{\sigma }_7+{\sigma }_8.\end{array}$$

(20)

$$\begin{array}{l}{\eta }_3{}^{\theta ,t}=\frac{{\partial }^{\theta }{\eta }_3}{\partial t^{\theta }}+\left({\eta }_{3w}-\theta \hspace{0.17em}{D}_t(\tau )\right)\frac{{\partial }^{\theta }w}{\partial t^{\theta }}-w\frac{{\partial }^{\theta }({\eta }_{3w})}{\partial t^{\theta }}+\left({\eta }_{3u}\frac{{\partial }^{\theta }u}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{3u})}{\partial t^{\theta }}\right)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left({\eta }_{3v}\frac{{\partial }^{\theta }v}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{3v})}{\partial t^{\theta }}\right)+\hspace{0.17em}\left({\eta }_{3z}\frac{{\partial }^{\theta }z}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{3z})}{\partial t^{\theta }}\right)\hspace{0.17em}+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\hspace{0.17em}\hspace{0.17em}\left[\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\right.}\frac{{\partial }^n{\eta }_{3w}}{\partial t^n}-\left(\begin{array}{c}\theta \\ \lambda +1\end{array}\right)D\left.{}_t^{\lambda +1}(\tau )\right]D_t^{\theta -\lambda }(w)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{3u})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(w)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{3v})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(v)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{3z})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(z)-\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}{D}_t^{\lambda }(\xi )D_t^{\theta -\lambda }(w_x)+{\sigma }_9+{\sigma }_{10}+{\sigma }_{11}+{\sigma }_{12}.\end{array}$$

(21)

and

$$\begin{array}{l}{\eta }_4{}^{\theta ,t}=\frac{{\partial }^{\theta }{\eta }_4}{\partial t^{\theta }}+\left({\eta }_{4z}-\theta \hspace{0.17em}{D}_t(\tau )\right)\frac{{\partial }^{\theta }z}{\partial t^{\theta }}-z\frac{{\partial }^{\theta }({\eta }_{4z})}{\partial t^{\theta }}+\left({\eta }_{4u}\frac{{\partial }^{\theta }u}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{4u})}{\partial t^{\theta }}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left({\eta }_{4w}\frac{{\partial }^{\theta }w}{\partial t^{\theta }}-u\frac{{\partial }^{\theta }({\eta }_{4w})}{\partial t^{\theta }}\right)+\hspace{0.17em}\left({\eta }_{4v}\frac{{\partial }^{\theta }z}{\partial t^{\theta }}-v\frac{{\partial }^{\theta }({\eta }_{4z})}{\partial t^{\theta }}\right)\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{\lambda =1}^{\infty }\hspace{0.17em}\hspace{0.17em}\left[\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\right.}\frac{{\partial }^n{\eta }_{4z}}{\partial t^n}-\left(\begin{array}{c}\theta \\ \lambda +1\end{array}\right)D\left.{}_t^{\lambda +1}(\tau )\right]D_t^{\theta -\lambda }(z)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{4u})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(u)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{4w})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(w)+{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}\frac{{\partial }^{\lambda }({\eta }_{4v})}{\partial t^{\lambda }}{D}_t^{\theta -\lambda }(v)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \sum _{\lambda =1}^{\infty }\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)}{D}_t^{\lambda }(\xi )D_t^{\theta -\lambda }(z_x)+{\sigma }_{13}+{\sigma }_{14}+{\sigma }_{15}+{\sigma }_{16}.\end{array}$$

(22)

$${\sigma }_i={\displaystyle \sum _{\lambda =2}^{\infty }{\displaystyle \sum _{m=2}^{\lambda }{\displaystyle \sum _{k=2}^m{\displaystyle \sum _{r=0}^{k-1}\left(\begin{array}{c}\theta \\ \lambda \end{array}\right)\left(\begin{array}{c}\lambda \\ m\end{array}\right)\left(\begin{array}{c}k\\ r\end{array}\right)}\frac{t^{\lambda -\theta }}{k!\mathrm{\Gamma }(\lambda +1-\theta )}}}}{(-u)}^r\frac{{\partial }^m}{\partial t^m}(u^{k-r})\frac{{\partial }^{\lambda -m+k}\eta }{\partial t^{\lambda -m}\partial u^k}\hspace{0.17em}.$$

(23)

For expressions ${\sigma }_i$; i = 5, 6, 7, …, 16 given by (23) vanishes.

4. Lie Symmetry Reduction in Nonlinear System of Hirota Equations

Applying prolongation on the set of Hirota systems of FPDEs (2), we obtain

$$\left\{\begin{array}{l}{\eta }_1^{\theta ,t}+{\eta }_1^{xxx}+6{\eta }_1^x\left({\displaystyle \sum u^2}\right)+12u_x\left(u{\eta }_1+v{\eta }_2+w{\eta }_3+z{\eta }_4\right)=0;\\ {\eta }_2^{\theta ,t}+{\eta }_2^{xxx}+6{\eta }_2^x\left({\displaystyle \sum u^2}\right)+12v_x\left(u{\eta }_1+v{\eta }_2+w{\eta }_3+z{\eta }_4\right)=0;\\ {\eta }_3^{\theta ,t}+{\eta }_3^{xxx}+6{\eta }_3^x\left({\displaystyle \sum u^2}\right)+12w_x\left(u{\eta }_1+v{\eta }_2+w{\eta }_3+z{\eta }_4\right)=0;\\ {\eta }_4^{\theta ,t}+{\eta }_4^{xxx}+6{\eta }_4^x\left({\displaystyle \sum u^2}\right)+12z_x\left(u{\eta }_1+v{\eta }_2+w{\eta }_3+z{\eta }_4\right)=0.\end{array}\right.$$

(24)

Substituting the values of extended infinitesimals and solving the system of the obtained fractional PDEs, we have the following infinitesimals:

$$\begin{array}{l}\xi =\frac{p_{1}x}{3}+p_2,\hspace{0.17em}\hspace{0.17em}\tau =\frac{p_{1}t}{\theta },\hspace{0.17em}\hspace{0.17em}{\eta }_1=\frac{-p_{1}u}{3}+p_{3}v+p_{4}w+p_{5}z,\hspace{0.17em}\hspace{0.17em}\\ {\eta }_2=-p_{3}u-\frac{p_{1}v}{3}+p_{6}w+p_{7}z\\ {\eta }_3=-p_{4}u-p_{6}v-\frac{p_{1}w}{3}+p_{8}z,\hspace{0.17em}\\ {\eta }_4=-p_{5}u-p_{7}v-p_{8}w-\frac{p_{1}z}{3},\end{array}$$

(25)

where $p_i(i=1\hspace{0.17em},2,\mathrm{...8})$ are components of the standard basis of a vector field

$$X_i=\xi {\partial }_x+\tau {\partial }_t+{\eta }_1{\partial }_u+{\eta }_2{\partial }_v+{\eta }_3{\partial }_w+{\eta }_4{\partial }_z.$$

(26)

Lie algebra has the following vectors

$$\begin{array}{l}{X}_1=\frac{x}{3}{\partial }_x+\frac{t}{\theta }{\partial }_t-\frac{u}{3}{\partial }_u-\frac{v}{3}{\partial }_v-\frac{w}{3}{\partial }_w-\frac{z}{3}{\partial }_z,\hspace{0.17em}\hspace{0.17em}\\ X_2={\partial }_x,\hspace{0.17em}\hspace{0.17em}{X}_3=v{\partial }_u-u{\partial }_v,\hspace{0.17em}\hspace{0.17em}{X}_4=w{\partial }_u-u{\partial }_w\\ X_5=z{\partial }_u-u{\partial }_z,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{X}_6=w{\partial }_v-v{\partial }_w,\hspace{0.17em}\\ X_7=z{\partial }_v-v{\partial }_z,\hspace{0.17em}\hspace{0.17em}{X}_8=z{\partial }_w-w{\partial }_z.\end{array}$$

(27)

Here, the optimal solution of a system of equations is evaluated by choosing the first infinitesimal generator with the help of a characteristic equation. These are:

$$\frac{3dx}{x}=\frac{\theta dt}{t}=\frac{-3du}{u}=\frac{-3dv}{v}=\frac{-3dw}{w}=\frac{-3dz}{z}.$$

(28)

The explicit solutions obtained with ‘α’ as the similarity variable are given by:

$$u=t^{-\theta /3}f(\alpha ),\hspace{0.17em}\hspace{0.17em}v=t^{-\theta /3}g(\alpha ),\hspace{0.17em}\hspace{0.17em}w=t^{-\theta /3}h(\alpha ),\hspace{0.17em}\hspace{0.17em}z=t^{-\theta /3}k(\alpha )\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\alpha =x{t}^{-\theta /3}.$$

(29)

5. Application of EK Differ-Integral Operators

Applying the Riemann–Liouville fractional derivative, we obtain:

$$\begin{array}{l}{\partial }_t^{\theta }u={\partial }_t^{\lambda }\left[\frac{1}{\mathrm{\Gamma }(\lambda -\theta )}{\displaystyle \underset{0}{\overset{t}{\int }}{(t-s)}^{\lambda -\theta -1}{s}^{-\theta /3}f(x{s}^{-\theta /3})ds}\right];\hspace{0.17em}\hspace{0.17em}q=t/s\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\partial }_t^{\lambda }\left[\frac{t^{\lambda -\frac{2\theta }{3}}}{\mathrm{\Gamma }(\lambda -\theta )}{\displaystyle \underset{1}{\overset{\infty }{\int }}{(q-1)}^{\lambda -\theta -1}{q}^{-\left(\lambda -\frac{2\theta }{3}+1\right)}f(\alpha q^{\theta /3})dq}\right].\end{array}$$

(30)

Applying E-K integral operator in (30), we get

$${\partial }_t^{\theta }u={\partial }_t^{\lambda }\left[t^{\lambda -\frac{2\theta }{3}}\left({\kappa }_{\frac{3}{\theta }}^{1-\frac{\theta }{3},\hspace{0.17em}\lambda -\theta }f\right)\alpha \right].$$

(31)

Additionally, using $t{\partial }_{t}f(\alpha )=tx\left(-\frac{\theta }{3}\right)t^{-\frac{\theta }{3}-1}f{\hspace{0.17em}}^{\prime }\hspace{0.17em}(\alpha )\hspace{0.17em}=-\frac{\theta }{3}\alpha \frac{d}{d\alpha }(f(\alpha ))$ in (31), we obtain

$$\begin{array}{l}{\partial }_t^{\theta }u={\partial }_t^{\lambda }\left({\partial }_t\left[t^{\lambda -\frac{2\theta }{3}}\left({\kappa }_{\frac{3}{\theta }}^{1-\frac{\theta }{3},\hspace{0.17em}\lambda -\theta }f\right)\alpha \right]\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\partial }_t^{\lambda -1}\left[t^{\lambda -\frac{2\theta }{3}-1}\left(\lambda -\frac{4\theta }{3}-\frac{\theta }{3}\alpha \frac{d}{d\alpha }\right)\left({\kappa }_{\frac{3}{\theta }}^{1-\frac{\theta }{3},\hspace{0.17em}\lambda -\theta }f\right)\alpha \right].\end{array}$$

(32)

Repeating the above steps λ − 1 times and applying E-K differential operator, we can obtain.

$${\partial }_t^{\theta }u=t^{-\frac{2\theta }{3}}\left(P_{\frac{3}{\theta }}^{1-\frac{4\theta }{3},\hspace{0.17em}\theta }f\right)\alpha .$$

(33)

System of FPDEs is reduced to the following system of FODEs:

$$\left\{\begin{array}{l}\left(P_{\frac{3}{\theta }}^{1-\frac{4\theta }{3},\hspace{0.17em}\theta }f\right)\alpha +f{\hspace{0.17em}}^{\prime \prime \prime }(\alpha )+6f{\hspace{0.17em}}^{\prime }(\alpha )\left({\displaystyle \sum f^2}\right)=0;\\ \left(P_{\frac{3}{\theta }}^{1-\frac{4\theta }{3},\hspace{0.17em}\theta }g\right)\alpha +g{\hspace{0.17em}}^{\prime \prime \prime }(\alpha )+6g{\hspace{0.17em}}^{\prime }(\alpha )\left({\displaystyle \sum f^2}\right)=0;\\ \left(P_{\frac{3}{\theta }}^{1-\frac{4\theta }{3},\hspace{0.17em}\theta }h\right)\alpha +h{\hspace{0.17em}}^{\prime \prime \prime }(\alpha )+6h{\hspace{0.17em}}^{\prime }(\alpha )\left({\displaystyle \sum f^2}\right)=0;\\ \left(P_{\frac{3}{\theta }}^{1-\frac{4\theta }{3},\hspace{0.17em}\theta }k\right)\alpha +k{\hspace{0.17em}}^{\prime \prime \prime }(\alpha )+6k{\hspace{0.17em}}^{\prime }(\alpha )\left({\displaystyle \sum f^2}\right)=0,\end{array}\right.$$

(34)

where ${\displaystyle \sum f^2=f^2+g^2+h^2+k^2}$.

6. Power Series Solution of the System and Its Convergence

Let us us assume the power series solution is

$$f(\alpha )={\displaystyle \sum _{n=1}^{\infty }{a}_n{\alpha }^n},\hspace{0.17em}\hspace{0.17em}g(\alpha )={\displaystyle \sum _{n=1}^{\infty }{b}_n{\alpha }^n},\hspace{0.17em}\hspace{0.17em}h(\alpha )={\displaystyle \sum _{n=1}^{\infty }{c}_n{\alpha }^n,\hspace{0.17em}}k(\alpha )={\displaystyle \sum _{n=1}^{\infty }{d}_n{\alpha }^n}.$$

(35)

From above Equations (34) and (35), we obtain

$$\left\{\begin{array}{l}{\displaystyle \sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}}{a}_n{\alpha }^n+{\displaystyle \sum _{n=0}^{\infty }(n+3)(n+2)(n+1)a_{n+3}{\alpha }^n+}6\left({\displaystyle \sum _{n=0}^{\infty }(n+1)}{a}_{n+1}{\alpha }^n\right)\left[S\right]=0;\\ {\displaystyle \sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}}{b}_n{\alpha }^n+{\displaystyle \sum _{n=0}^{\infty }(n+3)(n+2)(n+1)b_{n+3}{\alpha }^n+}6\left({\displaystyle \sum _{n=0}^{\infty }(n+1)}{b}_{n+1}{\alpha }^n\right)\left[S\right]=0;\\ {\displaystyle \sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}}{c}_n{\alpha }^n+{\displaystyle \sum _{n=0}^{\infty }(n+3)(n+2)(n+1)c_{n+3}{\alpha }^n+}6\left({\displaystyle \sum _{n=0}^{\infty }(n+1)}{c}_{n+1}{\alpha }^n\right)\left[S\right]=0;\\ {\displaystyle \sum _{n=0}^{\infty }\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}}{d}_n{\alpha }^n+{\displaystyle \sum _{n=0}^{\infty }(n+3)(n+2)(n+1)d_{n+3}{\alpha }^n+}6\left({\displaystyle \sum _{n=0}^{\infty }(n+1)}{d}_{n+1}{\alpha }^n\right)\left[S\right]=0,\end{array}\right.$$

(36)

$$\mathrm{where} S=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \sum _{n=1}^{\infty }{a}_n{\alpha }^n}\right)}^2+{\left({\displaystyle \sum _{n=1}^{\infty }{b}_n{\alpha }^n}\right)}^2+{\left({\displaystyle \sum _{n=1}^{\infty }{c}_n{\alpha }^n}\right)}^2+{\left({\displaystyle \sum _{n=1}^{\infty }{d}_n{\alpha }^n}\right)}^2$$

(37)

Substituting n = 0 in (36), we get

$$\begin{array}{l}\left\{\begin{array}{l}{a}_3=-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{a}_0-a_1{S}_0;\\ b_3=-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{b}_0-b_1{S}_0;\\ c_3=-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{c}_0-c_1{S}_0;\\ d_3=-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{d}_0-d_1{S}_0,\end{array}\right.\\ \mathrm{where} S_0=a_0{}^2+b_0{}^2+c_0{}^2+d_0{}^2,\end{array}$$

(38)

and

$$\left\{\begin{array}{l}{a}_{n+3}=\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}{a}_n-\frac{6}{(n+3)(n+2)}{\displaystyle \sum _{k=0}^n{a}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)};\\ b_{n+3}=\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}{b}_n-\frac{6}{(n+3)(n+2)}{\displaystyle \sum _{k=0}^n{b}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)};\\ c_{n+3}=\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}{c}_n-\frac{6}{(n+3)(n+2)}{\displaystyle \sum _{k=0}^n{c}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)};\\ d_{n+3}=\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}{d}_n-\frac{6}{(n+3)(n+2)}{\displaystyle \sum _{k=0}^n{d}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)}.\end{array}\right.$$

(39)

So explicit power series solution of system is given by:

$$\left\{\begin{array}{l}f(\alpha )=a_0+a_1\left(x{t}^{-\theta /3}\right)+a_2{\left(x{t}^{-\theta /3}\right)}^2+\left[-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{a}_0-a_1{S}_0\right]{\left(x{t}^{-\theta /3}\right)}^3+{\displaystyle \sum _{n=1}^{\infty }{a}_{n+3}{\left(x{t}^{-\theta /3}\right)}^{n+3}}\hspace{0.17em}\hspace{0.17em}\\ g(\alpha )=b_0+b_1\left(x{t}^{-\theta /3}\right)+b_2{\left(x{t}^{-\theta /3}\right)}^2+\left[-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{b}_0-b_1{S}_0\right]{\left(x{t}^{-\theta /3}\right)}^3+{\displaystyle \sum _{n=1}^{\infty }{b}_{n+3}{\left(x{t}^{-\theta /3}\right)}^{n+3}}\\ h(\alpha )=c_0+c_1\left(x{t}^{-\theta /3}\right)+c_2{\left(x{t}^{-\theta /3}\right)}^2+\left[-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{c}_0-c_1{S}_0\right]{\left(x{t}^{-\theta /3}\right)}^3+{\displaystyle \sum _{n=1}^{\infty }{c}_{n+3}{\left(x{t}^{-\theta /3}\right)}^{n+3}}\\ k(\alpha )=d_0+d_1\left(x{t}^{-\theta /3}\right)+d_2{\left(x{t}^{-\theta /3}\right)}^2+\left[-\frac{\mathrm{\Gamma }(2-4\theta /3)}{3!\mathrm{\Gamma }(2-\theta /3)}{d}_0-d_1{S}_0\right]{\left(x{t}^{-\theta /3}\right)}^3+{\displaystyle \sum _{n=1}^{\infty }{d}_{n+3}{\left(x{t}^{-\theta /3}\right)}^{n+3}}\end{array}\right.$$

(40)

$$\left\{\begin{array}{l}|a_{n+3}|=\left|\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}\right||a_n|-\left|\frac{6}{(n+3)(n+2)}\right|\left|{\displaystyle \sum _{k=0}^n{a}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)}\right|;\\ |b_{n+3}|=\left|\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}\right||b_n|-\left|\frac{6}{(n+3)(n+2)}\right|\left|{\displaystyle \sum _{k=0}^n{b}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)}\right|;\\ |c_{n+3}|=\left|\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}\right||c_n|-\left|\frac{6}{(n+3)(n+2)}\right|\left|{\displaystyle \sum _{k=0}^n{c}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)}\right|;\\ |d_{n+3}|=\left|\frac{n\hspace{0.17em}!}{(n+3)!}\frac{\mathrm{\Gamma }\left(2-\frac{4\theta }{3}+\frac{n\theta }{3}\right)}{\mathrm{\Gamma }\left(2-\frac{\theta }{3}+\frac{n\theta }{3}\right)}\right||d_n|-\left|\frac{6}{(n+3)(n+2)}\right|\left|{\displaystyle \sum _{k=0}^n{d}_{n+1-k}\left(a_{n-k}^2+b_{n-k}^2+c_{n-k}^2+d_{n-k}^2\right)}\right|.\end{array}\right.$$

(41)

For n < m, $\left|\frac{\mathrm{\Gamma }(n)}{\mathrm{\Gamma }(m)}\right|<1$, and ‘M’ is maximum of the arbitrary coefficients involved in the set of equations with the majorant series.

$$U(\chi )={\displaystyle \sum _{n=0}^{\infty }{p}_n{\chi }^n}\hspace{0.17em};\hspace{0.17em}V(\chi )={\displaystyle \sum _{n=0}^{\infty }{q}_n{\chi }^n}\hspace{0.17em};\hspace{0.17em}W(\chi )={\displaystyle \sum _{n=0}^{\infty }{r}_n{\chi }^n}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Y(\chi )={\displaystyle \sum _{n=0}^{\infty }{s}_n{\chi }^n},$$

(42)

p_n = |a_n|, q_n = |b_n|, r_n = |c_n|, s_n = |d_n|, n = 1, 2, 3, …, then, we can have

$$\left\{\begin{array}{l}{p}_{n+3}=M\left[p_n+{\displaystyle \sum _{k=0}^n{p}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right];\\ q_{n+3}=M\left[q_n+{\displaystyle \sum _{k=0}^n{q}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right];\\ r_{n+3}=M\left[r_n+{\displaystyle \sum _{k=0}^n{r}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right];\\ s_{n+3}=M\left[s_n+{\displaystyle \sum _{k=0}^n{s}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right].\end{array}\right.$$

(43)

Assuming the implicit functions of system with variable χ.

$$\begin{array}{l}{U}_1(\chi ,U)=U(\chi )-p_0-p_1\chi -p_2{\chi }^2-p_3{\chi }^3-M\left[p_n+{\displaystyle \sum _{k=0}^n{p}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right];\\ V_1(\chi ,V)=V(\chi )-q_0-q_1\chi -q_2{\chi }^2-q_3{\chi }^3-M\left[q_n+{\displaystyle \sum _{k=0}^n{q}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right];\\ W_1(\chi ,W)=W(\chi )-r_0-r_1\chi -r_2{\chi }^2-r_3{\chi }^3-M\left[r_n+{\displaystyle \sum _{k=0}^n{r}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right];\\ Y_1(\chi ,Y)=Y(\chi )-s_0-s_1\chi -s_2{\chi }^2-s_3{\chi }^3-M\left[s_n+{\displaystyle \sum _{k=0}^n{s}_{n+1-k}\left(p_{n-k}^2+q_{n-k}^2+r_{n-k}^2+s_{n-k}^2\right)}\right].\end{array}$$

(44)

Here, U₁, V₁, W₁ and Y₁ are analytic in a neighborhood of $(0,p_0),\hspace{0.17em}(0,q_0),(0,r_0)$ and $(0,s_0)$ respectively, where $U_1(0,\hspace{0.17em}{p}_0)=0$, $V_1(0,\hspace{0.17em}{q}_0)=0$, $W_1(0,\hspace{0.17em}{r}_0)=0$ and $Y_1(0,s_0)=0$ with $\frac{\partial }{\partial U}(U_1(0,p_0))\ne 0$ $\frac{\partial }{\partial V}(V_1(0,q_0))\ne 0$ $\hspace{0.17em}\frac{\partial }{\partial W}(W_1(0,r_0))\ne 0$ and $\hspace{0.17em}\frac{\partial }{\partial Y}(Y_1(0,s_0))\ne 0$ then, with Implicit function theorem [40,50], we secured the convergence of power series solution.

7. Conservation Laws

In this section, we devise conservation laws of the given nonlinear complex Hirota system of partial differential equations. Noether’s theorem [31,62] is employed to establish the conservation laws and it states that every law of the physical system is invariant under time, space translation and orientation. The stability, existence and uniqueness of the fractional-order and classical systems can be achieved with the construction of conservation laws or conserved quantities. The extension of conservation laws [23,25,44] from classical order to local order systems has been performed successfully.

Transport equation or continuity equation is used to define fluid flow problems in fluid dynamics for the conservation of mass of the physical system, electromagnetic fields for conservation law of charges. The conservation of probability can be obtained in quantum mechanics by the use of Equation (45).

The adjoint system of conserved quantities with respect to time and space are ${\omega }^t$ and ${\omega }^x$, which follows the continuity equation given by

$$D_t({\omega }^t)+D_x({\omega }^x)=0.$$

(45)

The Lagrangian of system of FPDEs is given in the form

$$\begin{array}{l}\ell =P\left({\partial }_t^{\theta }u+{\partial }_x^{3}u+6{\partial }_{x}u\left({\displaystyle \sum u^2}\right)\right)+Q\left({\partial }_t^{\theta }v+{\partial }_x^{3}v+6{\partial }_{x}v\left({\displaystyle \sum u^2}\right)\right)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}R\left({\partial }_t^{\theta }w+{\partial }_x^{3}w+6{\partial }_{x}w\left({\displaystyle \sum u^2}\right)\right)+S\left({\partial }_t^{\theta }z+{\partial }_x^{3}z+6{\partial }_{x}z\left({\displaystyle \sum u^2}\right)\right).\end{array}$$

(46)

Euler-Lagrange’s (E-L) equations or adjoint equations are described as:

$$\frac{\delta \hspace{0.17em}\ell }{\delta \hspace{0.17em}u}=\frac{\delta \hspace{0.17em}\ell }{\delta \hspace{0.17em}v}=\frac{\delta \hspace{0.17em}\ell }{\delta \hspace{0.17em}w}=\frac{\delta \hspace{0.17em}\ell }{\delta \hspace{0.17em}z}=0.$$

(47)

With E-L operators

$$\frac{\delta }{\delta u^i}=\frac{\partial }{\partial u^i}+{(D_t^{\theta })}^{\ast }\frac{\partial }{\partial D_t^{\theta }{u}^i}-D_x\left(\frac{\partial }{\partial u^i{}_x}\right)+D_{xx}\left(\frac{\partial }{\partial u^i{}_{xx}}\right)-D_{xxx}\left(\frac{\partial }{\partial u^i{}_{xxx}}\right).$$

(48)

Additionally, ${(D_t^{\theta })}^{\ast }$ represents the adjoint operator to $D_t^{\theta }$, which is defined as:

$${(D_t^{\theta })}^{\ast }={}_t{}^{C}D{}_T^{\theta }u=\frac{{(-1)}^n}{\mathrm{\Gamma }(n-\theta )}{\displaystyle \underset{t}{\overset{T}{\int }}{(\upsilon -t)}^{n-1-\theta }{D}_{\upsilon }^{\theta }u(\upsilon ,x)d\upsilon ;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=[\theta ]+1}.$$

(49)

The adjoint system of equations formed by system of FPDEs can be given as:

$$\begin{array}{l}\left\{\begin{array}{l}(D_t^{\theta }){ }^{\ast }P-P_{xxx}+6P_x\left({\displaystyle \sum u^2}\right)-12P\left({\displaystyle \sum u{u}_x}\right)=0;\\ (D_t^{\theta }){ }^{\ast }Q-Q_{xxx}+6Q_x\left({\displaystyle \sum u^2}\right)-12Q\left({\displaystyle \sum u{u}_x}\right)=0;\\ (D_t^{\theta }){ }^{\ast }R-R_{xxx}+6R_x\left({\displaystyle \sum u^2}\right)-12R\left({\displaystyle \sum u{u}_x}\right)=0;\\ (D_t^{\theta }){ }^{\ast }S-S_{xxx}+6S_x\left({\displaystyle \sum u^2}\right)-12S\left({\displaystyle \sum u{u}_x}\right)=0,\end{array}\right.\\ \mathrm{where}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum u^2}=u^2+v^2+w^2+z^2,\\ \mathrm{and}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum u{u}_x=u{u}_x+v{v}_x+w{w}_x+z{z}_x}.\end{array}$$

(50)

The components of conserved quantities are ${\omega }^t$ and ${\omega }^x$ and expressed as

$$\begin{array}{l}{\omega }^x=\xi \ell +W_j\left[\frac{\partial \ell }{\partial u_x^j}-D_x\left(\frac{\partial \ell }{\partial u_{xx}^j}\right)+D_x^2\left(\frac{\partial \ell }{\partial u_{xxx}^j}\right)\right]+D_x(W_j)\left[\frac{\partial \ell }{\partial u_{xx}^j}-D_x\left(\frac{\partial \ell }{\partial u_{xxx}^j}\right)\right]+D_x^2(W_j)\left[\frac{\partial \ell }{\partial u_{xxx}^j}\right];\\ {\omega }^t=\tau \ell +D_t^{\theta -1}(W_j)\frac{\partial \ell }{\partial D_t^{\theta }{u}^j}+I\left(W_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{D}_t\frac{\partial \ell }{\partial D_t^{\theta }{u}^j}\right).\end{array}$$

(51)

Here, $W_j={\eta }_j-{\xi }_j{u}_x-{\tau }_j{u}_t$, $\ell$ is defined in (46) and ‘I’ is the integral given by:

$$I(f,g)=\frac{1}{\mathrm{\Gamma }(1-\theta )}{\displaystyle {\int }_0^t{\displaystyle {\int }_t^T\frac{f(s,x)g(\mu ,x)}{{(\mu -s)}^{\theta }}}}d\mu ds.$$

(52)

We operated the Lagrangian from Equation (46) into conserved quantities in Equation (51) with the simultaneous use of Equations (52) and (29) to obtain the cases mentioned below. Conserved quantities are the invariants from the calculated symmetries of a nonlinear system. To describe any physical system, it allows anyone to consider all classes of a considered Lagrangian under given invariants. There exist one-to-one and onto correspondence between the conserved quantities and Lie algebra of the vectors given by Equation (27). For the sake of understanding the procedure of the obtained cases 1–8, the reader may refer to [33,34,35,44,62].

Case 1:

For X₁, the Lie characteristic functions are $W_1=-\frac{u}{3}-\frac{x}{3}{u}_x-\frac{t}{\theta }{u}_t\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}{W}_2=-\frac{v}{3}-\frac{x}{3}{v}_x-\frac{t}{\theta }{v}_t.$ $W_3=-\frac{w}{3}-\frac{x}{3}{w}_x-\frac{t}{\theta }{w}_t\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}{W}_4=-\frac{z}{3}-\frac{x}{3}{z}_x-\frac{t}{\theta }{z}_t$ and the components of conserved vector are obtained as:

$$\begin{array}{l}{\omega }^x=\frac{x}{3}\ell +\left(-\frac{u}{3}-\frac{x}{3}{u}_x-\frac{t}{\theta }{u}_t\right)\left[6P\left({\displaystyle \sum u^2}\right)+P_{xx}\right]+P_x\left(\frac{2u_x}{3}+\frac{x{u}_{xx}}{3}+\frac{t{u}_{tx}}{\theta }\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}P\left(\frac{u_{xx}}{3}+\frac{x{u}_{xxx}}{3}-\frac{t{u}_{txx}}{\theta }\right)+\left(-\frac{v}{3}-\frac{x}{3}{v}_x-\frac{t}{\theta }{v}_t\right)\left[6Q\left({\displaystyle \sum u^2}\right)+Q_{xx}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{Q}_x\left(\frac{2v_x}{3}+\frac{x{v}_{xx}}{3}+\frac{t{v}_{tx}}{\theta }\right)-Q\left(\frac{v_{xx}}{3}+\frac{x{v}_{xxx}}{3}-\frac{t{v}_{txx}}{\theta }\right)+\left(-\frac{w}{3}-\frac{x}{3}{w}_x-\frac{t}{\theta }{w}_t\right)\left[6R\left({\displaystyle \sum u^2}\right)+R_{xx}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{R}_x\left(\frac{2w_x}{3}+\frac{x{w}_{xx}}{3}+\frac{t{w}_{tx}}{\theta }\right)-R\left(\frac{w_{xx}}{3}+\frac{x{w}_{xxx}}{3}-\frac{t{w}_{txx}}{\theta }\right)+\left(-\frac{z}{3}-\frac{x}{3}{z}_x-\frac{t}{\theta }{z}_t\right)\left[6S\left({\displaystyle \sum u^2}\right)+S_{xx}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+S_x\left(\frac{2z_x}{3}+\frac{x{z}_{xx}}{3}+\frac{t{z}_{tx}}{\theta }\right)-S\left(\frac{z_{xx}}{3}+\frac{x{z}_{xxx}}{3}-\frac{t{z}_{txx}}{\theta }\right),\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}$$

(53)

$$\begin{array}{l}{\omega }^t=P{D}_t^{\theta -1}\left(-\frac{u}{3}-\frac{x}{3}{u}_x-\frac{t}{\theta }{u}_t\right)+I\left(-\frac{u}{3}-\frac{x}{3}{u}_x-\frac{t}{\theta }{u}_t,P_t\right)+Q{D}_t^{\theta -1}\left(-\frac{v}{3}-\frac{x}{3}{v}_x-\frac{t}{\theta }{v}_t\right)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}I\left(-\frac{v}{3}-\frac{x}{3}{v}_x-\frac{t}{\theta }{v}_t,Q_t\right)+R{D}_t^{\theta -1}\left(-\frac{w}{3}-\frac{x}{3}{w}_x-\frac{t}{\theta }{w}_t\right)+I\left(-\frac{w}{3}-\frac{x}{3}{w}_x-\frac{t}{\theta }{w}_t,R_t\right)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}S{D}_t^{\theta -1}\left(-\frac{z}{3}-\frac{x}{3}{z}_x-\frac{t}{\theta }{z}_t\right)+I\left(-\frac{z}{3}-\frac{x}{3}{z}_x-\frac{t}{\theta }{z}_t,S_t\right).\end{array}$$

(54)

Case 2:

For X₂, $W_1=-u_x,\hspace{0.17em}\hspace{0.17em}{W}_2=-v_x,\hspace{0.17em}\hspace{0.17em}{W}_3=-w_x,\hspace{0.17em}\hspace{0.17em}$ and $W_4=-z_x$, we have

$$\begin{array}{ll}{\omega }^x=& \ell -u_x\left[6P{\displaystyle \sum u^2+P_{xx}}\right]-u_{xx}(P_x)-u_{xxx}P-v_x\left[6Q{\displaystyle \sum u^2+Q_{xx}}\right]-v_{xx}(Q_x)-u_{xxx}Q\\ & -w_x\left[6R{\displaystyle \sum u^2+R_{xx}}\right]-w_{xx}(R_x)-w_{xxx}R-z_x\left[6S{\displaystyle \sum u^2+S_{xx}}\right]-z_{xx}(S_x)-z_{xxx}S,\end{array}$$

(55)

$$\begin{array}{ll}{\omega }^t=& -P{D}_t^{\theta -1}(u_x)+I(-u_x,\hspace{0.17em}{P}_t)-Q{D}_t^{\theta -1}(v_x)+I(-v_x,\hspace{0.17em}{Q}_t)\\ & -R{D}_t^{\theta -1}(w_x)+I(-w_x,\hspace{0.17em}{R}_t)-S{D}_t^{\theta -1}(z_x)+I(-z_x,\hspace{0.17em}{S}_t).\end{array}$$

(56)

Case 3:

For X₃, $W_1=v,\hspace{0.17em}\hspace{0.17em}{W}_2=-u,\hspace{0.17em}\hspace{0.17em}{W}_3=0,$ and $\hspace{0.17em}\hspace{0.17em}{W}_4=0$, we have

$$\begin{array}{l}{\omega }^x=v\left[6P\left({\displaystyle \sum u^2}\right)+P_{xx}\right]-v_x{P}_x+P{v}_{xx}-u\left[6Q\left({\displaystyle \sum u^2}\right)+Q_{xx}\right]+u_x{Q}_x+Q{u}_{xx},\\ {\omega }^t=P{D}_t^{\theta -1}(v)+I(v,P_t)-Q{D}_t^{\theta -1}(u)+I(-u,Q_t).\end{array}$$

(57)

Case 4:

For X₄, $W_1=w,\hspace{0.17em}\hspace{0.17em}{W}_2=0,\hspace{0.17em}\hspace{0.17em}{W}_3=-u,\hspace{0.17em}\hspace{0.17em}$ and $W_4=0$, we have

$$\begin{array}{l}{\omega }^x=w\left[6P\left({\displaystyle \sum u^2}\right)+P_{xx}\right]-w_x{P}_x+P{w}_{xx}-u\left[6R\left({\displaystyle \sum u^2}\right)+R_{xx}\right]+u_x{R}_x+R{u}_{xx},\\ {\omega }^t=P{D}_t^{\theta -1}(w)+I(w,P_t)-R{D}_t^{\theta -1}(u)+I(-u,R_t).\end{array}$$

(58)

Case 5:

For X₅, $W_1=z,\hspace{0.17em}\hspace{0.17em}{W}_2=0,\hspace{0.17em}\hspace{0.17em}{W}_3=0,$ and $\hspace{0.17em}\hspace{0.17em}{W}_4=-u$, we have

$$\begin{array}{l}{\omega }^x=z\left[6P\left({\displaystyle \sum u^2}\right)+P_{xx}\right]-z_x{P}_x+P{z}_{xx}-u\left[6S\left({\displaystyle \sum u^2}\right)+S_{xx}\right]+u_x{S}_x+S{u}_{xx},\\ {\omega }^t=P{D}_t^{\theta -1}(z)+I(z,P_t)-S{D}_t^{\theta -1}(u)+I(-u,S_t).\end{array}$$

(59)

Case 6:

For X₆, $W_1=0,\hspace{0.17em}\hspace{0.17em}{W}_2=w,\hspace{0.17em}\hspace{0.17em}{W}_3=-v,$ and $W_4=0$, we have

$$\begin{array}{l}{\omega }^x=w\left[6Q\left({\displaystyle \sum u^2}\right)+Q_{xx}\right]-w_x{Q}_x+Q{w}_{xx}-v\left[6R\left({\displaystyle \sum u^2}\right)+R_{xx}\right]+v_x{R}_x+R{v}_{xx},\\ {\omega }^t=Q{D}_t^{\theta -1}(w)+I(w,Q_t)-R{D}_t^{\theta -1}(v)+I(-v,R_t).\end{array}$$

(60)

Case 7:

For X₇, $W_1=0,\hspace{0.17em}\hspace{0.17em}{W}_2=z,\hspace{0.17em}\hspace{0.17em}{W}_3=0,$ and $W_4=-v$, we have

$$\begin{array}{l}{\omega }^x=z\left[6Q\left({\displaystyle \sum u^2}\right)+Q_{xx}\right]-z_x{Q}_x+Q{z}_{xx}-v\left[6S\left({\displaystyle \sum u^2}\right)+S_{xx}\right]+v_x{S}_x+S{v}_{xx},\\ {\omega }^t=Q{D}_t^{\theta -1}(w)+I(w,Q_t)-S{D}_t^{\theta -1}(v)+I(-v,S_t).\end{array}$$

(61)

Case 8:

For X₈, $W_1=0,\hspace{0.17em}\hspace{0.17em}{W}_2=0,\hspace{0.17em}\hspace{0.17em}{W}_3=z,$ and $W_4=-w$, we have

$$\begin{array}{l}{\omega }^x=z\left[6R\left({\displaystyle \sum u^2}\right)+R_{xx}\right]-w_x{R}_x+R{w}_{xx}-w\left[6S\left({\displaystyle \sum u^2}\right)+S_{xx}\right]+w_x{S}_x+S{w}_{xx},\\ {\omega }^t=R{D}_t^{\theta -1}(z)+I(z,R_t)-S{D}_t^{\theta -1}(w)+I(-w,S_t).\end{array}$$

(62)

8. Conclusions

In this study, the Lie symmetry reduction was applied to a fractional-order nonlinear coupled Hirota system of PDEs. The obtained infinitesimals and generators were used to reduce the Hirota system to nonlinear FODEs on operating E-K operators, in the sense of the Riemann–Liouville fractional. The series solution and its convergence were discussed by exploiting a majorant series and the implicit function theorem. We applied Euler–Lagrange operators to obtain the adjoint system. Noether’s theorem was exploited for conservation analysis to examine the stability and consistency of the given system. In addition, the solution and the conservation analysis procured in the work can be very advantageous for illuminating nonlinear evolution problems, including fractional fluid flow, sound propagation, single-mode-fiber studies, and others.