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Lie symmetries, exact solutions and conservation laws of time fractional Boussinesq–Burgers system in ocean waves

  • Jicheng Yu , 1, * ,
  • Yuqiang Feng 1, 2
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  • 1School of Science, Wuhan University of Science and Technology, Wuhan 430081, Hubei, China
  • 2Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan 430081, Hubei, China

*Author to whom any correspondence should be addressed.

Received date: 2024-07-01

  Revised date: 2024-08-14

  Accepted date: 2024-08-21

  Online published: 2024-10-22

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© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing. All rights, including for text and data mining, AI training, and similar technologies, are reserved.

Abstract

In this paper, the Lie symmetry analysis method is applied to the time-fractional Boussinesq–Burgers system which is used to describe shallow water waves near an ocean coast or in a lake. We obtain all the Lie symmetries admitted by the system and use them to reduce the fractional partial differential equations with a Riemann–Liouville fractional derivative to some fractional ordinary differential equations with an Erdélyi–Kober fractional derivative, thereby getting some exact solutions of the reduced equations. For power series solutions, we prove their convergence and show the dynamic analysis of their truncated graphs. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Cite this article

Jicheng Yu , Yuqiang Feng . Lie symmetries, exact solutions and conservation laws of time fractional Boussinesq–Burgers system in ocean waves[J]. Communications in Theoretical Physics, 2024 , 76(12) : 125002 . DOI: 10.1088/1572-9494/ad71ab

1. Introduction

Nonlinear partial differential equations are an important tool in the nonlinear modeling of nature phenomena. Among them, the following Boussinesq–Burgers system is considered [1]:
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{\beta -1}{2}{u}_{{xx}}+2{{uu}}_{x}+\displaystyle \frac{1}{2}{v}_{x},\\ {v}_{t}=(1-\displaystyle \frac{\beta }{2})\beta {u}_{{xxx}}+2{\left({uv}\right)}_{x}+\displaystyle \frac{1-\beta }{2}{v}_{{xx}},\end{array}\right.\end{eqnarray}$
where u and v denote the field of horizontal velocity and the height derivatives from the equilibrium position of water, respectively. Independent variables x and t represent the normalized space coordinate and time, and the real parameter β means the dispersive power. In the past few decades, many scholars have studied the Boussinesq–Burgers system (1.1) and its generalized forms using different methods (see [29] and the references therein). Recently, the classical Boussinesq–Burgers system has been extended to the fractional order form, attracting the interest of many scholars [1018]. For example, Kumar et al [10] successfully applied a residual power series method and modified homotopy analysis transform method to obtain the approximate solutions of fractional nonlinear coupled Boussinesq–Burger equations. Abu Irwaq et al [14] discussed the effect of the fractional order in the propagation of the obtained solutions by using an adaptation of both the time-spectrum function method and the homotopy perturbation method. Zafar et al [17] highlighted the shallow water wave patterns along the ocean shore or in lakes with the higher-order Boussinesq–Burgers system possessing a fractional derivative operator.
Fractional differential equations (FDEs), due to the nonlocality of fractional derivative [1922], exhibit genetic effects and long-range dependence, and are widely used in many fields of science and technology. Therefore, solving FDEs is of great significance. At present, there are only some specialized numerical and analytical solutions available, such as the Adomian decomposition method [23], finite difference method [24], homotopy perturbation method [25], the sub-equation method [26], the variational iteration method [27], invariant subspace method [28], Lie symmetry analysis method [29] and so on. The Lie symmetry analysis method has received increasing attention because it can treat differential equations uniformly regardless of their forms, transforming some solutions of these equations into other forms of solutions [30]. It was first extended to solve FDEs by Gazizov et al [29] in 2007, and then effectively applied to study various FDEs models occurring in different areas of applied science (see [3141]).
In this paper, the Lie symmetry analysis method is extended to the following time-fractional Boussinesq–Burgers equations:
$\begin{eqnarray}\left\{\begin{array}{l}{D}_{t}^{\alpha }u=\displaystyle \frac{\beta -1}{2}{u}_{{xx}}+2{{uu}}_{x}+\displaystyle \frac{1}{2}{v}_{x},\\ {D}_{t}^{\alpha }v=(1-\displaystyle \frac{\beta }{2})\beta {u}_{{xxx}}+2{\left({uv}\right)}_{x}+\displaystyle \frac{1-\beta }{2}{v}_{{xx}},\end{array}\right.\end{eqnarray}$
with 0 < α < 1. As we all know, there are many types of definitions for fractional derivatives, such as Riemann–Liouville type, Caputo type, Weyl type and so on. This paper adopts the Riemann–Liouville fractional derivative defined by [1922]
$\begin{eqnarray*}\begin{array}{l}{}_{a}{D}_{t}^{\alpha }f(t,x)={D}_{t}^{n}{\ }_{a}{I}_{t}^{n-\alpha }f(t,x)\\ \quad =\,\left\{\begin{array}{ll}\displaystyle \frac{1}{{\rm{\Gamma }}(n-\alpha )}\displaystyle \frac{{\partial }^{n}}{\partial {t}^{n}}{\displaystyle \int }_{a}^{t}\displaystyle \frac{f(s,x)}{{\left(t-s\right)}^{\alpha -n+1}}{\rm{d}}s,\ & n-1\lt \alpha \lt n,n\in {\mathbb{N}}\\ {D}_{t}^{n}f(t,x),\ & \alpha =n\in {\mathbb{N}}\end{array}\right.\end{array}\end{eqnarray*}$
for t > a. We denote the operator ${}_{0}{D}_{t}^{\alpha }$ as ${D}_{t}^{\alpha }$ throughout this paper, while ${D}_{t}^{-\alpha }={I}_{t}^{\alpha }$ is Riemann–Liouville fractional integral.
The aim of this paper is to find symmetry groups for equation (1.2) by using the Lie symmetry analysis method and use group generators to reduce equation (1.2) to some fractional ordinary differential equations. Moreover, we obtained some exact solutions for the reduced equations and proved the convergence of power series solutions. Finally, we constructed the conservation laws corresponding to each symmetry using the generalized Noether theorem [42, 43].
This paper is organized as follows. In section 2, Lie symmetry analysis of equation (1.2) is presented. In section 3, the similarity reductions and exact solutions for equation (1.2) are obtained. The conserved vectors for all Lie symmetries admitted by equation (1.2) are constructed in section 4. The conclusion is given in the last section.

2. Lie symmetry analysis of equation (1.2)

Consider time-fractional Boussinesq–Burgers equation (1.2), which is assumed to be invariant under the one-parameter (ε) Lie group of continuous point transformations, i.e.
$\begin{eqnarray}\begin{array}{rcl}{t}^{* } & = & t+\epsilon \tau (t,x,y,u,v)+o(\epsilon ),\\ {x}^{* } & = & x+\epsilon \xi (t,x,y,u,v)+o(\epsilon ),\\ {u}^{* } & = & u+\epsilon \eta (t,x,y,u,v)+o(\epsilon ),\\ {v}^{* } & = & v+\epsilon \zeta (t,x,y,u,v)+o(\epsilon ),\\ {D}_{{t}^{* }}^{\alpha }{u}^{* } & = & {D}_{t}^{\alpha }u+\epsilon {\eta }^{\alpha ,t}+o(\epsilon ),\\ {D}_{{t}^{* }}^{\alpha }{v}^{* } & = & {D}_{t}^{\alpha }v+\epsilon {\zeta }^{\alpha ,t}+o(\epsilon ),\\ {D}_{{x}^{* }}{u}^{* } & = & {D}_{x}u+\epsilon {\eta }^{x}+o(\epsilon ),\\ {D}_{{x}^{* }}{v}^{* } & = & {D}_{x}v+\epsilon {\zeta }^{x}+o(\epsilon ),\\ {D}_{{x}^{* }}^{2}{u}^{* } & = & {D}_{x}^{2}u+\epsilon {\eta }^{{xx}}+o(\epsilon ),\\ {D}_{{x}^{* }}^{2}{v}^{* } & = & {D}_{x}^{2}v+\epsilon {\zeta }^{{xx}}+o(\epsilon ),\\ {D}_{{x}^{* }}^{3}{u}^{* } & = & {D}_{x}^{3}u+\epsilon {\eta }^{{xxx}}+o(\epsilon ),\end{array}\end{eqnarray}$
where τ, ξ, θ and ζ are infinitesimals and ηα,t, ζα,t, ηx, ζx, ηxx, ζxx, ηxxx are the corresponding prolongations of orders α, 1, 2 and 3, respectively.
The corresponding group generator is defined by
$\begin{eqnarray}X=\tau \displaystyle \frac{\partial }{\partial t}+\xi \displaystyle \frac{\partial }{\partial x}+\eta \displaystyle \frac{\partial }{\partial u}+\zeta \displaystyle \frac{\partial }{\partial v}.\end{eqnarray}$
So the prolongation of the above group generator X has the form
$\begin{eqnarray}\begin{array}{rcl}{prX} & = & X+{\eta }^{\alpha ,t}\displaystyle \frac{\partial }{\partial {u}_{t}^{\alpha }}+{\zeta }^{\alpha ,t}\displaystyle \frac{\partial }{\partial {v}_{t}^{\alpha }}+{\eta }^{x}\displaystyle \frac{\partial }{\partial {u}_{x}}+{\zeta }^{x}\displaystyle \frac{\partial }{\partial {v}_{x}}\\ & & +{\eta }^{{xx}}\displaystyle \frac{\partial }{\partial {u}_{{xx}}}+{\zeta }^{{xx}}\displaystyle \frac{\partial }{\partial {v}_{{xx}}}+{\eta }^{{xxx}}\displaystyle \frac{\partial }{\partial {u}_{{xxx}}},\end{array}\end{eqnarray}$
where
$\begin{eqnarray}{\eta }^{x}={D}_{x}(\eta )-{u}_{t}{D}_{x}(\tau )-{u}_{x}{D}_{x}(\xi ),\end{eqnarray}$
$\begin{eqnarray}{\zeta }^{x}={D}_{x}(\zeta )-{v}_{t}{D}_{x}(\tau )-{v}_{x}{D}_{x}(\xi ),\end{eqnarray}$
$\begin{eqnarray}{\eta }^{{xx}}={D}_{x}({\eta }^{x})-{u}_{{xt}}{D}_{x}(\tau )-{u}_{{xx}}{D}_{x}(\xi ),\end{eqnarray}$
$\begin{eqnarray}{\zeta }^{{xx}}={D}_{x}({\zeta }^{x})-{v}_{{xt}}{D}_{x}(\tau )-{v}_{{xx}}{D}_{x}(\xi ),\end{eqnarray}$
$\begin{eqnarray}{\eta }^{{xxx}}={D}_{x}({\eta }^{{xx}})-{u}_{{xxt}}{D}_{x}(\tau )-{u}_{{xxx}}{D}_{x}(\xi ),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\eta }^{\alpha ,t} & = & {D}_{t}^{\alpha }(\eta )+\tau {D}_{t}^{\alpha }({u}_{t})\\ & & -{D}_{t}^{\alpha }(\tau {u}_{t})+\xi {D}_{t}^{\alpha }({u}_{x})-{D}_{t}^{\alpha }(\xi {u}_{x})\\ & = & \displaystyle \frac{{\partial }^{\alpha }\eta }{\partial {t}^{\alpha }}+({\eta }_{u}-\alpha {D}_{t}(\tau ))\displaystyle \frac{{\partial }^{\alpha }u}{\partial {t}^{\alpha }}-u\displaystyle \frac{{\partial }^{\alpha }{\eta }_{u}}{\partial {t}^{\alpha }}\\ & & +({\eta }_{v}\displaystyle \frac{{\partial }^{\alpha }v}{\partial {t}^{\alpha }}-v\displaystyle \frac{{\partial }^{\alpha }{\eta }_{v}}{\partial {t}^{\alpha }})\\ & & +\displaystyle \sum _{n=1}^{\infty }[\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\displaystyle \frac{{\partial }^{n}{\eta }_{u}}{\partial {t}^{n}}\\ & & -\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n+1}\right){D}_{t}^{n+1}(\tau )]{D}_{t}^{\alpha -n}(u)+{\mu }_{1}+{\mu }_{2}\\ & & +\displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\displaystyle \frac{{\partial }^{n}{\eta }_{v}}{\partial {t}^{n}}{D}_{t}^{\alpha -n}(v)\\ & & -\displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right){D}_{t}^{n}(\xi ){D}_{t}^{\alpha -n}({u}_{x}),\end{array}\end{eqnarray}$
with
$\begin{eqnarray*}\begin{array}{l}{\mu }_{1}=\displaystyle \sum _{n=2}^{\infty }\displaystyle \sum _{m=2}^{n}\displaystyle \sum _{k=2}^{m}\displaystyle \sum _{r=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\left(\displaystyle \genfrac{}{}{0em}{}{n}{m}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{r}\right)\\ \quad \times \displaystyle \frac{{t}^{n-\alpha }{\left(-u\right)}^{r}}{k!{\rm{\Gamma }}(n+1-\alpha )}\displaystyle \frac{{\partial }^{m}{u}^{k-r}}{\partial {t}^{m}}\displaystyle \frac{{\partial }^{n-m+k}\eta }{\partial {t}^{n-m}\partial {u}^{k}},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\mu }_{2}=\displaystyle \sum _{n=2}^{\infty }\displaystyle \sum _{m=2}^{n}\displaystyle \sum _{k=2}^{m}\displaystyle \sum _{r=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\left(\displaystyle \genfrac{}{}{0em}{}{n}{m}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{r}\right)\\ \quad \times \displaystyle \frac{{t}^{n-\alpha }{\left(-v\right)}^{r}}{k!{\rm{\Gamma }}(n+1-\alpha )}\displaystyle \frac{{\partial }^{m}{v}^{k-r}}{\partial {t}^{m}}\displaystyle \frac{{\partial }^{n-m+k}\eta }{\partial {t}^{n-m}\partial {v}^{k}},\end{array}\end{eqnarray*}$
and
$\begin{eqnarray}\begin{array}{rcl}{\zeta }^{\alpha ,t} & = & {D}_{t}^{\alpha }(\zeta )+\tau {D}_{t}^{\alpha }({v}_{t})-{D}_{t}^{\alpha }(\tau {v}_{t})\\ & & +\xi {D}_{t}^{\alpha }({v}_{x})-{D}_{t}^{\alpha }(\xi {v}_{x})\\ & = & \displaystyle \frac{{\partial }^{\alpha }\zeta }{\partial {t}^{\alpha }}+({\zeta }_{v}-\alpha {D}_{t}(\tau ))\displaystyle \frac{{\partial }^{\alpha }v}{\partial {t}^{\alpha }}\\ & & -v\displaystyle \frac{{\partial }^{\alpha }{\zeta }_{v}}{\partial {t}^{\alpha }}+\left({\zeta }_{u}\displaystyle \frac{{\partial }^{\alpha }u}{\partial {t}^{\alpha }}-u\displaystyle \frac{{\partial }^{\alpha }{\zeta }_{u}}{\partial {t}^{\alpha }}\right)\\ & & +\displaystyle \sum _{n=1}^{\infty }[\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\displaystyle \frac{{\partial }^{n}{\zeta }_{v}}{\partial {t}^{n}}\\ & & -\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n+1}\right){D}_{t}^{n+1}(\tau )]{D}_{t}^{\alpha -n}(v)+{\mu }_{3}+{\mu }_{4}\\ & & +\displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\displaystyle \frac{{\partial }^{n}{\zeta }_{u}}{\partial {t}^{n}}{D}_{t}^{\alpha -n}(u)\\ & & -\displaystyle \sum _{n=1}^{\infty }\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right){D}_{t}^{n}(\xi ){D}_{t}^{\alpha -n}({v}_{x}),\end{array}\end{eqnarray}$
with
$\begin{eqnarray*}\begin{array}{rcl}{\mu }_{3} & = & \displaystyle \sum _{n=2}^{\infty }\displaystyle \sum _{m=2}^{n}\displaystyle \sum _{k=2}^{m}\displaystyle \sum _{r=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\left(\displaystyle \genfrac{}{}{0em}{}{n}{m}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{r}\right)\\ \Space{0ex}{1.04em}{0ex} & \times & \displaystyle \frac{{t}^{n-\alpha }{\left(-u\right)}^{r}}{k!{\rm{\Gamma }}(n+1-\alpha )}\displaystyle \frac{{\partial }^{m}{u}^{k-r}}{\partial {t}^{m}}\displaystyle \frac{{\partial }^{n-m+k}\zeta }{\partial {t}^{n-m}\partial {u}^{k}},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{\mu }_{4} & = & \displaystyle \sum _{n=2}^{\infty }\displaystyle \sum _{m=2}^{n}\displaystyle \sum _{k=2}^{m}\displaystyle \sum _{r=0}^{k-1}\left(\displaystyle \genfrac{}{}{0em}{}{\alpha }{n}\right)\left(\displaystyle \genfrac{}{}{0em}{}{n}{m}\right)\left(\displaystyle \genfrac{}{}{0em}{}{k}{r}\right)\\ & \times & \Space{0ex}{3.04em}{0ex}\displaystyle \frac{{t}^{n-\alpha }{\left(-v\right)}^{r}}{k!{\rm{\Gamma }}(n+1-\alpha )}\displaystyle \frac{{\partial }^{m}{v}^{k-r}}{\partial {t}^{m}}\displaystyle \frac{{\partial }^{n-m+k}\zeta }{\partial {t}^{n-m}\partial {v}^{k}},\end{array}\end{eqnarray*}$
where Dt, Dx and Dy are the total derivative with respect to t, x and y, respectively.

Remark 1. The infinitesimal transformations (2.1) should conserve the structure of the Riemann–Liouville fractional derivative operator, of which the lower limit in the integral is fixed. Therefore, the manifold t = 0 should be invariant with respect to such transformations. The invariance condition arrives at

$\begin{eqnarray}\tau (t,x,y,u,v){| }_{t=0}=0.\end{eqnarray}$

Remark 2. From the expressions of ${\mu }_{1}$ and ${\mu }_{2}$, if the infinitesimals η and ζ are linear with respect to the variables u and v, then ${\mu }_{1}={\mu }_{2}=0$, that is,

$\begin{eqnarray}\displaystyle \frac{{\partial }^{2}\eta }{\partial {u}^{2}}=\displaystyle \frac{{\partial }^{2}\eta }{\partial {v}^{2}}=\displaystyle \frac{{\partial }^{2}\zeta }{\partial {u}^{2}}=\displaystyle \frac{{\partial }^{2}\zeta }{\partial {v}^{2}}=0.\end{eqnarray}$

The one-parameter Lie symmetry transformations (2.1) are admitted by the system (1.2), if the following invariance criterion holds:
$\begin{eqnarray}\left\{\begin{array}{l}{prX}\left({D}_{t}^{\alpha }u-\displaystyle \frac{\beta -1}{2}{u}_{{xx}}-2{{uu}}_{x}-\displaystyle \frac{1}{2}{v}_{x}\right){| }_{(1.2)}=0,\\ {prX}({D}_{t}^{\alpha }v-(1-\displaystyle \frac{\beta }{2})\beta {u}_{{xxx}}-2{\left({uv}\right)}_{x}-\displaystyle \frac{1-\beta }{2}{v}_{{xx}}){| }_{(1.2)}=0,\end{array}\right.\end{eqnarray}$
which can be rewritten as
$\begin{eqnarray}\left\{\begin{array}{l}\left({\eta }^{\alpha ,t}-\displaystyle \frac{\beta -1}{2}{\eta }^{{xx}}-2u{\eta }^{x}-2u{\zeta }^{x}-\displaystyle \frac{1}{2}{\zeta }^{x}-2{u}_{x}\eta \right){| }_{(1.2)}=0,\\ \left({\zeta }^{\alpha ,t}-(1-\displaystyle \frac{\beta }{2}){\eta }^{{xxx}}-\displaystyle \frac{1-\beta }{2}{\zeta }^{{xx}}-2v{\eta }^{x}-2u{\zeta }^{x}-2{v}_{x}\eta -2{u}_{x}\zeta \right){| }_{(1.2)}=0.\end{array}\right.\end{eqnarray}$
Putting ηα,t, ζα,t, ηx, ζx, ηxx, ζxx and ηxxx into (2.14) and letting coefficients of various derivatives of u and v to be zero, we can obtain the infinitesimals as follows:
$\begin{eqnarray}\begin{array}{rcl}\tau & = & {c}_{1}t,\ \xi =\displaystyle \frac{\alpha }{2}{c}_{1}x+{c}_{2},\\ \eta & = & \displaystyle \frac{\alpha }{2}{c}_{1}u,\ \zeta =-\alpha {c}_{1}v,\end{array}\end{eqnarray}$
where c1 and c2 are arbitrary constants. It means that equation (1.2) admit the two-dimension Lie algebra spanned by
$\begin{eqnarray}\begin{array}{rcl}{X}_{1} & = & \displaystyle \frac{\partial }{\partial x},\ {X}_{2}=t\displaystyle \frac{\partial }{\partial t}+\displaystyle \frac{\alpha }{2}x\displaystyle \frac{\partial }{\partial x}\\ & & -\displaystyle \frac{\alpha }{2}u\displaystyle \frac{\partial }{\partial u}-\alpha v\displaystyle \frac{\partial }{\partial v}.\end{array}\end{eqnarray}$

3. Similarity reductions and invariant solutions of equation (1.2)

In this section, the aimed equation (1.2) can be reduced to some fractional ordinary differential equations with the left-hand Erdélyi–Kober fractional derivative and some other solvable fractional ordinary differential equations with Riemann–Liouville fractional derivative.
Case 1: ${X}_{1}=\tfrac{\partial }{\partial x}$
The characteristic equation corresponding to the group generator X1 is
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}t}{0}=\displaystyle \frac{{\rm{d}}x}{1}=\displaystyle \frac{{\rm{d}}u}{0}=\displaystyle \frac{{\rm{d}}v}{0},\end{eqnarray}$
from which, we obtain the similarity variables t, u and v. So we get the invariant solutions of the system (1.2) as follows:
$\begin{eqnarray}u=f(t),\ v=g(t).\end{eqnarray}$
Substituting (3.2) into equation (1.2), we have the following reduced equations:
$\begin{eqnarray}\left\{\begin{array}{l}{D}_{t}^{\alpha }f=0,\\ {D}_{t}^{\alpha }g=0,\end{array}\right.\end{eqnarray}$
from which, we can easily get the following solutions:
$\begin{eqnarray}f=\displaystyle \frac{{k}_{1}}{{\rm{\Gamma }}(\alpha )}{t}^{\alpha -1},\ g=\displaystyle \frac{{k}_{2}}{{\rm{\Gamma }}(\alpha )}{t}^{\alpha -1},\end{eqnarray}$
where k1 = f(α−1)(0) and k2 = g(α−1)(0) are constants determined by initial conditions. In this case, we only obtain the above trivial solutions.
Case 2: ${X}_{2}=t\tfrac{\partial }{\partial t}+\tfrac{\alpha }{2}x\tfrac{\partial }{\partial x}-\tfrac{\alpha }{2}u\tfrac{\partial }{\partial u}-\alpha v\tfrac{\partial }{\partial v}$
The characteristic equation corresponding to the group generator X2 is
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}t}{t}=\displaystyle \frac{{\rm{d}}x}{\tfrac{\alpha }{2}x}=\displaystyle \frac{{\rm{d}}u}{-\tfrac{\alpha }{2}u}=\displaystyle \frac{{\rm{d}}v}{-\alpha v},\end{eqnarray}$
from which, we obtain the similarity variables ${{xt}}^{-\tfrac{\alpha }{2}}$, ${{ut}}^{\tfrac{\alpha }{2}}$ and vtα. So we get the invariant solutions of the system (1.2) as follows:
$\begin{eqnarray}\begin{array}{rcl}u(t,x) & = & {t}^{-\tfrac{\alpha }{2}}f(\omega ),\\ v(t,x) & = & {t}^{-\alpha }g(\omega ),\ \omega ={{xt}}^{-\tfrac{\alpha }{2}}.\end{array}\end{eqnarray}$

Theorem 3.1. The similarity transformations $u(t,x)\,={t}^{-\tfrac{\alpha }{2}}f(\omega )$, $v(t,x)={t}^{-\alpha }g(\omega )$ with the similarity variables $\omega ={{xt}}^{-\tfrac{\alpha }{2}}$ reduce the system (1.2) to the system of fractional ordinary differential equations given by

$\begin{eqnarray}\left\{\begin{array}{l}({{ \mathcal P }}_{\tfrac{2}{\alpha }}^{1-\tfrac{3\alpha }{2},\alpha }f)(\omega )=\displaystyle \frac{\beta -1}{2}f^{\prime\prime} +2{ff}^{\prime} +\displaystyle \frac{1}{2}g^{\prime} ,\\ ({{ \mathcal P }}_{\tfrac{2}{\alpha }}^{1-2\alpha ,\alpha }g)(\omega )=(1-\displaystyle \frac{\beta }{2})\beta {f}^{(3)}+2{fg}^{\prime} +2{gf}^{\prime} +\displaystyle \frac{1-\beta }{2}g^{\prime\prime} ,\end{array}\right.\end{eqnarray}$
where $({{ \mathcal P }}_{{\delta }_{1},{\delta }_{2}}^{\iota ,\kappa })$ is the left-hand Erdélyi–Kober fractional differential operator defined by
$\begin{eqnarray*}\begin{array}{l}({{ \mathcal P }}_{{\delta }_{1},{\delta }_{2}}^{\iota ,\kappa }\psi )({\omega }_{1},{\omega }_{2}):= \displaystyle \prod _{j=0}^{m-1}\left(\iota +j-\displaystyle \frac{1}{{\delta }_{1}}{\omega }_{1}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{\omega }_{1}}-\displaystyle \frac{1}{{\delta }_{2}}{\omega }_{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{\omega }_{2}}\right)\\ \quad \times \,({{ \mathcal K }}_{{\delta }_{1},{\delta }_{2}}^{\iota +\kappa ,m-\kappa }\psi )({\omega }_{1},{\omega }_{2}),\kappa \gt 0,\end{array}\end{eqnarray*}$
$\begin{eqnarray}m=\left\{\begin{array}{ll}[\kappa ]+1,\ & {if}\ \kappa \notin {\mathbb{N}},\\ \kappa ,\ & {if}\ \kappa \in {\mathbb{N}},\end{array}\right.\end{eqnarray}$
where
$\begin{eqnarray}\begin{array}{l}({{ \mathcal K }}_{{\delta }_{1},{\delta }_{2}}^{\iota ,\kappa }\psi )({\omega }_{1},{\omega }_{2}):= \,\left\{\begin{array}{ll}\displaystyle \frac{1}{{\rm{\Gamma }}(\kappa )}{\displaystyle \int }_{1}^{\infty }{\left(s-1\right)}^{\kappa -1}{s}^{-(\iota +\kappa )}\psi ({\omega }_{1}{s}^{\tfrac{1}{{\delta }_{1}}},{\omega }_{2}{s}^{\tfrac{1}{{\delta }_{2}}}){\rm{d}}s,\ & \kappa \gt 0,\\ \psi ({\omega }_{1},{\omega }_{2}),\ & \kappa =0,\end{array}\right.\end{array}\end{eqnarray}$
is the left-hand Erdélyi–Kober fractional integral operator.

Proof. For $0\lt \alpha \lt 1$, the Riemann–Liouville time-fractional derivative of $u(t,x)$ can be obtained as follows:

$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{{\partial }^{\alpha }u}{\partial {t}^{\alpha }} & = & \displaystyle \frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}({t}^{-\tfrac{\alpha }{2}}f(\omega ))=\displaystyle \frac{\partial }{\partial t}\left[\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\right.\\ & \times & \left.\,{\displaystyle \int }_{0}^{t}{\left(t-s\right)}^{-\alpha }{s}^{\tfrac{-\alpha }{2}}f({{xs}}^{-\tfrac{\alpha }{2}}){\rm{d}}s\right].\end{array}\end{eqnarray*}$
Assuming $r=\tfrac{t}{s}$, we have
$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{{\partial }^{\alpha }u}{\partial {t}^{\alpha }} & = & \displaystyle \frac{\partial }{\partial t}\left[\displaystyle \frac{{t}^{1-\tfrac{3\alpha }{2}}}{{\rm{\Gamma }}(1-\alpha )}{\displaystyle \int }_{1}^{\infty }{\left(r-1\right)}^{-\alpha }{r}^{\tfrac{3\alpha }{2}-2}f(\omega {r}^{\tfrac{\alpha }{2}}){\rm{d}}r\right]\\ & = & \displaystyle \frac{\partial }{\partial t}\left[{t}^{1-\tfrac{3\alpha }{2}}({{ \mathcal K }}_{\tfrac{2}{\alpha }}^{1-\tfrac{\alpha }{2},1-\alpha }f)(\omega )\right].\end{array}\end{eqnarray*}$
Because of $\omega ={{xt}}^{-\tfrac{\alpha }{2}}$, the following relation holds:
$\begin{eqnarray*}t\displaystyle \frac{\partial }{\partial t}\psi (\omega )=t(-\displaystyle \frac{\alpha }{2}{{xt}}^{-\tfrac{\alpha }{2}-1}{\psi }_{\omega })=-\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega }\psi .\end{eqnarray*}$
Hence, we arrive at
$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{{\partial }^{\alpha }u}{\partial {t}^{\alpha }} & = & {t}^{-\tfrac{3\alpha }{2}}\left[\left(1-\displaystyle \frac{3\alpha }{2}-\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega })({{ \mathcal K }}_{\tfrac{2}{\alpha }}^{1-\tfrac{\alpha }{2},1-\alpha }f)(\omega \right)\right]\\ & = & {t}^{-\tfrac{3\alpha }{2}}({{ \mathcal P }}_{\tfrac{2}{\alpha }}^{1-\tfrac{3\alpha }{2},\alpha }f)(\omega ).\end{array}\end{eqnarray*}$
Similarly,
$\begin{eqnarray*}\displaystyle \frac{{\partial }^{\alpha }v}{\partial {t}^{\alpha }}={t}^{-2\alpha }({{ \mathcal P }}_{\tfrac{2}{\alpha }}^{1-2\alpha ,\alpha }g)(\omega ).\end{eqnarray*}$
Meanwhile,
$\begin{eqnarray*}\begin{array}{rcl} & & \displaystyle \frac{\beta -1}{2}{u}_{{xx}}+2{{uu}}_{x}+\displaystyle \frac{1}{2}{v}_{x}\\ & = & {t}^{-\tfrac{3\alpha }{2}}(\displaystyle \frac{\beta -1}{2}f^{\prime\prime} +2{ff}^{\prime} +\displaystyle \frac{1}{2}g^{\prime} ),\\ & \times & \left(1-\displaystyle \frac{\beta }{2}\right)\beta {u}_{{xxx}}+2{\left({uv}\right)}_{x}+\displaystyle \frac{1-\beta }{2}{v}_{{xx}}\\ & = & {t}^{-2\alpha }((1-\displaystyle \frac{\beta }{2})\beta {f}^{(3)}\\ & + & 2{fg}^{\prime} +2{gf}^{\prime} +\displaystyle \frac{1-\beta }{2}g^{\prime\prime} ).\end{array}\end{eqnarray*}$
This completes the proof. □

Next we use the power series method to derive the power series solutions of the reduced equation (3.21). Let us assume
$\begin{eqnarray}f(\omega )=\displaystyle \sum _{k=0}^{\infty }{a}_{k}{\omega }^{k},\ g(\omega )=\displaystyle \sum _{k=0}^{\infty }{b}_{k}{\omega }^{k},\end{eqnarray}$
then
$\begin{eqnarray}\begin{array}{rcl}f^{\prime} (\omega ) & = & \displaystyle \sum _{k=0}^{\infty }(k+1){a}_{k+1}{\omega }^{k},\\ g^{\prime} (\omega ) & = & \displaystyle \sum _{k=0}^{\infty }(k+1){b}_{k+1}{\omega }^{k},\\ f^{\prime\prime} (\omega ) & = & \displaystyle \sum _{k=0}^{\infty }(k+2)(k+1){a}_{k+2}{\omega }^{k},\\ g^{\prime\prime} (\omega ) & = & \displaystyle \sum _{k=0}^{\infty }(k+2)(k+1){b}_{k+2}{\omega }^{k},\\ {f}^{(3)}(\omega ) & = & \displaystyle \sum _{k=0}^{\infty }(k+3)(k+2)(k+1){a}_{k+3}{\omega }^{k},\end{array}\end{eqnarray}$
and
$\begin{eqnarray}\begin{array}{rcl}({{ \mathcal P }}_{\tfrac{2}{\alpha }}^{1-\tfrac{3\alpha }{2},\alpha }f)(\omega ) & = & \left(1-\displaystyle \frac{3\alpha }{2}-\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega }\right)\left(\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\right.\\ & \times & \left.{\displaystyle \int }_{1}^{\infty }{\left(s-1\right)}^{-\alpha }{s}^{\tfrac{3\alpha }{2}-2}f(\omega {s}^{\tfrac{\alpha }{2}}){\rm{d}}s\right)\\ & = & \left(1-\displaystyle \frac{3\alpha }{2}-\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega }\right)\left(\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\right.\\ & \times & \left.{\displaystyle \int }_{1}^{\infty }{\left(s-1\right)}^{-\alpha }{s}^{\tfrac{3\alpha }{2}-2}\displaystyle \sum _{k=0}^{\infty }({a}_{k}{\omega }^{k}{s}^{\tfrac{k\alpha }{2}}){\rm{d}}s\right)\\ & = & \left(1-\displaystyle \frac{3\alpha }{2}-\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega }\right)\left(\displaystyle \sum _{k=0}^{\infty }{a}_{k}{\omega }^{k}\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\right.\\ & \times & \left.{\displaystyle \int }_{1}^{\infty }{\left(s-1\right)}^{-\alpha }{s}^{\tfrac{k+3}{2}\alpha -2}{\rm{d}}s\right)\\ & = & (1-\displaystyle \frac{3\alpha }{2}-\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega })(\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+1)\alpha }{2})}{{\rm{\Gamma }}(2-\tfrac{(k+3)\alpha }{2})}{a}_{k}{\omega }^{k})\\ & = & \displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+1)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+3)\alpha }{2})}{a}_{k}{\omega }^{k},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}({{ \mathcal P }}_{\tfrac{2}{\alpha }}^{1-2\alpha ,\alpha }g)(\omega ) & = & (1-2\alpha -\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega })(\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\\ & \times & {\displaystyle \int }_{1}^{\infty }{\left(s-1\right)}^{-\alpha }{s}^{2\alpha -2}g(\omega {s}^{\tfrac{\alpha }{2}}){\rm{d}}s)\\ & = & (1-2\alpha -\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega })(\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\\ & \times & {\displaystyle \int }_{1}^{\infty }{\left(s-1\right)}^{-\alpha }{s}^{2\alpha -2}\displaystyle \sum _{k=0}^{\infty }({b}_{k}{\omega }^{k}{s}^{\tfrac{k\alpha }{2}}){\rm{d}}s)\\ & = & (1-2\alpha -\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega })(\displaystyle \sum _{k=0}^{\infty }{b}_{k}{\omega }^{k}\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\\ & \times & {\displaystyle \int }_{1}^{\infty }{\left(s-1\right)}^{-\alpha }{s}^{\tfrac{k+4}{2}\alpha -2}{\rm{d}}s)\\ & = & (1-2\alpha -\displaystyle \frac{\alpha }{2}\omega \displaystyle \frac{{\rm{d}}}{{\rm{d}}\omega })(\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})}{{\rm{\Gamma }}(2-\tfrac{(k+4)\alpha }{2})}{b}_{k}{\omega }^{k})\\ & = & \displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+4)\alpha }{2})}{b}_{k}{\omega }^{k}.\end{array}\end{eqnarray}$
Substituting (3.10)–(3.13) into the reduced equations (3.7) and equating the coefficients of different powers of ω, we can obtain
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+1)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+3)\alpha }{2})}{a}_{k} & = & \displaystyle \frac{\beta -1}{2}(k+2)(k+1){a}_{k+2}\\ & + & \displaystyle \sum _{i+j=k}(j+1){a}_{i}{a}_{j+1}+\displaystyle \frac{1}{2}{b}_{k+1},\\ & \times & \displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+4)\alpha }{2})}{b}_{k}\\ & = & \left(1-\displaystyle \frac{\beta }{2}\right)\beta (k+3)(k+2)(k+1){a}_{k+3}\\ & + & \displaystyle \sum _{i+j=k}(j+1){a}_{i}{b}_{j+1}\\ & + & 2\displaystyle \sum _{i+j=k}(i+1){a}_{i+1}{b}_{j}+\displaystyle \frac{1-\beta }{2}(k+2)(k+1){b}_{k+2}.\end{array}\end{eqnarray}$
Let ${a}_{0}=f(0),{b}_{0}\,=\,g(0),{a}_{1}\,=\,f^{\prime} (0)$ and ${b}_{1}=g^{\prime} (0)$, from (3.8), we have
$\begin{eqnarray}{a}_{2}=\displaystyle \frac{1}{\beta -1}\left[\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{3\alpha }{2})}{a}_{0}-2{a}_{0}{a}_{1}-\displaystyle \frac{1}{2}{b}_{1}\right],\end{eqnarray}$
and for k ≥ 0,
$\begin{eqnarray}\begin{array}{l}{b}_{k+2}=\displaystyle \frac{1}{(k+2)(k+1)}\left[\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+4)\alpha }{2})}\right.\\ \quad \times \,\left(\beta (2-\beta )(k+1){a}_{k+1}-(\beta -1){b}_{k}\right)\\ \quad -2\beta (2-\beta )(k+1)\displaystyle \sum _{i+j=k}(j+1){a}_{i}{a}_{j+1}+2(\beta -1)\\ \quad \left.\times \,\displaystyle \sum _{i+j=k}\left[(j+1){a}_{i}{b}_{j+1}+(i+1){a}_{i+1}{b}_{j}\right]\right],\\ \quad {a}_{k+3}=\displaystyle \frac{2}{(\beta -1)(k+3)(k+2)}\left[\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+4)\alpha }{2})}{a}_{k+1}-2\right.\\ \quad \left.\times \,\displaystyle \sum _{i+j=k}(j+1){a}_{i}{a}_{j+1}-\displaystyle \frac{1}{2}(k+2){b}_{k+2}\right].\end{array}\end{eqnarray}$
Therefore, the power series solutions of equations (1.2) are obtained in the following forms:
$\begin{eqnarray}\begin{array}{rcl}u(t,x) & = & {a}_{0}{t}^{-\tfrac{\alpha }{2}}+{a}_{1}{{xt}}^{-\alpha }+\displaystyle \frac{{x}^{2}{t}^{-\tfrac{3\alpha }{2}}}{\beta -1}\left[\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{3\alpha }{2})}{a}_{0}-2{a}_{0}{a}_{1}-\displaystyle \frac{1}{2}{b}_{1}\right]\\ \Space{0ex}{1.04em}{0ex} & + & \displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{2{x}^{k+2}{t}^{-\tfrac{(k+3)\alpha }{2}}}{(\beta -1)(k+3)(k+2)}\\ \Space{0ex}{1.04em}{0ex} & \times & \left[\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+4)\alpha }{2})}{a}_{k+1}-2\displaystyle \sum _{i+j=k}(j+1){a}_{i}{a}_{j+1}\right.\\ \Space{0ex}{1.04em}{0ex} & & \left.-\,\displaystyle \frac{1}{2}(k+2){b}_{k+2}\right],\\ \Space{0ex}{1.04em}{0ex}v(t,x) & = & {b}_{0}{t}^{-\alpha }+{b}_{1}{{xt}}^{-\tfrac{3\alpha }{2}}+\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{x}^{k+1}{t}^{-\tfrac{(k+2)\alpha }{2}}}{(k+2)(k+1)}\\ \Space{0ex}{1.04em}{0ex} & \times & \left[\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{(k+4)\alpha }{2})}\left(\beta (2-\beta )(k+1){a}_{k+1}\right.\right.\\ \Space{0ex}{1.04em}{0ex} & & \left.-\,(\beta -1){b}_{k}\right)-2\beta (2-\beta )(k+1)\displaystyle \sum _{i+j=k}(j+1){a}_{i}{a}_{j+1}\\ & & \left.+\,2(\beta -1)\displaystyle \sum _{i+j=k}\left[(j+1){a}_{i}{b}_{j+1}+(i+1){a}_{i+1}{b}_{j}\right]\right].\end{array}\end{eqnarray}$

Theorem 3.2. The power series solutions (3.17) are convergent in a neighborhood of the point $(0,| {a}_{0}| ,| {b}_{0}| )$.

Proof. From (3.16), we can obtain

$\begin{eqnarray}\begin{array}{l}| {b}_{k+2}| \leqslant \displaystyle \frac{1}{(k+2)(k+1)}\left[\displaystyle \frac{\left|{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})\right|}{| }{\rm{\Gamma }}\right.\\ \quad \times \,\left(1-\displaystyle \frac{(k+4)\alpha }{2}\right)| =\left(| \beta (2-\beta )| (k+1)| {a}_{k+1}| +| (\beta -1)| | {b}_{k}| \right)\\ \quad +\,2| \beta (2-\beta )| (k+1)\displaystyle \sum _{i+j=k}(j+1)| {a}_{i}| | {a}_{j+1}| +2| (\beta -1)| \\ \quad \left.\times \,\displaystyle \sum _{i+j=k}\left[(j+1)| {a}_{i}| | {b}_{j+1}| +(i+1)| {a}_{i+1}| | {b}_{j}| \right]\right],\\ \quad | {a}_{k+3}| \leqslant \displaystyle \frac{2}{| \beta -1| (k+3)(k+2)}\left[\displaystyle \frac{\left|{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})\right|}{\left|{\rm{\Gamma }}\left(1-\tfrac{(k+4)\alpha }{2}\right)\right|}| {a}_{k+1}| +2\right.\\ \quad \left.\times \,\displaystyle \sum _{i+j=k}(j+1)| {a}_{i}| | {a}_{j+1}| +\displaystyle \frac{1}{2}(k+2)| {b}_{k+2}| \right].\end{array}\end{eqnarray}$
From the properties of the Gamma function, it can easily be found that $\tfrac{\left|{\rm{\Gamma }}(1-\tfrac{(k+2)\alpha }{2})\right|}{\left|{\rm{\Gamma }}(1-\tfrac{(k+4)\alpha }{2})\right|}\leqslant 1$ for arbitrary natural number k. Thus, (3.18) can be written as
$\begin{eqnarray}\begin{array}{rcl}| {b}_{k+2}| & \leqslant & M\left[| {a}_{k+1}| +| {b}_{k}| \right.\\ & & \left.+\displaystyle \sum _{i+j=k}\left(| {a}_{i}| | {a}_{j+1}| +| {a}_{i}| | {b}_{j+1}| +| {a}_{i+1}| | {b}_{j}| \right)\right],\\ | {a}_{k+3}| & \leqslant & M\left[\left|{a}_{k+1}| +\displaystyle \sum _{i+j=k}| {a}_{i}| | {a}_{j+1}| +| {b}_{k+2}\right|\right].\end{array}\end{eqnarray}$
where
$\begin{eqnarray*}M=\max \left\{\tfrac{2| \beta (2-\beta )| (k+1)}{k\,+\,2},\tfrac{2| \beta -1| }{k\,+\,2},\tfrac{4}{| \beta -1| (k+3)(k+2)},\tfrac{1}{| \beta -1| (k+3)}\right\}\end{eqnarray*}$
.

Consider another power series

$\begin{eqnarray}P(\omega )=\displaystyle \sum _{k=0}^{\infty }{p}_{k}{\omega }^{k},\ Q(\omega )=\displaystyle \sum _{k=0}^{\infty }{q}_{k}{\omega }^{k},\end{eqnarray}$
where ${p}_{0}=| {a}_{0}| ,{p}_{1}=| {a}_{1}| ,{q}_{0}=| {b}_{0}| ,{q}_{1}=| {b}_{1}| $ and
$\begin{eqnarray}\begin{array}{rcl}{p}_{2} & = & | {a}_{2}| =\displaystyle \frac{1}{\beta -1}\left[\displaystyle \frac{{\rm{\Gamma }}(1-\tfrac{\alpha }{2})}{{\rm{\Gamma }}(1-\tfrac{3\alpha }{2})}{a}_{0}\right.\\ & & \left.-2{a}_{0}{a}_{1}-\displaystyle \frac{1}{2}{b}_{1}\right],\ k\geqslant 0.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{k+2} & = & M\left[{p}_{k+1}+{q}_{k}+\displaystyle \sum _{i+j=k}\right.\\ & & \left.\times \left({p}_{i}{p}_{j+1}+{p}_{i}{q}_{j+1}+{p}_{i+1}{q}_{j}\right)\right],\ k\geqslant 0,\\ {p}_{k+3} & = & M\left[{p}_{k+1}+\displaystyle \sum _{i+j=k}{p}_{i}{p}_{j+1}+{q}_{k+2}\right],\ k\geqslant 0.\end{array}\end{eqnarray}$
Therefore, it is easily seen that $| {a}_{k}| \leqslant {p}_{k}$ and $| {b}_{k}| \leqslant {q}_{k}$ for $k=0,1,2,\ldots $, that is, the power series (3.20) are the majorant series of (3.10). We next show that the power series (3.20) are convergent. By simple calculation, we can get
$\begin{eqnarray}\begin{array}{l}P(\omega )={p}_{0}+{p}_{1}\omega +{p}_{2}{\omega }^{2}+M\left((P(\omega )-{p}_{0}){\omega }^{2}\right.\\ \quad \left.+\,P(\omega )(P(\omega )-{p}_{0}){\omega }^{2}+(B(\omega )-{p}_{0}-{p}_{1}\omega )\omega \right).\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}Q(\omega )={q}_{0}+{q}_{1}\omega +M\left((P(\omega )-{p}_{0})\omega +Q(\omega ){\omega }^{2}\right.\\ \quad +\,P(\omega )(P(\omega )-{p}_{0})\omega \\ \quad \left.+\,P(\omega )(Q(\omega )-{q}_{0})\omega +Q(\omega )(P(\omega )-{p}_{0})\omega \right).\end{array}\end{eqnarray}$
Consider the following implicit function with respect to the independent variable ω:
$\begin{eqnarray}\begin{array}{l}F(\omega ,P,Q)=P-{p}_{0}-{p}_{1}\omega -{p}_{2}{\omega }^{2}-M\left((P-{p}_{0})\omega \right.\\ \quad \left.+\,Q{\omega }^{2}+P(P-{p}_{0})\omega \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}G(\omega ,P,Q)=Q-{q}_{0}-{q}_{1}\omega -M\left((P-{p}_{0})\omega \right.\\ \quad \left.+\,Q{\omega }^{2}+P(P-{p}_{0})\omega +P(Q-{q}_{0})\omega +Q(P-{p}_{0})\omega \right),\end{array}\end{eqnarray}$
which are analytic in a neighborhood of $(0,{p}_{0},{q}_{0})$, and $F(0,{p}_{0},{q}_{0})=0$, $G(0,{p}_{0},{q}_{0})=0$, $\tfrac{\partial (F,G)}{\partial (P,Q)}{| }_{(0,{p}_{0},{q}_{0})}=1\ne 0$. Therefore, by the implicit function theorem, the power series (3.20) are analytic in neighborhood of the point $(0,{p}_{0},{q}_{0})$. It implies that the power series solution (3.17) are convergent in a neighborhood of the point $(0,| {a}_{0}| ,| {b}_{0}| )$. This completes the proof. □

Tables 1 and 2 show some values of an and bn in (3.16) with the condition β = 3, while figures 1 and 2 illustrate the dynamic behavior for the power series solutions (3.17) with the initial values a0 = a1 = 1, b0 = b1 = 1 and some different fractional orders α. It indicates that the fractional order α plays a crucial role in the local variation of the figures for the given initial and parameter values. Therefore, we can choose appropriate fractional order values in the model based on actual observation data and experimental data.
Figure 1. Dynamical profiles of the truncated power series solutions (3.17) with β = 1, a0 = a1 = 1, b0 = b1 = 1 and some different fractional orders α.
Figure 2. Dynamical profiles of the truncated power series solutions (3.17) with β = 1, a0 = a1 = 1, b0 = b1 = 1 and some different fractional orders α.
Table 1. Some of an for the truncated power series solution u(t, x).
a0 a1 a2 a3 a4
α = 0.15 1 1 −0.8104337241 −1.000060721 −0.6882762144
α = 0.30 1 1 −0.9058173903 −1.158636393 −0.6658972502
α = 0.45 1 1 −1.033054432 −1.400905199 −0.6084301533
α = 0.60 1 1 −1.181778300 −1.722280257 −0.1780747299
α = 0.75 1 1 −1.332280764 −2.096613308 −1.082694871
α = 0.90 1 1 −1.454233642 −2.467063398 −0.7228800966
Table 2. Some of bn for the truncated power series solution v(t, x).
b0 b1 b2 b3 b4
α = 0.15 1 1 4.857403092 5.884508535 0.7363849760
α = 0.30 1 1 5.537013110 6.140177533 0.4687129811
α = 0.45 1 1 6.575307994 6.369986771 −0.3327863383
α = 0.60 1 1 7.952629674 5.357040900 −2.696865021
α = 0.75 1 1 9.556914180 9.981749594 0.6981510628
α = 0.90 1 1 11.14455742 9.305861500 −1.100442082

4. Conservation laws of equations (1.2)

In this section, we will construct conservation laws of equations (1.2) by using the generalization of the Noether operators and the new conservation theorem [42, 43].
The system (1.2) is denoted as
$\begin{eqnarray}\left\{\begin{array}{l}{F}_{1}={D}_{t}^{\alpha }u-\displaystyle \frac{\beta -1}{2}{u}_{{xx}}-2{{uu}}_{x}-\displaystyle \frac{1}{2}{v}_{x}=0,\\ {F}_{2}={D}_{t}^{\alpha }v-(1-\displaystyle \frac{\beta }{2})\beta {u}_{{xxx}}-2{\left({uv}\right)}_{x}-\displaystyle \frac{1-\beta }{2}{v}_{{xx}}=0,\end{array}\right.\end{eqnarray}$
of which the formal Lagrangian is given by
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & p(t,x){F}_{1}+q(t,x){F}_{2}=p(t,x)\\ & & \times \left({D}_{t}^{\alpha }u-\displaystyle \frac{\beta -1}{2}{u}_{{xx}}-2{{uu}}_{x}-\displaystyle \frac{1}{2}{v}_{x}\right)\\ & & +q(t,x)\left({D}_{t}^{\alpha }v-(1-\displaystyle \frac{\beta }{2})\beta {u}_{{xxx}}\right.\\ & & \left.-2{\left({uv}\right)}_{x}-\displaystyle \frac{1-\beta }{2}{v}_{{xx}}\right),\end{array}\end{eqnarray}$
where p(t, x) and q(t, x) are new dependent variables. The Euler–Lagrange operators are
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\delta }{\delta u} & = & \displaystyle \frac{\partial }{\partial u}+{\left({D}_{t}^{\alpha }\right)}^{* }\displaystyle \frac{\partial }{\partial ({D}_{t}^{\alpha }u)}\\ & + & \displaystyle \sum _{s=1}^{\infty }{\left(-1\right)}^{s}{D}_{{i}_{1}}\cdots {D}_{{i}_{s}}\displaystyle \frac{\partial }{\partial {u}_{{i}_{1}\cdots {i}_{s}}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\delta }{\delta v} & = & \displaystyle \frac{\partial }{\partial v}+{\left({D}_{t}^{\alpha }\right)}^{* }\displaystyle \frac{\partial }{\partial ({D}_{t}^{\alpha }v)}\\ & + & \displaystyle \sum _{s=1}^{\infty }{\left(-1\right)}^{s}{D}_{{i}_{1}}\cdots {D}_{{i}_{s}}\displaystyle \frac{\partial }{\partial {v}_{{i}_{1}\cdots {i}_{s}}},\end{array}\end{eqnarray}$
where ${\left({D}_{t}^{\alpha }\right)}^{* }$ is the adjoint operator of ${D}_{t}^{\alpha }$. It is defined by the right-sided of Caputo fractional derivative, i.e.
$\begin{eqnarray*}\begin{array}{l}{\left({D}_{t}^{\alpha }\right)}^{* }f(t,x){\equiv }_{t}^{c}{D}_{T}^{\alpha }f(t,x)=\,\left\{\begin{array}{ll}\displaystyle \frac{1}{{\rm{\Gamma }}(n-\alpha )}{\displaystyle \int }_{t}^{T}\displaystyle \frac{1}{{\left(t-s\right)}^{\alpha -n+1}}\displaystyle \frac{{\partial }^{n}}{\partial {s}^{n}}f(s,x){\rm{d}}s, & n-1\lt \alpha \lt n,n\in {\mathbb{N}}\\ {D}_{t}^{n}f(t,x), & \alpha =n\in {\mathbb{N}}.\end{array}\right.\end{array}\end{eqnarray*}$
The system of adjoint equations to (4.1) is given by
$\begin{eqnarray}\left\{\begin{array}{l}{F}_{1}^{* }=\displaystyle \frac{\delta { \mathcal L }}{\delta u}={\left({D}_{t}^{\alpha }\right)}^{* }p+(1-\displaystyle \frac{\beta }{2})\beta {q}_{{xxx}}-\displaystyle \frac{\beta -1}{2}{p}_{{xx}}+4{{pu}}_{x}+4{{qv}}_{x}+2{{up}}_{x}+2{{vq}}_{x}=0,\\ {F}_{2}^{* }=\displaystyle \frac{\delta { \mathcal L }}{\delta v}={\left({D}_{t}^{\alpha }\right)}^{* }q-\displaystyle \frac{1-\beta }{2}{q}_{{xx}}+2{{qu}}_{x}+2{{uq}}_{x}+\displaystyle \frac{1}{2}{p}_{x}=0.\end{array}\right.\end{eqnarray}$
Next we will use the above adjoint equations and the new conservation theorem to construct conservation laws of equation (1.2). From the classical definition of the conservation laws, a vector C = (Ct, Cx) is called the conserved vector for the governing equations if it satisfies the conservation equation ${\left[{D}_{t}{C}^{t}+{D}_{x}{C}^{x}\right]}_{{F}_{1},{F}_{2}=0}=0$. We can obtain the components of the conserved vector by using the generalization of the Noether operators.
Firstly, from the fundamental operator identity, i.e.
$\begin{eqnarray}\begin{array}{rcl}{prX} & + & {D}_{t}\tau \cdot { \mathcal I }+{D}_{x}\xi \cdot { \mathcal I }={W}^{u}\cdot \displaystyle \frac{\delta }{\delta u}\\ & + & {W}^{v}\cdot \displaystyle \frac{\delta }{\delta v}+{D}_{t}{{ \mathcal N }}^{t}+{D}_{x}{{ \mathcal N }}^{x},\end{array}\end{eqnarray}$
where prX is mentioned in (2.3), ${ \mathcal I }$ is the identity operator, and Wu = ητutξux, Wv = ζτvtξvx are the characteristics for group generator X, we can get the Noether operators as follows:
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal N }}^{t} & = & \tau { \mathcal I }+\displaystyle \sum _{k=0}^{n-1}{\left(-1\right)}^{k}{D}_{t}^{\alpha -1-k}({W}^{u}){D}_{t}^{k}\displaystyle \frac{\partial }{\partial ({D}_{t}^{\alpha }u)}\\ \Space{0ex}{1.04em}{0ex} & & -{\left(-1\right)}^{n}J({W}^{u},{D}_{t}^{n}\displaystyle \frac{\partial }{\partial ({D}_{t}^{\alpha }u)})\\ \Space{0ex}{1.04em}{0ex} & & +\displaystyle \sum _{k=0}^{n-1}{\left(-1\right)}^{k}{D}_{t}^{\alpha -1-k}({W}^{v}){D}_{t}^{k}\displaystyle \frac{\partial }{\partial ({D}_{t}^{\alpha }v)}\\ \Space{0ex}{1.04em}{0ex} & & -{\left(-1\right)}^{n}J({W}^{v},{D}_{t}^{n}\displaystyle \frac{\partial }{\partial ({D}_{t}^{\alpha }v)}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal N }}^{x} & = & \xi { \mathcal I }+{W}^{u}\left(\displaystyle \frac{\partial }{\partial {u}_{x}}-{D}_{x}\displaystyle \frac{\partial }{\partial {u}_{{xx}}}+{D}_{x}^{2}\displaystyle \frac{\partial }{\partial {u}_{{xxx}}}\right)\\ & & +{W}^{v}\left(\displaystyle \frac{\partial }{\partial {v}_{x}}-{D}_{x}\displaystyle \frac{\partial }{\partial {v}_{{xx}}}\right)\\ & & +{D}_{x}({W}^{u})\left(\displaystyle \frac{\partial }{\partial {u}_{{xx}}}-{D}_{x}\displaystyle \frac{\partial }{\partial {u}_{{xxx}}}\right)\\ & & +{D}_{x}({W}^{v})\displaystyle \frac{\partial }{\partial {v}_{{xx}}}+{D}_{x}^{2}({W}^{u})\displaystyle \frac{\partial }{\partial {u}_{{xxx}}},\end{array}\end{eqnarray}$
where n = [α] + 1 and J is given by
$\begin{eqnarray}J(f,g)=\displaystyle \frac{1}{{\rm{\Gamma }}(n-\alpha )}{\int }_{0}^{t}{\int }_{t}^{T}\displaystyle \frac{f(\tau ,x,y)g(\theta ,x,y)}{{\left(\theta -\tau \right)}^{\alpha +1-n}}{\rm{d}}\theta {\rm{d}}\tau .\end{eqnarray}$
The components of conserved vector are defined by
$\begin{eqnarray}{C}^{t}={{ \mathcal N }}^{t}{ \mathcal L },\ {C}^{x}={{ \mathcal N }}^{x}{ \mathcal L }.\end{eqnarray}$
Case 1: ${X}_{1}=\tfrac{\partial }{\partial x}$
The characteristics of X1 are
$\begin{eqnarray}{W}^{u}=-{u}_{x},\ {W}^{v}=-{v}_{x}.\end{eqnarray}$
Therefore, for 0 < α < 1,
$\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & -{{pD}}_{t}^{\alpha -1}{u}_{x}-J({u}_{x},{p}_{t})\\ & = & -{{pD}}_{t}^{\alpha -1}{v}_{x}-J({v}_{x},{p}_{t}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{C}^{x} & = & \left(2{up}+2{vq}-\displaystyle \frac{\beta -1}{2}{p}_{x}+(1-\displaystyle \frac{\beta }{2})\beta {q}_{{xx}}\right){u}_{x}\\ & & +\left(\displaystyle \frac{1}{2}p+2{{qu}}_{x}-\displaystyle \frac{1-\beta }{2}{q}_{x}\right){v}_{x}\\ & & +\left(\displaystyle \frac{\beta -1}{2}p-(1-\displaystyle \frac{\beta }{2})\beta {q}_{x}\right){u}_{{xx}}\\ & & +\displaystyle \frac{1-\beta }{2}{{qv}}_{{xx}}+(1-\displaystyle \frac{\beta }{2})\beta {{qu}}_{{xxx}}.\end{array}\end{eqnarray}$
Case 2: ${X}_{2}=t\tfrac{\partial }{\partial t}+\tfrac{\alpha }{2}x\tfrac{\partial }{\partial x}-\tfrac{\alpha }{2}u\tfrac{\partial }{\partial u}-\alpha v\tfrac{\partial }{\partial v}$
The characteristics of X2 are
$\begin{eqnarray}\begin{array}{rcl}{W}^{u} & = & -\displaystyle \frac{\alpha }{2}u-{{tu}}_{t}-\displaystyle \frac{\alpha }{2}{{xu}}_{x},\\ {W}^{v} & = & -\alpha v-{{tv}}_{t}-\displaystyle \frac{\alpha }{2}{{xv}}_{x}.\end{array}\end{eqnarray}$
Therefore, for 0 < α < 1,
$\begin{eqnarray}\begin{array}{rcl}{C}^{t} & = & -{{pD}}_{t}^{\alpha -1}\left(\displaystyle \frac{\alpha }{2}u+{{tu}}_{t}+\displaystyle \frac{\alpha }{2}{{xu}}_{x}\right)\\ & & -J\left(\displaystyle \frac{\alpha }{2}u+{{tu}}_{t}+\displaystyle \frac{\alpha }{2}{{xu}}_{x},{p}_{t}\right)\\ & & -{{pD}}_{t}^{\alpha -1}\left(\alpha v+{{tv}}_{t}+\displaystyle \frac{\alpha }{2}{{xv}}_{x}\right)\\ & & -J\left(\alpha v+{{tv}}_{t}+\displaystyle \frac{\alpha }{2}{{xv}}_{x},{p}_{t}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{C}^{x} & = & \left(2{up}+2{vq}-\displaystyle \frac{\beta -1}{2}{p}_{x}+\left(1-\displaystyle \frac{\beta }{2}\right)\beta {q}_{{xx}}\right)\\ & & \times \left(\displaystyle \frac{\alpha }{2}u+{{tu}}_{t}+\displaystyle \frac{\alpha }{2}{{xu}}_{x}\right)+\left(\displaystyle \frac{1}{2}p+2{{qu}}_{x}-\displaystyle \frac{1-\beta }{2}{q}_{x}\right)\\ & & \times (\alpha v+{{tv}}_{t}+\displaystyle \frac{\alpha }{2}{{xv}}_{x})+\left(\displaystyle \frac{\beta -1}{2}p-\left(1-\displaystyle \frac{\beta }{2}\right)\beta {q}_{x}\right)\\ & & \times \left(\alpha {u}_{x}+{{tu}}_{{xt}}+\displaystyle \frac{\alpha }{2}{{xu}}_{{xx}}\right)\\ & & +\displaystyle \frac{1-\beta }{2}q\left(\displaystyle \frac{3\alpha }{2}{v}_{x}+{{tv}}_{{xt}}+\displaystyle \frac{\alpha }{2}{{xv}}_{{xx}}\right)\\ & & +\left(1-\displaystyle \frac{\beta }{2}\right)\beta q\left(\displaystyle \frac{3\alpha }{2}{u}_{{xx}}+{{tu}}_{{xxt}}+\displaystyle \frac{\alpha }{2}{{xu}}_{{xxx}}\right).\end{array}\end{eqnarray}$

5. Conclusion

This paper shows that the Lie symmetry analysis method is effective to solve nonlinear fractional partial differential equations. We obtained all the Lie symmetries of nonlinear time-fractional Boussinesq–Burgers equations and used them to reduce the equations, thereby getting some solutions, including the convergent power series solutions. Inspired by this, our next step is to apply the Lie symmetry analysis method to high-dimensional nonlinear fractional partial differential equations and stochastic fractional partial differential equations.

Declarations

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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