We state some notations and theorems. First, let us recall essentials of fractional calculus. The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order. It unifies and generalizes the notions of integer-order differentiation and *n*-fold integration. Besides the R-L definition of fractional derivatives, there are several other different definitions, such as the modified R-L (mR-L) derivative, the Grünwald-Letnikov derivative (G-L) and Caputo’s fractional derivative, and so on. For different circumstances, they can be used for dealing with different properties of physical models. For example, Caputo’s fractional derivative is related to initial value problems; generally speaking, the mR-L derivative is used to investigate exact and explicit solutions of some FDEs [36–38]. The modified Riemann-Liouville derivative was defined by Jumarie [39, 40].

**Definition 2.1**
${D}_{t}^{\alpha}f(t)=\{\begin{array}{cc}\frac{1}{\mathrm{\Gamma}(1-\alpha )}\frac{d}{dt}{\int}_{0}^{t}{(t-\xi )}^{-\alpha}(f(\xi )-f(0))\phantom{\rule{0.2em}{0ex}}d\xi ,\hfill & 0<\alpha <1,\hfill \\ {[{f}^{(n)}(t)]}^{(\alpha -n)},\hfill & n\le \alpha <n+1,n\ge 1,\hfill \end{array}$

(2)

where the Euler gamma function

$\mathrm{\Gamma}(z)$ is defined by the integral

$\mathrm{\Gamma}(z)={\int}_{0}^{\mathrm{\infty}}{e}^{-t}{t}^{z-1}\phantom{\rule{0.2em}{0ex}}dt,$

(3)

which converges in the right half of the complex plane $Re(z)>0$.

**Properties 2.1**
${D}_{t}^{\alpha}{t}^{\gamma}=\frac{\mathrm{\Gamma}(\gamma +1)}{\mathrm{\Gamma}(\gamma +1-\alpha )}{t}^{\gamma -\alpha},\phantom{\rule{1em}{0ex}}\gamma >0,$

(4)

${D}_{t}^{\alpha}[u(t)v(t)]=u(t){D}_{t}^{\alpha}v(t)+v(t){D}_{t}^{\alpha}u(t),$

(5)

${D}_{t}^{\alpha}\left[f(u(t))\right]={f}_{u}^{\prime}[u(t)]{D}_{t}^{\alpha}u(t)={D}_{u}^{\alpha}f[u(t)]{\left({u}_{t}^{\prime}\right)}^{\alpha},$

(6)

which will be used in the following sections.

In what follows, considering the general case of a nonlinear system of partial differential equations of order

*n* in

*p* independent and

*q* dependent variables [

1,

2,

4–

7]

${\mathrm{\Delta}}_{v}(x,{u}^{(n)})=0,\phantom{\rule{1em}{0ex}}v=1,2,3,\dots ,l,$

(7)

here

$x=({x}^{1},{x}^{2},\dots ,{x}^{p})$,

$u=({u}^{1},{u}^{2},\dots ,{u}^{q})$, and the derivatives of

*u* in reference to

*x* up to

*n*, where

${u}^{(n)}$ represents all the derivatives of

*u* of all orders from 0 to

*n*. Consider a one-parameter Lie group of infinitesimal transformations acting on system (7),

$\begin{array}{r}{\overline{x}}^{i}={x}^{i}+\u03f5{\xi}^{i}(x,u)+O\left({\u03f5}^{2}\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,p,\\ {\overline{u}}^{j}={u}^{j}+\u03f5{\eta}^{j}(x,u)+O\left({\u03f5}^{2}\right),\phantom{\rule{1em}{0ex}}j=1,\dots ,q,\end{array}$

(8)

where

*ϵ* is the parameter. The vector field

*V* is associated with the above group of transformations as follows:

$V=\sum _{i=1}^{p}{\xi}^{i}(x,u){\partial}_{{x}^{i}}+\sum _{\alpha =1}^{q}{\eta}^{\alpha}(x,u){\partial}_{u}^{\alpha}.$

(9)

The invariance of system (7) under the infinitesimal transformations leads to the invariance conditions

${pr}^{(n)}V\left[{\mathrm{\Delta}}_{v}(x,{u}^{(n)})\right]=0,\phantom{\rule{1em}{0ex}}v=1,2,3,\dots ,l,$

(10)

whenever

${\mathrm{\Delta}}_{v}(x,{u}^{(n)})=0$, where

${pr}^{(n)}V$ is called the

*n* th order prolongation of the infinitesimal generator given by

${pr}^{(n)}V=V+\sum _{\alpha =1}^{q}\sum _{J}{\eta}_{\alpha}^{J}(x,{u}^{(n)}){\partial}_{{u}_{j}^{\alpha}},$

(11)

where

$J=({j}_{1},{j}_{2},\dots ,{j}_{k})$, with

$1\le {j}_{k}\le p$,

$1\le k\le n$. The coefficient functions

${\eta}_{\alpha}^{J}$ of

${pr}^{(n)}V$ are given by the following formula:

${\eta}_{\alpha}^{J}(x,{u}^{(n)})={D}_{J}({\eta}_{\alpha}-\sum _{i=1}^{p}{\xi}^{i}{u}_{i}^{\alpha})+\sum _{i=1}^{p}{\xi}^{i}{u}_{J,i}^{\alpha},$

(12)

where ${u}_{i}^{\alpha}=\partial {u}^{\alpha}/\partial {x}^{i}$ and ${u}_{J,i}^{\alpha}=\partial {u}_{J}^{\alpha}/\partial {x}^{i}$.

However, for a nonlinear system of fractional partial differential equations (FPDEs), the prolongation formula will be different. Now, we present below brief details of the Lie symmetry analysis to FPDEs with respect to two independent variables.

Consider a scalar time FPDE having the following form [

21,

22]:

$\frac{{\partial}^{\alpha}u}{\partial {t}^{\alpha}}=F(x,t,u,{u}_{x},{u}_{xx},\dots ).$

(13)

If (13) is invariant under a one-parameter Lie group of point transformations

$\begin{array}{r}{t}^{\ast}=t+\u03f5\tau (x,t,u)+O\left({\u03f5}^{2}\right),\\ {x}^{\ast}=x+\u03f5\xi (x,t,u)+O\left({\u03f5}^{2}\right),\\ {u}^{\ast}=u+\u03f5\eta (x,t,u)+O\left({\u03f5}^{2}\right),\\ \frac{{\partial}^{\alpha}\overline{u}}{\partial {\overline{t}}^{\alpha}}=\frac{{\partial}^{\alpha}u}{\partial {t}^{\alpha}}+\u03f5{\eta}_{\alpha}^{0}(x,t,u)+O\left({\u03f5}^{2}\right),\\ \frac{\partial \overline{u}}{\partial \overline{x}}=\frac{\partial u}{\partial x}+\u03f5{\eta}^{x}(x,t,u)+O\left({\u03f5}^{2}\right),\\ \frac{{\partial}^{2}\overline{u}}{\partial {\overline{x}}^{2}}=\frac{{\partial}^{2}u}{\partial {x}^{2}}+\u03f5{\eta}^{xx}(x,t,u)+O\left({\u03f5}^{2}\right),\\ \vdots \end{array}$

(14)

where

$\begin{array}{rl}{\eta}_{\alpha}^{0}=& {D}_{t}^{\alpha}(\eta )+\xi {D}_{t}^{\alpha}({u}_{x})-{D}_{t}^{\alpha}(\xi {u}_{x})+{D}_{t}^{\alpha}({D}_{t}(\tau )u)-{D}_{t}^{\alpha +1}(\tau u)+\tau {D}_{t}^{\alpha +1}(u)\\ =& \frac{{\partial}^{\alpha}\eta}{\partial {t}^{\alpha}}+({\eta}_{u}-\alpha {D}_{t}(\tau ))\frac{{\partial}^{\alpha}u}{\partial {t}^{\alpha}}-u\frac{{\partial}^{\alpha}{\eta}_{u}}{\partial {t}^{\alpha}}+\mu \\ +\sum _{n=1}^{\mathrm{\infty}}[\left(\genfrac{}{}{0ex}{}{a}{n}\right)\frac{{\partial}^{\alpha}{\eta}_{u}}{\partial {t}^{\alpha}}-\left(\genfrac{}{}{0ex}{}{a}{n+1}\right){D}_{t}^{n+1}(\tau )]{D}_{t}^{\alpha -n}(u)\\ -\sum _{n=1}^{\mathrm{\infty}}\left(\genfrac{}{}{0ex}{}{a}{n}\right){D}_{t}^{n}(\xi ){D}_{t}^{\alpha -n}({u}_{x}),\end{array}$

(15)

where

$\mu =\sum _{n=2}^{\mathrm{\infty}}\sum _{m=2}^{n}\sum _{k=2}^{m}\sum _{r=0}^{k-1}\left(\genfrac{}{}{0ex}{}{a}{n}\right)\left(\genfrac{}{}{0ex}{}{n}{m}\right)\left(\genfrac{}{}{0ex}{}{k}{r}\right)\frac{1}{k!}\frac{{t}^{n-\alpha}}{\mathrm{\Gamma}(n+1-\alpha )}{[-u]}^{r}\frac{{\partial}^{m}}{\partial {t}^{m}}\left[{u}^{k-r}\right]\frac{{\partial}^{n-m+k}\eta}{\partial {t}^{n-m}\partial {u}^{k}}$

(16)

and

$\begin{array}{r}{\eta}^{x}={D}_{x}(\eta )-{u}_{x}{D}_{x}(\xi )-{u}_{t}{D}_{x}(\tau ),\\ {\eta}^{xx}={D}_{x}\left({\eta}^{x}\right)-{u}_{xt}{D}_{x}(\tau )-{u}_{xx}{D}_{x}(\xi ),\\ {\eta}^{xxx}={D}_{x}\left({\eta}^{xx}\right)-{u}_{xxt}{D}_{x}(\tau )-{u}_{xxx}{D}_{x}(\xi ),\\ \vdots \end{array}$

(17)

Here,

${D}_{x}$ denotes the total derivative operator and it is defined by

${D}_{x}=\frac{\partial}{\partial x}+{u}_{x}\frac{\partial}{\partial u}+{u}_{xx}\frac{\partial}{\partial {u}_{x}}+\cdots $

(18)

with the associated vector field of the form

$V=\tau (x,t,u)\frac{\partial}{\partial t}+\xi (x,t,u)\frac{\partial}{\partial x}+\eta (x,t,u)\frac{\partial}{\partial u},$

(19)

where the coefficient functions $\xi (x,t,u)$, $\tau (x,t,u)$, and $\eta (x,t,u)$ of the vector field are to be determined.

If vector field (19) generates a symmetry of (1), then

*V* must satisfy Lie’s symmetry condition

${pr}^{(n)}V({\mathrm{\Delta}}_{1}){|}_{{\mathrm{\Delta}}_{1}=0}=0,$

(20)

where ${\mathrm{\Delta}}_{1}=\frac{{\partial}^{\alpha}u}{\partial {t}^{\alpha}}-F(x,t,u,{u}_{x},{u}_{xx},\dots )$.