Abstract Finding exact solutions for Riemann–Liouville (RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions. Keywords：Riemann–Liouville derivative;exact solution;fractional differential equation;separation of variables method
PDF (221KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Cheng-Shi Liu. Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables. Communications in Theoretical Physics, 2020, 72(5): 055006- doi:10.1088/1572-9494/ab7ed1
1. Introduction
Although there have been a large number of studies on fractional calculus (see, for example [1, 2], and the references therein), it is still very difficult to find exact solutions for Riemann–Liouville (RL) fractional differential equations. Among all fractional derivative definitions, the RL derivative is the most basic and important. Some authors try to give new definitions of the fractional derivatives to satisfy the routine rules in classical analysis (a detailed analysis can be seen in [3]). However, there are counterexamples to show that for these new definitions the corresponding rules such as the Jumarie formulas [4] are not correct [5, 6]. Therefore, those results on exact solutions for the fractional differential equations under the meaning of Jumarie’s derivative are also wrong (see the references in [5]). An important result is Tarasov’s theorem, which shows that no violation of the Leibnitz rule will be no fractional derivative for differentiable functions [7]. This also means that we need to study every non-differentiable function and ask whether or not they satisfy the Leibnitz rule under some new definitions of the fractional derivatives. Among these, based on the Riemann–Liouville definition, Kolvankar and Gangal defined a local fractional derivative [8, 9] for which the usual derivative rules hold. However, we have proven strictly the nonexistence of the nontrivial local fractional derivative of non-differentiable functions on an interval [10]. Therefore, the local fractional derivative is in general nonapplicable in practice. But, recently, many authors have tried to use Jumarie’s formulas to solve the fractional differential equations, and also used the local fractional derivative to simulate the practice problems and solve the exact solutions for these models.
For RL fractional differential equations, there are few results on exact solutions. This is because the RL derivatives of general elementary functions are no longer elementary functions except for a few functions such as the power function. The invariant subspace method is a routine approach to solving exact solutions. In [11–15], the authors used invariant subspace and the Lie group method to provide some interesting and important results. In [16], Wu and Rui used a special method of separation of variables and a homogenous balance method to solve RL fractional differential equations. In [17] and [18], Rui used an invariant subspace and first integral method and a homogenous balance method to solve more fractional dynamical systems. In [19], He and Zhao proposed an expansion method based on the sub-equation and homogenous balance principle and got some interesting exact solutions. For more analysis and applications of the invariant space method, we can read Ma’s papers [20] and [21]. Essentially, the invariant subspace method is a kind of separation of variables method for nonlinear differential equations. However, we do not have a first principle to construct an invariant subspace for a concrete equation. In practice, we need a trial-and-error procedure to find such an invariant subspace. Thus a more convenient way is to use a direct separation of variables method to find the exact solutions for considered fractional differential equations. Perhaps we can say that the separation of variables method is an almost unique method to find exact solutions for RL fractional differential equations. However, for nonlinear RL fractional differential equations, the scope of application of the separation of variables method is still an open problem which needs further study.
In the paper, we propose a general method of separation of variables for solving the RL fractional differential equations and differential-difference equations, and then provide several general results using the method. Our separation of variable method can be considered as an extension of the method in [16]. In particular, we find that the separation of variables method is more suitable to solve high-dimensional problems. As an application, we give nontrivial exact solutions to some interesting RL fractional equations such as the (2+1)-dimensional fractional Kadomtsev–Petviashvili (KP) equation, the fractional Langmuir chain equation and so forth. Our given solutions include not only rational functions solutions, but also implicit function solutions in terms of trigonometric function and logarithmic function. Compared with the methods in [16] and [19], our method provides a general variable separation scheme by which more complicated solutions can be found directly without assuming the forms of solutions. For example, for a nonlinear RL action–diffusion equation, we get two solutions with the forms of implicated function in terms of logarithm and arctangent functions (see application 1.1).
The paper is organized as follows. In section 2, we give the outline of a general separation of variables method for RL fractional differential equations. In section 3, we study exact solutions of equation ${D}_{t}^{\alpha }u=H(u,{u}_{x},{u}_{{xx}},\cdots )$ where H is a homogenous polynomial. In section 4, we study the equation ${D}_{t}^{\alpha }u={H}_{1}(u,{u}_{x},{u}_{{xx}},\cdots )+{H}_{2}(u,{u}_{x},{u}_{{xx}},\cdots )$ where H1 and H2 are two homogenous polynomials with different orders. In section 5, we study exact solutions of fractional differential-difference equations. In the last section, we give a short discussion.
2. Outline of the separation of variables method
In the whole paper, for brevity, we take the RL fractional derivative for $0\lt \alpha \lt 1$ as follows$\begin{eqnarray}{D}^{\alpha }f(t)=\displaystyle \frac{1}{{\rm{\Gamma }}(1-\alpha )}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\int }_{0}^{t}\displaystyle \frac{f(s)}{{\left(t-s\right)}^{\alpha }}{\rm{d}}s.\end{eqnarray}$Consider the fractional evolution equation$\begin{eqnarray}\begin{array}{rcl}{D}_{t}^{\alpha }u & = & {H}_{1}(u,{u}_{x},{u}_{{xx}},\cdots )+{H}_{2}(u,{u}_{x},{u}_{{xx}},\cdots )\\ & & +\cdots +{H}_{m}(u,{u}_{x},{u}_{{xx}},\cdots ),\end{array}\end{eqnarray}$where Hi for $i=1,\cdots ,\,m$ are homogenous polynomials with different orders ${n}_{i},x$ is in general a vector(for brevity, we take x as a real variable). We take the solution as $u=f(t)v(x)$. Then the equation can be reduced to the following form$\begin{eqnarray}\begin{array}{rcl}{D}_{t}^{\alpha }f(t)v(x) & = & {f}^{{n}_{1}}(t){H}_{1}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )\\ & & +{f}^{{n}_{2}}(t){H}_{2}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )\\ & & +\cdots +{f}^{{n}_{m}}(t){H}_{m}(v,v^{\prime} ,v^{\prime\prime} ,\cdots ).\end{array}\end{eqnarray}$Now we take for some i$\begin{eqnarray}{D}_{t}^{\alpha }f(t)=\lambda {f}^{{n}_{i}}(t),\end{eqnarray}$$\begin{eqnarray}{H}_{i}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )=\lambda v,\end{eqnarray}$$\begin{eqnarray}{H}_{j}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )=0,j\ne i.\end{eqnarray}$By using the formula ${D}^{\alpha }{t}^{\gamma }=\tfrac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{t}^{\gamma -\alpha }$, from (4) and (5), we get$\begin{eqnarray}f(t)={{At}}^{\tfrac{\alpha }{1-{n}_{i}}},\lambda =\displaystyle \frac{{\rm{\Gamma }}\left(1+\tfrac{\alpha }{1-{n}_{i}}\right)}{{\rm{\Gamma }}\left(1+\tfrac{\alpha }{1-{n}_{i}}-\alpha \right)}{A}^{1-{n}_{i}}.\end{eqnarray}$In general, we can usually take A=1. Then what we need is to solve v from equations (5) and (6). In the next sections, we will give concrete examples to show these computations and applications of the separation of variables method.
3. Homogenous polynomial cases: exact solutions for the nonlinear fractional reaction–diffusion equation
Consider the fractional evolution equation$\begin{eqnarray}{D}_{t}^{\alpha }u=H(u,{u}_{x},{u}_{{xx}},\cdots ),\end{eqnarray}$where $H$ is a homogenous polynomial of $n$ th order. Then the equation can be reduced to$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}\left(1-\tfrac{\alpha }{n-1}\right)}{{\rm{\Gamma }}\left(1-\tfrac{n}{n-1}\alpha \right)}v(x)=H(v,v^{\prime} ,v^{\prime\prime} ,\cdots ).\end{eqnarray}$
By separation of variables method, letting$\begin{eqnarray}u(x,t)=f(t)v(x)\end{eqnarray}$and substituting it into equation (2), we get $f(t)={t}^{\gamma }$ and$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{t}^{\gamma -\alpha }v(x)={t}^{n\gamma }H(v,v^{\prime} ,v^{\prime\prime} ,\cdots ).\end{eqnarray}$We take $\gamma -\alpha =n\gamma $, that is $\gamma =\tfrac{\alpha }{1-n}$, and eliminate t from two sides of the above equation, we get the conclusion.
Therefore, for these kinds of equations, we only need to solve the usual ordinary differential equation (9).
As a nontrivial example, we consider the nonlinear time-fractional reaction–diffusion equation$\begin{eqnarray}{D}_{t}^{\alpha }u={{au}}^{3}{u}_{{xx}}.\end{eqnarray}$This kind of fractional equation can be used to describe anomalous diffusion based on a continuous time random walk model with incorporated memory effects [22].
Taking $u=f(t)v(x)$ and substituting it into equation gives $f(t)={t}^{\gamma }$ and$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{t}^{\gamma -\alpha }v={{at}}^{4\gamma }{v}^{2}v^{\prime\prime} .\end{eqnarray}$We take$\begin{eqnarray}\gamma -\alpha =4\gamma ,\end{eqnarray}$from which we get $\gamma =-\tfrac{\alpha }{3}$ and hence$\begin{eqnarray}v^{\prime\prime} =\displaystyle \frac{{\rm{\Gamma }}\left(1-\tfrac{\alpha }{3}\right)}{a{\rm{\Gamma }}\left(1-\tfrac{4\alpha }{3}\right){v}^{2}}.\end{eqnarray}$Integrating it yields$\begin{eqnarray}{\left(v^{\prime} \right)}^{2}=-\displaystyle \frac{2{\rm{\Gamma }}\left(1-\tfrac{\alpha }{3}\right)}{a{\rm{\Gamma }}\left(1-\tfrac{4\alpha }{3}\right)v}+{c}_{0}.\end{eqnarray}$Solving this equation, we give three solutions according to cases of ${c}_{0}=0,{c}_{0}\gt 0,{c}_{0}\lt 0$ as follows:$\begin{eqnarray}{u}_{1}={t}^{-\tfrac{\alpha }{2}}{\left\{-\displaystyle \frac{9{\rm{\Gamma }}\left(1-\tfrac{\alpha }{3}\right)}{2a{\rm{\Gamma }}\left(1-\tfrac{4\alpha }{3}\right)}\right\}}^{\tfrac{1}{3}}{\left(x-{x}_{0}\right)}^{\tfrac{2}{3}},\end{eqnarray}$$\begin{eqnarray}{u}_{k}={t}^{-\tfrac{\alpha }{2}}\displaystyle \frac{{\rm{\Gamma }}\left(1-\tfrac{\alpha }{3}\right)}{a{\rm{\Gamma }}\left(1-\tfrac{4\alpha }{3}\right)({w}_{k}^{2}-{c}_{0})},\end{eqnarray}$where wk are the implicated function forms which satisfy respectively for k=2, 3$\begin{eqnarray}\displaystyle \frac{2{w}_{2}}{{w}_{2}^{2}-{c}_{0}}+\displaystyle \frac{1}{\sqrt{{c}_{0}}}\mathrm{ln}\left|\displaystyle \frac{{w}_{2}-\sqrt{{c}_{0}}}{{w}_{2}+\sqrt{{c}_{0}}}\right|=\pm \displaystyle \frac{{c}_{0}a{\rm{\Gamma }}\left(1-\tfrac{4\alpha }{3}\right)}{{\rm{\Gamma }}\left(1-\tfrac{\alpha }{3}\right)}(x-{x}_{0})\end{eqnarray}$and$\begin{eqnarray}\arctan \displaystyle \frac{{w}_{3}}{\sqrt{-{c}_{0}}}+\displaystyle \frac{\sqrt{-{c}_{0}}{w}_{3}}{{w}_{3}^{2}-{c}_{0}}=\pm \displaystyle \frac{{\left(-{c}_{0}\right)}^{\tfrac{3}{2}}a{\rm{\Gamma }}\left(1-\tfrac{4\alpha }{3}\right)}{{\rm{\Gamma }}\left(1-\tfrac{\alpha }{3}\right)}(x-{x}_{0}).\end{eqnarray}$
The above exact solutions are the forms of non-power functions which cannot be easily obtained by other methods in general.
4. Non-homogenous polynomial cases: exact solutions for the fractional KP equation
If H is not a homogenous polynomial, we cannot use the conclusion in the section above. However, we can give a similar result.
Consider the fractional evolution equation$\begin{eqnarray}{D}_{t}^{\alpha }u={H}_{1}(u,{u}_{x},{u}_{{xx}},\cdots )+{H}_{2}(u,{u}_{x},{u}_{{xx}},\cdots ),\end{eqnarray}$where ${H}_{1}$ and ${H}_{2}$ are two homogenous polynomials with different orders $n$ and $m,x$ is in general a vector (for brevity, we take $x$ as a real variable). Then the equation can be reduced to the following equation system$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}\left(1-\tfrac{\alpha }{n-1}\right)}{{\rm{\Gamma }}\left(1-\tfrac{n}{n-1}\alpha \right)}v(x)={H}_{1}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )\end{eqnarray}$and$\begin{eqnarray}{H}_{2}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )=0\end{eqnarray}$or$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}\left(1-\tfrac{\alpha }{m-1}\right)}{{\rm{\Gamma }}\left(1-\tfrac{m}{m-1}\alpha \right)}v(x)={H}_{2}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )\end{eqnarray}$and$\begin{eqnarray}{H}_{1}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )=0.\end{eqnarray}$
By separation of variables method, letting$\begin{eqnarray}u(x,t)=f(t)v(x)\end{eqnarray}$and substituting it into equation (21), we get $f(t)={t}^{\gamma }$ and$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{t}^{\gamma -\alpha }v(x) & = & {t}^{n\gamma }{H}_{1}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )\\ & & +\,{t}^{m\gamma }{H}_{2}(v,v^{\prime} ,v^{\prime\prime} ,\cdots ).\end{array}\end{eqnarray}$We take $\gamma -\alpha =n\gamma $ or $\gamma -\alpha =m\gamma $, that is $\gamma =\tfrac{\alpha }{1-n}$ or $\gamma =\tfrac{\alpha }{1-m}$. Then we let$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{t}^{\gamma -\alpha }v(x)={t}^{n\gamma }{H}_{1}(v,v^{\prime} ,v^{\prime\prime} ,\cdots ),\end{eqnarray}$$\begin{eqnarray}{t}^{m\gamma }{H}_{2}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )=0\end{eqnarray}$or$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{t}^{\gamma -\alpha }v(x)={t}^{m\gamma }{H}_{2}(v,v^{\prime} ,v^{\prime\prime} ,\cdots ),\end{eqnarray}$$\begin{eqnarray}{t}^{n\gamma }{H}_{1}(v,v^{\prime} ,v^{\prime\prime} ,\cdots )=0\end{eqnarray}$and eliminate t from the above equations, we get the conclusion.
We consider the general fractional KP equation$\begin{eqnarray}{\left({D}_{t}^{\alpha }u+{{auu}}_{x}+{{bu}}_{{xxx}}\right)}_{x}={{cu}}_{{yy}},\end{eqnarray}$where $a,b,c$ are parameters. For $a=-6,b=1,c=\pm 3$, the corresponding equation is the usual fractional KP equation, while for $\alpha =1$, it becomes the KP equation. It is well known that the usual KP equation describes soliton motion in two-dimensional shallow water waves. Here the fractional KP equation shows the nonlocal time action which perhaps can be applied to describe wave propagation in a complex medium.
Taking $u=f(t)v(x,y)$ and substituting it into equation (32) gives $f(t)={t}^{\gamma }$ and$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{t}^{\gamma -\alpha }{v}_{x}+{{at}}^{2\gamma }{\left({{vv}}_{x}\right)}_{x}+{{bt}}^{\gamma }{v}_{{xxxx}}={{ct}}^{\gamma }{v}_{{yy}}.\end{eqnarray}$This is not a homogenous polynomial, so we cannot use theorem 1. By theorem 2, we can take$\begin{eqnarray}\gamma -\alpha =2\gamma \end{eqnarray}$and$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{{\rm{\Gamma }}(1+\gamma -\alpha )}{v}_{x}+a{\left({{vv}}_{x}\right)}_{x}=0,\end{eqnarray}$$\begin{eqnarray}{{bv}}_{{xxxx}}={{cv}}_{{yy}}.\end{eqnarray}$From equation (36), we have a solution$\begin{eqnarray}v=-\displaystyle \frac{{\rm{\Gamma }}(1+\gamma )}{a{\rm{\Gamma }}(1+\gamma -\alpha )}x+g(y),\end{eqnarray}$where g(y) is an arbitrary function. Here we assume $g(y)\in {C}^{2}$. Then substituting solution (37) into equation (35) gives$\begin{eqnarray}g^{\prime\prime} (y)=0.\end{eqnarray}$Therefore, we have$\begin{eqnarray}g(y)={c}_{0}y+{d}_{0},\end{eqnarray}$where c0 and d0 are two arbitrary constants. By equation (34), we get $\gamma =-\alpha $. Now we can give an exact solution for equation (32),$\begin{eqnarray}u(x,y,t)={t}^{-\alpha }\left\{-\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{a{\rm{\Gamma }}(1-2\alpha )}x+{c}_{0}y+{d}_{0}\right\}.\end{eqnarray}$
It is easily to see that theorem 2 can be generalized to the following form$\begin{eqnarray}\begin{array}{rcl}{D}_{t}^{\alpha }u & = & {H}_{1}(u,{u}_{x},{u}_{{xx}},\cdots )+{H}_{2}(u,{u}_{x},{u}_{{xx}},\cdots )\\ & & +\cdots +{H}_{N}(u,{u}_{x},{u}_{{xx}},\cdots ),\end{array}\end{eqnarray}$where ${H}_{i}^{{\prime} }s$ are homogenous polynomials with different orders.
From the above application, we can see that for high-dimensional space nonlinear problem, the separation of variables method is a more suitable tool to solve exact solutions because we have more freedom to find v.
5. Exact solutions to fractional differential-difference equations
It is more difficult to exactly solve differential-difference equations [23] because exact solutions of nonlinear difference equations cannot be represented in terms of elementary functions in general [24]. In this section, we give exact solutions for some nonlinear fractional differential-difference equations. Similarly, we can prove the following result.
Consider the fractional differential-difference equation$\begin{eqnarray}{D}_{t}^{\alpha }{u}_{n}=H(\cdots ,{u}_{n-1},{u}_{n},{u}_{n+1},\cdots ),\end{eqnarray}$where $H$ is a homogenous polynomial of $m$ order. Then the equation can be reduced to the difference equation$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}\left(1-\tfrac{1}{m-1}\right)}{{\rm{\Gamma }}\left(1-\tfrac{m}{m-1}\alpha \right)}{v}_{n}=H(\cdots ,{v}_{n-1},{v}_{n},{v}_{n+1},\cdots ).\end{eqnarray}$
We consider the fractional Langmuir chain equation$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{\alpha }{u}_{n}}{{\rm{d}}{t}^{\alpha }}={u}_{n}({u}_{n+1}-{u}_{n-1}).\end{eqnarray}$Taking ${u}_{n}=f(t){v}_{n}$, we get $f(t)={t}^{\gamma }$ and $\gamma =-\alpha $, and$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{{\rm{\Gamma }}(1-2\alpha )}={v}_{n+1}-{v}_{n-1}.\end{eqnarray}$We assume that ${v}_{n}={kn}+h$ where $k$ and $h$ are two undetermined constants. Substituting it into the above equation gives$\begin{eqnarray}k=\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{2{\rm{\Gamma }}(1-2\alpha )}.\end{eqnarray}$So the exact solution is given by$\begin{eqnarray}{u}_{n}(t)={t}^{-\alpha }\left(\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{2{\rm{\Gamma }}(1-2\alpha )}n+h\right),\end{eqnarray}$where $h$ is an arbitrary constant.
When $\alpha =1$, some exact solutions of the usual Langmuir chains equation were obtained in [15].
We consider fractional Toda lattice system$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{\alpha }{u}_{n}}{{\rm{d}}{t}^{\alpha }}={v}_{n-1}-{v}_{n},\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{\alpha }{v}_{n}}{{\rm{d}}{t}^{\alpha }}={v}_{n}({u}_{n}-{u}_{n+1}).\end{eqnarray}$Taking ${u}_{n}=f(t){A}_{n},{v}_{n}=g(t){B}_{n}$, we get $f(t)={t}^{{\gamma }_{1}}$ and $g(t)={t}^{{\gamma }_{2}}$, and ${\gamma }_{1}=-\alpha ,{\gamma }_{2}=-2\alpha $, and$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{{\rm{\Gamma }}(1-2\alpha )}{A}_{n}={B}_{n+1}-{B}_{n},\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{\rm{\Gamma }}(1-2\alpha )}{{\rm{\Gamma }}(1-3\alpha )}={A}_{n}-{A}_{n+1}.\end{eqnarray}$Solving the above system, we get$\begin{eqnarray}{A}_{n}(t)=-\displaystyle \frac{{\rm{\Gamma }}(1-2\alpha )}{{\rm{\Gamma }}(1-3\alpha )}n+h,\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{B}_{n}(t) & = & c+\left(\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{{\rm{\Gamma }}(1-2\alpha )}h+\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{2{\rm{\Gamma }}(1-3\alpha )}\right)n\\ & & -\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{2{\rm{\Gamma }}(1-3\alpha )}{n}^{2},\end{array}\end{eqnarray}$where c is an arbitrary constant. Therefore, the exact solutions are given by$\begin{eqnarray}{u}_{n}(t)={t}^{-\alpha }\left\{-\displaystyle \frac{{\rm{\Gamma }}(1-2\alpha )}{{\rm{\Gamma }}(1-3\alpha )}n+h\right\},\end{eqnarray}$$\begin{eqnarray}\begin{array}{rcl}{v}_{n}(t) & = & {t}^{-2\alpha }\left\{c+\left(\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{{\rm{\Gamma }}(1-2\alpha )}h+\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{2{\rm{\Gamma }}(1-3\alpha )}\right)n\right.\\ & & \left.-\displaystyle \frac{{\rm{\Gamma }}(1-\alpha )}{2{\rm{\Gamma }}(1-3\alpha )}{n}^{2}\right\}.\end{array}\end{eqnarray}$
In [13], some exact solutions for the fractional Toda lattice system were given by the invariant subspace method.
6. Conclusion
The separation of variables method is a powerful tool by which we study the exact solutions for fractional differential equations and fractional differential-difference equations under the meaning of the Riemann–Liouville derivative. By the properties of homogenous polynomials and the separation of variables method, we construct the nontrivial exact solutions for some interesting fractional equations. Of course, many similar equations can be dealt with by the method. Our results include more abundant types of functions than the existed results. In particular, we find that for high-dimensional nonlinear cases the separation of variables method is more suitable since we have more freedom to solve the corresponding equations system.
When using the separation of variables method for concrete nonlinear differential equations, in general, we can assume that the solution has the form of $u(x,t)={f}_{1}(t){g}_{1}(x)+\cdots +{f}_{m}(t){g}_{m}(x)$ where fi(t) and gi(x) are to be determined by some balance principles. This is often a difficult problem although we have achieved certain successes. On the other hand, we can firstly give concrete fi(t) and gi(x), and then study which nonlinear differential equations have such solutions $u(x,t)$ (see, for example [25], and references therein). For RL fractional differential equations, we can also study such classifications.
Recently, some further interesting applications of the invariant subspace method have been given in [26, 27] in which some complicated fractional partial differential equation systems were solved. It is worth using the direct separation of variables method to solve these equations in a further study.
Up to now, we can perhaps say that the separation of variables method (or its some modified versions) is the only known method for solving exact solutions for the RL equation. To find a new method will be challenging.
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,Department of Mathematics, 
