New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations

Abstract

The main purpose of this paper is to present a new approach to achieving analytical solutions of parameter containing fractional-order differential equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries of these equations are also obtained via a new formulation of an improved form of the Noether’s theorem. It is indicated that invariant solutions, reduced equations, perturbed or unperturbed symmetries and conservation laws can be obtained by applying a nonlinear self-adjoint notion. The method is applied to the time fractional-order Fokker–Planck equation. We obtained new results in a highly efficient and elegant manner.

1. Introduction

Fractional partial differential equations are a generalization of classical ordinary calculus with utilizations of integrals and derivatives with an arbitrary order. In the last decade, these equations were employed in various scientific and engineering phenomena including fluid mechanics, gas dynamics, nonlinear acoustics, biology, control theory, earthquake modeling, traffic flow models. There are several different types of fractional-order derivative and integral operators including the Riesz, Riemann–Liouville, Grünwald–Letnikov and Caputo fractional derivatives [1].

We are concerned with approximations using a small parameter of the Caputo and Riemann–Liouville type fractional derivative operators. Using this approximation, a fractional-order differential equation may be converted into an integer-order equation [2,3,4,5,6,7].

By the Lie symmetry techniques [8,9,10], we can obtain analytical solutions of many perturbed differential equations. Noether’s theorem which was introduced by Emmy Noether in 1918 describing general concepts related to symmetry groups and conservation laws is a useful tool in the solutions of perturbed differential equations, see, e.g., [11,12,13]. Finding approximate symmetries of perturbed partial dofferential equations was first introduced by Fushchich, Shtelen and Baikov [14,15]. Because of the importance of perturbed systems to describe the natural phenomena, they generalized the Noether’s theorem to approximated version. This generalization helps to find approximate conservation laws of a given system including the related topics [16,17]. For a system, approximate conservation laws is determined by approximate formal Lagrange and nonlinear self-adjointness for approximate equations [18]. We present conservation laws of fractional partial differential equations [19,20] with an effective method based on nonlinear self-adjointness.

The Fokker–Planck equations play an important role in fluid mechanics, control theory, astrophysics and quantum [21,22]. We are concerned with the perturbed fractional-order Fokker–Planck equation

$$\begin{array}{c}\hfill D_t^{\alpha }u-\frac{1}{2}{a}^2{u}_{xx}-bu-bx{u}_x+\epsilon u_t=0.\end{array}$$

(1)

In which a, b are constants and $D_t^{\alpha }$ is fractional derivative of order $\alpha$.

2. Approximation of Fractional-Order Operators

Definition 1.

The left and right-sided Riemann–Liouville fractional partial derivatives are defined as

$$\left({}_{a}D_{x^1}^{\alpha +k}u\right)\left(x\right)=\frac{1}{\Gamma (1-\alpha )}{\left(\frac{\partial }{\partial x^1}\right)}^{k+1}{\int }_a^{x^1}\frac{u(\xi ,x^2,\dots ,x^n)}{{(x^1-\xi )}^{\alpha }}d\xi ,$$

(2)

$$\left({}_{x^1}D_b^{\alpha +k}u\right)\left(x\right)=\frac{{(-1)}^{k+1}}{\Gamma (1-\alpha )}{\left(\frac{\partial }{\partial x^1}\right)}^{k+1}{\int }_{x^1}^b\frac{u(\xi ,x^2,\dots ,x^n)}{{(x^1-\xi )}^{\alpha }}d\xi .$$

(3)

Respectively in which $\Gamma (·)$ denotes the Gamma function and $\alpha \in (0,1), k=0,1,\dots ,m, m\in \mathbb{N}$.

Definition 2.

The left and right-sided Caputo type fractional partial derivative are defined as

$$\left({}_a{}^{C}D_{x^1}^{\alpha +k}u\right)\left(x\right)=\frac{1}{\Gamma (1-\alpha )}{\int }_a^{x^1}\frac{1}{{(x^1-\xi )}^{\alpha }}\frac{{\partial }^{k+1}u(\xi ,x^2,\dots ,x^n)}{\partial {\xi }^{k+1}}d\xi ,$$

(4)

$$\left({}_{x^1}{}^{C}D_b^{\alpha +k}u\right)\left(x\right)=\frac{{(-1)}^{k+1}}{\Gamma (1-\alpha )}{\int }_{x^1}^b\frac{1}{{(\xi -x^1)}^{\alpha }}\frac{{\partial }^{k+1}u(\xi ,x^2,\dots ,x^n)}{\partial {\xi }^{k+1}}d\xi .$$

(5)

Respectively in which $\Gamma (·)$ denotes the Gamma function and $\alpha \in (0,1), k=0,1,\dots ,m, m\in \mathbb{N}$.

For the natural numbers, $k,c,d,$ let $u\left(x\right):=u$ be the function of $x=(x^1,x^2,\dots ,x^n)\in {\mathbf{R}}^n,$ we consider an fractional differential equation in the form of

$$P\left(x,u,u_{\left(1\right)},\dots ,u_{\left(k\right)},{}_{a}D_{x^1}^{\alpha 0},{}_{a}D_{x^1}^{\alpha 1},\dots ,{}_{a}D_{x^1}^{{\alpha }_{\mathrm{d}}},{}_{x^1}D_b^{{\beta }_0},{}_{x^1}D_b^{{\beta }_1},\dots ,{}_{x^1}D_b^{{\beta }_c}\right)=0,$$

(6)

$0<{\alpha }_0<{\alpha }_1<\dots <{\alpha }_d,$ $0<{\beta }_0<{\beta }_1<\dots <{\beta }_c$.

The partial derivative of u is denoted as

$$u_{\left(s\right)}\equiv \left\{u_{i_1\dots i_s}\right\}=\left\{\frac{{\partial }^{s}u\left(x\right)}{\partial x^{i_1}\dots \partial x^{i_s}}\right\}, (i_1,\dots ,i_s=1,\dots n,s=1,\dots ,k).$$

If the orders of fractional differential Equation (6) are all nearly integers, then it is possible to approximate Equation (6):

$$P\left(x,u,u_{\left(1\right)},\dots ,u_{\left(r\right)},{}_{a}D_{x^1}^{\alpha },{}_{a}D_{x^1}^{\alpha +1},\dots ,{}_{a}D_{x^1}^{\alpha +d}{}_{x^1}D_b^{\alpha },{}_{x^1}D_b^{\alpha +1},\dots ,{}_{x^1}D_b^{\alpha +d}\right)=0.$$

(7)

In which $\alpha \in (0,1)$. Assuming $\alpha =\epsilon$ or $\alpha =1-\epsilon$ in Equation (7), we can turn the right and left-sided Riemann–Liouville fractional partial derivatives into a Taylor expansion having arbitrarily small parameter, $1>\epsilon >0.$

Supposing the existence of each derivative ${}_{a}D_{x^1}^{k+\epsilon }u, {}_{x^1}D_b^{k+\epsilon }u (k=0,1,\dots )$ or ${}_{a}D_{x^1}^{k-\epsilon }u,{}_{x^1}D_b^{k-\epsilon }u$ $(k=1,2,\dots )$ at arbitrary point $x^1\in (a,b)$, we have

$$\begin{array}{l}{}_{a}D_{x^1}^{k\pm \epsilon }u={\displaystyle \sum _{s=0}^{\infty }}\left(\genfrac{}{}{0pt}{}{k\pm \epsilon }{s}\right)\frac{{(x^1-a)}^{s-k\mp \epsilon }}{\Gamma (1-k+s\mp \epsilon )}\frac{{\partial }^{s}u(\xi ,x^2,\dots ,x^n)}{\partial {\xi }^s}\\ \hspace{1em}\hspace{1em}\hspace{1em}=\frac{{\partial }^{k}u}{\partial {\left(x^1\right)}^k}\pm \epsilon ([\psi (k+1)-ln(x^1-a)]\frac{{\partial }^{k}u}{\partial {\left(x^1\right)}^k}\\ \hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \sum _{s=0,s\ne k}^{\infty }}\frac{{(-1)}^{s-k}}{(s-k)}\frac{k!}{s!}{(x^1-a)}^{s-k}\frac{{\partial }^{s}u}{\partial {\left(x^1\right)}^s})+o\left(\epsilon \right),\end{array}$$

(8)

$$\begin{array}{l}{}_{x^1}D_b^{k\pm \epsilon }u=\frac{{\partial }^{k}u}{\partial {\left(x^1\right)}^k}\pm \epsilon ([\psi (k+1)-ln(b-x^1)]\frac{{\partial }^{k}u}{\partial {\left(x^1\right)}^k}\\ \hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \sum _{s=0,s\ne k}^{\infty }}\frac{{(-1)}^{s-k}}{(s-k)}\frac{k!}{s!}{(b-x^1)}^{s-k}\frac{{\partial }^{s}u}{\partial {\left(x^1\right)}^s})+o\left(\epsilon \right).\end{array}$$

(9)

Here, $\psi \left(z\right)=\frac{{\Gamma }^{\prime }\left(z\right)}{\Gamma \left(z\right)}$ is the digamma function and $\left(\genfrac{}{}{0pt}{}{k\pm \epsilon }{s}\right)=\frac{\Gamma (1+k\pm \epsilon )}{\Gamma (1+k-s\pm \epsilon )s!}$ is a binomial coefficient.

For the Caputo fractional derivative

$$\begin{array}{c}{}_a{}^{C}D_{x^1}^{k\pm \epsilon }u={}_{a}D_{x^1}^{k\pm \epsilon }u\mp \epsilon \sum _{s=0}^{k-1}{(-1)}^{s-k}(k-s-1)!{(x^1-a)}^{s-k}\frac{{\partial }^{s}u}{\partial {\left(x^1\right)}^s}{|}_{x^1=a}+p(x,a),\end{array}$$

(10)

$$\begin{array}{c}\hfill {}_{x^1}{}^{C}D_b^{k\pm \epsilon }u={}_{x^1}D_b^{k\pm \epsilon }u\mp \epsilon \sum _{s=0}^{k-1}{(-1)}^{s-k}(k-s-1)!{(b-x^1)}^{s-k}\frac{{\partial }^{s}u}{\partial {\left(x^1\right)}^s}{|}_{x^1=b}+q(x,b).\end{array}$$

(11)

In which

$$\begin{array}{c}\hfill p(x,a)=\left\{\begin{array}{cc}-[1+\epsilon (\psi \left(1\right)-ln(x^1-a))]\frac{{\partial }^{k}u}{\partial {\left(x^1\right)}^k}{|}_{x^1=a};\hfill & for {}_a{}^{C}D_{x^1}^{k+\epsilon }u,\hfill \\ 0;\hfill & for {}_a{}^{C}D_{x^1}^{k-\epsilon }u,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill q(x,b)=\left\{\begin{array}{cc}-[1+\epsilon (\psi \left(1\right)-ln(b-x^1))]\frac{{\partial }^{k}u}{\partial {\left(x^1\right)}^k}{|}_{x^1=b};\hfill & for {}_{x^1}{}^{c}D_b^{k+\epsilon }u,\hfill \\ 0;\hfill & for {}_{x^1}{}^{c}D_b^{k-\epsilon }u.\hfill \end{array}\right.\end{array}$$

Proposition 1.

Let F be a continuously differentiable function with respect to ${}_{a}D_{x^1}^{\alpha \pm k}u$ and ${}_{x^1}D_b^{\alpha \pm k}u$ $(k=0,1,\dots ,d)$. Then, for $\alpha =\epsilon$ or $\alpha =1-\epsilon$, we can approximate Equation (7) as follows:

$$\begin{array}{c}\hfill P_{\left(0\right)}(x,u,u_{\left(1\right)},\dots )+\epsilon P_{\left(1\right)}(x,u,u_{\left(1\right)},\dots ,D_{x^1}^{c+1}u,D_{x^1}^{c+2}u,\dots )\approx 0,\end{array}$$

(12)

in which $c=max\{d,r\}$ for $\alpha =1-\epsilon$ and $c=max\{d-1,r\}$ for $\alpha =\epsilon$.

3. Lie Group Analysis

We consider a differential operator of first order defined as

$$\begin{array}{l}X\approx X_{\left(0\right)}+\epsilon X_{\left(1\right)}\\ \hspace{1em}\hspace{1em}\equiv \left({\zeta }_{\left(0\right)}^i(x,u)+\epsilon {\zeta }_{\left(1\right)}^i(x,u)\right)\frac{\partial }{\partial x^i}+\left({\theta }_{\left(0\right)}(x,u)+\epsilon {\theta }_{\left(1\right)}(x,u)\right)\frac{\partial }{\partial u},\end{array}$$

(13)

in which

$$\begin{array}{c}{\zeta }_{\left(0\right)}^i(x,u)=\frac{\partial g_{\left(0\right)}^i(x,u,a)}{\partial a}{{|}_{a=0}, \hspace{1em}{\zeta }_{\left(1\right)}^i(x,u)=\frac{\partial g_{\left(1\right)}^i(x,u,a)}{\partial a}|}_{a=0},\hfill \\ {\theta }_{\left(0\right)}(x,u)=\frac{\partial h_{\left(0\right)}(x,u,a)}{\partial a}{{|}_{a=0},\hspace{1em} {\theta }_{\left(1\right)}(x,u)=\frac{\partial h_{\left(1\right)}(x,u,a)}{\partial a}|}_{a=0}.\hfill \end{array}$$

Calculating the solutions of

$$\begin{array}{c}\hfill X\left(P_{\left(0\right)}+\epsilon P_{\left(1\right)}\right){|}_{\left(12\right)}\approx 0,\end{array}$$

(14)

exact symmetries of the perturbed Equation (7) can be achieved.

$$\begin{array}{ccc}\hfill {\overline{x}}^i& \approx & g^i(x,u,a,\epsilon )\equiv g_{\left(0\right)}^i(x,u,a)+\epsilon g_{\left(1\right)}^i(a,u,a),\hfill \\ \hfill \overline{u}& \approx & h(x,u,a,\epsilon )\equiv h_{\left(0\right)}(x,u,a)+\epsilon h_{\left(1\right)}(a,u,a),\hfill \end{array}$$

(15)

with

$$\begin{array}{c}\hfill {\overline{x}}^i{|a=0\approx x^i, \overline{u}|}_{a=0}=u,\end{array}$$

are group of Lie point transformations under the group conditions

$$\begin{array}{c}{g}^i\left(g^1(x,u,a,\epsilon ),\dots ,g^n(x,u,a,\epsilon ),h(x,u,a,\epsilon ),b,\epsilon \right)\approx g^i(x,a+b,\epsilon );\hfill \\ h\left(g^1(x,u,a,\epsilon ),\dots ,g^n(x,u,a,\epsilon ),h(x,u,a,\epsilon ),b,\epsilon \right)\approx h(x,a+b,\epsilon ),\hfill \end{array}$$

by $o\left(\epsilon \right)$.

4. Classification of Group-Invariant Solution

We present the optimal system of approximate Fokker–Planck equation symmetries [23] by employing the fact that every s-dimensional subalgebra is equivalent to a unique member of the optimal system with an adjoint representation. If we know the infinitesimal adjoint action $ad\mathbf{g}$ of a Lie algebra $\mathbf{g}$ on itself, we can reconstruct the adjoint representation $AdG$ of the underlying Lie group.

$$\begin{array}{c}\hfill \frac{dX}{d\epsilon }{=adY|}_X,\hspace{1em}X\left(0\right)=X_0,\end{array}$$

with solution

$$\begin{array}{c}\hfill X\left(\epsilon \right)=Ad\left(exp\left(\epsilon Y\right)\right)X_0,\end{array}$$

where

$$\begin{array}{c}\hfill Ad\left(exp\left(\epsilon Y\right)\right)X_0=\sum _{n=0}^{\infty }\frac{{\epsilon }^n}{n!}{\left(adY\right)}^n\left(X_0\right)=X_0-\epsilon [Y,X_0]+\frac{{\epsilon }^2}{2}[Y,[Y,X_0]]-\dots .\end{array}$$

It is clear that $[Xi,Xj]$ is the usual commutator and $\epsilon$ is a parameter.

Optimal System and Exact Solutions

Consider the perturbed fractional-order Fokker–Planck equation

$$\begin{array}{c}\hfill {}_{0}D_t^{\alpha }u=\frac{1}{2}{a}^2{u}_{xx}+bu+bx{u}_x-\epsilon u_t, u=u(x,t), \alpha \in (0,1).\end{array}$$

(16)

In order to calculate the approximate symmetries of the perturbed fractional equation, we apply the extension of Equation (8) to Equation (16). Setting $\alpha =1-\epsilon$, we can write Equation (16) as

$$\begin{array}{l}{P}_{\left(0\right)}+\epsilon P_{\left(1\right)}=u_t-\frac{1}{2}{a}^2{u}_{xx}-bu-bx{u}_x\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\epsilon \left[(lnt+\nu )u_t+u+\sum _{k=1}^{\infty }\frac{{(-t)}^k}{k(k+1)!}{u}_t^{(k+1)}\right]+\epsilon u_t=0.\end{array}$$

(17)

We get symmetries of perturbed equation Equation (17) using the Maple software.

$$\begin{array}{c}{X}_1={\partial }_t, X_2=u{\partial }_u, X_3=e^{-bt}{\partial }_x, X_4=e^{bt}{\partial }_x-\frac{2bxu}{a^2}{e}^{bt}{\partial }_u,\hfill \\ X_5=e^{-2bt}{\partial }_t-bx{e}^{-2bt}{\partial }_x+bu{e}^{-2bt}{\partial }_u, X_6=bx{e}^{2bt}{\partial }_t+bx{e}^{2bt}{\partial }_x-\frac{2b^2{x}^{2}u}{a^2}{e}^{2bt}{\partial }_u,\hfill \\ X_7=e^{bt+\frac{c_{1}t}{2}-\frac{b}{a^2}{x}^2}KummerM(\frac{4b+c_1}{4b},\frac{3}{2},\frac{b}{a^2}{x}^2){\partial }_u,\hfill \\ X_8=e^{bt+\frac{c_{1}t}{2}-\frac{b}{a^2}{x}^2}KummerU(\frac{4b+c_1}{4b},\frac{3}{2},\frac{b}{a^2}{x}^2){\partial }_u.\hfill \\ Y_1=\frac{\epsilon }{a^2}{\partial }_t, Y_2=\epsilon u{\partial }_u, Y_3=\frac{-\epsilon }{a^2}{e}^{-bt}{\partial }_x, Y_4=\frac{-\epsilon }{a^2}{e}^{bt}\left({\partial }_x+2bxu{\partial }_u\right),\hfill \\ Y_5=\frac{\epsilon }{a^2}{e}^{-2bt}\left({\partial }_t+bx{\partial }_x+b{a}^{2}u{\partial }_u\right), Y_6=\frac{\epsilon }{a^2}{e}^{2bt}\left({\partial }_t-bx{\partial }_x-2b^2{x}^{2}u{\partial }_u\right),\hfill \\ Y_7=x\epsilon e^{t{c}_1-\frac{b{x}^2}{a^2}}KummerM(\frac{b+c_1}{2b},\frac{3}{2},\frac{b}{a^2}{x}^2){\partial }_u,\hfill \\ Y_8=x\epsilon e^{t{c}_1-\frac{b{x}^2}{a^2}}KummerU(\frac{b+c_1}{2b},\frac{3}{2},\frac{b}{a^2}{x}^2){\partial }_u.\hfill \end{array}$$

(18)

where the Kummer functions, $KummerM(\mu ,\nu ,z)$ and $KummerU(\mu ,\nu ,z)$ solve the differential equation $z{y}^{″}+(\nu -z)y^{\prime }-\mu y=0$.

By the possession of infinitesimal generators (18), a number of adjoint representations are given as

$$\begin{array}{c}Ad[X_1,X_j]=X_j, j=1,\dots ,5, Ad[X_i,X_i]=X_i, i=1,\dots ,5,\hfill \\ Ad[X_2,X_1]=X_1-\epsilon b{X}_3, Ad[X_2,X_4]=\frac{2\epsilon b}{a^2}{X}_2+X_4,\hfill \\ Ad[X_3,X_1]=X_1-\epsilon b{X}_4, Ad[X_3,X_4]=\frac{2\epsilon b}{a^2}{X}_2+X_4,\hfill \\ Ad[X_4,X_1]=X_1+\epsilon b{X}_4, Ad[X_4,X_3]=-\frac{2\epsilon b}{a^2}{X}_2+X_3,\hfill \\ Ad[X_4,X_5]=-\frac{2{\epsilon }^2{b}^2}{a^2}{X}_2+2\epsilon b{X}_3+X_5, Ad[X_5,X_1]=X_1-2\epsilon b{X}_5,\hfill \\ Ad[Y_1,Y_j]=Y_j, j=1,\dots ,4, Ad[Y_i,Y_i]=Y_i, i=1,\dots ,4,\hfill \\ Ad[Y_2,Y_1]=Y_1-\frac{\epsilon b}{a^2}{Y}_3, Ad[Y_2,Y_4]=-\frac{2\epsilon b}{a^4}{Y}_2+Y_4,\hfill \\ Ad[Y_3,Y_1]=Y_1-\frac{\epsilon b}{a^2}{Y}_3, Ad[Y_3,Y_4]=-\frac{2\epsilon b}{a^4}{Y}_3+Y_4,\hfill \\ Ad[Y_4,Y_1]=Y_1+\frac{\epsilon b}{a^2}{Y}_4, Ad[Y_4,Y_3]=\frac{2\epsilon b}{a^4}{Y}_2+Y_3, \dots \hfill \end{array}$$

Suppose that $V={\sum }_{i=1}^8{X}_i$ and $\tilde{V}={\sum }_{i=1}^8{Y}_i$ are the most general element. Eventually, we will obtain one-dimensional optimal system of Equation (18). The following symmetries are just a few members of optimal system of the perturbed Fokker–Planck equation

$$\begin{array}{c}{V}_1=X_1, V_2=X_2, V_3=X_3, V_4=X_4 V_5=X_2+X_3,\hfill \\ V_6=X_5, V_7=X_6, V_8=X_7, V_9=X_8 V_{10}=X_0,\hfill \\ V_{11}=X_3+X_5, V_{12}=X_2+X_5, V_{13}=X_2+X_4, V_{14}=X_1+X_4,\hfill \\ V_{15}=X_2+X_3+X_4, V_{16}=X_2+X_3+X_5, V_{17}=X_1+X_2+X_4,\hfill \\ V_{18}=X_1+X_3+X_4, V_{19}=X_1+X_4+X_5, V_{20}=X_2+X_3+X_4+X_5,\dots \hfill \\ {\tilde{V}}_1=Y_1, {\tilde{V}}_2=Y_2, {\tilde{V}}_3=Y_3, {\tilde{V}}_4=Y_4 {\tilde{V}}_5=Y_2+Y_4,\hfill \\ {\tilde{V}}_6=Y_1+Y_3+Y_4, {\tilde{V}}_7=Y_1+Y_2+Y_3,\dots \hfill \end{array}$$

Case 1:

For the symmetry of $V_1=X_1$, corresponding characteristic equation is given as:

$$\begin{array}{c}\hfill \frac{dt}{1}=\frac{dx}{0}=\frac{du}{0},\end{array}$$

(19)

integration of Equation (19) yields the following similarity variable and function

$$\begin{array}{c}\hfill u=g\left(x\right),\end{array}$$

(20)

thus we have

$$\begin{array}{c}\hfill u_t=0, u_x=g^{\prime }\left(x\right), u_{xx}=g^{″}\left(x\right).\end{array}$$

(21)

Substituting Equations (20) and (21) into Equation (17), we can get the reduced equation:

$$\begin{array}{c}\hfill -\frac{1}{2}{a}^2{g}^{″}-bg-bx{g}^{\prime }+\epsilon \left[\frac{g}{t}\right]=0,\end{array}$$

where solution of unperturbed part of reduced equation will be in the form

$$u=e^{(-\frac{b{x}^2}{a^2})}erf\left(-\frac{\sqrt{b}{c}_{1}x}{a}+c_2\right).$$

Case 2:

For $V_3=X_3$, using the corresponding characteristic equation and change of variables, we write

$$\begin{array}{ccc}\hfill \frac{dt}{0}& =& \frac{dx}{e^{-bt}}=\frac{du}{0}, u=g\left(t\right),\hfill \\ \hfill u_t& =& g^{\prime }\left(t\right), u_x=u_{xx}=0.\hfill \end{array}$$

We reduce the perturbed equation Equation (17) to a first order equation:

$$\begin{array}{c}\hfill g^{\prime }\left(t\right)-bg\left(t\right)+\epsilon \left[(lnt+\nu )g^{\prime }\left(t\right)+g\left(t\right)+\sum _{k=1}^{\infty }\frac{{(-t)}^k}{k(k+1)!}\frac{{\partial }^{k+1}g}{\partial t^{k+1}}\right]=0.\end{array}$$

$u=c_1{e}^{bt}$ is a solution of unperturbed equation $g^{\prime }\left(t\right)-bg\left(t\right)=0$.

Case 3:

For $V_5=X_2+X_3$, the reduced equation is:

$$\begin{array}{c}\hfill g^{\prime }-\frac{1}{2}{a}^2{e}^{2bt}g-bg-\epsilon \left[(lnt+\nu )g^{\prime }+g(1+bx{e}^{bt})+\sum _{k=1}^{\infty }\frac{{(-t)}^k}{k(k+1)!}\frac{{\partial }^{k+1}\left(g{e}^{x{e}^{bt}}\right)}{\partial t^{k+1}}\right]=0.\end{array}$$

where $u=exp\left(\frac{a^2{e}^{2bt}+4b^{2}t}{4b}\right)$ is a solution of unperturbed equation.

Case 4:

For component of one-dimensional optimal system $V_4,V_6$ and $V_7$, solutions of unperturbed part of Equation (17) are given in

Table 1. Solutions for unperturbed part of equation Equation (17).

${\mathit{V}}_{\mathit{i}}$	u
$V_4=X_4$	$u=c_1{e}^{\frac{-b}{a^2}{x}^2}$
$V_6=X_5$	$u=e^{bt}(c_1+c_{2}x{e}^{bt})$
$V_7=X_6$	$u=e^{\frac{-b}{a^2}{x}^2}(c_1+c_{2}x{e}^{-bt})$

.

5. Approximate Conservation Laws

We consider approximate nonlinear self-adjointness for a system of perturbed PDEs, see, e.g., [24,25] for details. In the rest of this section, we present a formal Lagrange of perturbed Equation (12) and obtain conservation laws.

5.1. Basic Definitions for Constructing Conservation Laws

Let $\mathcal{L}$ be the formal Lagrange of Equation (12):

$$\begin{array}{c}\hfill \mathcal{L}\approx {\mathcal{L}}_{\left(0\right)}+\epsilon {\mathcal{L}}_{\left(1\right)}\equiv v{P}_{\left(0\right)}+\epsilon v{P}_{\left(1\right)},\end{array}$$

(22)

hence, the adjoint equations of Equation (12) are defined as

$$\begin{array}{c}\frac{\delta \mathcal{L}}{\delta u}=P_{\left(0\right)}^{*}(x,u,v,u_{\left(1\right)},v_{\left(1\right)},\dots )\hfill \\ +\epsilon P_{\left(1\right)}^{*}(x,u,v,\dots ,D_{x^1}^{c+1}u,D_{x^1}^{c+1}v,D_{x^1}^{c+2}u,D_{x^1}^{c+3}v,\dots )\approx 0,\hfill \end{array}$$

(23)

where $v_i$ represents all $i^{th}$-order derivatives of variable v with respect to x, $\frac{\delta }{\delta u}$ is the variational derivative written in terms of the total derivative operator $D_i$:

$$\begin{array}{c}\hfill \frac{\delta }{\delta u}=\frac{\partial }{\partial u}+\sum _{s=1}^{\infty }{(-1)}^s{D}_{i_1}\dots D_{i_s}\frac{\partial }{\partial u_{i_1\dots i_s}}.\end{array}$$

$D_i$ indicates the operator of total differentiation with respect to $x^i$:

$$\begin{array}{c}\hfill D_i=\frac{\partial }{\partial x^i}+u_i\frac{\partial }{\partial u}+v_i\frac{\partial }{\partial v}+\sum _{s=1}^{\infty }\left[u_{i{i}_1\dots i_s}\frac{\partial }{\partial u_{i_1\dots i_s}}+v_{i{i}_1\dots i_s}\frac{\partial }{\partial v_{i_1\dots i_s}}\right].\end{array}$$

If we consider

$$\begin{array}{c}\hfill v\approx {\phi }_{\left(0\right)}(x,u)+\epsilon {\phi }_{\left(1\right)}(x,u)\ne 0,\end{array}$$

(24)

we have

$$\mathcal{L}\approx {\phi }_{\left(0\right)}{P}_{\left(0\right)}+\epsilon \left({\phi }_{\left(1\right)}{P}_{\left(0\right)}+{\phi }_{\left(0\right)}{P}_{\left(1\right)}\right),$$

and if it satisfies the nonlinear self adjoint condition:

$$\begin{array}{c}\hfill P_0^{*}{{|}_{v\approx {\phi }_{\left(0\right)}+\epsilon {\phi }_{\left(1\right)}}+\epsilon P_{\left(1\right)}^{*}|}_{v\approx {\phi }_{\left(0\right)}}\approx {\gamma }_{\left(0\right)}{P}_{\left(0\right)}+\epsilon \left({\gamma }_{\left(1\right)}{P}_{\left(0\right)}+{\gamma }_{\left(0\right)}{P}_{\left(1\right)}\right).\end{array}$$

(25)

In which ${\gamma }_{\left(0\right)}$ and ${\gamma }_{\left(1\right)}$ are to be determined coefficients.

Any approximate symmetry Equation (13) of Equation (12) leads to a conservation law

$$\begin{array}{c}\hfill D_i\left(C^i\right)=0, C^i\approx C_{\left(0\right)}^i+\epsilon C_{\left(1\right)}^i,\end{array}$$

where the components $C^i$ are obtained by

$$\begin{array}{l}{C}_{\left(0\right)}^i=W_{\left(0\right)}\left(\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_i}+{\displaystyle \sum _{s=1}^{c-1}}{(-1)}^s{D}_{i_1}\dots D_{i_s}\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_{i{i}_1\dots i_s}}\right)\\ \hspace{1em}\hspace{1em}+{\displaystyle \sum _{r=1}^{c-1}}{D}_{k_1}\dots D_{k_r}\left(W_{\left(0\right)}\right)\left[\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_{i{k}_1\dots k_r}}+{\displaystyle \sum _{s=1}^{c-r-1}}{(-1)}^s{D}_{i_1}\dots D_{i_s}\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_{i{k}_1\dots k_r{i}_1\dots i_s}}\right],\hfill \end{array}$$

(26)

$$\begin{array}{l}{C}_{\left(1\right)}^i=W_{\left(1\right)}\left(\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_i}+{\displaystyle \sum _{s=1}^{c-1}}{(-1)}^s{D}_{i_1}\dots D_{i_s}\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_{i{i}_1\dots i_s}}\right)\\ \hspace{1em}\hspace{1em}+{\displaystyle \sum _{r=1}^{c-1}}{D}_{k_1}\dots D_{k_r}\left(W_{\left(1\right)}\right)\left[\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_{i{k}_1\dots k_r}}+{\displaystyle \sum _{s=1}^{c-r-1}}{(-1)}^s{D}_{i_1}\dots D_{i_s}\frac{\partial {\mathcal{L}}_{\left(0\right)}}{\partial u_{i{k}_1\dots k_r{i}_1\dots i_s}}\right]\hfill \\ \hspace{1em}\hspace{1em}+W_{\left(0\right)}\left(\frac{\partial {\mathcal{L}}_{\left(1\right)}}{\partial u_i}+{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^s{D}_{i_1}\dots D_{i_s}\frac{\partial {\mathcal{L}}_{\left(1\right)}}{\partial u_{i{i}_1\dots i_s}}\right)\hfill \\ \hspace{1em}\hspace{1em}+{\displaystyle \sum _{r=1}^{\infty }}{D}_{k_1}\dots D_{k_r}\left(W_{\left(0\right)}\right)\left[\frac{\partial {\mathcal{L}}_{\left(1\right)}}{\partial u_{i{k}_1\dots k_r}}+{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^s{D}_{i_1}\dots D_{i_s}\frac{\partial {\mathcal{L}}_{\left(1\right)}}{\partial u_{i{k}_1\dots k_r{i}_1\dots i_s}}\right].\hfill \end{array}$$

(27)

In which $W_{\left(0\right)}={\theta }_{\left(0\right)}-{\zeta }_{\left(0\right)}^i{u}_i$, $W_{\left(1\right)}={\theta }_{\left(1\right)}-{\zeta }_{\left(1\right)}^i{u}_i$.

5.2. Approximate Conservation Laws for pfPE

By choosing approximate formal Lagrange

$$\begin{array}{l}\mathcal{L}\hspace{1em}\hspace{1em}v(x,t,u)\left(P_{\left(0\right)}+\epsilon P_{\left(1\right)}\right)=\approx v(x,t,u)[u_t-\frac{1}{2}{a}^2{u}_{xx}-bu-bx{u}_x\\ \hspace{1em}\hspace{1em}+\epsilon \left((lnt+\nu )u_t+\frac{u}{t}+{\displaystyle \sum _{k=1}^{\infty }}\frac{{(-t)}^k}{k(k+1)!}{u}_t^{(k+1)}\right)],\hfill \end{array}$$

(28)

where

$$\begin{array}{c}\hfill v={\phi }_0(x,t,u)+\epsilon {\phi }_1(x,t,u),\end{array}$$

(29)

we obtain adjoint equation using Equation (23) as:

$$\begin{array}{c}\hfill P^{*}\approx -v_t+bx{v}_x-\frac{1}{2}{a}^2{v}_{xx}-\epsilon \left[v_t(lnt+\nu )+\sum _{k=1}^{\infty }\frac{{(-t)}^k}{k(k+1)!}{D}_t^{(k+1)}\left(v{t}^k\right)\right].\end{array}$$

(30)

It is easy to achieve an approximate formal Lagrange by placing Equation (29) into Equation (30), and solving characteristic equation of the Equation (25) with the Maple software, we have

$$\begin{array}{l}v=(c_{1}x{e}^{bt}+c_2)+\epsilon c_{3}x{e}^{c_{1}t}(c_{4}KummerM(\frac{b-c_1}{2b},\frac{3}{2},\frac{b{x}^2}{a^2})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+c_{5}KummerU(\frac{b-c_1}{2b},\frac{3}{2},\frac{b{x}^2}{a^2})),\hfill \end{array}$$

(31)

and

$$\begin{array}{c}\hfill \mathcal{L}\approx {\mathcal{L}}_{\left(0\right)}+\epsilon {\mathcal{L}}_{\left(1\right)},\end{array}$$

(32)

where

$$\begin{array}{l}{\mathcal{L}}_{\left(0\right)}=(c_{1}x{e}^{bt}+c_2)(u_t-\frac{1}{2}{a}^2{u}_{xx}-bu-bx{u}_x),\\ {\mathcal{L}}_{\left(1\right)}=\epsilon \left[c_{3}x{e}^{c_{1}t}\right(c_{4}KummerM(\frac{b-c_1}{2b},\frac{3}{2},\frac{b{x}^2}{a^2})\hfill \\ \hspace{1em}\hspace{1em}+c_{5}KummerU(\frac{b-c_1}{2b},\frac{3}{2},\frac{b{x}^2}{a^2}))(u_t-\frac{1}{2}{a}^2{u}_{xx}-bu-bx{u}_x)\hfill \\ \hspace{1em}\hspace{1em}+(c_{1}x{e}^{bt}+c_2)\left((lnt+\nu )u_t+\frac{u}{t}+{\displaystyle \sum _{k=1}^{\infty }}\frac{{(-t)}^k}{k(k+1)!}{u}_t^{(k+1)}\right)].\hfill \end{array}$$

(33)

Here, $c_1,c_2,c_3,c_4,c_5,a$ and b are arbitrary constants. Applying the formula Equations (26) and (27), we perform all computations to approximate conservation laws. Finally, we obtain

$$\begin{array}{l}{C}_{\left(0\right)}^x=W_{\left(0\right)}\left(-bx{\phi }_{\left(0\right)}+\frac{1}{2}{a}^2{D}_x\left({\phi }_{\left(0\right)}\right)\right)-\frac{1}{2}{a}^2{\phi }_{\left(0\right)}{D}_x\left(W_{\left(0\right)}\right),\\ C_{\left(0\right)}^t=W_{\left(0\right)}{\phi }_{\left(0\right)},\\ C_{\left(1\right)}^x=W_{\left(1\right)}\left(-bx{\phi }_{\left(0\right)}+\frac{1}{2}{a}^2{D}_x\left({\phi }_{\left(0\right)}\right)\right)-\frac{1}{2}{a}^2{\phi }_{\left(0\right)}{D}_x\left(W_{\left(1\right)}\right)\\ \hspace{1em}\hspace{1em}+W_{\left(0\right)}\left(-bx{\phi }_{\left(1\right)}+\frac{1}{2}{a}^2{D}_x\left({\phi }_{\left(1\right)}\right)\right)-\frac{1}{2}{a}^2{\phi }_{\left(1\right)}{D}_x\left(W_{\left(0\right)}\right),\end{array}$$

$$\begin{array}{l}{C}_{\left(1\right)}^t=W_{\left(1\right)}{\phi }_{\left(0\right)}+W_{\left(0\right)}\left[c_{3}x{e}^{c_{1}t}{\phi }_{\left(1\right)}+{\phi }_{\left(0\right)}\left(lnt+\nu +{\displaystyle \sum _{k=1}^{\infty }}\frac{{(-t)}^k}{k(k+1)!}\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\phi }_{\left(0\right)}{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^{(s+1)}{D}_{st}\left(W_{\left(0\right)}\right)\left[{\displaystyle \sum _{k=s}^{\infty }}{D}_{(k-s)t}\frac{t^k}{k(k+1)!}\right].\end{array}$$

where

$$\begin{array}{c}\hfill C^x=C_{\left(0\right)}^x+\epsilon C_{\left(1\right)}^x,\hspace{1em}\hspace{1em}{C}^t=C_{\left(0\right)}^t+\epsilon C_{\left(1\right)}^t.\end{array}$$

• For $X_1={\partial }_t$, we have $W_{\left(0\right)}=-u_t,$ $W_{\left(1\right)}=0$, the components of approximate conservation laws are:

  $$\begin{array}{l}{C}^x=u_t\left(bx{\phi }_{\left(0\right)}-\frac{1}{2}{a}^2{c}_1{e}^{bt}\right)+\frac{1}{2}{a}^2{u}_{xt}{\phi }_{\left(0\right)}\\ \hspace{1em}\hspace{1em}+\epsilon \left[u_t\left(bx{\phi }_{\left(1\right)}-\frac{1}{2}{a}^2{D}_x{\phi }_{\left(1\right)}\right)+\frac{1}{2}{a}^2{\phi }_{\left(1\right)}{u}_{xt}\right],\hfill \end{array}$$

  $$\begin{array}{l}{C}^t=-u_t{\phi }_{\left(0\right)}+\epsilon [-u_t{\phi }_{\left(1\right)}+(lnt+\nu ){\phi }_{\left(0\right)}{\displaystyle \sum _{k=1}^{\infty }}{D}_{kt}\frac{{(-t)}^k}{k(k+1)!}\\ \hspace{1em}\hspace{1em}-{\phi }_{\left(0\right)}{\displaystyle \sum _{s=1}^{\infty }}{D}_{st}\left(u_t\right)\left({\displaystyle \sum _{k=s}^{\infty }}{D}_{(k-s)t}\frac{t^k}{k(k+1)!}\right)].\end{array}$$
• For $X_2=u{\partial }_u$, $W_{\left(0\right)}=u$ and $W_{\left(1\right)}=0$, we have:

  $$\begin{array}{l}{C}^x=+u\left(-bx{\phi }_{\left(0\right)}+\frac{1}{2}{c}_1{a}^2{e}^{bt}\right)-\frac{1}{2}{a}^2{u}_x{\phi }_{\left(0\right)}\\ \hspace{1em}\hspace{1em}+\epsilon \left[u\left(-bx{\phi }_{\left(1\right)}+\frac{1}{2}{a}^2{D}_x{\phi }_{\left(1\right)}\right)-\frac{1}{2}{a}^2{u}_x{\phi }_{\left(1\right)}\right],\end{array}$$

  $$\begin{array}{l}{C}^t=+u{\phi }_{\left(0\right)}+\epsilon [u\left({\phi }_{\left(1\right)}+{\phi }_{\left(0\right)}(lnt+\nu +{\displaystyle \sum _{k=1}^{\infty }}{D}_{kt}\frac{{(-t)}^k}{k(k+1)!})\right)\\ \hspace{1em}\hspace{1em}+{\phi }_{\left(0\right)}{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^{s+1}{D}_{st}\left(u\right)\left({\displaystyle \sum _{k=s}^{\infty }}{D}_{(k-s)t}\frac{t^k}{k(k+1)!}\right)].\end{array}$$
• For $X_3=e^{-bt}{\partial }_x$, $W_{\left(0\right)}=-e^{-bt}{u}_x$ and $W_{\left(1\right)}=0$, we have:

  $$\begin{array}{l}{C}^x=+e^{-bt}{u}_x(bx{\phi }_{\left(0\right)}-\frac{1}{2}{c}_1{a}^2{e}^{bt})+\frac{1}{2}{a}^2{e}^{-bt}{u}_{xx}{\phi }_{\left(0\right)}\\ \hspace{1em}\hspace{1em}+e^{-bt}{u}_x(bx{\phi }_{\left(1\right)}-\frac{1}{2}{a}^2{D}_x{\phi }_{\left(1\right)})+\frac{1}{2}{a}^2{e}^{-bt}{u}_{xx}{\phi }_{\left(1\right)},\end{array}$$

  $$\begin{array}{l}{C}^t=-u_x{e}^{-bt}{\phi }_{\left(0\right)}-\epsilon [u_x{e}^{-bt}\left({\phi }_{\left(1\right)}+{\phi }_{\left(0\right)}(lnt+\nu +{\displaystyle \sum _{k=1}^{\infty }}\frac{{(-t)}^k}{k(k+1)!})\right)\\ \hspace{1em}\hspace{1em}+{\phi }_{\left(0\right)}{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^{s+1}{D}_{st}\left(u_x{e}^{-bt}\right)\left({\displaystyle \sum _{k=s}^{\infty }}{D}_{(k-s)t}\frac{t^k}{k(k+1)!}\right)].\end{array}$$
• For $X_4=e^{bt}{\partial }_x-\frac{2b}{a^2}xu{e}^{bt}{\partial }_u$, $W_{\left(0\right)}=-\frac{2b}{a^2}xu{e}^{bt}-e^{bt}{u}_x$ and $W_{\left(1\right)}=0$, therefore:

  $$\begin{array}{l}{C}^x=e^{bt}[(\frac{2b}{a^2}xu+u_x)(bx{\phi }_{\left(0\right)}-\frac{1}{2}{c}_1{a}^2{e}^{bt})+(\frac{2b}{a^2}u+u_{xx})\left(\frac{1}{2}{a}^2{\phi }_{\left(0\right)}\right)\\ \hspace{1em}\hspace{1em}+\epsilon (\frac{2b}{a^2}xu+u_x)(bx{\phi }_{\left(1\right)}-\frac{1}{2}{a}^2{D}_x{\phi }_{\left(1\right)})+(\frac{2b}{a^2}u+u_{xx})\left(\frac{1}{2}{a}^2{\phi }_{\left(1\right)}\right)],\end{array}$$

  $$\begin{array}{l}{C}^t=-e^{bt}[(\frac{2b}{a^2}xu+u_x){\phi }_{\left(0\right)}\\ \hspace{1em}\hspace{1em}+\epsilon ((\frac{2b}{a^2}xu+u_x)\left({\phi }_{\left(1\right)}+{\phi }_{\left(0\right)}(lnt+\nu +{\displaystyle \sum _{k=1}^{\infty }}{D}_{kt}\frac{{(-t)}^k}{k(k+1)!})\right)\\ \hspace{1em}\hspace{1em}+{\phi }_{\left(0\right)}{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^{s+1}{D}_{st}(\frac{2b}{a^2}xu{e}^{bt}-e^{bt}{u}_x)\left({\displaystyle \sum _{k=s}^{\infty }}{D}_{(k-s)t}\frac{t^k}{k(k+1)!}\right)\left)\right].\end{array}$$
• For $X_5=e^{-2bt}({\partial }_t-bx{\partial }_x+bu{\partial }_u)$, $W_{\left(0\right)}=e^{-2bt}(bu-bx{u}_x-u_t)$ and $W_{\left(1\right)}=0$, so we have:

  $$\begin{array}{l}{C}^x=-e^{-2bt}[(bu-bx{u}_x-u_t)(bx{\phi }_{\left(0\right)}+\frac{1}{2}{c}_1{a}^2{e}^{bt})-\frac{1}{2}{a}^2{\phi }_{\left(0\right)}(bx{u}_{xx}+u_{xt})\\ \hspace{1em}\hspace{1em}+\epsilon \left((bu-bx{u}_x-u_t)(bx{\phi }_{\left(1\right)}-\frac{1}{2}{a}^2{D}_x{\phi }_{\left(1\right)})-\frac{1}{2}{a}^2{\phi }_{\left(1\right)(bx{u}_{xx}+u_{xt})}\right)],\end{array}$$

  $$\begin{array}{l}{C}^t=e^{-2bt}{\phi }_{\left(0\right)}(bu-bx{u}_x-u_t)\\ \hspace{1em}\hspace{1em}+\epsilon [e^{-2bt}(bu-bx{u}_x-u_t)\left({\phi }_{\left(1\right)}+{\phi }_{\left(0\right)}(lnt+\nu )({\displaystyle \sum _{k=1}^{\infty }}{D}_{kt}\frac{{(-t)}^k}{k(k+1)!})\right)\\ \hspace{1em}\hspace{1em}+{\phi }_{\left(0\right)}{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^{s+1}{D}_{st}\left(e^{-2bt}(bu-bx{u}_x-u_t)\right)\left({\displaystyle \sum _{k=s}^{\infty }}{D}_{(k-s)t}\frac{t^k}{k(k+1)!}\right)].\end{array}$$
• For $X_6=e^{2bt}({\partial }_t+bx{\partial }_x-\frac{2b^2}{a^2}{x}^{2}u{\partial }_u)$, $W_{\left(0\right)}=-e^{2bt}(\frac{2b^2}{a^2}{x}^{2}u+u_t+bx{u}_x)$ and $W_{\left(1\right)}=0$, we have:

  $$\begin{array}{l}{C}^x=e^{2bt}[(\frac{2b^2}{a^2}{x}^{2}u+u_t+bx{u}_x)(\frac{1}{2}{c}_1{a}^2{e}^{bt}-bx{\phi }_{\left(0\right)})\\ \hspace{1em}\hspace{1em}+\epsilon \frac{1}{2}{a}^2{\phi }_{\left(0\right)}\left(\frac{4b^2}{a^2}xu+\frac{2b^2{x}^2}{a^2}{u}_x+b{u}_x+bx{u}_{xx}+u_{xt}\right)\\ \hspace{1em}\hspace{1em}+\epsilon ((\frac{2b^2}{a^2}{x}^{2}u+u_t+bx{u}_x)(bx{\phi }_{\left(1\right)}-\frac{1}{2}{a}^2{D}_x{\phi }_{\left(1\right)})\\ \hspace{1em}\hspace{1em}+\frac{1}{2}{a}^2{\phi }_{\left(1\right)}(\frac{4b^2}{a^2}xu+\frac{2b^2{x}^2}{a^2}{u}_x+b{u}_x+bx{u}_{xx}+u_{xt})\left)\right],\end{array}$$

  $$\begin{array}{l}{C}^t=-e^{2bt}(\frac{2b^2}{a^2}{x}^{2}u+u_t+bx{u}_x){\phi }_{\left(0\right)}\\ \hspace{1em}\hspace{1em}-\epsilon [e^{2bt}(\frac{2b^2}{a^2}{x}^{2}u+u_t+bx{u}_x)\left({\phi }_{\left(1\right)}+{\phi }_{\left(0\right)}(lnt+\nu )({\displaystyle \sum _{k=1}^{\infty }}{D}_{kt}\frac{{(-t)}^k}{k(k+1)!})\right)\\ \hspace{1em}\hspace{1em}+{\phi }_{\left(0\right)}{\displaystyle \sum _{s=1}^{\infty }}{(-1)}^{s+1}{D}_{st}\left(e^{2bt}(\frac{2b^2}{a^2}{x}^{2}u+u_t+bx{u}_x)\right)\left({\displaystyle \sum _{k=s}^{\infty }}{D}_{(k-s)t}\frac{t^k}{k(k+1)!}\right)].\end{array}$$
• For $Y_1=\frac{1}{a^2}\epsilon {\partial }_t$, $W_{\left(0\right)}=0$ and $W_{\left(1\right)}=\frac{-1}{a^2}\epsilon u_t$, we have:

  $$\begin{array}{ccc}\hfill C^x& =& \epsilon \left[\frac{1}{a^2}{u}_t(bx{\phi }_{\left(0\right)}-\frac{1}{2}{c}_1{a}^2{e}^{bt})+\frac{1}{2}{u}_{xt}{\phi }_{\left(0\right)}\right],\hfill \\ \hfill C^t& =& -\frac{1}{a^2}\epsilon u_t{\phi }_{\left(0\right)}.\hfill \end{array}$$
• For $Y_2=\epsilon u{\partial }_u$, $W_{\left(0\right)}=0$ and $W_{\left(1\right)}=\epsilon u$, we have:

  $$\begin{array}{ccc}\hfill C^{\left(x\right)}& =& \epsilon \left[u(\frac{1}{2}{c}_1{a}^2{e}^{bt}-bx{\phi }_{\left(0\right)})-\frac{1}{2}{a}^2{u}_x{\phi }_{\left(0\right)}\right],\hfill \\ \hfill C^{\left(t\right)}& =& \epsilon u{\phi }_{\left(0\right)}.\hfill \end{array}$$
• For $Y_3=\frac{-1}{a^2}\epsilon e^{-bt}{\partial }_x$, $W_{\left(0\right)}=0$ and $W_{\left(1\right)}=\frac{1}{a^2}\epsilon e^{-bt}{u}_x$, we have:

  $$\begin{array}{ccc}\hfill C^x& =& -\epsilon \left[\frac{1}{a^2}{e}^{-bt}{u}_x\left(bx{\phi }_{\left(0\right)}-\frac{1}{2}{c}_1{a}^2{e}^{bt}\right)+\frac{1}{2}{e}^{-bt}{u}_{xx}{\phi }_{\left(0\right)}\right],\hfill \\ \hfill C^t& =& \frac{1}{a^2}\epsilon e^{-bt}{u}_x{\phi }_{\left(0\right)}.\hfill \end{array}$$
• For $Y_4=\frac{-1}{a^2}\epsilon e^{bt}({\partial }_x+2bxu{\partial }_u)$, $W_{\left(0\right)}=0$ and $W_{\left(1\right)}=\frac{1}{a^2}\epsilon e^{bt}(u_x-2bxu)$, we have:

  $$\begin{array}{l}{C}^x=\frac{1}{a^2}\epsilon e^{bt}[(u_x-2bxu)\left(\frac{1}{2}{c}_1{a}^2{e}^{bt}-bx{\phi }_{\left(0\right)}\right)\\ \hspace{1em}\hspace{1em}+\frac{a^2}{2}{\phi }_{\left(0\right)}(2bu+2bx{u}_x-u_{xx})],\end{array}$$

  $$\begin{array}{c}\hfill C^t=\frac{1}{a^2}\epsilon e^{bt}(u_x-2bxu){\phi }_{\left(0\right)}.\end{array}$$
• For $Y_5=\frac{1}{a^2}\epsilon e^{-2bt}({\partial }_t+bx{\partial }_x+bu{\partial }_u)$, $W_{\left(0\right)}=0$ and $W_{\left(1\right)}=\frac{1}{a^2}\epsilon e^{-2bt}(bu-u_t-bx{u}_x)$, we have:

  $$\begin{array}{l}{C}^x=\frac{1}{a^2}\epsilon e^{-2bt}[(bu-u_t-bx{u}_x)\left(\frac{1}{2}{c}_1{a}^2{e}^{bt}-bx{\phi }_{\left(0\right)}\right)\\ \hspace{1em}\hspace{1em}+\frac{1}{2}{a}^2{\phi }_{\left(0\right)}(u_{xt}+bx{u}_{xx})],\end{array}$$

  $$\begin{array}{c}\hfill C^t=\frac{1}{a^2}\epsilon e^{-2bt}(bu-u_t-bx{u}_x){\phi }_{\left(0\right)}.\end{array}$$
• For $Y_6=\frac{1}{a^2}\epsilon e^{2bt}({\partial }_t-bx{\partial }_x-2b^2{x}^{2}u{\partial }_u)$, $W_{\left(0\right)}=0$ and
  $W_{\left(1\right)}=\frac{1}{a^2}\epsilon e^{2bt}(bx{u}_x-u_t-2b^2{x}^{2}u)$, we have:

  $$\begin{array}{l}{C}^x=\frac{1}{a^2}\epsilon e^{2bt}(bx{u}_x-u_t-2b^2{x}^{2}u)\left(\frac{1}{2}{c}_1{a}^2{e}^{bt}-bx{\phi }_{\left(0\right)}\right)\\ \hspace{1em}\hspace{1em}+\frac{1}{a^2}{\phi }_{\left(0\right)}(4b^{2}xu+2b^2{x}^2{u}_x+u_{xt}-b{u}_x-bx{u}_{xx}),\\ C^t=\frac{1}{a^2}\epsilon e^{2bt}(bx{u}_x-u_t-2b^2{x}^{2}u){\phi }_{\left(0\right)}.\end{array}$$

6. Conclusions and Outlook

We presented a new approach for calculating new exact analytical solutions of parameter containing fractional-order equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries for these equations are obtained. Computational results indicate the strength of new method. We will apply the method to fractional-stochastic differential equations in a future work.