﻿ 满足变系数方程的G展开法及RLW-Burgers方程的新精确解 The G Expansion Method of Satisfying a Variable Coefficient Equation and New Exact Solutions of Rlw-Burgers Equation

Vol.07 No.01(2018), Article ID:23551,11 pages
10.12677/AAM.2018.71010

The G Expansion Method of Satisfying a Variable Coefficient Equation and New Exact Solutions of Rlw-Burgers Equation

Xin Wang

College of Information Science and Technology, Hainan University, Haikou Hainan

Received: Dec. 21st, 2017; accepted: Jan. 18th, 2018; published: Jan. 25th, 2018

ABSTRACT

Based on the basic idea of the $\left({G}^{\prime }/G\right)$ expansion method, we construct a kind of New G method, and make the function G satisfy a class of variable coefficient Bernoulli equation. The RLW-Burgers equation is solved by this method, and several new explicit traveling wave solutions of the equation are obtained. It has been proved that this kind of satisfying variable coefficient equation G expansion method for solving nonlinear partial differential equations solutions is feasible and effective.

Keywords:RLW-Burgers Equation, Variable Coefficient Bernoulli Equation, G Expansion Method, Exact Solutions

1. 引言

RLW-Burgers方程

${u}_{t}+{u}_{x}-\theta {u}_{xx}+u{u}_{x}-\delta {u}_{xxt}=0$ (1)

2. 满足变系数方程的G展开法

$P\left(u,{u}_{x},{u}_{t},{u}_{xx},{u}_{xt},{u}_{tt},\cdots \right)=0$ (2)

$P\left(u,{u}^{\prime },{u}^{″},\cdots \right)=0$

$u\left(\xi \right)=\sum _{i=0}^{l}{a}_{i}{\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{i}$ (3)

${G}^{\prime }\left(\xi \right)+p\left(\xi \right)G\left(\xi \right)+q\left(\xi \right){G}^{2}\left(\xi \right)=0$ (4)

$G\left(\xi \right)=\frac{{\text{e}}^{{\int }_{1}^{\xi }-p\left(\tau \right)d\tau }}{{C}_{1}-{\int }_{1}^{\xi }\left(-{\text{e}}^{{\int }_{1}^{\varsigma }-p\left(\tau \right)d\tau }q\left(\varsigma \right)\right)d\varsigma }$ (5)

3. RLW-Burgers方程新的精确解

$\left(c+1\right){u}^{\prime }+u{u}^{\prime }-\theta {u}^{″}-c\delta {u}^{‴}=0$

$\left(c+1\right)u+\frac{1}{2}{u}^{2}-\theta {u}^{\prime }-c\delta {u}^{″}=M$(6)

${G}^{\prime }\left(\xi \right)+p\left(\xi \right)G\left(\xi \right)+q\left(\xi \right)G{\left(\xi \right)}^{2}=0$

$u\left(\xi \right)={a}_{2}{\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{2}+{a}_{1}\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)+{a}_{0}$(7)

$\begin{array}{c}{u}^{\prime }\left(\xi \right)=-\frac{2{a}_{2}\left(q\left(\xi \right)\left(-{m}_{2}^{2}+p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)+{n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right){q}^{\prime }\left(\xi \right)\right)}{q\left(\xi \right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}{\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{q\left(\xi \right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}\left(q\left(\xi \right)\left({a}_{1}\left({m}_{2}^{2}-p\left(\xi \right){m}_{2}{n}_{2}-{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)+2{a}_{2}\left({m}_{1}\left(-2{m}_{2}+p\left(\xi \right){n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}\left(p\left(\xi \right){m}_{2}+2{n}_{2}{p}^{\prime }\left(\xi \right)\right)\right)\right)+\left({a}_{1}{n}_{2}\left(-{m}_{2}+p\left(\xi \right){n}_{2}\right)+2{a}_{2}\left({m}_{2}{n}_{1}+\left({m}_{1}-2p\left(\xi \right){n}_{1}\right){n}_{2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{q}^{\prime }\left(\xi \right)\right){\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{2}+\frac{1}{q\left(\xi \right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}\left(q\left(\xi \right)\left(2{a}_{2}\left({m}_{1}^{2}-p\left(\xi \right){m}_{1}{n}_{1}-{n}_{1}^{2}{p}^{\prime }\left( \xi \right) \right)\end{array}$

$\begin{array}{l}+{a}_{1}\left({m}_{1}\left(p\left(\xi \right){n}_{2}-2{m}_{2}\right)+{n}_{1}\left(p\left(\xi \right){m}_{2}+2{n}_{2}{p}^{\prime }\left(\xi \right)\right)\right)\right)\\ +\left(2{a}_{2}{n}_{1}\left(p\left(\xi \right){n}_{1}-{m}_{1}\right)+{a}_{1}\left({m}_{2}{n}_{1}+\left({m}_{1}-2p\left(\xi \right){n}_{1}\right){n}_{2}\right)\right){q}^{\prime }\left(\xi \right)\right)\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)\\ +\frac{{a}_{1}\left(q\left(\xi \right)\left({m}_{1}^{2}-p\left(\xi \right){m}_{1}{n}_{1}-{n}_{1}^{2}{p}^{\prime }\left(\xi \right)\right)+{n}_{1}\left(-{m}_{1}+p\left(\xi \right){n}_{1}\right){q}^{\prime }\left(\xi \right)\right)}{q\left(\xi \right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}\end{array}$

$\begin{array}{c}{u}^{″}\left(\xi \right)=\frac{6{a}_{2}}{q{\left(\xi \right)}^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}{\left(q\left(\xi \right)\left({m}_{2}^{2}-p\left(\xi \right){m}_{2}{n}_{2}-{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)+{n}_{2}\left(-{m}_{2}+p\left(\xi \right){n}_{2}\right){q}^{\prime }\left(\xi \right)\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{4}+\frac{2}{q{\left(\xi \right)}^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\left({n}_{2}\left(p\left(\xi \right){n}_{2}-{m}_{2}\right)\left({a}_{1}{n}_{2}\left(p\left(\xi \right){n}_{2}-{m}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{a}_{2}\left(2{m}_{2}{n}_{1}+\left(3{m}_{1}-5p\left(\xi \right){n}_{1}\right){n}_{2}\right)\right){q}^{\prime }{\left(\xi \right)}^{2}+q{\left(\xi \right)}^{2}\left({a}_{1}{\left(-{m}_{2}^{2}+p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{2}\left({m}_{1}\left(10{m}_{2}^{3}-15p\left(\xi \right){m}_{2}^{2}{n}_{2}+{m}_{2}{n}_{2}^{2}\left(5p{\left(\xi \right)}^{2}-11{p}^{\prime }\left(\xi \right)\right)+{n}_{2}^{3}\left(5p\left(\xi \right){p}^{\prime }\left(\xi \right)-{p}^{″}\left(\xi \right)\right)\right)+{n}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(5p{\left(\xi \right)}^{2}{m}_{2}^{2}{n}_{2}-5p\left(\xi \right)\left({m}_{2}^{3}-3{m}_{2}{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)+{n}_{2}\left(-9{m}_{2}^{2}{p}^{\prime }\left(\xi \right)+10{n}_{2}^{2}{p}^{\prime }{\left(\xi \right)}^{2}+{m}_{2}{n}_{2}{p}^{″}\left(\xi \right)\right)\right)\right)\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+q\left(\xi \right)\left(\left(2{a}_{1}{n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left(-{m}_{2}^{2}+p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)+{a}_{2}\left(5\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left({m}_{2}^{2}{n}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3{m}_{2}\left({m}_{1}-p\left(\xi \right){n}_{1}\right){n}_{2}-p\left(\xi \right){m}_{1}{n}_{2}^{2}\right)+2{n}_{2}^{2}\left(-7{m}_{2}{n}_{1}+\left(-3{m}_{1}+10p\left(\xi \right){n}_{1}\right){n}_{2}\right){p}^{\prime }\left(\xi \right)\right)\right){q}^{\prime }\left(\xi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{2}{n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right){q}^{″}\left(\xi \right)\right)\right){\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{3}+\frac{1}{q{\left(\xi \right)}^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(2\left({a}_{2}\left({m}_{2}^{2}{n}_{1}^{2}+2{m}_{2}{n}_{1}\left(4{m}_{1}-5p\left(\xi \right){n}_{1}\right){n}_{2}+\left(3{m}_{1}^{2}-14p\left(\xi \right){m}_{1}{n}_{1}+12p{\left(\xi \right)}^{2}{n}_{1}^{2}\right){n}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{1}{n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left(2{m}_{1}{n}_{2}+{n}_{1}\left({m}_{2}-3p\left(\xi \right){n}_{2}\right)\right)\right){q}^{\prime }{\left(\xi \right)}^{2}+q{\left(\xi \right)}^{2}\left(2{a}_{2}\left({m}_{1}^{2}\left(12{m}_{2}^{2}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-12p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}\left(2p{\left(\xi \right)}^{2}-5{p}^{\prime }\left(\xi \right)\right)\right)+{n}_{1}^{2}\left(2p{\left(\xi \right)}^{2}{m}_{2}^{2}+12p\left(\xi \right){m}_{2}{n}_{2}{p}^{\prime }\left(\xi \right)-3{p}^{\prime }\left(\xi \right){m}_{2}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-4{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)+2{m}_{2}{n}_{2}{p}^{″}\left(\xi \right)\right)+2{m}_{1}{n}_{1}\left(4p{\left(\xi \right)}^{2}-{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)-{n}_{2}\left(8{m}_{2}{p}^{\prime }\left(\xi \right)+{n}_{2}{p}^{″}\left(\xi \right)\right)\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{a}_{1}\left({m}_{1}\left(6{m}_{2}^{3}-9p\left(\xi \right){m}_{2}^{2}{n}_{2}+{m}_{2}{n}_{2}^{2}\left(3p{\left(\xi \right)}^{2}-7{p}^{\prime }\left(\xi \right)\right)+{n}_{2}^{3}\left(3p\left(\xi \right){p}^{\prime }\left(\xi \right)-{p}^{″}\left(\xi \right)\right)\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}\left(3p{\left(\xi \right)}^{2}{m}_{2}^{2}{n}_{2}-3p\left(\xi \right)\left({m}_{2}^{3}-3{m}_{2}{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)+{n}_{2}\left(-5{m}_{2}^{2}{p}^{\prime }\left(\xi \right)+6{n}_{2}^{2}{p}^{\prime }{\left(\xi \right)}^{2}\left({a}_{1}\left(\left(3\left({m}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-p\left(\xi \right){n}_{2}\right)\left({m}_{2}^{2}{n}_{1}+3{m}_{2}\left({m}_{1}-p\left(\xi \right){n}_{1}\right){n}_{2}-p\left(\xi \right){m}_{1}{n}_{2}^{2}\right)-4{n}_{2}^{2}\left(2{m}_{2}{n}_{1}+\left({m}_{1}-3p\left(\xi \right){n}_{1}\right){n}_{2}\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right){q}^{″}\left(\xi \right)\right)+2{a}_{2}\left(-2\left(2{m}_{1}^{2}{n}_{2}\left(3{m}_{2}-2p\left(\xi \right){n}_{2}\right)+{m}_{1}{n}_{1}\left(6{m}_{2}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-16p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}\left(6p{\left(\xi \right)}^{2}-7{p}^{\prime }\left(\xi \right)\right)\right)+{n}_{1}^{2}\left(2p\left(\xi \right){m}_{2}\left(-2{m}_{2}+3p\left(\xi \right){n}_{2}\right)+{n}_{2}\left(-5{m}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+12p\left(\xi \right){n}_{1}-{m}_{1}{n}_{2}\right)\left({m}_{2}{n}_{1}+\left({m}_{1}-2p\left(\xi \right){n}_{1}\right){n}_{2}\right){q}^{″}\left(\xi \right)\right)\right)\right){\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{2}+\frac{1}{q{\left(\xi \right)}^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(-2{n}_{1}\right)\left({a}_{1}{n}_{2}\left(-2{m}_{2}{n}_{1}-\left({m}_{1}-3p\left(\xi \right){n}_{1}\right){n}_{2}\right)+2{a}_{2}{n}_{1}\left(2{m}_{1}{n}_{2}+{n}_{1}\left({m}_{2}-3p\left(\xi \right){n}_{2}\right)\right)\right){q}^{\prime }{\left(\xi \right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+q{\left(\xi \right)}^{2}\left(-2{a}_{2}\left({m}_{1}^{3}\left(6{m}_{2}-3p\left(\xi \right){n}_{2}\right)+{m}_{1}^{2}{n}_{1}\left(3p{\left(\xi \right)}^{2}{n}_{2}-9p\left(\xi \right){m}_{2}-7{n}_{2}{p}^{\prime }\left( \xi \right) \right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}^{3}\left(3p\left(\xi \right){m}_{2}{p}^{\prime }\left(\xi \right)+6{n}_{2}{p}^{\prime }{\left(\xi \right)}^{2}+{m}_{2}{p}^{″}\left(\xi \right)\right)+{m}_{1}{n}_{1}^{2}\left(3p{\left(\xi \right)}^{2}{m}_{2}-5{m}_{2}{p}^{\prime }\left(\xi \right)+9p\left(\xi \right){n}_{2}{p}^{\prime }\left(\xi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{n}_{2}{p}^{″}\left(\xi \right)\right)\right)+{a}_{1}\left({m}_{1}^{2}\left(6{m}_{2}^{2}-6p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}\left(p{\left(\xi \right)}^{2}-3{p}^{\prime }\left(\xi \right)\right)\right)+{n}_{1}^{2}\left(p{\left(\xi \right)}^{2}{m}_{2}^{2}-{m}_{2}^{2}{p}^{\prime }\left(\xi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+6p\left(\xi \right){p}^{\prime }\left(\xi \right){m}_{2}{n}_{2}+6{n}_{2}^{2}{p}^{\prime }{\left(\xi \right)}^{2}+2{m}_{2}{n}_{2}{p}^{″}\left(\xi \right)\right)+2{m}_{1}{n}_{1}\left(2p{\left(\xi \right)}^{2}{m}_{2}{n}_{2}-3p\left(\xi \right)\left({m}_{2}^{2}-{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{n}_{2}\left(4{p}^{\prime }\left(\xi \right){m}_{2}+{n}_{2}{p}^{″}\left(\xi \right)\right)\right)\right)\right)+q\left(\xi \right)\left(2{a}_{2}\left(\left(3\left({m}_{1}-p\left(\xi \right){n}_{1}\right)\left({m}_{2}{n}_{1}\left(3{m}_{1}-p\left(\xi \right){n}_{1}\right)+{m}_{1}\left({m}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-3p\left(\xi \right){n}_{1}\right){n}_{2}\right)-2{n}_{1}^{2}\left(5{m}_{1}{n}_{2}+{m}_{1}{n}_{2}\right){q}^{″}\left(\xi \right)\right)+{a}_{1}\left(2\left({m}_{1}^{2}{n}_{2}\left(2p\left(\xi \right){n}_{2}-3{m}_{2}\right)+{n}_{1}^{2}\left(p\left(\xi \right){m}_{2}\left(2{m}_{2}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-3p\left(\xi \right){n}_{2}\right)+2{n}_{2}\left({m}_{2}-3p\left(\xi \right){n}_{2}\right){p}^{\prime }\left(\xi \right)\right)+{m}_{1}{n}_{1}\left(8p\left(\xi \right){m}_{2}{n}_{2}-3{m}_{2}^{2}+{n}_{2}^{2}\left(4{p}^{\prime }\left(\xi \right)-3p{\left(\xi \right)}^{2}\right)\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{q}^{\prime }\left(\xi \right)+\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left({m}_{2}{n}_{1}+\left({m}_{1}-2p\left(\xi \right){n}_{1}\right){n}_{2}\right){q}^{″}\left(\xi \right)\right)\right)\right)\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{q{\left(\xi \right)}^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}{\left(2{n}_{1}-p\left(\xi \right){n}_{1}\right)}^{2}\left({a}_{2}{n}_{1}-{a}_{1}{n}_{2}\right){q}^{\prime }{\left(\xi \right)}^{2}+q{\left(\xi \right)}^{2}\left(2{a}_{2}\left(-{m}_{1}^{2}+p\left(\xi \right){m}_{1}{n}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{{n}_{1}^{2}{p}^{\prime }\left(\xi \right)\right)}^{2}-{a}_{1}\left({m}_{1}^{3}\left({m}_{1}\left(2{m}_{2}\left(2{m}_{2}-p\left(\xi \right){n}_{2}\right)+{m}_{1}^{2}{n}_{1}\left(-3p\left(\xi \right){m}_{2}+p{\left(\xi \right)}^{2}{n}_{2}-3{n}_{2}{p}^{\prime }\left( \xi \right) \right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}^{3}\left(p\left(\xi \right){m}_{2}{p}^{\prime }\left(\xi \right)+2{n}_{2}{p}^{\prime }{\left(\xi \right)}^{2}+{m}_{2}{p}^{″}\left(\xi \right)\right)+{m}_{1}{n}_{1}^{2}\left(p{\left(\xi \right)}^{2}{m}_{2}-{m}_{2}{p}^{\prime }\left(\xi \right)+3p\left(\xi \right){n}_{2}{p}^{\prime }\left(\xi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{n}_{2}{p}^{″}\left(\xi \right)\right)\right)\right)+q\left(\xi \right)\left({m}_{1}-p\left(\xi \right){n}_{1}\right)\left(4{a}_{2}{n}_{1}\left(-{m}_{1}^{2}+p\left(\xi \right){m}_{1}{n}_{1}+{n}_{1}^{2}{p}^{\prime }\left(\xi \right)\right){q}^{\prime }\left(\xi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{a}_{1}\left(\left({m}_{1}^{2}{n}_{2}+3{m}_{1}{n}_{1}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)-{n}_{1}^{2}\left(p\left(\xi \right){m}_{2}+4{p}^{\prime }\left(\xi \right){n}_{2}\right)\right){q}^{\prime }\left(\xi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{n}_{1}\left(-{m}_{2}{n}_{1}+{m}_{1}{n}_{2}\right){q}^{″}\left(\xi \right)\right)\right)\right)\end{array}$

${\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{0}:$

$\begin{array}{l}q{\left(\xi \right)}^{2}\left(-2M+2\left(1+c\right){a}_{0}+{a}_{0}^{2}\right){\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}-8c\delta q\left(\xi \right){a}_{2}{n}_{1}\left({m}_{1}-p\left(\xi \right){n}_{1}\right)\left(-{m}_{1}^{2}\\ +p\left(\xi \right){m}_{1}{n}_{1}+{n}_{1}^{2}{p}^{\prime }\left(\xi \right)\right){q}^{\prime }\left(\xi \right)-2c\delta q\left(\xi \right){a}_{1}\left({m}_{1}-p\left(\xi \right){n}_{1}\right)\left({m}_{1}^{2}{n}_{2}+3{m}_{1}{n}_{1}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\\ -{n}_{1}^{2}\left(p\left(\xi \right){m}_{2}+4{n}_{2}{p}^{\prime }\left(\xi \right)\right)\right){q}^{\prime }\left(\xi \right)-4c\delta {n}_{1}{\left({m}_{1}-p\left(\xi \right){n}_{1}\right)}^{2}\left({a}_{2}{n}_{1}-{a}_{1}{n}_{2}\right){q}^{\prime }{\left(\xi \right)}^{2}\\ -2c\delta q{\left(\xi \right)}^{2}\left(2{a}_{2}{\left(-{m}_{1}^{2}+p\left(\xi \right){m}_{1}{n}_{1}+{n}_{1}^{2}{p}^{\prime }\left(\xi \right)\right)}^{2}-{a}_{1}\left({m}_{1}^{3}\left(2{m}_{2}-p\left(\xi \right){n}_{2}\right)+{m}_{1}^{2}{n}_{1}\end{array}$

$\begin{array}{l}×\left(-3p\left(\xi \right){m}_{2}+p{\left(\xi \right)}^{2}{n}_{2}-3{n}_{2}{p}^{\prime }\left(\xi \right)\right)+{n}_{1}^{3}\left(p\left(\xi \right){m}_{2}{p}^{\prime }\left(\xi \right)+2{n}_{2}{p}^{\prime }{\left(\xi \right)}^{2}+{m}_{2}{p}^{″}\left(\xi \right)\right)\\ +{m}_{1}{n}_{1}^{2}-{n}_{2}{p}^{″}\left(\xi \right)\right)\right)\right)+2q\left(\xi \right){a}_{1}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left(\theta q\left(\xi \right)\left(-{m}_{1}^{2}+p\left(\xi \right){m}_{1}{n}_{1}+{n}_{1}^{2}{p}^{\prime }\left(\xi \right)\right)\\ +{n}_{1}\left({m}_{1}-p\left(\xi \right)\left(p{\left(\xi \right)}^{2}{m}_{2}-{m}_{2}{p}^{\prime }\left(\xi \right)+3p\left(\xi \right){n}_{2}{p}^{\prime }\left(\xi \right){n}_{1}\right)\left(\theta {q}^{\prime }\left(\xi \right)+c\delta {q}^{″}\left(\xi \right)\right)\right)=0\end{array}$

${\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{1}:$

$\begin{array}{l}2\left(2c\delta \left({m}_{1}-p\left(\xi \right){n}_{1}\right)\left({a}_{1}{n}_{2}\left(-2{m}_{2}{n}_{1}-\left({m}_{1}-3p\left(\xi \right){n}_{1}\right){n}_{2}\right)+2{a}_{2}{n}_{1}\left(2{m}_{1}{n}_{2}+{n}_{1}\\ ×\left({m}_{2}-3p\left(\xi \right){n}_{2}\right)\right)\right){q}^{\prime }{\left(\xi \right)}^{2}+q{\left(\xi \right)}^{2}\left(2{a}_{2}\left({m}_{1}^{3}\left(6c\delta {m}_{2}+\left(\theta -3c\delta p\left(\xi \right)\right){n}_{2}\right)\\ -{m}_{1}^{2}{n}_{1}\left(\left(\theta +9c\left(p\left(\xi \right)\left(\theta -3c\delta p\left(\xi \right)\right)+7c\delta {p}^{\prime }\left(\xi \right)\right)\right)+{m}_{1}{n}_{1}^{2}\left(3c\delta p{\left(\xi \right)}^{2}{m}_{2}\\ -\left(5c\delta {m}_{2}+\theta {n}_{2}\right){p}^{\prime }\left(\xi \right)+p\left(\xi \right)\left(\theta {m}_{2}+9c\delta {n}_{2}{p}^{\prime }\left(\xi \right)\right)-c\delta {n}_{2}{p}^{″}\left( \xi \right) \right)\end{array}$

$\begin{array}{l}+{n}_{1}^{3}\left(6c\delta {n}_{2}{p}^{\prime }{\left(\xi \right)}^{2}+{m}_{2}\left(\left(\theta +3c\delta p\left(\xi \right)\right){p}^{\prime }\left(\xi \right)+{p}^{″}\left(\xi \right)c\delta \right)\right)\right)+{a}_{1}\\ ×\left({m}_{1}^{2}\left(-6c\delta {m}_{2}^{2}-2\left(\theta -3c\delta p\left(\xi \right)\right){m}_{2}{n}_{2}+{n}_{2}^{2}\left(1+c+p\left(\xi \right)\left(\theta -c\delta p\left(\xi \right)\right)\\ +{a}_{0}+3c\delta {p}^{\prime }\left(\xi \right)\right)\right)+2{m}_{1}{n}_{1}\left(\left(\theta +3c\delta p\left(\xi \right)\right){m}_{2}^{2}-{m}_{2}{n}_{2}\left(1+c+{a}_{0}+2c\delta \\ ×\left(p{\left(\xi \right)}^{2}-2{p}^{\prime }\left(\xi \right)\right)\right)+{n}_{2}^{2}\left(\left(-3c\delta p\left(\xi \right)+\theta \right){p}^{\prime }\left(\xi \right)+c\delta {p}^{″}\left(\xi \right)\right)\right)+{n}_{1}^{2}\end{array}$

$\begin{array}{l}×\left({a}_{0}+{m}_{2}^{2}\left(1+c+c\delta {p}^{\prime }\left(\xi \right)\right)-2{m}_{2}{n}_{2}\left(\left(\theta +3c\delta p\left(\xi \right)\right){p}^{\prime }\left(\xi \right)+c\delta {p}^{″}\left(\xi \right)\right)\right)\right)\right)\\ +q\left(\xi \right)\left(-2{a}_{2}\left(\left(\left({m}_{1}-p\left(\xi \right){n}_{1}\right)\left(-{m}_{2}{n}_{1}\left(-9c\delta {m}_{1}+\left(\theta +3c\delta p\left(\xi \right)\right){n}_{1}\right)+{m}_{1}\\ ×\left(3c\delta {m}_{1}+\left(\theta -9c\delta p\left(\xi \right)\right){n}_{1}\right){n}_{2}\right)-2c\delta {n}_{1}^{2}\left(5{m}_{1}{n}_{2}+{n}_{1}\left({m}_{2}-6p\left(\xi \right){n}_{2}\right)\right){p}^{\prime }\left(\xi \right)\right)\\ ×{q}^{\prime }\left(\xi \right)+c\delta {n}_{1}\left({m}_{1}-p\left(\xi \right){n}_{1}\right)\left(-{m}_{2}{n}_{1}+{m}_{1}{n}_{2}\right){q}^{″}\left(\xi \right)\right)+{a}_{1}\left({m}_{1}^{2}{n}_{2}\end{array}$

$\begin{array}{l}×\left(\left(6c\delta {m}_{2}+\left(\theta -4c\delta p\left(\xi \right)\right){n}_{2}\right){q}^{\prime }\left(\xi \right)+c\delta {n}_{2}{q}^{″}\left(\xi \right)\right)+2{m}_{1}{n}_{1}\left(\left(3c\delta {m}_{2}^{2}\\ -8c\delta p\left(\xi \right){m}_{2}{n}_{2}-{n}_{2}^{2}\left(p\left(\xi \right)\left(\theta -3c\delta p\left(\xi \right)\right)+4c\delta {p}^{\prime }\left(\xi \right)\right)\right){q}^{\prime }\left(\xi \right)\\ -c\delta p\left(\xi \right){q}^{″}\left(\xi \right){n}_{2}^{2}\right)+{n}_{1}^{2}\left(\left({m}_{2}\left(-\left(\theta +4c\delta p\left(\xi \right)\right){m}_{2}+2p\left(\xi \right)\left(\theta +3c\delta p\left(\xi \right)\right){n}_{2}\right)\\ -4c\delta {n}_{2}\left(-3p\left(\xi \right){n}_{2}+{m}_{2}\right){p}^{\prime }\left(\xi \right)\right){q}^{\prime }\left(\xi \right)-c\delta {m}_{2}\left(-p\left(\xi \right){n}_{2}+{m}_{2}\right){q}^{″}\left(\xi \right)\right)\right)\right)\right)=0\end{array}$

${\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{2}:$

$\begin{array}{l}\left(4c\delta \left(-{a}_{2}\left({m}_{2}^{2}{n}_{1}^{2}+2{m}_{2}{n}_{1}\left(4{m}_{1}-5p\left(\xi \right){n}_{1}\right){n}_{2}+\left(3{m}_{1}^{2}-14p\left(\xi \right){m}_{1}{n}_{1}+12p{\left(\xi \right)}^{2}{n}_{1}^{2}\right){n}_{2}^{2}\right)\\ +{a}_{1}{n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left(2{m}_{1}{n}_{2}+{n}_{1}\left({m}_{2}-3p\left(\xi \right){n}_{2}\right)\right)\right){q}^{\prime }{\left(\xi \right)}^{2}+q{\left(\xi \right)}^{2}\left(2{a}_{2}\left({m}_{1}^{2}\left(-c{m}_{2}^{2}24\delta \\ -4\left(-6c\delta p\left(\xi \right)+\theta \right){m}_{2}{n}_{2}+{n}_{2}^{2}\left(1+c+2p\left(\xi \right)\left(\theta -2c\delta p\left(\xi \right)\right)+{a}_{0}+10{p}^{\prime }\left(\xi \right)c\delta \right)\right)\\ +2{m}_{1}{n}_{1}\left(2\left(\theta +6c\delta p\left(\xi \right)\right){m}_{2}^{2}-{m}_{2}{n}_{2}\left(1+c+{a}_{0}+8c\delta \left(p{\left(\xi \right)}^{2}-2{p}^{\prime }\left(\xi \right)\right)\right)+2{n}_{2}^{2}\end{array}$

$\begin{array}{l}×\left(\left(\theta -6c\delta p\left(\xi \right)\right){p}^{\prime }\left(\xi \right)+{p}^{″}\left(\xi \right)c\delta \right)\right)+{n}_{1}^{2}\left(-24c\delta {n}_{2}^{2}{p}^{\prime }{\left(\xi \right)}^{2}+{m}_{2}^{2}\left(1+c-2p\left(\xi \right)\\ ×\left(2c\delta p\left(\xi \right)+\theta \right)+{a}_{0}+6c\delta {p}^{\prime }\left(\xi \right)\right)-4{m}_{2}{n}_{2}\left(\left(\theta +6c\delta p\left(\xi \right)\right){p}^{\prime }\left(\xi \right)+c\delta {p}^{″}\left(\xi \right)\right)\right)\right)\\ +{a}_{1}\left({a}_{1}{m}_{1}^{2}{n}_{2}^{2}+2{m}_{1}\left(6c\delta {m}_{2}^{3}+\left(\theta -9c\delta p\left(\xi \right)\right){m}_{2}^{2}{n}_{2}-{m}_{2}{n}_{2}\left({a}_{1}{n}_{1}+{n}_{2}\left(p\left(\xi \right)\left(\theta -3c\delta p\left(\xi \right)\right)\\ +7c\delta {p}^{\prime }\left(\xi \right)\right)\right)-{n}_{2}^{3}\left(\left(\theta -3c\delta p\left(\xi \right)\right){p}^{\prime }\left(\xi \right)+c\delta {p}^{″}\left(\xi \right)\right)\right)+{n}_{1}\left(-2\left(\theta +3c\delta p\left(\xi \right)\right){m}_{2}^{3}\end{array}$

$\begin{array}{l}+12c\delta {n}_{2}^{3}{p}^{\prime }{\left(\xi \right)}^{2}+{m}_{2}^{2}\left({a}_{1}{n}_{1}+2{n}_{2}\left(p\left(\xi \right)\left(\theta +3c\delta p\left(\xi \right)\right)-5c\delta {p}^{\prime }\left(\xi \right)\right)\right)+2{m}_{2}{n}_{2}^{2}\\ ×\left(\left(\theta +9c\delta p\left(\xi \right)\right){p}^{\prime }\left(\xi \right)+c\delta {p}^{″}\left(\xi \right)\right)\right)\right)\right)+2q\left(\xi \right)\left({a}_{1}\left(\left(-\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left(3c\delta {m}_{2}^{2}{n}_{1}\\ +{m}_{2}\left(9c\delta {m}_{1}-\left(\theta +9c\delta p\left(\xi \right)\right){n}_{1}\right){n}_{2}+\left(\theta -3c\delta p\left(\xi \right)\right){m}_{1}{n}_{2}^{2}\right)+4c\delta {n}_{2}^{2}\left(2{m}_{2}{n}_{1}\\ +\left({m}_{1}-3p\left(\xi \right){n}_{1}\right){n}_{2}\right){p}^{\prime }\left(\xi \right)\right){q}^{\prime }\left(\xi \right)+c\delta {n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right){q}^{″}\left( \xi \right) \right)\end{array}$

$\begin{array}{l}+2{a}_{2}\left({m}_{1}^{2}{n}_{2}\left(\left(12c\delta {m}_{2}+\left(\theta -8c\delta p\left(\xi \right)\right){n}_{2}\right){q}^{\prime }\left(\xi \right)+c\delta {n}_{2}{q}^{″}\left(\xi \right)\right)\\ +2{m}_{1}{n}_{1}\left(\left(6c\delta {m}_{2}^{2}-16c\delta p\left(\xi \right){m}_{2}{n}_{2}-{n}_{2}^{2}\left(p\left(\xi \right)\left(\theta -6c\delta p\left(\xi \right)\right)+7c\delta {p}^{\prime }\left(\xi \right)\right)\right){q}^{\prime }\left(\xi \right)\\ -c\delta p\left(\xi \right){n}_{2}^{2}{q}^{″}\left(\xi \right)\right)+{n}_{1}^{2}\left(\left({m}_{2}\left(-\left(\theta +8c\delta p\left(\xi \right)\right){m}_{2}+2p\left(\xi \right)\left(\theta +6p\left(\xi \right)c\delta \right){n}_{2}\right)\\ +2c\delta {n}_{2}\left(-5{m}_{2}+12p\left(\xi \right){n}_{2}\right){p}^{\prime }\left(\xi \right)\right){q}^{\prime }\left(\xi \right)-c\delta {m}_{2}\left({m}_{2}-2p\left(\xi \right){n}_{2}\right){q}^{″}\left(\xi \right)\right)\right)\right)=0\end{array}$

${\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{3}:$

$\begin{array}{l}\left(4c\delta {n}_{2}\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left({a}_{1}{n}_{2}\left(-{m}_{2}+p\left(\xi \right){n}_{2}\right)+2{a}_{2}\left(2{m}_{2}{n}_{1}+\left(3{m}_{1}-5p\left(\xi \right){n}_{1}\right){n}_{2}\right)\right){q}^{\prime }{\left(\xi \right)}^{2}\\ +2q{\left(\xi \right)}^{2}\left(-2c\delta {a}_{1}{\left(-{m}_{2}^{2}+p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)}^{2}+{a}_{2}\left({a}_{1}{m}_{1}^{2}{n}_{2}^{2}+2{m}_{1}\left(10c\delta {m}_{2}^{3}\\ +\left(-15c\delta p\left(\xi \right)+\theta \right){m}_{2}^{2}{n}_{2}-{m}_{2}{n}_{2}\left({a}_{1}{n}_{1}+{n}_{2}\left(p\left(\xi \right)\left(\theta -5c\delta p\left(\xi \right)\right)+11c\delta {p}^{\prime }\left(\xi \right)\right)\right)\\ -{n}_{2}^{3}\left(\left(\theta -5c\delta p\left(\xi \right)\right){p}^{\prime }\left(\xi \right)+c\delta {p}^{″}\left(\xi \right)\right)\right)+{n}_{1}\left(-2\left(\theta +5c\delta p\left(\xi \right)\right){m}_{2}^{3}+20c\delta {n}_{2}^{3}{p}^{\prime }{\left(\xi \right)}^{2}\\ +{m}_{2}^{2}\left({a}_{1}{n}_{1}+2{n}_{2}\left(p\left(\xi \right)\left(\theta +5p\left(\xi \right)c\delta \right)-9c\delta {p}^{\prime }\left(\xi \right)\right)\right)+2{m}_{2}{n}_{2}^{2}\left(\left(\theta +15c\delta p\left(\xi \right)\right){p}^{\prime }\left( \xi \right)\end{array}$

$\begin{array}{l}+c\delta {p}^{″}\left(\xi \right)\right)\right)\right)\right)+4q\left(\xi \right)\left(2c\delta {a}_{1}{n}_{2}\left(p\left(\xi \right){n}_{2}-{m}_{2}\right)\left(-{m}_{2}^{2}+p\left(\xi \right){m}_{2}{n}_{2}+{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right){q}^{\prime }\left(\xi \right)\\ +{a}_{2}\left(\left(-\left({m}_{2}-p\left(\xi \right){n}_{2}\right)\left(5c\delta {m}_{2}^{2}{n}_{1}+{m}_{2}\left(15c\delta {m}_{1}-\left(\theta +15c\delta p\left(\xi \right)\right){n}_{1}\right){n}_{2}\\ +\left(-5p\left(\xi \right)c\delta +\theta \right){m}_{1}{n}_{2}^{2}\right)+2c\delta {n}_{2}^{2}\left(7{m}_{2}{n}_{1}+\left(3{m}_{1}-10p\left(\xi \right){n}_{1}\right){n}_{2}\right){p}^{\prime }\left(\xi \right)\right){q}^{\prime }\left(\xi \right)\\ +c\delta {n}_{2}\left({m}_{2}-{n}_{2}p\left(\xi \right)\right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right){q}^{″}\left(\xi \right)\right)\right)\right)=0\end{array}$

${\left(\frac{{m}_{1}G+{n}_{1}{G}^{\prime }}{{m}_{2}G+{n}_{2}{G}^{\prime }}\right)}^{4}$ :

$\begin{array}{l}q{\left(\xi \right)}^{2}{a}_{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}-12c\delta \left(q\left(\xi \right)\left({m}_{2}^{2}-p\left(\xi \right){m}_{2}{n}_{2}-{n}_{2}^{2}{p}^{\prime }\left(\xi \right)\right)\\ +{{n}_{2}\left(-{m}_{2}+{n}_{2}p\left(\xi \right)\right){q}^{\prime }\left(\xi \right)\right)}^{2}=0\end{array}$

1) ${a}_{2}=\frac{3{\left(10c\delta {m}_{2}+\left(\theta -10c\delta {C}_{3}\right){n}_{2}\right)}^{2}{\left(-10c\delta {m}_{2}+\left(\theta +10c\delta {C}_{3}\right){n}_{2}\right)}^{2}}{2500{c}^{3}{\delta }^{3}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}$

${a}_{1}=\frac{3\left(10c\delta {m}_{1}-\left(\theta +10c\delta {C}_{3}\right){n}_{1}\right){\left(10c\delta {m}_{2}+\left(\theta -10c\delta {C}_{3}\right){n}_{2}\right)}^{2}\left(-10c\delta {m}_{2}+\left(\theta +10c\delta {C}_{3}\right){n}_{2}\right)}{1250{c}^{3}{\delta }^{3}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}$

$\begin{array}{c}{a}_{0}=\frac{1}{2500{c}^{3}{\delta }^{3}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\left(3{\theta }^{3}{n}_{1}{n}_{2}\left(20c\delta {m}_{2}{n}_{1}+\left(-20c\delta {m}_{1}+\theta {n}_{1}\right){n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-300{c}^{2}{\delta }^{2}{\theta }^{2}\left({m}_{2}^{2}{n}_{1}^{2}-2{C}_{3}{m}_{2}{n}_{1}^{2}{n}_{2}+\left({m}_{1}^{2}-2{C}_{3}{m}_{1}{n}_{1}+2{C}_{3}^{2}{n}_{1}^{2}\right){n}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+500{c}^{3}{\delta }^{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left(-{n}_{1}\left(\left(5+5c-12\theta {C}_{3}\right){m}_{2}+12\theta {C}_{3}^{2}{n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{m}_{1}\left(-12\theta {m}_{2}+\left(5+5c+12\theta {C}_{3}\right){n}_{2}\right)\right)\right)\end{array}$

$M=\frac{18{\theta }^{4}}{625{c}^{2}{\delta }^{2}}-\frac{{c}^{2}}{2}-c-\frac{1}{2}$$p\left(\xi \right)={C}_{3}+\frac{\theta }{10c\delta }\mathrm{tanh}\left(\frac{\theta \left(\xi +5c\delta {C}_{1}\right)}{10c\delta }\right)$

$q\left(\xi \right)={C}_{1}{\text{e}}^{{C}_{3}\xi +{C}_{2}}\mathrm{sech}\left(\frac{\theta \left(\xi +5c\delta {C}_{1}\right)}{10c\delta }\right)$

$\begin{array}{c}u\left(\xi \right)=\frac{3{\left(10c\delta {m}_{2}+\left(\theta -10c\delta {C}_{3}\right){n}_{2}\right)}^{2}{\left(-10c\delta {m}_{2}+\left(\theta +10c\delta {C}_{3}\right){n}_{2}\right)}^{2}}{2500{c}^{3}{\delta }^{3}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \frac{{\left(10c\delta \theta \left({C}_{1}{n}_{1}{\text{e}}^{{C}_{2}}+{C}_{3}{C}_{4}{n}_{1}-{C}_{4}{m}_{1}\right)+\left({C}_{4}{\theta }^{2}{n}_{1}-100{C}_{1}{c}^{2}{\delta }^{2}{\text{e}}^{{C}_{2}}\left({m}_{1}-{C}_{3}{n}_{1}\right)\right)\mathrm{tanh}\left(\frac{\theta \left(\xi +5c\delta {C}_{1}\right)}{10c\delta }\right)\right)}^{2}}{{\left(10c\delta \theta \left({C}_{1}{n}_{2}{\text{e}}^{{C}_{2}}+{C}_{3}{C}_{4}{n}_{2}-{C}_{4}{m}_{2}\right)+\left({C}_{4}{\theta }^{2}{n}_{2}-100{C}_{1}{c}^{2}{\delta }^{2}{\text{e}}^{{C}_{2}}\left({m}_{2}-{C}_{3}{n}_{2}\right)\right)\mathrm{tanh}\left(\frac{\theta \left(\xi +5c\delta {C}_{1}\right)}{10c\delta }\right)\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{3\left(10c\delta {m}_{1}-\left(\theta +10c\delta {C}_{3}\right){n}_{1}\right){\left(10c\delta {m}_{2}+\left(\theta -10c\delta {C}_{3}\right){n}_{2}\right)}^{2}\left(-10c\delta {m}_{2}+\left(\theta +10c\delta {C}_{3}\right){n}_{2}\right)}{1250{c}^{3}{\delta }^{3}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \frac{10c\delta \theta \left({C}_{1}{n}_{1}{\text{e}}^{{C}_{2}}+{C}_{3}{C}_{4}{n}_{1}-{C}_{4}{m}_{1}\right)+\left({C}_{4}{\theta }^{2}{n}_{1}-100{C}_{1}{c}^{2}{\delta }^{2}{\text{e}}^{{C}_{2}}\left({m}_{1}-{C}_{3}{n}_{1}\right)\right)\mathrm{tanh}\left(\frac{\theta \left(\xi +5c\delta {C}_{1}\right)}{10c\delta }\right)}{10c\delta \theta \left({C}_{1}{n}_{2}{\text{e}}^{{C}_{2}}+{C}_{3}{C}_{4}{n}_{2}-{C}_{4}{m}_{2}\right)+\left({C}_{4}{\theta }^{2}{n}_{2}-100{C}_{1}{c}^{2}{\delta }^{2}{\text{e}}^{{C}_{2}}\left({m}_{2}-{C}_{3}{n}_{2}\right)\right)\mathrm{tanh}\left(\frac{\theta \left(\xi +5c\delta {C}_{1}\right)}{10c\delta }\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2500{c}^{3}{\delta }^{3}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\left(3{\theta }^{3}{n}_{1}{n}_{2}\left(20c\delta {m}_{2}{n}_{1}+\left(-20c\delta {m}_{1}+\theta {n}_{1}\right){n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-300{c}^{2}{\delta }^{2}{\theta }^{2}\left({m}_{2}^{2}{n}_{1}^{2}-2{C}_{3}{m}_{2}{n}_{1}^{2}{n}_{2}+\left({m}_{1}^{2}-2{C}_{3}{m}_{1}{n}_{1}+2{C}_{3}^{2}{n}_{1}^{2}\right){n}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+500{c}^{3}{\delta }^{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left(-{n}_{1}\left(\left(5+5c-12\theta {C}_{3}\right){m}_{2}+12\theta {C}_{3}^{2}{n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{m}_{1}\left(-12\theta {m}_{2}+\left(5+5c+12\theta {C}_{3}\right){n}_{2}\right)\right)\right)\end{array}$

2) ${a}_{2}=\frac{245\theta {n}_{2}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}{54{n}_{1}^{3}}$${a}_{1}=\frac{70\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}{27{n}_{1}^{2}}$${a}_{0}=\frac{10\theta }{27}\left(\frac{{m}_{1}}{{n}_{1}}-\frac{{m}_{2}}{{n}_{2}}\right)-1$

$c=\frac{44}{135}\theta \left(\frac{{m}_{1}}{{n}_{1}}-\frac{{m}_{2}}{{n}_{2}}\right)$$\delta =-\frac{243{n}_{1}^{2}{n}_{2}^{2}}{352{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}$

$M=\frac{3760{\theta }^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}+2376\theta {n}_{1}{n}_{2}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)-3645{n}_{1}^{2}{n}_{2}^{2}}{7290{n}_{1}^{2}{n}_{2}^{2}}$

$p\left(\xi \right)=\frac{\left(-2{m}_{2}{n}_{1}+5{m}_{1}{n}_{2}\right){\text{e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +24{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}+\left(2{m}_{2}{n}_{1}+7{m}_{1}{n}_{2}\right){\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{m}_{1}{n}_{1}{n}_{2}^{2}}}{3{n}_{1}{n}_{2}{\text{e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +24{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}+9{n}_{1}{n}_{2}{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{m}_{1}{n}_{1}{n}_{2}^{2}}}$

$q\left(\xi \right)=\frac{{C}_{2}{\text{e}}^{\left(\frac{5{m}_{1}}{3{n}_{1}}+\frac{2{m}_{2}}{9{n}_{2}}\right)\xi }}{{\text{e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +24{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}+3{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{m}_{1}{n}_{1}{n}_{2}^{2}}}$

$G\left(\xi \right)=\frac{8\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right){e}^{\frac{1}{9}\left(-\frac{7{m}_{1}}{{n}_{1}}+\frac{6{m}_{2}}{{n}_{2}}\right)\xi }}{9{C}_{2}{n}_{1}{n}_{2}{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\left(\xi -27{C}_{1}{n}_{1}^{2}{n}_{2}^{2}\right)}+8{C}_{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left({\text{e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +24{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}+3{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{m}_{1}{n}_{1}{n}_{2}^{2}}\right)}$

$\begin{array}{c}u\left(\xi \right)=\frac{490\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right){\left(\text{3}{C}_{2}{n}_{1}{n}_{2}{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi }-8{C}_{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left({\text{e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +48{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}-{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{n}_{1}{n}_{2}\left({m}_{2}{n}_{1}+{m}_{1}{n}_{2}\right)}\right)\right)}^{2}}{27{n}_{1}{n}_{2}{\left(\text{21}{C}_{2}{n}_{1}{n}_{2}{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi }+8{C}_{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left({\text{5e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +48{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}+7{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{n}_{1}{n}_{2}\left({m}_{2}{n}_{1}+{m}_{1}{n}_{2}\right)}\right)\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{140\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left(\text{3}{C}_{2}{n}_{1}{n}_{2}{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi }-8{C}_{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left({\text{e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +48{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}-{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{n}_{1}{n}_{2}\left({m}_{2}{n}_{1}+{m}_{1}{n}_{2}\right)}\right)\right)}{27{n}_{1}{n}_{2}\left(\text{21}{C}_{2}{n}_{1}{n}_{2}{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi }+8{C}_{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)\left({\text{5e}}^{\frac{8{m}_{1}}{9{n}_{1}}\xi +48{C}_{1}{m}_{2}{n}_{1}^{2}{n}_{2}}+7{\text{e}}^{\frac{8{m}_{2}}{9{n}_{2}}\xi +24{C}_{1}{n}_{1}{n}_{2}\left({m}_{2}{n}_{1}+{m}_{1}{n}_{2}\right)}\right)\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{10\theta }{27}\left(\frac{{m}_{1}}{{n}_{1}}-\frac{{m}_{2}}{{n}_{2}}\right)-1\end{array}$

3) ${a}_{2}=\frac{12{\left({m}_{2}-{C}_{1}{n}_{2}\right)}^{2}{\left(5c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)\mp \theta {n}_{2}\right)}^{2}}{25c\delta {\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}$

$\begin{array}{c}{a}_{1}=\frac{6}{125{c}^{2}{\delta }^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\left(10c\delta \left({m}_{1}-{C}_{1}{n}_{1}\right)-\left(1±1\right)\theta {n}_{1}\right)\left({m}_{2}-{C}_{1}{n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left(5c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)\mp \theta {n}_{2}\right)\left(10c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)+\left(1\mp 1\right)\theta {n}_{2}\right)\end{array}$

$\begin{array}{c}{a}_{0}=\frac{1}{125{c}^{2}{\delta }^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\left({n}_{1}^{2}\left(5c\delta \left(25{c}^{2}\delta \left(12{C}_{1}^{2}\delta -1\right)±6{\theta }^{2}+5c\delta \left(-5+12{C}_{1}\theta \left(1±1\right)\right)\right){m}_{2}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-12{\left(5c\delta {C}_{1}+\theta \right)}^{2}\left(10c\delta {C}_{1}+\left(-1±1\right)\theta \right){m}_{2}{n}_{2}+60c\delta {C}_{1}^{2}{\left(5c\delta {C}_{1}±\theta \right)}^{2}{n}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{m}_{1}{n}_{1}\left(300{c}^{2}{\delta }^{2}\left(10c\delta {C}_{1}+\left(1±1\right)\theta \right){m}_{2}^{2}-10c\delta \left(25{c}^{2}\delta \left(1+24{C}_{1}^{2}\delta \right)+6{\theta }^{2}+5c\delta \left(5±24{C}_{1}\theta \right)\right){m}_{2}{n}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3{\left(10c\delta {C}_{1}+\left(-1±1\right)\theta \right)}^{2}\left(10c\delta {C}_{1}+\left(1±1\right)\theta \right){n}_{2}^{2}\right)+5c\delta {m}_{1}^{2}\left(300{c}^{2}{\delta }^{2}{m}_{2}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-60c\delta \left(10c\delta {C}_{1}+\left(-1±1\right)\theta \right){m}_{2}{n}_{2}+\left(25{c}^{2}\delta \left(12\delta {C}_{1}^{2}-1\right)+5c\delta \left(-5+12\theta {C}_{1}\left(-1±1\right)\right)\mp 6{\theta }^{2}\right){n}_{2}^{2}\right)\right)\end{array}$

$M=\frac{18{\theta }^{4}}{625{c}^{2}{\delta }^{2}}-\frac{{c}^{2}}{2}-c-\frac{1}{2}$$p\left(\xi \right)={C}_{1}±\frac{\theta }{5c\delta }$$q\left(\xi \right)={C}_{2}{\text{e}}^{{C}_{1}\xi }$

$G\left(\xi \right)=\frac{\theta {e}^{-{C}_{1}\xi }}{{C}_{3}\theta {e}^{±\frac{\theta }{5c\delta }\xi }\mp 5c\delta {C}_{2}}$

$u\left(\xi \right)=\frac{12{\left({m}_{2}-{C}_{1}{n}_{2}\right)}^{2}}{25c\delta {\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\frac{{\left({C}_{3}\theta {\text{e}}^{{}^{{}^{±\frac{\theta }{5c\delta }\xi }}}\left(-5c\delta \left({m}_{1}-{C}_{1}{n}_{1}\right)±\theta {n}_{1}\right)±25{c}^{2}{\delta }^{2}{C}_{2}\left({m}_{1}-{C}_{1}{n}_{1}\right)\right)}^{2}}{{\left({C}_{3}\theta {\text{e}}^{{}^{±\frac{\theta }{5c\delta }\xi }}\left(-5c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)±\theta {n}_{2}\right)±25{c}^{2}{\delta }^{2}{C}_{2}\left({m}_{2}-{C}_{1}{n}_{2}\right)\right)}^{2}}$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}×{\left(5c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)\mp \theta {n}_{2}\right)}^{2}+\frac{6}{125{c}^{2}{\delta }^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\left(10c\delta \left({m}_{1}-{C}_{1}{n}_{1}\right)-\left(1±1\right)\theta {n}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\left({m}_{2}-{C}_{1}{n}_{2}\right)\left(5c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)\mp \theta {n}_{2}\right)\left(10c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)+\left(1\mp 1\right)\theta {n}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\frac{\left({C}_{3}\theta {\text{e}}^{{}^{±\frac{\theta }{5c\delta }\xi }}\left(-5c\delta \left({m}_{1}-{C}_{1}{n}_{1}\right)±\theta {n}_{1}\right)±25{c}^{2}{\delta }^{2}{C}_{2}\left({m}_{1}-{C}_{1}{n}_{1}\right)\right)}{\left({C}_{3}\theta {\text{e}}^{{}^{{}^{±\frac{\theta }{5c\delta }\xi }}}\left(-5c\delta \left({m}_{2}-{C}_{1}{n}_{2}\right)±\theta {n}_{2}\right)±25{c}^{2}{\delta }^{2}{C}_{2}\left({m}_{2}-{C}_{1}{n}_{2}\right)\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{125{c}^{2}{\delta }^{2}{\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)}^{2}}\left({n}_{1}^{2}\left(5c\delta \left(25{c}^{2}\delta \left(12{C}_{1}^{2}\delta -1\right)±6{\theta }^{2}+5c\delta \left(-5+12{C}_{1}\theta \left(1±1\right)\right)\right){m}_{2}^{2}\end{array}$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-12{\left(5c\delta {C}_{1}+\theta \right)}^{2}\left(10c\delta {C}_{1}+\left(-1±1\right)\theta \right){m}_{2}{n}_{2}+60c\delta {C}_{1}^{2}{\left(5c\delta {C}_{1}±\theta \right)}^{2}{n}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{m}_{1}{n}_{1}\left(300{c}^{2}{\delta }^{2}\left(10c\delta {C}_{1}+\left(1±1\right)\theta \right){m}_{2}^{2}-10c\delta \left(25{c}^{2}\delta \left(1+24{C}_{1}^{2}\delta \right)+6{\theta }^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+5c\delta \left(5±24{C}_{1}\theta \right)\right){m}_{2}{n}_{2}+3{\left(10c\delta {C}_{1}+\left(-1±1\right)\theta \right)}^{2}\left(10c\delta {C}_{1}+\left(1±1\right)\theta \right){n}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+5c\delta {m}_{1}^{2}\left(300{c}^{2}{\delta }^{2}{m}_{2}^{2}-60c\delta \left(10c\delta {C}_{1}+\left(-1±1\right)\theta \right){m}_{2}{n}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(25{c}^{2}\delta \left(12\delta {C}_{1}^{2}-1\right)+5c\delta \left(-5+12\theta {C}_{1}\left(-1±1\right)\right)\mp 6{\theta }^{2}\right){n}_{2}^{2}\right)\right)\end{array}$

4) ${a}_{2}=-\frac{2{\left(2\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)-{n}_{1}{n}_{2}\right)}^{2}}{5{n}_{1}^{4}}$${a}_{1}=0$${a}_{0}=\frac{2\left({\theta }^{2}-\delta \right)}{15\delta }$$c=-\frac{2{\theta }^{2}}{15\delta }$$M=0$

$p\left(\xi \right)=\frac{\left(2\theta {m}_{1}+{n}_{1}\right){e}^{\frac{\xi }{2\theta }}-{m}_{1}{e}^{{C}_{1}{n}_{1}^{2}}}{2{n}_{1}\theta {e}^{\frac{\xi }{2\theta }}-{n}_{1}{e}^{{C}_{1}{n}_{1}^{2}}}$$q\left(\xi \right)=\frac{{C}_{2}{e}^{\left(\frac{1}{2\theta }+\frac{{m}_{1}}{{n}_{1}}\right)\xi }}{-2\theta {e}^{\frac{\xi }{2\theta }}+{e}^{{C}_{1}{n}_{1}^{2}}}$

$G\left(\xi \right)=\frac{{e}^{-\frac{{m}_{1}}{{n}_{1}}\xi }}{{C}_{2}+{C}_{3}{e}^{{C}_{1}{n}_{1}^{2}}-2\theta {C}_{3}{e}^{\frac{\xi }{2\theta }}}$

$u\left(\xi \right)=-\frac{2{C}_{3}^{2}{\left(2\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)-{n}_{1}{n}_{2}\right)}^{2}{e}^{\frac{\xi }{\theta }}}{5{\left(\left({C}_{2}+{C}_{3}{e}^{{C}_{1}{n}_{1}^{2}}\right)\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)-{C}_{3}\left(2\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)-{n}_{1}{n}_{2}\right){e}^{\frac{\xi }{2\theta }}\right)}^{2}}+\frac{2\left({\theta }^{2}-\delta \right)}{15\delta }$

5) ${a}_{2}=\frac{2{\left(2\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)+{n}_{1}{n}_{2}\right)}^{2}}{5{n}_{1}^{4}}$${a}_{1}=0$${a}_{0}=-\frac{2\left({\theta }^{2}+14\delta \right)}{15\delta }$$c=\frac{2{\theta }^{2}}{15\delta }$$M=0$

$p\left(\xi \right)=\frac{{m}_{1}{e}^{\frac{\xi }{2\theta }}-\left(2\theta {m}_{1}-{n}_{1}\right){e}^{{C}_{1}{n}_{1}^{2}}}{{n}_{1}{e}^{\frac{\xi }{2\theta }}-2{n}_{1}\theta {e}^{{C}_{1}{n}_{1}^{2}}}$$q\left(\xi \right)=\frac{{C}_{2}{e}^{\frac{{m}_{1}}{{n}_{1}}\xi }}{{e}^{\frac{\xi }{2\theta }}-2\theta {e}^{{C}_{1}{n}_{1}^{2}}}$

$G\left(\xi \right)=\frac{{e}^{\frac{1}{2}\left(\frac{1}{\theta }-\frac{2{m}_{1}}{{n}_{1}}\right)\xi }}{{C}_{3}{e}^{\frac{\xi }{2\theta }}-2\theta {C}_{3}{e}^{{C}_{1}{n}_{1}^{2}}-2\theta {C}_{2}}$

$u\left(\xi \right)=\frac{2{\left(2\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)+{n}_{1}{n}_{2}\right)}^{2}{\left({C}_{2}+{C}_{3}{e}^{{C}_{1}{n}_{1}^{2}}\right)}^{2}}{5{\left(\left(2\theta \left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right)+{n}_{1}{n}_{2}\right)\left({C}_{2}+{C}_{3}{e}^{{C}_{1}{n}_{1}^{2}}\right)-{C}_{3}\left({m}_{2}{n}_{1}-{m}_{1}{n}_{2}\right){e}^{\frac{\xi }{2\theta }}\right)}^{2}}-\frac{2\left({\theta }^{2}+14\delta \right)}{15\delta }$

4. 总结

The G Expansion Method of Satisfying a Variable Coefficient Equation and New Exact Solutions of Rlw-Burgers Equation[J]. 应用数学进展, 2018, 07(01): 80-90. http://dx.doi.org/10.12677/AAM.2018.71010

1. 1. Wang, M.L., Li, X.Z. and Zhang, J.L. (2008) The (G’/G)-Expansion Method and Travelling Wave Solutions of Non-linear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423.
https://doi.org/10.1016/j.physleta.2007.07.051

2. 2. 王明新. 非线性抛物形方程[M]. 北京: 科学出版社, 1993.

3. 3. 黄正洪, 夏莉. RLW-Burgers方程行波解的性质[J]. 重庆师范学院学报(自然科学版), 1998, 15(1): 24-28.

4. 4. 谈骏渝. RLW-Burgers方程的一类解析解[J]. 数学的实践与认识, 2001, 31(5): 545-549.

5. 5. 刘金枝, 吴爱祥. RLW-Burgers方程的显式行波解[J]. 南华大学学报(自然科学版), 2004, 18(3): 18-20.

6. 6. 鲍春玲, 苏道毕力格, 韩雁清. RLW-Burgers方程的势对称及其精确解[J]. 应用数学进展, 2016, 5(1): 112-120.

7. 7. 王鑫. RLW-Burgers方程的新显式行波解[J]. 应用数学进展, 2017, 6(4): 619-626.