基于有理变换的改进辅助方程法在变系数非线性发展方程中的应用  

Application of modified auxiliary equation method based on rational transformation in nonlinear evolution equation with variable coefficient

作　　者：尹天乐 庞晶[1,2] Yin Tian-le;Pang Jing(College of Science,Inner Mongolia University of Technology,Hohhot 010051,China;Inner Mongolia Key Laboratory of Statistical Analysis Theory for Life Data and Neural Network Modeling,Inner Mongolia University of Technology,Hohhot 010051,China)

机构地区：[1]内蒙古工业大学,理学院,内蒙古呼和浩特010051 [2]内蒙古自治区生命数据统计分析理论与神经网络建模重点实验室,内蒙古呼和浩特010051

出　　处：《高校应用数学学报（A辑）》2022年第3期297-307,共11页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(11561051)。

摘　　要：首次提出了基于有理变换的改进辅助方程法,成功地解决了变系数非线性发展方程的求解问题,并且验证了三种辅助方程适用于该方法.该方法与传统方法的主要区别是率先对给定方程进行了有理变换,其优势在于可以得到一个较为统一的初步的解,进而再依据不同的辅助方程等将初步的解转化为最终的解.同时,由此法得到的解省去了多个待定未知量,仅需要确定k,ω(t)即可,使计算过程更加方便简洁.简言之,改进辅助方程法将多种辅助方程法的步骤统一化,用统一的方式得到方程的解,无需依据各种辅助方程法的特性去设解,进一步完善辅助方程法的研究理论.The modified auxiliary equation method based on rational transformation is proposed for the first time,and author successfully solves the problem of nonlinear evolution equation with variable coefficient,and three auxiliary equations are suitable for this method.Compared with the traditional method,The main difference of the method is that takes the lead in rational transformation of the given equation,and its advantage is that it can obtain a relatively uniform initial solution,then transform the initial solution into the final solution according to different auxiliary equations.At the same time,the solution obtained by the method eliminates many unknown quantities,only k andω(t)is needed,which makes the calculation process more convenient and concise.In short,the modified auxiliary equation method will unify the steps of various auxiliary equation methods,and obtain the solution of the equation in a unified way.There is no need to set the solution according to the characteristics of various auxiliary equation methods,and further improve the research theory of the auxiliary equation method.

关 键 词：有理变换 辅助方程 变系数非线性发展方程 精确解 

分 类 号：O175.2[理学—数学]