基于辅助方程法求解分数阶非线性偏微分方程

发布时间：2018-08-08 15:37

【摘要】：随着计算机技术的不断应用与发展,以及非线性科学理论的进一步完善,求解非线性偏微分方程的精确解已经越来越成为一项富有重要意义的科研工作。近几年,分数阶非线性偏微分方程相比于整数阶更具有一般性,有许多学者都在致力于对求解分数阶非线性偏微分方程,故而该研究方向已成为研究热点。目前已经有许多方法可以求解出分数阶非线性偏微分方程的精确解,如:齐次平衡法,B?cklund变换法等等。文章主要在总结前人研究的基础上,利用辅助方程进一步研究B?cklund变换以及非线性叠加公式在非线性偏微分方程中的应用,以及求出分数阶非线性偏微分方程的精确解和各种形式的无穷序列解。文章对分数阶非线性偏微分方程进行了大量研究和叙述,内容安排如下:第一章叙述了非线性科学、孤立子理论研究与发展、分数阶偏微分方程研究与发展以及研究中所用到的方法简介。第二章用Riccati作辅助方程研究了时间-空间分数阶非线性偏微分方程PKP方程和Gardner方程。第三章研究了分别用Riccati方程和第一种椭圆方程作辅助方程时间-空间分数阶非线性偏微分方程mBBM方程并分析两种辅助方程的优缺点。第四章对全文进行了总结性概括,并对文章研究课题以后的发展做出了展望。
[Abstract]:With the continuous application and development of computer technology and the further improvement of nonlinear scientific theory, the exact solution of nonlinear partial differential equations has become an important research work. In recent years, fractional nonlinear partial differential equations are more general than integer order. Many scholars are devoted to solving fractional nonlinear partial differential equations. At present, there are many methods to solve the exact solutions of fractional nonlinear partial differential equations, such as the homogeneous equilibrium method and Bcklund transform method and so on. In this paper, the application of B?cklund transform and nonlinear superposition formula in nonlinear partial differential equations is studied by using auxiliary equations on the basis of summarizing previous studies. The exact solutions of fractional nonlinear partial differential equations and various forms of infinite sequence solutions are obtained. In this paper, the fractional order nonlinear partial differential equations are studied and described. The contents are arranged as follows: in chapter one, the nonlinear science, the research and development of soliton theory are described. The research and development of fractional partial differential equation and the methods used in the study. In chapter 2, the PKP equation and Gardner equation of fractional partial differential equation in time space are studied by using Riccati as auxiliary equation. In chapter 3, the mBBM equations of fractional partial differential equations with Riccati equation and the first elliptic equation are studied, and the advantages and disadvantages of the two auxiliary equations are analyzed. The fourth chapter summarizes the whole paper and prospects the future development of the research topic.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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