基于Riccati方程的非线性微分方程并行求解及在线方程知识库的研发

发布时间：2018-04-13 01:29

本文选题：非线性演化方程 + 行波解　；参考：《华东师范大学》2017年硕士论文

【摘要】：非线性演化方程在非线性科学领域的研究中起着非常重要的作用,这些方程的解析解,特别是行波解,可以准确地描述许多物理现象的内在规律,例如振动、传播波以及孤立子等。随着计算机科学和符号计算方法的快速发展,涌现了很多构造非线性演化方程解析解的方法,如双曲正切方法,椭圆函数方法等。这些方法被称为直接代数方法。然而,由于符号计算的精确性及表达式快速膨胀等原因,直接代数方法普遍存在计算效率较低的问题。为了有效提高求解效率,本文将并行计算的思想应用到了直接代数方法求解非线性演化方程的过程中。基于Riccati方程方法和并行计算的理念,本文提出了一种并行化构造非线性演化方程精确行波解的新算法。特别是在并行求解的过程中,将多项式因式分解与负载均衡技术有机结合,有效地提高了多个CPU的利用率。而且,相比于其他已有的方法,通过分解算法和运行时间限制,我们可以获得更多的解。本文提出的并行算法已经在Maple 18上进行了实现,并封装成一个具有灵活接口和输入输出形式的软件包PREM。通过将该软件应用于一些具体实例及对结果进行比较分析,验证了本文提出的并行算法不仅在效率上比串行求解算法有了显著的提高,而且求解能力也强于原有算法。从自然科学到社会科学,几乎所有的学科领域都越来越倾向于采用微分方程方法求解问题。为了便于不同领域的学者及工程技术人员学习微分方程知识、进行学术交流和合作,本文研发了一个在线的开放的方程知识库系统。该知识库的研发综合使用了多种较新的技术来实现方程的存储、编辑和展示。特别地,系统中数学公式的可视化显示均采用了二维形式。另外,对每一个方程,除了用一条记录保存其重要信息,如方程类型、方程名称、方程表达式等,还为每个方程建立了一个方程页面,在其中展示该方程的基本信息、方程背景知识、方程的相关研究成果等。在方程页面中也内置了评论区域以便学术讨论。该系统不仅功能完备,而且能快速响应各种操作,具有良好的用户体验。
[Abstract]:Nonlinear evolution equations play an important role in the research of nonlinear science. The analytical solutions of these equations, especially the traveling wave solutions, can accurately describe the inherent laws of many physical phenomena, such as vibration.Propagation waves and solitons, etc.With the rapid development of computer science and symbolic computing methods, there are many methods to construct analytical solutions of nonlinear evolution equations, such as hyperbolic tangent method, elliptic function method and so on.These methods are called direct algebraic methods.However, due to the accuracy of symbolic computation and the rapid expansion of expressions, direct algebraic methods generally have the problem of low computational efficiency.In order to improve the efficiency of solution, the idea of parallel computing is applied to the direct algebraic method for solving nonlinear evolution equations.Based on the Riccati equation method and the idea of parallel computing, a new algorithm for constructing exact traveling wave solutions of nonlinear evolution equations is proposed in this paper.Especially in the process of parallel solution, combining polynomial factorization with load balancing technology can effectively improve the utilization rate of multiple CPU.Moreover, compared with other existing methods, we can obtain more solutions by decomposing algorithms and running time constraints.The parallel algorithm proposed in this paper has been implemented on Maple 18 and encapsulated into a software package PREMwith flexible interface and input and output forms.By applying the software to some concrete examples and comparing and analyzing the results, it is verified that the proposed parallel algorithm is not only more efficient than the serial algorithm, but also better than the original algorithm.From natural science to social science, almost all disciplines are more and more inclined to use differential equations to solve problems.In order to facilitate scholars and engineers in different fields to learn the knowledge of differential equations and to carry out academic exchanges and cooperation, an online and open knowledge base system of equations is developed in this paper.The research and development of the knowledge base uses a variety of new technologies to store, edit and display equations.In particular, the visualization of mathematical formulas in the system is in two-dimensional form.In addition, for each equation, in addition to keeping its important information with a record, such as the type of equation, the name of the equation, the expression of the equation, and so on, an equation page is established for each equation, in which the basic information of the equation is displayed.Equation background knowledge, equation related research results and so on.Comment areas are also built into the equation page for academic discussion.The system not only has complete functions, but also can quickly respond to various operations and has a good user experience.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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