差分唯一性与两类微分方程的精确解

发布时间：2018-06-09 01:09

  本文选题：差分 + 唯一性　； 参考：《南京大学》2017年博士论文

【摘要】：科学的最大挑战就是描述和预测。当观察到某一现象后,我们总是希望能够表述现在看到的现象以及将要发生的事情。在许多重要的情况下,我们总会得到一些微分方程或差分方程。而且在经过一些努力研究后,很多关于几何学、物理学、工程学及经济学方面的重要信息都可以经过分析方程得到。对于这些方程我们最基本的问题就是它们的解的存在性和唯一性。当然,我们目前已经可以应用计算机得到数值解。尽管可以由计算机得到我们所关心的方程的部分结果,我们依然对于求解方程很感兴趣。对于给定的方程,一个具体的解更有利于应用。对于具体解的研究面临很多困难。由于在复数邻域的展开式可以帮助我们获得更多的认知,差分和微分方程从实数域向复数域的发展是一个不可避免的趋势。我们已经对于求解方程的亚纯解做出了很多努力,并且得到了一些可以处理复方程的方法,其中局部定理是这些方法中研究最多的一种。在本文中,我们从解决方程的最关心的问题出发,探讨了函数在差分情况下唯一的一个充要条件,并且得到了两类Briot-Bouquet微分方程的精确亚纯解。在第一章中,我们主要介绍了起源于上世纪20年代的Nevanlinna理论,它在复方程的研究中具有非常重要的作用。之后我们给出了在研究整函数中起到重要作用的Wiman-Valiron理论。当然我们也需要介绍差分的一些结论。最后,我们应用Kowalevski-Gambier方法介绍了Painleve测试并且引入了一些椭圆函数的内容。在第二章中,我们介绍了关于Brück猜想的一个差分版本,并且对于超越整函数差分多项式分担小函数给出了其唯一性的一个充要条件。进一步,我们还能够确定多项式。第三章和第四章中分别探讨了两类Briot-Bouquet微分方程,其目标是应用Kowalevski-Gambier 方法得到这两类方程的精确亚纯解。
[Abstract]:The greatest challenge of science is to describe and predict. When we observe a phenomenon, we always want to be able to express what we see and what will happen. In many important cases, we always get some differential equations or difference equations. And after some hard work, a lot of important information about geometry, physics, engineering, and economics can be obtained by analytical equations. The most basic problem for these equations is the existence and uniqueness of their solutions. Of course, we can now use the computer to obtain numerical solutions. Although some results of the equation we are concerned with can be obtained by computer, we are still interested in solving the equation. For a given equation, a specific solution is more convenient for application. There are many difficulties in the study of concrete solutions. Because the expansion in complex neighborhood can help us to gain more cognition, the development of difference and differential equations from real field to complex field is an inevitable trend. We have made a lot of efforts to solve the meromorphic solutions of equations, and we have obtained some methods to deal with complex equations, among which the local theorem is one of the most studied methods. In this paper, we discuss a necessary and sufficient condition for the function to be unique in the case of difference, and obtain two kinds of exact meromorphic solutions of the Briot-Bouquet differential equation. In the first chapter, we mainly introduce Nevanlinna theory, which originated in 1920s, which plays an important role in the study of complex equations. Then we give the Wiman-Valiron theory which plays an important role in the study of whole functions. Of course, we also need to introduce some conclusions of the difference. Finally, we introduce the Painleve test using Kowalevski-Gambier method and introduce some elliptic functions. In the second chapter, we introduce a difference version of Br 眉 ck conjecture, and give a necessary and sufficient condition for the difference polynomial of transcendental whole function to share a small function. Furthermore, we can determine the polynomial. In chapter 3 and chapter 4, we discuss two classes of Briot-Bouquet differential equations, the purpose of which is to obtain exact meromorphic solutions of these two kinds of equations by using Kowalevski-Gambier method.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O174.52;O175
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本文编号：1998099