本文選題：差分 + 唯一性��； 參考：《南京大學(xué)》2017年博士論文

【摘要】：科學(xué)的最大挑戰(zhàn)就是描述和預(yù)測。當(dāng)觀察到某一現(xiàn)象后,我們總是希望能夠表述現(xiàn)在看到的現(xiàn)象以及將要發(fā)生的事情。在許多重要的情況下,我們總會得到一些微分方程或差分方程。而且在經(jīng)過一些努力研究后,很多關(guān)于幾何學(xué)、物理學(xué)、工程學(xué)及經(jīng)濟學(xué)方面的重要信息都可以經(jīng)過分析方程得到。對于這些方程我們最基本的問題就是它們的解的存在性和唯一性。當(dāng)然,我們目前已經(jīng)可以應(yīng)用計算機得到數(shù)值解。盡管可以由計算機得到我們所關(guān)心的方程的部分結(jié)果,我們依然對于求解方程很感興趣。對于給定的方程,一個具體的解更有利于應(yīng)用。對于具體解的研究面臨很多困難。由于在復(fù)數(shù)鄰域的展開式可以幫助我們獲得更多的認(rèn)知,差分和微分方程從實數(shù)域向復(fù)數(shù)域的發(fā)展是一個不可避免的趨勢。我們已經(jīng)對于求解方程的亞純解做出了很多努力,并且得到了一些可以處理復(fù)方程的方法,其中局部定理是這些方法中研究最多的一種。在本文中,我們從解決方程的最關(guān)心的問題出發(fā),探討了函數(shù)在差分情況下唯一的一個充要條件,并且得到了兩類Briot-Bouquet微分方程的精確亞純解。在第一章中,我們主要介紹了起源于上世紀(jì)20年代的Nevanlinna理論,它在復(fù)方程的研究中具有非常重要的作用。之后我們給出了在研究整函數(shù)中起到重要作用的Wiman-Valiron理論。當(dāng)然我們也需要介紹差分的一些結(jié)論。最后,我們應(yīng)用Kowalevski-Gambier方法介紹了Painleve測試并且引入了一些橢圓函數(shù)的內(nèi)容。在第二章中,我們介紹了關(guān)于Brück猜想的一個差分版本,并且對于超越整函數(shù)差分多項式分擔(dān)小函數(shù)給出了其唯一性的一個充要條件。進(jìn)一步,我們還能夠確定多項式。第三章和第四章中分別探討了兩類Briot-Bouquet微分方程,其目標(biāo)是應(yīng)用Kowalevski-Gambier 方法得到這兩類方程的精確亞純解。
[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.
【學(xué)位授予單位】：南京大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O174.52;O175
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本文編號：1998099