一类奇异非线性偏微分方程形式解的研究

发布时间：2018-03-11 01:12

本文选题：非线性偏微分方程　切入点：形式幂级数解　出处：《渤海大学》2017年硕士论文　论文类型：学位论文

【摘要】：近来,人们发现一类双奇异常微分方程(组)的形式解关于一个双变量的单项式是可和的,很多奇异偏微分方程的形式解是多重可和的,可见,多变量的形式幂级数的可和性理论对于研究偏微分方程的形式解具有举足轻重的作用,尤其是二变量的形式幂级数的单项可和性理论的建立,更加方便了人们对于偏微分方程形式解的可和性的研究.本文建立了一类偏微分方程,并论证其形式解的单项可和性,丰富了微分方程形式解的研究方面上的成果,是形式幂级数的单项可和性理论的一个应用.以下为本文的主要研究工作:首先,给出一类偏微分方程,做出适当的假设,使其具有特定形式的形式幂级数解.并给出一个具体的例子,计算其形式解,指出它关于一单项式的Gevrey阶数,说明此类偏微分方程具备这类关于一单项式可和的形式解.其次,通过形式上的变换,将偏微分方程化为两列常微分方程,根据其解,选定其特殊的存在区域,利用不动点原理,论证偏微分方程在该类区域上解析有界解的存在唯一性.最后,应用可和性理论中的一重要结论,论证偏微分方程形式解的单项可和性.
[Abstract]:Recently, it has been found that the formal solutions of a class of bisingular ordinary differential equations (systems) are summable with respect to a bivariate monomial expression, and many singular partial differential equations are multifold summable. The summability theory of multivariable formal power series plays an important role in the study of formal solutions of partial differential equations, especially the establishment of the monomial summability theory of two-variable formal power series. It is more convenient for people to study the summability of formal solutions of partial differential equations. In this paper, a class of partial differential equations is established, and the monomial summability of their formal solutions is proved, which enriches the achievements in the study of formal solutions of differential equations. It is an application of the theory of monomial summability of formal power series. The following is the main research work of this paper: firstly, we give a class of partial differential equations and make appropriate assumptions. We give a concrete example to calculate its formal solution, point out its Gevrey order with respect to a monomial, and prove that this kind of partial differential equation has this kind of formal solution for a monomial summation. By means of formal transformation, the partial differential equation is transformed into two series of ordinary differential equations. According to its solution, its special existence region is selected, and the existence and uniqueness of analytic bounded solution of partial differential equation in this kind of region are proved by using the fixed point principle. The monomial summability of formal solutions of partial differential equations is proved by applying an important conclusion in summability theory.
【学位授予单位】：渤海大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2

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