本文選題：非線性偏微分方程　切入點：形式冪級數(shù)解　出處：《渤海大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：近來,人們發(fā)現(xiàn)一類雙奇異常微分方程(組)的形式解關(guān)于一個雙變量的單項式是可和的,很多奇異偏微分方程的形式解是多重可和的,可見,多變量的形式冪級數(shù)的可和性理論對于研究偏微分方程的形式解具有舉足輕重的作用,尤其是二變量的形式冪級數(shù)的單項可和性理論的建立,更加方便了人們對于偏微分方程形式解的可和性的研究.本文建立了一類偏微分方程,并論證其形式解的單項可和性,豐富了微分方程形式解的研究方面上的成果,是形式冪級數(shù)的單項可和性理論的一個應(yīng)用.以下為本文的主要研究工作:首先,給出一類偏微分方程,做出適當(dāng)?shù)募僭O(shè),使其具有特定形式的形式冪級數(shù)解.并給出一個具體的例子,計算其形式解,指出它關(guān)于一單項式的Gevrey階數(shù),說明此類偏微分方程具備這類關(guān)于一單項式可和的形式解.其次,通過形式上的變換,將偏微分方程化為兩列常微分方程,根據(jù)其解,選定其特殊的存在區(qū)域,利用不動點原理,論證偏微分方程在該類區(qū)域上解析有界解的存在唯一性.最后,應(yīng)用可和性理論中的一重要結(jié)論,論證偏微分方程形式解的單項可和性.
[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.
【學(xué)位授予單位】：渤海大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.2

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本文編號：1595911