Singular Nonlinear Partial Differential Equations
Singular Nonlinear Partial Differential Equations
复制标题
奇异非线性偏微分方程
DOI：
10.1007/978-3-322-80284-2
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
H. Tahara
中科院分区：
文献类型：
--
作者：
R. Gérard;H. Tahara
这本书的目的是把所有已知的结果放在一起关于奇异非线性偏微分方程的形式解、全纯解和奇异解的存在性，我们研究奇异非线性偏微分方程的形式幂级数解、全纯解和奇异解的存在性。在第一章中，我们在单变量情形下引入了正则奇异算子，并给出了经典代数微分方程Maillet定理的一个新的简单证明。在第二章中，我们将这一理论推广到多元算子。第三章利用迭代法和牛顿法研究了一类具有线性部分的奇异偏微分方程的形式幂级数解的收敛性。作为前面结果的应用，我们在第四章中研究了xy ′ = 1（x，y）形式的微分方程的局部理论，特别地，我们说明了当1（0，0）= 0时，找到这样一个方程的经典结果是多么容易，并给出了当1（0，0）= 0时，这样一个方程的研究。0）#-0，这是以前从未给出过的，可以扩展到形式为Ty= F（x，y）的方程，其中T是任意向量场。
The aim of this book is to put together all the results that are known about the existence of formal, holomorphic and singular solutions of singular non linear partial differential equations.We study the existence of formal power series solutions, holomorphic solutions, and singular solutions of singular non linear partial differential equations. In the first chapter, we introduce operators with regular singularities in the one variable case and we give a new simple proof of the classical Maillet's theorem for algebraic differential equations. In chapter 2, we extend this theory to operators in several variables. The chapter 3 is devoted to the study of formal and convergent power series solutions of a class of singular partial differential equations having a linear part, using the method of iteration and also Newton's method. As an application of the former results, we look in chapter 4 at the local theory of differential equations of the form xy'= 1 (x, y) and, in particular, we show how easy it is to find the classical results on such an equation when 1 (0, 0)= 0 and give also the study of such an equation when 1 (0, 0)#-0 which was never given before and can be extended to equations of the form Ty= F (x, y) where T is an arbitrary vector field.