微分方程解的相似结构初探与展望  

Preliminary Exploration and Prospects of the Similar Structure of Solutions of Differential Equations

作　　者：李顺初[1] 

机构地区：[1]西华大学应用数学研究所,四川成都610039

出　　处：《西华大学学报（自然科学版）》2010年第2期223-226,238,共5页Journal of Xihua University:Natural Science Edition

基　　金：国家"九五"重点攻关项目(96-111-03-01-07)

摘　　要：针对二阶齐次线性常微分方程(组)的边值问题和二阶齐次线性偏微分方程(组)的混合问题,综述了分别对其解和Laplace空间解的相似结构理论的初步探讨的一些成果;分析了解的相似思想的形成及相关研究的意义,剖析了解的相似结构与定解方程和边值条件间的联系,发现了一种利用相似结构式和相似核函数构造定解问题的解的方法,介绍了它在解决渗流力学问题中的现有应用成果;最后,对微分方程解的相似结构理论的研究与应用前景进行了展望。Aiming at the boundary-value problem for second-order homogeneous linear ordinary differential equations（systems） and the mixed problem for second-order homogeneous linear partial differential equations （systems） , this paper reviews some results of preliminary explorations with regard to the similar structure theory of their solutions or their Laplace space solutions; analyzes the formation idea of the similar solution and its significance of research ; takes apart the relation between the similar structure of it＇ s solution and definite equation and the boundary conditions ; finds a way to construct the solution of definite problems by making use of the formula of similar structure together with the similar core function; introduces the existing achievements which have been applied to permeation fluid mechanics. Finally, the Research and application prospect of the Similar Structure theory of differential equation solution is presented. It can be predicted that quite the same as the study of other properties of differential equation solution, the exploration of similar structure theory of differential equation solution will inevitably arouse great interest among Mathematics and Applied Mathematics science and Engineering technology, and this will inspire people to explore more of its deeper theoretical results and applications.

关 键 词：微分方程 渗流力学 边值问题 相似结构 LAPLACE空间解 

分 类 号：O29[理学—应用数学] O357.3[理学—数学]