地下水资源研究中的变分原理和广义变分原理的推导

Deduction of Variational Principles and Generalized Variational Principles in Groundwater Resource Research

本文着重讨论建立地下水资源研究中的经典变分原理和广义变分原理的方法,并以岩层渗流问题中附有齐次和非齐次边界条件的二维Laplace方程为例,介绍了如下四种推求变分原理的方法: 1.根据物理学中已确定的定理来推求泛函; 2.用数学运算推导变分原理; 3.利用数学领域中已有的经典定理来提出变分原理; 4.利用Fréchet导数验证变分原理的存在和提出近似变分原理。 然后,作者进一步叙述了把附有约束条件的变分原理转化为广义变分原理的方法。此外,在本文的最后部分还对郭友中所建立的互补变分原理作了简略的叙述。

In this paper, the author laid special stress on diacussing the methods which are used to establish classical variational principles and generalized variational principles in groundwater resource research. By means of the example of two-dimensional Laplace equation with homogeneous and non-homogeneous boundary condition in the seepage flow problems of strata, the author discribes the following methods for deduction of variational principles.1 . To deduce the function according to some established theorems of physics ;2. To dedece the variational principles by means of mathematical calculation ;3. The variational principle is presented by means of some present classical theorems in the area of mathematics;4 . To verify the variational principle and to present the approximate variational principle Frechet derivative is used.Then the author further discribes the methods of translating variational principle with the restrictive conditions into generalized variational principle, In addition, a concise introduction to the complementary variational principles established by Guo Youzhong is given.