偏微分方程的函数论方法及应用和计算(英文)

Function Theoretic Methods for Partial Differential Equations With Their Applications and Numerical Analysis

本文主要介绍了偏微分方程一些边值问题的函数论方法。首先给出了边值问题的适定提法;其次研究了多复变函数、Clifford代数、某类抛物型方程、一些复合型方程组和双曲型方程组各种边值问题的可解性;进而使用一阶椭圆型方程组间断边值问题的结果,解决了渗流理论、空气动力学与弹性力学中提出的若干自由边界问题;最后还讨论了某些椭圆边值问题与拟共形映射的近似解法。从此文可以看出;函数论方法在处理偏微分方程时的一些优点。

In this paper, we attempt to introduce mainly the solvability of several boundary value problems for partial differential equations by using function theoretic methods. We first give the proper formulations of some elliptic boundary value problems and solution's stability, and then investigate various boundary value problems for several complex variables and in Clifford analysis. Meanwhile, we examine some parabolic equations and systems of hyperbolic equations as well as composite type equations. By applying the results of discontinuous boundary value problems for elliptic systems of first order, we solve the free boundary problems occuring such as in filtrations, gas dynamics and elastico-plastic mechanics. Furthermore, by using the finite element method and the Newton imbedding method in numerical analysis, we obtain the approximate solutions of some elliptic boundary value problems and quasiconformal mappings. Throughout this paper, in handling partial differential equations, we can see the advantage of function theoretic methodsi however, there remain many questions not yet solved.