摘要
为了求解物理化学生物材料和金融中的微分方程,提出了一种总体(Global)和局部(Local)场方法.微分方程的求解区域可以是有限域,无限域,或具曲面边界的部分无限域.其无限域包括有限有界不均匀介质区域.其不均匀介质区域被分划为若干子区域之和.在这含非均匀介质的无限区域,将微分方程的解显式地表示为在若干非均匀介质子区域上和局部子曲面的积分的递归和.把正反算的非线性关系递归地显式化.在无限均匀区域,微分方程的解析解被称为初始总体场.微分方程解的总体场相继地被各个非均匀介质子区域的局部散射场所修正.这种修正过程是一个子域接着另个子域逐步相继地进行的.一旦所有非均匀介质子区域被散射扫描和有限步更新过程全部完成后,微分方程的解就获得了.称其为总体和局部场的方法,简称为GL方法.GL方法完全地不同于有限元及有限差方法,GL方法直接地逐子域地组装逆矩阵而获得解.GL方法无需求解大型矩阵方程,它克服了有限元大型矩阵解的困难.用有限元及有限差方法求解无限域上的微分方程时,人为边界及其上的吸收边界条件是必需的和困难的,人为边界上的吸收边界条件的不精确的反射会降低解的精确度和毁坏反算过程.GL方法又克服了有限元和有限差方法的人为边界的困难.GL方法既不需要任何人为边界又不需要任何吸收边界条件就可以子域接子域逐步精确地求解无限域上的微分方程.有限元和有限差方法都仅仅是数值的方法,GL方法将解析解和数值方法相容地结合起来.提出和证明了三角的格林函数积分方程公式.证明了当子域的直经趋于零时,波动方程的GL方法的数值解收敛于精确解.GL方法解波动方程的误差估计也获得了.求解椭圆型,抛物线型,双曲线型方程的GL模拟计算结果显示出我们的GL方法具有准确,快速,稳定的许多优
We propose a new GL method for solving the ordinary and the partial differential equation. These equations govern the electromagnetic field etc. macro and micro physical, chemical, financial problems in the sciences and engineering. The domain can be finite, infinite, or part of the infinite domain with a curve surface. The differential equation is held in an infinite domain which includes a finite inhomogeneous domain. The inhomogeneous domain is divided into finite sub domains. We present the solution of the differential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains. We discover the explicit relationship between forward modeling and inversion. The analytical solution of the equation in the infinite homogeneous domain is called as an initial global field. The global field is updated by local scattering field successively subdomain by subdomain. Once all subdomains are scattered and the finite updating process is finished in all the sub domains, the solution of the equation is obtained. We call our method as Global and Local field method, in short GL method. The GL method is totally different from Finite Element Method （FEM） method and Finite Difference Method （FD）, solution successively subdomain by the GL method directly assemble inverse matrix and get subdomain. There is no big matrix equation needs to solve in the GL method which overcome FEMI and FDIs difficult for solving big matrix equation. When the FEM and FD are used to solve the differential equation in the infinite domain, the artificial boundary and absorption boundary condition are necessary and difficult. The error reflections from the artificial absorption boundary condition downgrade the accuracy of the forward solution and damage the inversion resolution. The GL method resolves the artificial boundary difficulty in FEM and FD methods. There is no artificial boundary and no absorption boundary condition for infinite domain in the GL method. We proposed a triangle integral equation of the Green＇
出处
《数学的实践与认识》
CSCD
北大核心
2007年第22期98-109,共12页
Mathematics in Practice and Theory
关键词
总体和局部场方法
正反算
非均匀介质
无限域
无大型矩阵
无人为边界
电磁场
弹塑性场
地震波场
声波场
流场
量子场
Global and local field method
modeling and inversion
infinite domain
no largematrix
no artificial boundary
elastic and plastic field
seismic field
acoustic field
flow field
quantum field
