基于递归公式求解某些椭圆型偏微分方程之特解(英文)

Particular solutions of some elliptic partial differential equations via recursive formulas

对某些具有多项式右端项的非齐次椭圆型偏微分方程,利用基于待定系数法原理而得到的一些直接迭代程式,就可以快速得到精确的多项式函数特解.我们对对流-反应方程、轴对称Poisson方程、轴对称Helmholtz型方程等给出了显式迭代公式,它们本质上等价于解对应的决定特解多项式系数的上三角型线性方程组.这些特解可用于工程上常用的"基本解方法"来数值求解有关的偏微分方程边值问题.

Using the principle of the method of undetermined coefficients and the technique of upper triangular systems of linear algebraic equations, a simple and direct iterative numerical procedure was proposed to obtain particular solutions for various types of inhomogeneous elliptic partial differential equations. This procedure employs the power series expansion of both the source function of the partial differential equation and the solution, and the problem of finding a particular solution is equivalent to solving a triangular system of linear algebraic equations. In the special case of polynomial source functions, we are often led to solving a finite triangular system. To demonstrate the effectiveness of the proposed scheme, recursive formulas for some elliptic equations were developed. Coupled with existing boundary methods for solving boundary value problems of homogeneous equations, the proposed method can be used to solve various types of partial differential equations.