奇异摄动非局部问题的高精度数值方法

HIGH ACCURATE NUMERICAL METHOD FOR A SINGULLARLY PERTURBED NONLOCAL PROBLEM

在许多物理现象的模型问题中会出现如（其中０＜ε＜＜１）的奇异振动非局部问题，Ｂｉｃａｄｚｅ和Ｓａｍａｒｓｋｉｉ［１］指出条件Ａ．ｂ（ｘ）∈Ｃ２（ｘ），０＜β２）保证了问题（１）存在唯一解，并且给出了求数值解的方法，但是它的精度较低。本文把问题（１）分解成两个奇异摄动常微分方程边值问题，利用Ｌｉｏｕｖｉｌｌｅ－Ｇｒｅｅｎ变换我们得到了这两个微分方程的近似微分方程。进而在非均匀网格上建立了具有三阶精度的一致收敛差分格式．最后给出了数值例子，计算结果比理论分析更好。

The singularly perturbed nonlocal problem (when 0<ε<<1) often Occurs in the model problem of many physical phenomena. Bicadze and Samarskii[1] pointed out that conditions A. b(x) ∈C2(I), 0<β2 ≤ b (x), x∈ I, and B. there exists the solution for problem (1). They established a method of numerical solution for problem (1). But the accuracy of the method is low. In this paper, we decompose problem (1) into two boundary value problems of differetial epuation and using Liouvile-Green transformation, we obtain two approxmate differential iquations of these two differential equations. Further, we establish the third order accuracy uniform convergence difference shceme on a nonuniform mesh.