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GAUSS格式的拟渐近性(英文0714).
ON SPURIOUS ASYMPTOTIC NUMERICAL SOLUTIONS OF GAUSS SCHEME
Liu Wenhai
Fujian International Business and Economic College
Abstract
In this paper, we discuss characteristics of Gauss scheme for numerical ordinary differential equations, which is a one-step high-order accurate difference scheme. We apply such scheme to some equations, and analyze the fixed point and stability accordingly.
Key words: Ordinary differential equations,difference scheme,stability ,fixed point
1 Introduction
For most of the ordinary differential equations, their analytical expressions are not easily identified. Occasionally, even if the closed form of solutions can be found, these solutions turn out to be impractical because of the large amount of calculation involved. Under such circumstances numerical methods are introduced to solve ordinary differential equations.
With regard to the numerical scheme of ordinary differential equations, []the solution of different scheme may become periodic, chaotic, and divergent once the step length exceeds its stable boundary. For example, using an explicit Euler difference scheme to solve the following ordinary differential equations
(1.1)
We obtain different equations
(1.2)
Here we let h time step of different scheme , . If r 2 and (1.2) is convergent, the solutions to (1.2) are convergent to 1, which is the fixed point of the ordinary differential equation (1.1). If 2?r?2.43, the solutions of (1.2) possess double periodicity; If 2.43 r3, the solution(1.2) have multiple periodicity chaos phenomenon. If r?3, the solutions to (1.2) dissipates as a result.
This paper aims to construct the one-step explicit different scheme of ordinary differential equation (1.1), using GAUSS type integration formula. Furthermore, the different scheme obtained is applied to the ecological community growth model ,[] and discussion on its fixed point and stability is conducted accordingly.
2 GAUSS d
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