广东工业大学学报 ›› 2020, Vol. 37 ›› Issue (04): 69-74.doi: 10.12052/gdutxb.190162

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一类延迟Gompertz方程的数值解的振动性分析

阳倩, 王琦   

  1. 广东工业大学 应用数学学院，广东 广州 510520
  • 收稿日期:2019-12-25 出版日期:2020-07-11 发布日期:2020-07-02
  • 通信作者: 王琦(1978-)，男，教授，主要研究方向为微分方程数值解，E-mail：bmwzwq@126.com E-mail:bmwzwq@126.com
  • 作者简介:阳倩(1997-)，女，硕士研究生，主要研究方向为常微分方程数值解的振动性
  • 基金资助:
    广东省自然科学基金资助项目(2017A030313031)

An Oscillation Analysis of Numerical Solution of A Class of Delayed Gompertz Equations

Yang Qian, Wang Qi   

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2019-12-25 Online:2020-07-11 Published:2020-07-02

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摘要： Gompertz方程常用于描述种群动态和肿瘤生长, 本文研究了一类延迟Gompertz方程的振动性。首先利用泰勒公式线性化该方程, 再对线性方程应用线性θ方法得到其差分格式。其次, 运用振动理论分别分析线性化后的方程和所得差分格式。在研究方程数值解的振动性时, 把差分方程中θ的取值范围分为2部分, 通过分析差分方程的特征方程的解的性质, 得到延迟Gompertz方程的解析解和数值解振动的充分条件，最后进行数值实验验证。

关键词: 非线性延迟微分方程, 振动性, 数值解, Gompertz

Abstract: Oscillation of numerical solutions is studied with regard to a class of delayed Gompertz equations, which have been widely used in description of the population dynamics and tumour growth. Firstly, the linearized equations are obtained by Taylor formula and its corresponding difference equations by linear θ method. Secondly, the oscillation theory is applied to analyze those obtained equations. In the process, the oscillation is primarily discussed through studying the properties of roots for the corresponding characteristic equations. For requirement, the oscillation of numerical solutions is discussed while the variable θ belongs in different scopes. Accordingly, the sufficient conditions under which numerical solutions oscillate are acquired. To verify the results, some numerical experiments are given. The first three experiments validate the conditions of the delayed Gompertz equation which has three delay terms. And the rest of the experiments check on another which has two delay terms.

Key words: nonlinear delay differential equation, oscillation, numerical solutions, Gompertz

中图分类号: 

  • O241
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[1] 谭满春； . 非线性差分方程的振动性充分判则[J]. 广东工业大学学报, 2002, 19(1): 106-109.
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