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一类生物模型方程组的差分特征列方法及精确解

蒋鲲1,2,王志科1,李文婷1,2   

  1. 1. 黑龙江大学数学科学学院,  哈尔滨 150080; 2. 黑龙江大学 黑龙江省复杂系统与计算重点实验室, 哈尔滨 150080
  • 出版日期:2019-08-25 发布日期:2019-12-05

蒋鲲,王志科,李文婷. 一类生物模型方程组的差分特征列方法及精确解[J]. 系统科学与数学, 2019, 39(8): 1322-1335.

JIANG Kun, WANG Zhike, LI Wenting. Difference Characteristic Method and Exact Solution for a Class of Biological Model Equation[J]. Journal of Systems Science and Mathematical Sciences, 2019, 39(8): 1322-1335.

Difference Characteristic Method and Exact Solution for a Class of Biological Model Equation

JIANG Kun 1,2 , WANG Zhike1 , LI Wenting 1,2   

  1. 1. School of Mathematical Science, Heilongjiang University, Harbin 150080; 2. Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Heilongjiang University, Harbin 150080
  • Online:2019-08-25 Published:2019-12-05

文章应用差分特征列方法研究一类具有生物性质的非线性差分方程组. 首先介绍差分特征列及$Z$变换的重要定义、定理, 接下来通过将生物模型方程组\uppercase\expandafter{\romannumeral1}, \uppercase\expandafter{\romannumeral2}~按照有效化、求特征列、判断一致性及不可约性等步骤获得方程组\uppercase\expandafter{\romannumeral1}, \uppercase\expandafter{\romannumeral2}~的特征列集和零点集, 最后结合 ~$Z$ 变换法分别得到方程组\uppercase\expandafter{\romannumeral1}, \uppercase\expandafter{\romannumeral2}~的零点集的一组精确解.

In this paper, the difference characteristic set method is used to study a class of nonlinear differential equations \uppercase\expandafter{\romannumeral1} and \uppercase\expandafter{\romannumeral2} who have special biological properties. First of all, the relevant definitions and important theorems of the difference characteristic set method are proposed. After that, the $Z$-transformation method are introduced by the definitions and properties. And then, in the third party of the paper, two differential equations \uppercase\expandafter{\romannumeral1}, \uppercase\expandafter{\romannumeral2} with biological properties are studied by the difference characteristic set method step by step, such as changing the difference equations, finding the characteristic sets and zero sets, judging the consistency and irreducibility of the sets, etc. In addition, combing with the given initial conditions, the $Z$-transformation methods are used to solve the zero sets of the differential equations \uppercase\expandafter{\romannumeral1}, \uppercase\expandafter{\romannumeral2}. At last, two groups of the exact solution of equations \uppercase\expandafter{\romannumeral1}, \uppercase\expandafter{\romannumeral2} are obtained respetively.

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