时间非齐次离散Fisher-KPP方程的行波解

发布时间：2018-05-03 14:13

本文选题：离散Fisher-KPP方程 + 行波解　；参考：《南京航空航天大学》2017年硕士论文

【摘要】：近年来,关于反应扩散方程的研究受到越来越广泛的关注,尤其是动力学方面的研究,这对于分析许多物理学、化学和生物学中的数学模型起到了重要的作用。其中一个很重要的研究课题是分析反应扩散方程的解的长期性态,而行波解理论就是其中典型的一类,它反映了方程的解具有波动的性质,一般用来刻画空间平移不变的解。1937年,Fisher和Kolmogorov,Petrovsky,Piskunov在研究基因传播模型时首次提出了反应扩散方程行波解的概念。此后,各类反应扩散方程的行波解及其相关性质被广泛地研究。在本文中,我们研究时间非齐次空间离散Fisher-KPP方程的行波解。首先,我们构造合适的全局上解和下解。其次,我们构造方程的逼近解序列,且为单调序列。最后,基于上述结论和逼近解序列的收敛性,我们可以得到行波解的存在性。另外,我们指出,当空间变量在给定的方向上趋于无穷时,这些行波解具有精确的衰减速度。
[Abstract]:In recent years, more and more attention has been paid to the study of reaction-diffusion equations, especially in the field of kinetics, which plays an important role in the analysis of many mathematical models in physics, chemistry and biology. One of the most important research topics is to analyze the long-term behavior of the solution of the reaction diffusion equation, and the traveling wave solution theory is a typical one, which reflects the wave property of the solution of the equation. In 1937, Fisher and Kolmogorovski Petrovsky Piskiev proposed for the first time the concept of traveling wave solution of the reaction diffusion equation when studying the gene propagation model. Since then, the traveling wave solutions of various reaction-diffusion equations and their related properties have been extensively studied. In this paper, we study the traveling wave solutions of discrete Fisher-KPP equations in time inhomogeneous spaces. First, we construct suitable global upper and lower solutions. Secondly, we construct a sequence of approximate solutions of the equation, which is a monotone sequence. Finally, based on the above conclusions and the convergence of the approximate solution sequence, we can obtain the existence of the traveling wave solution. In addition, we point out that these traveling wave solutions have an accurate attenuation rate when space variables tend to be infinite in a given direction.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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