【作者】 董文佳；

【导师】 徐阳；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2009， 硕士

【摘要】 本文主要研究了自变量分段连续微分方程数值解的稳定性和振动性。第一章,概要地介绍了自变量分段连续时滞微分方程的研究目的和意义,以及近些年来在该领域的研究现状。这类方程广泛存在于应用科学的各个领域中,其数值解的稳定性分析具有重要的理论价值和实际意义。振动性的研究是时滞微分方程定性理论研究的一个重要方面,而时滞微分方程又可以刻画生活中的许多模型,因此,研究时滞微分方程的振动性是很有必要的。第二章,针对分段连续时滞微分方程构造出Rosenbrock方法的计算格式,并将该方法应用于具体的分段连续时滞微分方程,讨论得到数值解渐近稳定的充分条件,同时考虑解析解的渐近稳定性条件,加以改进,得到最后的结论。然后利用特征方程的方法对Rosenbrock方法进行分析,最后得到Rosenbrock方法保持数值解振动性的充要条件。第三章,将分离变量法应用于分段连续抛物型微分方程,进而得到解析解的稳定性条件,在此条件下,针对具体的纯量抛物微分方程的Crank-Nicholson差分格式进行分析,得出关于方程参数的稳定性条件。对于每一部分的讨论,都给出了相应的数值算例,这些算例验证了理论上推得的结论的正确性。更多还原

【Abstract】 This paper deals with the numerical stability of the differential equations with piecewise continuous arguments mainly.First, this paper presents the purpose and the significance of researching about the delay differential equations with piecewise continuous arguments in brief, what’s more, it introduces the research actuality in this field. This class of equations appears in many fields of applied sciences, and the stability analysis of the numerical solution has important theoretical and practical significance. The research about oscillation is an important part of the research about the differential equations. Also, the delay differential equations can show many phenomena, so it is necessary to research about the oscillation of the delay differential equations.Secondly, it applies the Rosenbrock method to the delay differential equations with piecewise continuous arguments, then discusses the stability conditions of numerical solution when Rosenbrock method is applied to the concrete delay differential equations with piecewise continuous arguments. Also, the preservation of oscillatory property of Rosenbrock method is considered.Last but not least, we apply the Separation of variables method to the parabolic partial differential equation with piecewise continuous arguments, and obtain the stability condition of the concrete scalar parabolic partial equation using Crank-Nicholson scheme, and the stability condition is about the parameters of the equation.Moreover, the relevant numerical examples are given in every part. These experiments verify the conclusion obtained in the theoretical analysis.更多还原