【作者】 王雪芹；

【导师】 余越昕；

【作者基本信息】 湘潭大学 ， 计算数学， 2011， 硕士

【摘要】 泛函微分方程(FDEs)在生物学、物理学、化学、经济学、控制理论等众多领域有广泛应用。由于其理论解很难获得，只能用数值方法进行近似计算，因而其算法理论的研究具有毋庸置疑的重要性。近几十年，众多学者对问题的适定性及数值方法的稳定性和收敛性进行了大量研究，取得了丰硕成果。这些研究大多局限于讨论有限维内积空间中的初值问题，是以内积范数和单边Lipschitz条件为基础的。然而，科学与工程技术中还存在大量刚性问题，尽管问题本身是整体良态的，但当使用内积范数时，其最小单边Lipschitz常数却不可避免地取非常巨大的正值。因此有必要突破内积范数和单边Lipschitz常数的局限，直接在Banach空间研究数值方法的理论。在此方面，文立下、王晚生等做了许多有益的工作，取得了一系列研究成果。泛函微分与泛函方程(FDFEs)是较泛函微分方程更广泛的一类混合系统，是由泛函微分方程与泛函方程耦合而成，其理论解和数值方法的研究更具复杂性，目前仅有少量文献在内积空间基于Lipschitz条件对数值方法的稳定性进行了研究。有鉴于此，本文直接在Banach空间中对一类非线性泛函微分与泛函方程研究Runge—Kutta法的数值稳定性，获得了方法稳定的条件，数值试验也验证了所获理论的正确性。更多还原

【Abstract】 Functional Di?erential Equations(FDEs) arise widely in physics, biology,chemistry, economics, control theory and so on. Their exact solutions are hardlyobtained. Thus, it is meaningful to investigate the theory and application ofnumerical methods for FDEs. In the last few decades, the theory of numericalmethods for FDEs have been widely discussed by many authors, and a seriesof stability and convergence results have been obtained. The existing resultsfor FDEs are mainly based on the inner product and corresponding norm inEuclidean space of finite dimension. But there exists lots of sti? problems inscience and engineering, it may happen that the one-sided Lipschitz constantis very large. Therefore, it is urgent and meaningful to break the restriction ofthe inner product and the corresponding norm to study the numerical theory inBanach space. In this field, Liping Wen and Wansheng Wang have done a lotof work and obtained a series of results.Functional di?erential and functional equations(FDFEs) are more complexthan FDEs. They are a class of hybird problem formed by functional di?erentialequations and functional equations. In the present paper, we study the numericalstability of Runge-Kutta methods for a kind of nonlinear FDFEs in Banachspace. The stability conditions of the methods are derived. Numerical testshave given to confirm the theoretical results in the end.更多还原