几类随机泛函微分方程数值方法的收敛性、稳定性和散逸性

项目名称： 几类随机泛函微分方程数值方法的收敛性、稳定性和散逸性

项目编号： No.10871207

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 甘四清

作者单位： 中南大学

项目金额： 27万元

中文摘要： 本项目研究随机泛函微分方程数值方法的收敛性和稳定性。对中立型随机延迟微分方程，建立了单步方法的局部截断误差与整体截断误差之间的关系，获得了单步方法收敛阶的判别准则，进一步将这些判别准则推广到带跳的中立型随机延迟微分方程，直接获得了一些半隐和全隐方法的收敛性结论。系统深入地研究了随机延迟微分方程，随机比例方程，中立型随机延迟微分方程数值方法的稳定性，提出了分别求解非线性随机微分方程、非线性变延迟随机微分方程和带跳的非线性随机微分方程的有效数值格式，在不降低精度的前提下，我们所构造的算法在稳定性方面明显优于文献中相应的数值格式，这对今后构造高稳定和高精度的数值格式，对高效求解刚性随机微分方程都具有非常重要的指导意义。作为全隐方法的代表，平衡隐式方法在数值求解随机微分方程中具有特别重要的地位，本项目还研究了随机泛函微分方程平衡隐式方法的收敛性和稳定性。 随机微分方程、随机延迟微分方程和随机中立型延迟微分方程是科学与工程中常见的几类随机泛函微分方程，深入开展上述几类随机泛函微分方程数值方法的收敛性和稳定性研究具有重要的理论意义和广泛的应用前景。

中文关键词： 随机泛函微分方程；数值方法；收敛性；稳定性

英文摘要： This program focused on the convergence and the stability of numerical methods for stochastic functional differenrial equations (SFDEs). The relationship between the global error and local error of one- step methods for neutral stochastic delay differential equations (NSDDEs) is established, and a criteria for order of convergence of one-step method is obtained. Further, the criteria is extended to NSDDEs with jumps and convergence results of semi-implicit and fully implicit schemes are derived directly. The stability of numerical methods for stochastic delay differential equations (SDDEs), stochastic pantograph equations and neutral stochastic delay differential equations is investigated. Effective numerical schemes are proposed for nonlinear stochastic differential equations (SDEs), nonlinear SDDEs with variable delays and nonlinear SDEs with jumps, respectively. The proposed schemes have the same order of convergence as the existing related schemes do, but the proposed schemes have an advantage over the existing related schemes in the stability. This will be very helpful for us to construct numerical methods with higher accuracy and better stability and to solve stiff stochastic differential equations efficiently. As a presentative of fully implicit methods, the balanced implicit method gives its status in solving SDEs. This program also studied the convergence and stability of the balanced implicit methods. SDEs, SDDEs and NSDDEs are three classes of important SFDEs. The models arise widely in many areas of science and engineering. It is of great theoritical significance and a wide range of applications to study numerical methods for them.

英文关键词： stochastic functional differential equation; numerical method;convergence;stability.