【作者】 孙喜堂；

【导师】 肖爱国；

【作者基本信息】 湘潭大学 ， 计算数学， 2009， 硕士

【摘要】 刚性延迟微分方程广泛地出现于自动控制、电力工程、生态学等领域。目前,国内外文献大都集中对求解这类问题的Runge-Kutta方法、Rosenbrock方法、线性多步法及更一般的线性方法进行研究。在实际问题中,一些刚性延迟系统往往可分解为两个耦合的子系统:刚性延迟子系统和非刚性延迟子系统,自然可采用组合方法进行求解,且往往更为高效。因此,针对求解刚性延迟微分方程的组合方法的研究具有重要的理论和实际意义。本文第二章介绍了三种求解刚性微分方程和刚性延迟微分方程的Rosenbrock型方法;第三章对求解刚性延迟微分方程的组合RK-Rosenbrock方法的稳定性进行了分析,给出了求解该方程的组合RK-Rosenbrock方法GP-稳定的充要条件;第四章对求解刚性延迟微分方程的组合连续RK-Rosenbrock方法的稳定性进行了分析,给出了求解该方程的组合连续RK-Rosenbrock方法GP-稳定的充要条件;第五章对求解刚性延迟微分方程的连续Rosenbrock方法和连续并行Rosenbrock方法的稳定性进行了分析,利用B-级数理论得到了方法的阶条件方程,且构造了一类方法,并证明了该方法具有GP-稳定性;第六章通过数值试验验证了Rosenbrock型方法对刚性延迟系统的有效性。更多还原

【Abstract】 Delay differential equations(DDEs) arise widely in many fields such as control engineering,power engineering,ecology etc.In recent years,much discussion about Runge-Kutta methods,Rosenbrock methods,linear multistep methods and general linear methods for DDEs is found in majority relevant literatures.In the practical application,stiff delay systems could be often partitioned into two coupling subsystems:stiff delay subsystems and non-stiff subsystems,thus we can use combined methods to solve these systems,and they are often more efficient. Therefore,it is significant in theory and practice to study the combined methods for stiff DDEs.In Chapter 2,we present three kinds of Rosenbrock methods for stiff differential equations and stiff delay differential equations.In Chapter 3,we analyze the stability properties of combined RK-Rosenbrock methods for stiff delay systems,and give the sufficient and necessary conditions under which the combined RK-Rosenbrock methods are GP-stable.In Chapter 4,we analyze the stability properties of combined continuous RK-Rosenbrock methods for stiff delay systems,get the methods’ order condition equations,and give the sufficient and necessary conditions under which the combined continuous RK-Rosenbrock methods are GP-stable.In Chapter 5,we analyze the stability properties of continuous Rosenbrock methods and continuous parallel Rosenbrock methods for stiff delay differential equations,use B-series to get the order conditions,and give the sufficient and necessary conditions under which the methods are GP-stable.Numerical experiments show that the methods are efficient.更多还原