【作者】 何耀耀；

【导师】 张诚坚；

【作者基本信息】 华中科技大学 ， 计算数学， 2006， 硕士

【摘要】 Volterra型延迟积分微分方程(VDIDEs)广泛运用于物理学,生物学,生态学及控制论等科学领域,延迟积分微分方程通常很难获得理论解的解析式,因此研究这类方程的数值方法是十分必要的。多步方法是利用前面几步得到的关于方程解的信息来构造解在新的节点上的近似值。已有的多步方法中,BDF方法及其扩展的方法已经被很多学者研究,但是他们的工作主要集中在常微分方程和延迟微分方程(ODDEs)中,很少涉及延迟积分微分方程。由于BDF方法及其扩展的方法对ODDEs十分有效,因此扩展上述方法用于求解VDIDEs是值得研究的。本文首先研究了求解刚性多滞量Volterra型积分微分方程的BDF方法的非线性稳定性和计算有效性。经典BDF方法被改造用于求解一类刚性多滞量Volterra型积分微分方程,数值试验表明所给出的方法是高度有效的。此外,证明了:在适当条件下,其扩展的BDF方法是渐近稳定和整体稳定的。其次本文对刚性中立型延迟积分微分方程的单支方法的非线性稳定性进行了系统的研究,单支方法被改造用于求解一类刚性中立型延迟积分微分方程,证明了单支方法在适当的条件下是渐近稳定和整体稳定的。最后我们构造了求解刚性Volterra型延迟积分微分方程的EBDF和MEBDF方法,分析了EBDF和MEBDF方法用于刚性Volterra型延迟积分微分方程时阶降低的原因,给出了它们收敛的条件,并给出若干数值试验与传统的BDF方法进行了比较。更多还原

【Abstract】 Volterra delay-integro-differential equations(VDIDEs) arise widely in scientific fieldssuch as physics ,biology,ecology,control theory and so on.Usually,it is difficult to obtain ananalytical solution of VDIDEs ,so one turn to study the numerical solutions of VDIDEs.A multistep method is the method to obtain the current approximation solution bythe information from several previous steps .Among the existed multistep methods, BDFmethods,and their extension have been investigated by many authors. But their contributionare main for ordinary and delay differential equations(ODDEs) and few for VDIDEs .SinceBDF methods and their extension are quite effect for ODDEs,it is promising to extend theabove methods to solve VDIDEs.Firstly,this paper deals with nonlinear stability and computational effectiveness of BDFmethods for solving stiff multidelay-integro- differential equations of Volterra type. Theclassical BDF methods are adapted to a class of nonlinear stiff multidelay-integro- differ-ential equations of Volterra type, and the numerical experiments show that the presentedmethods are highly effective.Moreover, it is proven under the suitable conditions that theextended BDF methods are globally and asymptotically stable.Secondly ,this paper deals with nonlinear stability of one-leg methods for solving stiffNDIDEs systemically. One leg methods are adapted to a class of stiff NDIDEs. It is proventhat the extended one leg methods are globally and asymptotically stable under suitableconditions.Finally, We construct EBDF and MEBDF methods for Solving stiff VDIDEs , analysethe reason of order reduced when EBDF and MEBDF methods are applied to stiff VDIDEs,and give convergence conditions . They are compared with classical BDF methods byseveral numerical experiments .更多还原