Several Classes Of Multistep Methods For Stiff Delay Integro-differential Equations

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 .