Linear Multistep Methods For Singularly Perturbed Delay-integro-differential Equations

Singular perturbation problems widely arise in many practical applications, such aschemical kinetics, automatic control, and electrical circuits et.al. By view of the specialcomplexity and importance of this problems, many scientists from all over the worldhave paid extensive attention to develop and deepen quantitative convergence theory ofnumerical methods for these important typical subclasses of stiff problems.Initial value problems of ordinary differential equations in singular perturbation formare of the class of stiff problems which can't be satisfactorily covered by B-theory becauseof their very special structure. They mainly include two subclasses: singly stiff singularperturbation problems (their stiffness is only caused by some small parameters) and multi-ply stiff singular perturbation problems (their stiffness is caused by some small parametersand some other factors). At present, there exist many researches into numerical methodsfor typical singular perturbation problems. Many important and interesting results on theconvergence of linear multistep methods, RungeffKutta methods, Rosenbrock methodsand general linear methods applied to the classical singular perturbation problems havebeen given by many authors by means of two different approaches ff direct approach and-expansion approach.But up to now, there exits no results of numerical methods for singular perturbationintegral differential problems with delays which often arise in many practical applications,such as chemical kinetics, automatic control, electrical circuits and nuclear reactor kineticset.al. Therefore, a deep research into this field is of important theoretical meanings andvast applied prospects.In this paper, we first proved the stability property of a class of singular perturba- tion delay volterra integral differential systems(SPDVIDs). Then, we have gained somequantitative convergence results for linear multistep methods applied to both singly stiffsingular perturbation problems and multiply stiff singular perturbation problems in chap-ter 3 and 4. In the end of each chapter, we also gave some numerical test which confirmedour theoretical results.