Highly Efficient Boundary Value Methods For Several Classes Of Delay Differential Equations

Delay differential equations always play an important role in modeling a great variety of phenomena arising from physics,mechanical engineering,biomedicine,chemical reac-tion kinetics,economy and other scientific fields.However,in fact,it is almost impossible to obtain analytic solutions of these problems.As a consequence,based on the characteristics of equations themselves,it is very important to give an insight into the theoretical properties and numerical methods for such equations.In Chapter 1,we give a brief review to the applicational background of delay differen-tial equations and associative theoretical and numerical results.Furthermore,the research work in this context is outlined.In Chapter 2,exponential stability for second order nonlinear functional differential equations with time-variable delays is investigated.We derive a stability criterion under which second order functional differential equations can preserve the exponential stability of the corresponding ordinary differential equations.This stability result is also illustrated with a numerical approach.In Chapter 3,we deal with numerical solutions of boundary value problems for second order differential equations with discrete delay.The generalized St?rmer-Cowell methods are extended to solve such problems.The existence and uniqueness criterion of the result-ing methods is derived.It is proved,under appropriate conditions,the resulting scheme is stable,and convergent of order p whenever this method has the consistent order p.The nu-merical examples illustrate efficiency and accuracy of the methods.Moreover,a comparison between the proposed methods and Generalized Backward Differential Formulae for corre-sponding equivalent first-order boundary value problems is given.The numerical results show that the proposed methods are comparable.In Chapter 4,our attention is further paid to boundary value problems for second order differential equations with distributed delay.The generalized St?rmer-Cowell methods cou-pled with compound quadrature formulas are extended to deal with such case.Under some suitable conditions,it is proved that the extended generalized St?rmer-Cowell methods are uniquely solvable,stable and have convergent order min{p,q},where p,q are consistent order of the generalized St?rmer-Cowell methods and convergent order of the compound quadrature rules,respectively.Furthermore,some numerical examples are performed to illustrate the validity and accuracy of the proposed methods.In Chapter 5,functional differential equations with piecewise continuous arguments are under consideration.Block boundary value methods are applied to solve this class of problems.It is shown,under certain conditions,the convergent order of an extended block boundary value method coincides with its order of consistency.Moreover,we study the linear stability of the extended methods and give the corresponding asymptotical stability criterion.In the end,with several numerical examples,the theoretical results and the com-putational effectiveness of the methods are further demonstrated.In the last chapter,a brief conclusion is given and some future work is proposed.