Analysis of Delay-dependent Stability of Boundary Value Methods for Delay Differential Equations
|School||Central South University|
|Keywords||delay differential equations boundary value meth-ods delay-dependent stability region Tk1，k2 （0）-stability Tk1，k2-stability|
Delay differential equations (DDEs) arise widely in physics, en-gineering, biology, medical science, economics and so on. Yet, the number of instances where an exact solution can be found by ana-lytical means is very limited. Hence, it is meaningful to investigate the efficient numerical methods for delay differential equations. The stability of numerical methods plays an important role in the numer-ical solution of DDEs. In the last decades, many papers have dealt with this topic and a significant number of important results have been found. This thesis deals with the delay-dependent stability of boundary value methods (BVMs) for delay differential equations.In the first Chapter, many applications of delay differential equa-tions in different fields are presented. The development of the stability theory of DDEs in the past decades is introduced. Moreover, we give a brief introduction to boundary value methods.Secondly, background materials for this paper are presented. Fur-thermore, we introduce new stability concepts, i. e.(?)k1,k2(0)-stability and (?)k1,k2-stability. They are the analogues of the concepts of (?)(0)-stability and (?)-stability.In Chapter 3, we consider the delay-dependent stability of sym-metric schemes in BVMs for delay differential equations. In the case of real coefficients model, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. A necessary and sufficient condition, which guarantees symmetric schemes are (?)v,v-1(0)-stable is obtained. Several symmetric schemes under con-sideration are verified to be (?)v,v-1(0)-stable. Moreover, we find the numerical stability region if the model coefficients are complex. It is proved that all the symmetric schemes in BVMs are not (?)v,v-1-stable. Chapter 4 is concerned with the study of the stability analysis of symmetric schemes in BVMs for a class of second order delay differen-tial equations. The delay-dependent stability region of the symmetric schemes is analyzed and their boundaries are found. Then, it is shown that a symmetric scheme can completely preserve the delay-dependent stability for the underlying system if it is (?)v,v-1(0)-stable.In the fifth Chapter, we deal with the delay-dependent stability properties of generalized backward differentiation formulae (GBDFe) for linear scaler delay differential model. The delay-dependent stabil-ity region of the GBDFe is analyzed and their boundaries are found. Then, we prove that a k-step GBDF is (?)v,k-v(0)-stable if and only if k=1,2,4,6.Finally, we consider the delay-dependent stability of boundary value methods for delay differential equations. A unified stability cri-terion for (?)k1,k2(0)-stability is obtained. Three types of BVMs. In par-ticularly, the stability analysis of generalized Adams methods (GAMs) is considered. Some GAMs are verified to be (?)v,k-v(0)-stable.